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Learning Objectives
• Learn how to prepare an elevator pitch for your brand.
Now you know about an elevator pitch and how it can help you in your sales approach. But do you have an elevator pitch for your personal brand? If the answer is no, now is the time to craft it. Just as in selling, your personal elevator pitch should be less than a few minutes and should be a way for you to tell someone who you are, what you’ve done, and what you’re looking for. Your elevator pitch will serve as the approach for your internship and job search in several different ways.
Elevator Pitch 101: Be Prepared
Your elevator pitch is critical because it tells a prospective employer or someone in your network what you have to offer, what makes you different, and what you want to do. You’ll use your elevator pitch in many different situations; you may even use it in situations when you least expect it. Chris O’Leary, author of Elevator Pitch Essentials, suggests that many people are not prepared to take advantage of relationships and opportunities that come their way simply because they are not prepared with a compelling statement about who they are and what they are looking for.Chris O’Leary, “Elevator Pitch 101,” Elevator Pitch Essentials, January 27, 2009, http://www.elevatorpitchessentials.com/essays/ElevatorPitch.html (accessed July 26, 2009).
Creating Your Personal Elevator Pitch
See how to craft your personal elevator pitch.
How to Create Your Elevator Pitch
Before you can deliver your elevator pitch, you have to write it first. Start by reviewing your brand positioning points that you identified in the Selling U section in Chapter 1. As you recall, your brand positioning points are the foundation of your résumé and cover letter and now your elevator pitch. You can see how you are building a consistent brand story by always focusing on the same key selling points about yourself.
To craft your elevator pitch, keep the following points in mind:
• Who are you?
• What experience and skills do you have?
• What makes you unique?
• What problem can you help your prospective employer solve?
• What are you looking for?Michelle Dumas, “How to Create a Compelling Branded Elevator Speech for Your Job Search,” EzineArticles, April 23, 2008, ezinearticles.com/?How-to-Create-a-Compelling,-Branded-Elevator-Pitch-for-Your-Job-Search&id=1128958 (accessed July 26, 2009).
Here’s an example of how an elevator pitch comes together from Jobstar.org:
Hello, my name is Melinda Stevens. I’m a graduating senior from Southton College. I got your name from the alumni office, where they said you were an alumna from 1983. I understand you’re now a CPA and audit manager in Chicago. My minor was in business, and I’m interested in positions in accounting. I’d like to know how you got where you are today and what advice you’d have for a college graduate just coming into the job market today. Do you have a moment right now?Don Asher, “Sample 30 Second Speeches” JobStar, April 14, 2009, http://jobstar.org/hidden/asher2.php (accessed July 26, 2009).
This is an example of a telephone approach. You can see that it is concise and to the point. If you are networking, at a job interview, or talking with someone, you might have the time for one or two more sentences, but not much more. The secret to an effective elevator pitch is to intrigue the listener so that he wants to hear more. If your elevator pitch is compelling and brief, the listener will respond by asking a question, and you will get the conversation started.
Elevator Pitch Critiques
(click to see video)
This video provides some sample elevator pitches and constructive feedback about how the pitches can be improved.
When to Use Your Elevator Pitch
One of the most common uses for an elevator pitch is networking. For example, if you attend a professional event you’ll have the opportunity to meet many new people. And you’ll want to tell each one a little bit about yourself. This is a perfect opportunity to use your elevator pitch; it’s not too long and gives you the perfect way to start a conversation and give the person to whom you are speaking the chance to ask a question. You might even find something or someone in common as a result of the information in your elevator pitch: “You were an intern at Classic Architects? My brother used to work there. His name is Jeremy Slater. Do you know him?”
Another opportunity to use your elevator pitch is in an interview. Although you will need more preparation than simply your elevator pitch for an informational interview or a job interview, you will have a head start on your preparation with a strong elevator pitch. It’s the perfect response to what is commonly the first question that is asked at almost every job interview: “So tell me about yourself.” It’s important to be ready with a clear, concise, and compelling statement. If you think you can wing it, you will probably start your interview off on the wrong foot. On the other hand, a good elevator pitch allows you to direct the conversation to the things you want to talk about (your brand positioning points).
You’ve Got the Power: Tips for Your Job Search
Make Your Elevator Pitch Work for You
It might be challenging to think about communicating your brand story in only sixty seconds, but don’t forget your objective: you want to get the internship or job. While there’s a long way between your elevator pitch and an internship or job, keep your eye on the prize; always have a call to action as part of your elevator pitch. For example, ask for a business card from everyone with whom you speak or meet. That means that whether you are at a networking event or on a job interview, it’s always appropriate to ask the person for their business card. (You might want to brush up and review the business card etiquette covered in Chapter 5.)
Then, follow-up is key. After you meet someone, follow up with an e-mail or phone call within twenty-four hours (or on the appropriate date after an interview). Tell the person how much you enjoyed meeting her and mention something specific about your conversation. It’s a good idea to include a link to an interesting article or video in your e-mail; that will help you stand out in the person’s mind.
Be Yourself
Your elevator pitch is a reflection of you, so when you are creating your elevator pitch, write it down, and then say it out loud in front of a mirror until you are comfortable with it. It’s important to rehearse it so that you are comfortable with communicating this brand message in just a few minutes without rambling or stumbling.Laura Raines, “Making Your Pitch,” The Atlanta Journal-Constitution, Jobs, www.ajc.com/hotjobs/content/hotjobs/careercenter/articles/2007_0225_elevatorsp.html (accessed July 26, 2009). But you don’t want to have your elevator pitch down cold; in other words, you want to deliver it with ease and with a natural tone and pacing, as if you were saying it for the first time. It’s hard to get the balance between preparation and spontaneity, which is why it’s a good idea to use your elevator pitch frequently. That way you will be able to feel natural saying it and make adjustments based on how it sounds and feels. And don’t forget to smile!
Key Takeaways
• An elevator pitch is a concise description of a product or service that should take no longer than an average elevator ride and is designed to get conversation started.
• An elevator pitch requires preparation, and you should always be prepared because you never know when you might have an opportunity to use it.
• Your elevator pitch should be approximately sixty seconds long and should use your brand positioning points as the foundation to answer the following questions:
• Who are you?
• What experience and skills do you have?
• What makes you unique?
• What problem can you help your prospective employer solve?
• What are you looking for?
• You can use your elevator pitch in many situations including networking and informational or job interviews.
• Write down your elevator pitch and rehearse it out loud in front of a mirror. But deliver it naturally, as if it were being said for the first time, and always with a smile.
• Don’t forget to make your elevator pitch work for you by asking for a business card and following up with each person individually within twenty-four hours with a thank-you note or follow-up e-mail.
Exercise \(1\)
1. Write your elevator pitch. Give your pitch to the person next to you and then listen to hers. How long was each elevator pitch? What elements did she include that you didn’t? What elements could you include if time permits? What is your call to action (what you want the person to do at the end of your elevator pitch)?
2. Name three situations in which you could use your elevator pitch.
3. Create a short video of your elevator pitch and post it to YouTube (keep in mind that it should not take longer than the average elevator ride).
4. Create your elevator pitch in two PowerPoint slides (use only two slides). Post the “pitch” to Slideshare.net and share it with your class. | textbooks/biz/Marketing/The_Power_of_Selling/09%3A_The_Approach-_The_Power_of_Connecting/9.06%3A_Selling_U_-_Whats_Your_Elevator_Pitch_for_Your_Brand.txt |
Power Wrap-Up
Now that you have read this chapter, you should be able to understand how to approach a prospect.
• You understand the importance of your first impression.
• You learned the elements of making contact.
• You can describe the role of an elevator pitch in the approach.
• You can list the dos and don’ts of making contact via phone and in person.
• You can describe the different types of sales approaches.
• You can understand how to create an elevator pitch for your personal brand to use during your approach for networking, interviews, and other contacts.
TEST YOUR POWER KNOWLEDGE (AnswerS ARE BELOW)
1. Name the six Cs of the sales approach.
2. Identify one way of demonstrating active listening.
3. What is the 70/30 rule of listening?
4. What is an elevator pitch, and why is it important in a sales approach?
5. Why should you prepare a script for your opening statement for a telephone approach?
6. Describe an effective e-mail approach.
7. Why are social networks an effective way to approach prospects?
8. List two opening lines you should avoid in a sales approach.
9. Describe the customer benefit approach.
10. What is a gatekeeper?
11. What kind of information should be included in the elevator pitch for your personal brand?
POWER (ROLE) PLAY
Now it’s time to put what you’ve learned into practice. The following are two roles that are involved in the same selling situation—one role is the customer, and the other is the salesperson. This will give you the opportunity to think about this selling situation from the point of view of both the customer and the salesperson.
Read each role carefully along with the discussion questions. Then, be prepared to play either of the roles in class using the concepts covered in this chapter. You may be asked to discuss the roles and do a role-play in groups or individually.
A Good Sport
Role: Operations manager, Trident Office Equipment
You are responsible for all the operations for a major office equipment distributor. Trident counts hundreds of businesses among its B2B customers. As part of building relationships with customers, the company entertains its B2B customers by taking them to professional sporting events, dinner, and other activities. The company is currently a season ticket holder for the local professional football team. However, given the state of the economy, you are reconsidering the company’s investment in season tickets. Your time is valuable to you, so you don’t want to take the time to meet with a sales rep from each of the teams.
• What will you say when a sales rep from one of the sports teams approaches you?
• What type of approach will you find compelling enough to take the time to meet with a sales rep?
Role: Sales rep for the stadium that hosts the city’s minor league baseball team
You have qualified your prospect as someone who is responsible for the decisions for purchases of season tickets to entertain customers. While he has traditionally purchased season tickets for the local professional football team, you believe that you can approach him with an opportunity to save money and have an excellent opportunity to entertain clients and support the local baseball team. The baseball season is longer and offers more opportunities for Trident to entertain its customers, and the cost per game is less for your baseball tickets than what Trident has been paying for football tickets, although the total cost for season tickets is greater. You are preparing your approach to make an appointment on the phone.
• What will you say to approach this prospect?
• What type of approach will you use?
• What is your elevator pitch for the season tickets?
ACTIVITIES
1. Ask your professor or another professional to share his elevator pitch with you. Deliver your elevator pitch to him and ask him to critique it.
2. Visit your career center and ask one of the counselors to provide feedback to you on your elevator pitch.
3. Use your elevator pitch in a professional situation such as your internship, class, or interview. What elements do you think work in your elevator pitch? What elements are not as effective? What modifications will you make as a result?
TEST YOUR POWER KNOWLEDGE AnswerS
1. Confidence, credibility, contact, communication, customization, and collaboration.
2. Eye contact, lean forward, take notes, and repeat key points to check for understanding.
3. You should be listening 70 percent of the time and asking questions 30 percent of the time to engage the prospect.
4. An elevator pitch is a concise description of a product, service, project, or person that should take no longer than the average elevator ride. It’s an important part of the sales approach because it is a good way to give your prospect an overview and get conversation started.
5. You need to get your prospect’s attention in the first twenty seconds; you don’t want to stumble over your words or sound like you’re rambling. A script is a good way to stay focused and communicate effectively.
6. Personalized e-mails that address a prospect’s needs can be very effective. An e-mail should be well written and interesting to read and include proper spelling and grammar.
7. You can network, get referrals, and add value to the conversation on social networks.
8. “Would you be interested in saving money?”; “You’re probably a busy person, so I promise I’m not about to waste your time”; “I just happened to be in the area visiting another customer so I thought I’d drop by”; and “I’ve heard that you’ve been having trouble in your customer service department.”
9. Opening the sales call by directing your prospect’s attention to a specific benefit of your products or services.
10. The secretary or assistant whose job it is to screen calls or “guard” the entrance to an executive’s office. It’s the person you have to do through first before seeing your prospect.
11. Who are you, what experience and skills do you have, what makes you unique, what problem can you help your prospective employer solve, and what are you looking for? | textbooks/biz/Marketing/The_Power_of_Selling/09%3A_The_Approach-_The_Power_of_Connecting/9.07%3A_Review_and_Practice.txt |
The Presentation: The Power of Solving Problems
Video Ride-Along with Paul Blake, Vice President of Sales at Greater Media Philadelphia
You met Paul Blake in Chapter 4 when he talked about ethics and doing the right thing. Now hear his tips for making a successful sales presentation. While most salespeople find this step of the selling process to be their favorite, it takes a lot more homework than meets the eye.
Listen to Paul’s advice for learning about what makes the customer tick and delivering value “in a big way.”
10.02: Preparation - Your Key to Success
Learning Objectives
• Learn how to prepare for a sales presentation.
You’ve made it! After all your hard work you have reached the point in the selling process where the qualifying, researching, and planning stages pay off. Finally, your story and the customer’s story are about to connect in an exciting way. Most salespeople think of the presentation as the best part of the selling process. It’s the opportunity to show the prospect that you know your stuff—and the chance to deliver value by putting your problem solving skills to work. So get ready, visualize the best possible outcome to your sales presentation, and take the necessary steps to make this outcome a reality.
Keep Your Eye on the Prize
As excited as you might be about your product, or as eager as you are to demonstrate your solution, keep in mind that your sales presentation is primarily about building a relationship and beginning a partnership, especially in the business-to-business (B2B) arena. When Selena Lo, CEO of Ruckus Wireless, is gearing up for a sales presentation, she focuses her final preparations on making it personal. Lo’s company specializes in wireless routers that handle video, voice, and data capabilities for businesses. When she identifies a prospect, Lo’s first priority is finding the person she refers to as “the fox”: her ally in the prospect company who wants to see technological changes take place in his organization. Lo gives this relationship special attention, often inviting this individual out to dinner before the presentation to win his loyalty and get any additional details about his company.
Several days before the presentation, Lo researches everyone who will be in the meeting. She reads their bios and googles them to find out their employment histories. “You don’t want someone to think you checked out their entire past,” says Lo, but “you try to strike up more links between you and that person.” She prepares the seating arrangement for the sales meeting strategically, making sure that she will be sitting directly across from the highest-ranking person there so that she can make eye contact. On the day of the presentation, she asks a member of her sales team to write down each person’s name when they walk in the door—and to make a point of using the names during the presentation.Stephanie Clifford, “Find the Fox,” Inc., February 1, 2007, www.inc.com/magazine/20070201/features-sales-performance-lo.html (accessed May 16, 2010). Lo’s efforts to give the sales presentation a personal touch are a reminder that in relationship selling, you can never lose sight of the most important thing: your customer. Coach yourself on this on the day of your presentation and keep it in mind in the days leading up to it. What can you do to personalize this presentation and show your customers that it’s all about their organization?
Taking a customer-centric approach lies at the heart of delivering value. In these terms, value isn’t about offering a good price. It’s not just about solving the customer’s problems either. As Tom Reilly, author of Value-Added Selling: How to Sell More Profitably, Confidently, and Professionally by Competing on Value, Not Price, explains it, delivering value means that you “define value in customer terms, ask questions, listen to customers, and put the spotlight on customer-centric solutions.”Tom Reilly, Value-Added Selling: How to Sell More Profitably, Confidently, and Professionally by Competing on Value, Not Price, 2nd ed. (New York: McGraw-Hill, 2002), 23–24. This might mean that it takes more than one meeting to close your sale; you might need several visits to adequately respond to your customer’s needs. According to one study, “Today’s presentations typically are conducted over several meetings, with the salesperson often doing more listening than talking.”William C. Moncrief and Greg W. Marshall, “The Evolution of the Seven Steps of Selling,” Industrial Marketing Management 34, no. 1 (2005): 18. Make it your goal to see that you and your prospect get what you want out of the meeting.
It’s a good idea to visualize this outcome before going into the meeting. Review your precall objectives. What will it look like to achieve these objectives? What steps will you and your prospect have to take? How will it feel when you both have achieved your goals? This isn’t just about calming your nerves; visualizing the outcome you want is actually a powerful tool to help you achieve that outcome. For one thing, it’s another form of planning. If you mentally run through a “movie” of the sales presentation, allowing yourself to picture your reactions and the steps you will take to close in on your objective, you will be better prepared when the meeting takes place.Richard White, “Déjà Vu,” Pro Excellence, www.pro-excellence.com/html/resources.html (accessed May 16, 2010). Each step of the presentation will come naturally to you because you have already mentally rehearsed, and you will be better positioned to sell adaptively because you have already imagined a number of possible scenarios and customer responses.
For another thing, mental rehearsal fools your subconscious mind into believing you have already achieved your goals. Sales trainer and CEO Brian Tracy says, “Your subconscious mind cannot tell the difference between a real experience and one that you vividly imagine,” so if you imagine a successful presentation and its outcome several times before your actual presentation, you will be as calm and confident as if you had already closed the sale. You will smile more easily, you will speak more slowly and clearly, and you will command attention. In addition, if your subconscious mind believes you have already been in this situation before, it will direct you to say and do the things you need to achieve your objective.Brian Tracy, Advanced Selling Strategies (New York: Simon & Schuster, 1996), 80.
The Power to Adapt
The sales presentation is where adaptive selling makes all the difference. Up until this point, you have researched and prepared and developed a solution that you think will meet your prospect’s needs, but walking into the presentation and delivering on that preparation requires a different set of skills. Among other things, it requires flexibility and the ability to think on your feet. The best salespeople adapt their presentations to their prospect’s reactions, and they go in knowing they may have to adapt to surprises for which they were unable to prepare (maybe the building has a power outage during the slideshow, for instance, or maybe one of the people from the customer organization decides to send another employee in his place at the last minute). These top-performing salespeople know that keeping a customer-centric focus, visualizing a successful outcome, and mentally rehearsing your presentation before you deliver it will give you the power to adapt with confidence and ease.
Adapting is all about listening. As Paul Blake noted in the video ride-along at the beginning of the chapter, your sales presentation is really a compilation of all the listening you have done to this point. And listening doesn’t stop there. It’s impossible to adapt if you’re not listening. When you are creating your presentation, keep in mind that it is not a one-way communication. Presentations are for listening, adapting, and solving problems.
Listen and Sell
(click to see video)
This video highlights the power of listening and tips to listen effectively during your presentation.
Logistics Matter
There’s nothing worse than putting hours into preparing a killer sales presentation, only to blow your chances because you forgot to bring an important part of your demonstration or because you got lost on your way to the meeting. Don’t let disorganization hold you back: take charge of the details so that your only concern on the day of the presentation is the delivery.
The Night Before
The evening before your meeting, read over your precall objectives; practice your presentation a number of times out loud; and walk through your mental rehearsal, visualizing success. You can’t practice too many times. The content of your presentation should be second nature by the time you get up in front of your audience so that you can focus your energy on your prospect. Rehearsal is one of the best ways to calm your nerves so that you can focus on delivering your presentation naturally and connecting with your prospect.
Power Player: Lessons in Selling from Successful Salespeople
Rehearse Your Way
Andres Mendes, global CIO of Special Olympics International, says that rehearsing out loud makes him too nervous; he likes to leave room for spontaneity and adaptation. Mendes develops the big themes of the presentation and maps these out into PowerPoint slides that tell the whole story. “I time the slides to move exactly at my pace, so I rehearse the mechanics and make sure those are right,” he says.Maryfran Johnson, “Rehearsing Success,” CIO Magazine, June 10, 2009, www.cio.com/article/494729/Why_Even_Successful_Speakers_Need_To_Practice (accessed May 16, 2010).
CIO Magazine columnist Martha Heller, on the other hand, likes to rehearse in the traditional style, delivering the presentation out loud and pacing the room as if she were in front of an audience. She never rehearses the opening though. She likes to adapt her comments to the immediate situation and energy in the room.Maryfran Johnson, “Rehearsing Success,” CIO Magazine, June 10, 2009, www.cio.com/article/494729/Why_Even_Successful_Speakers_Need_To_Practice (accessed May 16, 2010).
The bottom line? While nearly all top-performing salespeople rehearse, not all approach rehearsal in the same way. Find the style of rehearsal that works best for you. Additionally, don’t let your rehearsal lock you into delivering a rigidly defined set of remarks. You have to leave room for flexibility and adaptation.
The night before, you should also get together all the materials you’ll need for your presentation—handouts, files, product samples, and contracts—and have them ready to go for the following morning. This will save you time tracking down loose supplies at the last minute, when you’re trying to get out the door to make it to your meeting. It’s also a good idea to set out your clothes the night before for the same reason.
If you are planning to use multimedia equipment in your presentation, make sure in advance that your prospect will have everything you’ll need to make it run. If you aren’t sure, bring everything (e.g., cables, adapters, remotes) with you. And of course, make sure you know how to use all your equipment. When Keith Waldon, CEO of Earth Preserv, was preparing for a meeting with JCPenney, one of his biggest prospects, he spent hours rehearsing with his multimedia equipment. The technology was a key element of his presentation, and he wanted to make sure everything would work perfectly for the big day. “I had to learn how to use all the remote-control equipment,” he says. Waldon also brought a technical assistant with him as backup to safeguard against any glitches.Susan Greco, “Anatomy of a Launch: The Five-Hour Multimedia Sales Presentation,” Inc., October 1, 1995, www.inc.com/magazine/19951001/2441.html (accessed May 16, 2010).
Getting There
It might surprise you to know how often salespeople show up late to their own presentations because they get lost on the way to the meeting. When you are traveling to an unfamiliar place for your appointment, get directions in advance, and allow extra travel time in case of traffic delays or wrong turns. Make sure you also research the parking situation beforehand. If your prospect is a large corporation with its own complex, are there reserved employee lots and visitor lots? Will you have to walk a considerable distance from your car to the meeting room? If you’ll be meeting in an urban area, is street parking available, or will you have to find a parking garage? You don’t want to arrive on time only to get delayed because you spent twenty minutes driving around in search of a parking spot. It’s a good idea to make a “test” trip in advance of your meeting. That will help avoid surprises with traffic, parking, security, or other areas that might cause a delay. If something unavoidable does come up to set you back, make sure you call ahead to let your customer know you will be arriving late.
Besides the extra time you allow for travel, plan to arrive at the meeting a little early. Not only does this convey professionalism, but it also gives you the time to mentally prepare once you arrive and to set up any equipment you’ll be using. It’s a good idea to allow time to stop in the restroom and take one last look to be sure you’re at your best (and it’s a good time to use a breath mint). Finally, bring something to read in case you have to wait: a business magazine, a newspaper like the Wall Street Journal, or maybe a Kindle.
Key Takeaways
• When preparing for your sales presentation, stay focused on the essentials: your relationship with the prospect and your precall objectives.
• Practice mental rehearsal by visualizing the best possible outcome to the sales presentation.
• Delivering value to the customer means practicing adaptive selling and listening to the customer to understand her needs. Keep this in mind before and during the presentation.
• The night before your presentation, make sure you have all the logistics worked out: your equipment, your wardrobe, directions to the location, and parking information.
Exercise \(1\)
1. You are preparing for a presentation with three executives to be considered for the internship or job you really want. List the steps you would take to rehearse your sales presentation, making sure to leave room for adaptability.
2. You are preparing a presentation for representatives from a large department store who are considering buying your line of men’s shoes. There will be six representatives present, none of whom you have met in person before. You have heard from your original contact at the company that one person in the group is against purchasing your product because he believes he already has something in the line that has the same look. List some things you can do to prepare for this presentation that will address the prospect’s concerns.
3. Assume you are a real estate agent and you are selling the dorm room, apartment, or home in which you live. Create a short sales presentation. Rehearse it so that the presentation takes only three minutes. What is the way that works best for you to rehearse?
4. Assume you are sales rep for a major telecommunications company and you are preparing a presentation for a buying group at a national retailer. Identify four sources you would use to personalize the presentation to the people in the room. How would you research each of the appropriate people? | textbooks/biz/Marketing/The_Power_of_Selling/10%3A_The_Presentation_-_The_Power_of_Solving_Problems/10.01%3A_Introduction.txt |
Learning Objectives
• Discuss how to dress for success for a sales presentation.
Your appearance communicates volumes about you before you ever open your mouth.
Tom Reilly tells the story of a salesperson that showed up to one of his recent seminars dressed in flip-flops and a T-shirt. “I thought he was there to clean the windows,” Reilly says.Tom Reilly, “Dress for Success,” Tom Reilly Training, 2009, www.tomreillytraining.com/Ezine%207-07%20DressforSuccess.htm (accessed May 16, 2010). You want your prospective customers to take you seriously at first glance, so pay careful attention to what you wear on your sales call. Think about it this way, when you are buying a product off the shelf in a store, isn’t packaging the first thing that catches your attention? Marketers know that packaging can influence a consumer’s decision to buy before she ever even researches the product or reads about its features. In the same way, your prospect will make a judgment about you based on the way you “package” yourself; a professionally dressed salesperson can have a huge influence on a prospect’s perception of him, his company, and the product he represents.“Dress for Success,” Sales Success Blog, November 29, 2006, salesuccess.blogspot.com/2006/11/dress-for-success.html (accessed May 16, 2010). Your appearance should convey professionalism, competence, and success. Most important, regardless of the dress code at your prospect’s business, be sure your appearance includes a smile. A smile is an instant rapport builder. No one wants to buy from someone who isn’t excited about the company or product he’s representing. Show your prospect that this isn’t just a job; it’s a passion.
Business Casual or Business?
When you are making a sales presentation at a company, remember the advice from Chapter 9 and dress one step above what you would wear if you worked at the organization.Ross Macpherson, “6 Keys to Making the Right Impression in an Interview,” A Career in Sales, 2002, www.acareerinsales.com/careerToolsDress4Success.aspx (accessed May 16, 2010). If you are ever unsure about a company’s standard dress code, always dress up. It’s easier to take off a jacket and tie than to put them on at the last minute.Geoffrey James, “Is ‘Dress for Success’ Still Mandatory?” BNET, January 22, 2009, blogs.bnet.com/salesmachine/?p=732 (accessed May 16, 2010). However, if your prospect tells you the dress code beforehand, here are some general guidelines to follow.
Business Attire
For most of your business-to-business (B2B) sales situations, business attire will be the norm. For a while in the ’90s there was a trend toward more casual clothing in the workplace, but that trend is mostly on the way out. “I see a return to more traditional business wear,” says Gary Brody, president of the Marcraft Apparel Group.Paul Burnham Finney, “Redefining Business Casual,” New York Times, October 23, 2007, query.nytimes.com/gst/fullpage.html?res=9405EEDD1F39F930A15753C 1A9619C8B63&sec=&spon=&pagewanted=all (accessed May 16, 2010). For that matter, even if your customer says business casual is the standard in his workplace, if you are aiming to dress a notch up from that standard, you might decide that business attire is the way to go. As Mark-Evan Blackman of the Fashion Institute of Technology says, suits “universally project an air of authority.”Paul Burnham Finney, “Redefining Business Casual,” New York Times, October 23, 2007, query.nytimes.com/gst/fullpage.html?res=9405EEDD1F39F930A15753C 1A9619C8B63&sec=&spon=&pagewanted=all (accessed May 16, 2010).
For men, business attire means a suit (matching pants and jacket), a necktie, a long-sleeved shirt, and lace-up shoes.Andy Gilchrist, “Cracking the Dress Code,” Ask Andy about Clothes, www.askandyaboutclothes.com/Clothes%20Articles/cracking_the_dress_code.htm (accessed May 16, 2010). Go for conservative, dark colors such as gray, black, or dark blue for the suit; white or light blue for the shirt. For women, business means a suit (skirt or pants and matching jacket), shoes with moderate heels in a basic pump style (closed-toe), a blouse, and tan or light pantyhose.
Business Casual
Business casual can sometimes be tricky because it’s less clearly defined than business attire. According to Monster.com, business casual “means dressing professionally, looking relaxed, yet neat and pulled together.”Paul Burnham Finney, “Redefining Business Casual,” New York Times, October 23, 2007, query.nytimes.com/gst/fullpage.html?res=9405EEDD1F39F930A15753C 1A9619C8B63&sec=&spon=&pagewanted=all (accessed May 16, 2010). For men, a bare minimum approach to business casual means dress pants and a collared shirt. Women can wear skirts or pants, but skirts should be a conservative length, and pants should be well tailored: not too tight or too loose. On the top, a blouse or a tailored knit sweater are good choices, and for footwear, make sure to wear closed-toe shoes.Virginia Tech University Career Services, “Business Casual Attire,” Virginia Tech University, www.career.vt.edu/Jobsearc/BusCasual.htm (accessed May 16, 2010). Business casual for men or women does not include workout clothes or shoes, wrinkled clothing, worn blue jeans, shorts, miniskirts, athletic socks, or overly revealing clothing.Paul Burnham Finney, “Redefining Business Casual,” New York Times, October 23, 2007, query.nytimes.com/gst/fullpage.html?res=9405EEDD1F39F930A15753C 1A9619C8B63&sec=&spon=&pagewanted=all (accessed May 16, 2010).
Best-Dressed Men
(click to see video)
This video provides tips for men to dress in business casual.
What Not to Wear
(click to see video)
These videos include tips for what not to wear to work.
Details Matter
Getting the clothes right but missing the mark on the details will create a poor impression just as much as underdressing for the occasion can, so make sure everything from your nails to your hair and choice of accessories conveys professionalism.
• All clothes should be cleaned and pressed. Wrinkled or stained clothing looks very unprofessional. Take the time to review your wardrobe days before your presentation to be sure everything is cleaned and pressed. A trip to the dry cleaner is money well spent.
• If the garment has belt loops, wear a belt. Belts should be dark leather.
• Make sure your briefcase or handbag is professional, not casual.Gloria Starr, “The New Dress for Success Look,” EvanCarmichael.com, www.evancarmichael.com/Business-Coach/2445/The-New-Dress-for-Success-Look.html (accessed May 16, 2010).
• Men should avoid sports watches, and women should wear conservative jewelry—nothing flashy.
• Make sure your hair looks professional and well groomed.Tom Reilly, “Dress for Success,” Tom Reilly Training, 2009, www.tomreillytraining.com/Ezine%207-07%20DressforSuccess.htm (accessed May 16, 2010).
• Carry a good quality portfolio or notebook and a nice pen.
• Women should wear hosiery if they are wearing a skirt. Avoid wearing perfume or cologne.Ross Macpherson, “6 Keys to Making the Right Impression in an Interview,” A Career in Sales, 2002, www.acareerinsales.com/careerToolsDress4Success.aspx (accessed May 16, 2010).
And don’t forget good grooming. Body odor, bad breath, poorly manicured fingernails, and messy hair can be a deal breaker.
Dress to Impress
(click to see video)
This video provides some good advice on how to dress for interviews and in the office.
The Image Your Customer Wants
When employees whose businesses rent space in the Coca-Cola building on New York’s Fifth Avenue want to bring a canned or bottled beverage to work, they have a list of drinks to choose from. Vermont Pure Water is OK, but Evian is definitely out. Food and drink orders coming into the building are scanned, and anything with non-Coca-Cola brand products gets sent away.BNET Advertising Industry, “Adds New Meaning to ‘Always Coca-Cola,’” BNET, findarticles.com/p/articles/mi_m0BDW/is_12_40/ai_54233838 (accessed May 16, 2010). While this rule is on the extreme side, it’s true that even the products you use reflect an image, and when you’re doing business with a potential customer, you want that image to be the right one. This is something worth researching before you go into your sales call. If you know who your prospect’s customers are, use those company’s products. Does the prospect do advertising for Apple? Don’t listen to your Zune while you’re waiting for the appointment. If your prospect is a publishing house, read some of their books before you go to your meeting. If they have a radio station or record label, listen to it. Knowing the prospect’s products, or their customers’ products, is part of your credibility.
Key Takeaways
• When you prepare for a sales presentation, pay careful attention to your appearance because this is an important part of your first impression.
• Always dress more formally than you think your customer will be dressed. When in doubt, dress up.
• Give careful attention to detail, such as accessories and grooming.
• Make sure to convey an image that’s in line with your customer’s products and values.
Exercise \(1\)
1. Review the clothes that are currently in your closet. Do you own a suit and accessories that would be appropriate for business attire? Do you have several pieces you could wear at a business casual event? If not, what will you need to purchase to dress for success?
2. Assume you are a salesperson for a financial services company and you are making a presentation to the vice president of operations and her staff about your corporate financial services. What would you wear?
3. Your prospect is sponsoring a team-building happy hour and dinner that is being held at a local restaurant and sports bar on a Thursday evening and has invited you to attend. What would you wear? Would you consider wearing jeans? Why or why not?
4. You have a meeting with your prospect on Friday at his office. The office is very casual, and your client usually wears jeans. What would you wear? Would you wear jeans? Why or why not? | textbooks/biz/Marketing/The_Power_of_Selling/10%3A_The_Presentation_-_The_Power_of_Solving_Problems/10.03%3A_Dress_for_Success.txt |
Learning Objectives
• Learn how to deliver your message in a powerful and effective way.
When deciding on the structure of your presentation, there are a number of things to consider. Will you present to a group or to an individual? Where will you be giving your presentation? What tools will you use? Sometimes these options are under your control, but often in business-to-business (B2B) sales, you will have to adapt your presentation to your prospect’s needs. In either situation, you can maximize your presentation if you know what to avoid, what to prepare for, and how to make your solution come to life with the tools you have.
Elements of an Effective Presentation
(click to see video)
“Effective Presentations,” featuring author Terri Sjodin, highlights how a great sales presentation comes together.
The Right Size
A good salesperson can read group dynamics as skillfully as she can read an individual prospect’s verbal and nonverbal cues and is comfortable in one-on-one and in group presentation situations. This is critical because as a salesperson sometimes you have control over the kind of presentation you will deliver (group versus individual), but in many situations, the size of the audience to which you will present is determined by the needs and structure of your prospect’s organization. In many organizations large purchasing decisions are the responsibility of purchasing committees or of a combination of individual and group decision makers.Gary M. Grikscheit, Harold C. Cash, and Clifford E. Young, The Handbook of Selling: Psychological, Managerial, and Marketing Dynamics, 2nd ed. (Hoboken, NJ: Wiley-Blackwell, 1993), 152. You might find that you begin with several individual presentations to decision makers in an organization and then are asked to give a follow-up group presentation to a purchasing committee.
Presenting to Individuals
In one-on-one presentations, of course, you only have one person’s needs, preferences, and background to research and adapt to, so customization is usually an easy task. You can closely observe your prospect’s nonverbal communication and listen to her stated needs and concerns and respond accordingly. Does he look worried when you tell him that your company’s integrated marketing plan usually takes four months to develop? You can explain that for preferred prospects you are sometimes able to turn around a faster solution. Does he seem distracted when you begin discussing product features? You can back off and begin asking more questions.
As you learned in Chapter 3 in the discussion about social styles, you will be in a better position to deliver value during your sales presentation if you know something about your buyer’s personality before going into the meeting: Is your prospect conversational and people oriented, or is he task oriented and businesslike? Does your prospect care about details and thorough descriptions, or does he prefer to see the “big picture”? Is he competitive? How does he feel about change? Understanding these things about your prospect will help you to favorably position your product and plan your presentation so that you can put emphasis on the things that matter most to the individual. If you know your prospect is highly competitive, for instance, he will probably be interested in learning about the features that set your product apart from others on the market and the ways in which your product can give him or his company a competitive edge.
Writing up a customer trait description before your meeting can be very helpful so that you can use the information as a guideline in preparing your presentation.Gary M. Grikscheit, Harold C. Cash, and Clifford E. Young, The Handbook of Selling: Psychological, Managerial, and Marketing Dynamics, 2nd ed. (Hoboken, NJ: Wiley-Blackwell, 1993), 127. If you’re working with an existing customer or if you’ve interacted with your prospect prior to the presentation, you can use your observations to write a trait description. If you haven’t met the prospect before, try asking other salespeople in your organization, noncompetitive salespeople at other companies, or other contacts you have who might have met your prospect and who can tell you something about her personality.Gary M. Grikscheit, Harold C. Cash, and Clifford E. Young, The Handbook of Selling: Psychological, Managerial, and Marketing Dynamics, 2nd ed. (Hoboken, NJ: Wiley-Blackwell, 1993), 136. Also, use the company resources including the CRM system to gather as much information as possible about the company and your contact. In addition, it is also a good idea to send a precall questionnaire to your contact to gather information such as the names and titles of the people who will be attending the presentation, how much time has been allotted for you, objectives for the meeting, and any other information that will help you plan the meeting. This information can provide valuable information and help you create an agenda, which is a good idea to send to the prospect before the meeting.
In adapting to an individual buyer, it’s also important to consider his motivation.Gary M. Grikscheit, Harold C. Cash, and Clifford E. Young, The Handbook of Selling: Psychological, Managerial, and Marketing Dynamics, 2nd ed. (Hoboken, NJ: Wiley-Blackwell, 1993), 128. What are his responsibilities in the organization? What pressures does he face? Is he on a strict budget? Is he concerned with his status in the company? If you have two buyers who purchase the same product, chances are they’ll be doing it for different reasons:Gary M. Grikscheit, Harold C. Cash, and Clifford E. Young, The Handbook of Selling: Psychological, Managerial, and Marketing Dynamics, 2nd ed. (Hoboken, NJ: Wiley-Blackwell, 1993), 135. one person might buy a car from you because he sees it as a status symbol, while another person might buy the same car because it gets good gas mileage and is well built and reliable. Keep in mind that delivering value isn’t only about meeting a prospect’s needs; it’s also about showing him that you understand his specific motivations and concerns. The best salespeople present themselves as advisors their customers can trust.Ian Brodie, “Becoming a Trusted Advisor,” Ian Brodie: Business Growth for Professional Service Firms, blog post, July 5, 2008, http://www.ianbrodie.com/blog/becoming-trusted-advisor (accessed May 16, 2010). Is a prospect worried about proving herself in a new role in her company? Show him how your product can help him perform her role better, or demonstrate how people in similar positions at other companies have used your product with success.
Sell with Success Stories
(click to see video)
Listen to how Rachel Gordon, account manager at WMGK, uses success stories with other customers as a selling tool in her presentations to new prospects.
Presenting to Groups
If customization is that straightforward with an individual buyer, why would you ever choose to sell to a group? Besides the fact that sometimes the nature of the sale demands it, selling to groups is also more efficient than selling to individuals. If you’re selling accounting software to a number of departments in an organization, rather than meeting individually with a decision maker from each department, you can save time by giving your sales presentation to a number of decision makers at once. Group presentations can also help you identify the decision makers in an organization if you aren’t yet sure who they are. By keeping an eye on group dynamics during the presentation you can usually observe the “pecking order” among members and identify the individuals in the group whose opinions hold the most leverage.
Additionally, group presentations can be a way to win greater support for your sale. If you know one or two people in an organization who are excited about your product, you can allow their enthusiasm to influence others in a group setting.Gary M. Grikscheit, Harold C. Cash, and Clifford E. Young, The Handbook of Selling: Psychological, Managerial, and Marketing Dynamics, 2nd ed. (Hoboken, NJ: Wiley-Blackwell, 1993), 165. Recall Selena Lo of Ruckus Wireless, who finds the “fox” within each of her target organizations and leverages his support of her product to sway the group buying decision.
If you know what is at stake for each member of the group, you will be able to facilitate the discussion during your presentation much more effectively. This is why it’s important to gather information about everyone who will attend your sales meeting. Again, think of Lo’s method, where she reads each group member’s bio and googles their names before going into a group meeting. Find out the individual’s needs within the organization. What is her status? How does she perceive the urgency of the problem you want to solve? Does she have any ego involvement in the product or service?Barton A. Weitz, Stephen Byron Castleberry, and John F. Tanner, Selling: Building Partnerships, 5th ed. (New York: McGraw-Hill, 2003), 264. (For instance, an accountant in the organization might feel threatened by new accounting software if it replaces part of her current role.) This will help you understand the most important concerns you will need to address in the presentation, and if certain parts of your presentation apply more directly to certain members of the group, you can direct those parts specifically at those individuals.
Keep in mind that people act differently in group settings than they do when you are interacting with them alone, so finding out about individual members’ personalities is less important in group presentations. Instead, adjust your presentation to the dynamics in the room. Watch the group for nonverbal cues; when one member is talking, observe how others react to see whether or not they support what she’s saying.Barton A. Weitz, Stephen Byron Castleberry, and John F. Tanner, Selling: Building Partnerships, 5th ed. (New York: McGraw-Hill, 2003), 265. If the energy in the room feels low, or if you get the sense that the group is getting restless, consider moving on to the next part of your presentation or changing tactics.
Sometimes you won’t know who or how many people you will be presenting to beforehand, so you won’t be able to research the individuals. However, it’s always a good idea to ask when you call to schedule your meeting. You may be able to find out information that your contact at the organization wouldn’t otherwise volunteer.
Group Presentations
(click to see video)
Hear about how to use a group presentation to your “unfair” advantage in this video:
The Right Place
You also might not know where your presentation will happen. If you know you’ll be presenting to your prospect at his office or in a conference room at his company, you won’t have control over the environment. What happens if your prospect has reserved a meeting room and when you arrive there are no empty walls on which you can project the PowerPoint presentation you brought along? When you know you’ll be presenting in an unfamiliar environment, make sure to have a contingency plan in place. If slides or other multimedia equipment are central to your presentation, talk to someone at the company to make sure you’ll be able to use the equipment. And if this fails, be ready to rely on your handouts, product samples, or the good old whiteboard to carry the presentation through.
Of course, in other situations, you will have control over the environment. In real estate, for instance, the presentation takes place inside the product. In retail, the presentation generally happens at your store. And there are other selling situations in which the prospect will come to your office or a conference room at your company or where you will meet at a “neutral” location like a rented meeting space.John Chapin, “Sales Presentations—How Location Can Affect Your Presentation and What to Do,” CompleteSelling.com, blog post, March 14, 2008, www.completeselling.com/members/completeselling/blog/VIEW/00000009/00000076/Sales-Presentations---How-Location-can-affect-Your-Sales-Presentation- and-What-to-Do.html (accessed May 16, 2010). Here are a few guidelines to follow, depending on the environment in which you’ll be presenting.
Your Place of Business
When the prospect comes to you, treat her like you would treat a guest in your home. Make sure you set up any presentation materials well in advance and have refreshments set out in the conference room or your office. Think about ways you can add personal touches—for instance, a sign with the prospect’s name on it (“[Your company name] welcomes [prospect’s company name]”), or, for a group presentation, information packets at each person’s seat with his or her name on the front. Sales professional John Chapin suggests having small items on hand that you can give to your prospect, such as pens or calculators with your company logo on them.John Chapin, “Sales Presentations—How Location Can Affect Your Presentation and What to Do,” CompleteSelling.com, blog post, March 14, 2008, www.completeselling.com/members/completeselling/blog/VIEW/00000009/00000076/Sales-Presentations---How-Location-can-affect-Your-Sales-Presentation- and-What-to-Do.html (accessed May 16, 2010). Small, thoughtful details can make an important difference.
A Neutral Location
If you are giving your presentation in a neutral location like a rented conference room you have the freedom to set up and work out any technical bugs well beforehand. When Keith Waldon of Earth Preserv was preparing for the presentation that secured his biggest customer, JCPenney, he rented a boardroom in a building near JCPenney’s corporate headquarters. He opted for the rented space so that he could pull out all the stops for the presentation. “I wanted to catch JCPenney by surprise,” Waldon says.
When the five executives arrived, Waldon had set up multimedia equipment for video, sound, and slides. He had placed a thick binder of presentation materials (including television storyboards, magazine advertisements, and product comparisons) at each executive’s seat with his name and the JCPenney corporate logo embossed on the front. Besides the conference room, Waldon had also rented an empty storefront in the same building, and halfway through the presentation, he took his customers to see the retail window display he had created there to look like one JCPenney might use to display Earth Preserv products in their stores.Susan Greco, “Anatomy of a Launch: The Five-Hour Multimedia Sales Presentation,” Inc., October 1, 1995, www.inc.com/magazine/19951001/2441.html (accessed May 16, 2010).
Since you will have time to set up beforehand at a rented location, you can treat the presentation the way you would treat a presentation at your home office. Bring refreshments, set up any multimedia equipment well in advance, and arrive early to make sure everything is in working order at the facility. Make sure you know the name of the facility’s contact person; you can call her several days ahead of time to find out what equipment she has at on hand and what you will need to bring.John Chapin, “Sales Presentations—How Location Can Affect Your Presentation and What to Do,” CompleteSelling.com, blog post, March 14, 2008, www.completeselling.com/members/completeselling/blog/VIEW/00000009/00000076/Sales-Presentations---How-Location-can-affect-Your-Sales-Presentation- and-What-to-Do.html (accessed May 16, 2010).
Your Prospect’s Place of Business
When you deliver your presentation at your prospect’s location, you won’t have the luxury of extensive setup time, and you may find that you have to adapt to the space and resources on hand. However, there are a few things you can do to make a good impression and ensure that things go as smoothly as possible:
• Arrive early and set up any technology you plan to use so that you can minimize the chance of something going wrong.
• When it’s possible, call ahead to find out about the space in which you will be presenting and the materials that will be available to you.
• Let your prospect know how long you will need to set up—particularly if you are using multimedia equipment.
• When you arrive, the first person you interact with will probably be the receptionist. Introduce yourself and let her know that the customer is expecting you.
• In addition to your presentation items, consider bringing food, coffee, or small giveaway items.
• In B2B sales, if your presentation will be around the lunch hour, it’s often customary to offer to take your prospect to lunch before or after the meeting.John Chapin, “Sales Presentations—How Location Can Affect Your Presentation and What to Do,” CompleteSelling.com, blog post, March 14, 2008, www.completeselling.com/members/completeselling/blog/VIEW/00000009/00000076/Sales-Presentations---How-Location-can-affect-Your-Sales-Presentation- and-What-to-Do.html (accessed May 16, 2010).
Webinars and Video Conferences
So how do you give a sales presentation if your prospect lives across the country, but you have a limited budget for travel? Unless there is a good chance that a prospect will become a key customer, it usually isn’t practical for a salesperson to travel long distances to make one presentation. However, thanks to improved technology, it’s becoming increasingly common for salespeople to address this problem using Webinars, video conferences, and online meetings. These technologies are allowing companies to reach more prospects in less time and to reach prospects internationally and across long distances.
Video Clip
Remote Presentations
Learn more about how and why salespeople are now using the Web to make sales presentations.
www.webex.com/overview/index.html
Of course, there are some drawbacks to giving sales presentations through video conferencing rather than in person. For one thing, it’s always easier to establish rapport with your prospect if you’re able to have a face-to-face interaction. Video conferences offer the benefit of visuals, so you and your prospect can read one another’s body language and visual cues, but this is not a complete substitute for sitting in the same room with someone. Additionally, since the presentation relies entirely on technology—both on your end and on the prospect’s end—there is a greater chance that a technological malfunction could prevent the presentation from working.
In-person presentations are still the most effective and personal method, so whenever you are able (and when it is practical) to give a face-to-face presentation, this is your best option. However, technology keeps improving, and online meetings and video teleconferences are becoming more successful as an alternative method all the time.“Sales Trends: Electronic Sales Presentations,” KnowThis.com, www.knowthis.com/principles-of-marketing-tutorials/personal-selling/selling-trends-electronic-sales-presentations (accessed May 16, 2010). Depending on your selling situation, this is something you might consider. As online sales strategist Joanna Lees Castro points out, video conferencing can be almost as effective as an in-person meeting in a number of selling situations, and it is certainly a better, more personal approach than e-mail or telephone.Joanna Lees Castro, “Using Video Conferencing to Host an Effective Online Sales Presentation—6 Best Practice Tips,” EzineArticles, ezinearticles.com/?id=1316495 (accessed May 16, 2010).
Even though video conferencing feels different from in-person communications, you should essentially treat your online meetings the way you would treat any sales call. Keep in mind that nonverbal communication has a strong influence on interactions—and, especially with good technology, your customer can see you clearly. Pay attention to your body language and facial expressions, and avoid personal gestures (like playing with your hair or scratching an itch).“Video Conferencing Etiquette Checklist,” Manage Smarter, June 8, 2009, www.presentations.com/msg/content_display/training/e3i0fe06f39ca140432cc75be4595e2c6e1 (accessed May 16, 2010). Dress professionally, plan your agenda carefully, and make sure to prepare and get your materials set up ahead of time. If you are conferencing from a location other than your office, arrive early to make sure the technology is set up to run smoothly for your presentation.
It is also important to resist the temptation to multitask during your video conference. Close down any other applications you might have open on your computer, clear off your desk, and make sure you will not be interrupted until the call is over. Mute any cell phones and close the door to the room in which you are presenting. Give your customer your full attention. While this level of focus is a given on your end, unfortunately, you can’t always be certain that your prospect will give a video conference meeting his full attention by minimizing distractions. For this reason, it is especially important to have a clear agenda that you follow closely. Keep your presentation brief, and be aware that you will have to work harder to hold your prospect’s attention. Live interaction from your audience is critical to make sure your participants are engaged.
Besides a greater likelihood of distraction, there are a few other extra considerations to keep in mind in a video conference situation. Sales and Management magazine notes that privacy is expected during a video conference, so if you want to record part of your presentation, it is important to ask your prospect for permission.“Video Conferencing Etiquette Checklist,” Manage Smarter, June 8, 2009, www.presentations.com/msg/content_display/training/e3i0fe06f39ca140432cc75be4595e2c6e1 (accessed May 16, 2010). When the presentation is over, Joanna Lees Castro suggests closing the meeting with a clear call to action in which you include a wrap-up and well-defined next steps that you and your prospects should take. At the end of a conventional sales presentation, Lees Castro points out, next-step discussions can happen more organically, as the customer is walking you to the door, but this is obviously impossible in an online situation.Joanna Lees Castro, “Using Video Conferencing to Host an Effective Online Sales Presentation—6 Best Practice Tips,” EzineArticles, ezinearticles.com/?id=1316495 (accessed May 16, 2010).
The Right Tools
In the best sales presentations, the product or service comes alive. Try to see the presentation through your prospect’s eyes. What is the best way to capture his imagination? How will you tell the story that will make your product or service compelling? In what ways can you delight or surprise your customer? Few people know how to do this better than Dann Ilicic, CEO of Wow Branding. Wow, a small start-up, frequently outperforms big name competitors when vying for a prospect. Ilicic approaches each presentation with the same mind-set: you can’t bore your customer into buying from you, so why not dazzle them? One customer said the presentation Ilicic put together for his company couldn’t have been better: “Dann unquestionably knocked it out of the park compared with the other firms, and they were really high-end firms with spectacular portfolios.”Stephanie Clifford, “Fasten Your Seatbelts,” Inc., February 1, 2007, www.inc.com/magazine/20070201/features-sales-performance-ilicic.html (accessed May 16, 2010).
So how does Wow Branding wow its prospects? Ilicic’s approach offers three lessons:
1. Take customization to a new level. Ilicic says he and his team spend about fifty hours preparing for a sales presentation. They call low-level employees in the customer company, the company’s past customers, and companies that have chosen not to do business with the prospect to learn things the prospects might not even know about themselves. Glumac, an engineering firm in Portland, Oregon, and one of Wow’s customers, said Ilicic’s technique “was a brilliant move…because he wasn’t asking what our imagery should be”; instead, he researched to find out what the image already was.
2. Never miss an opportunity to delight. Ilicic likes to surprise his customers with the small things: like stamping green thumbprints throughout a proposal for an agricultural company—or, for a pharmaceutical company, handing out vitamin bottles on which he has replaced the label with a message about Wow. Sometimes he brings in a cake on which he reveals the suggested name for a new company. Because Ilicic’s intensive research allows him to understand his customers so well, he is able to perfectly match the wow factor to each prospect and make the product come alive.
3. Always make the presentation creative and fun. This technique engages the customer, even when the meeting agenda isn’t exciting itself. It also allows Wow to get around difficult or sensitive parts of the presentation. Rather than talking about Wow’s successes, Ilicic records customer testimonials about his company and plays these for his prospects. On another occasion, rather than potentially putting a prospect on the defensive by telling the company what their image should be, Ilicic told them that Wow had been assigned a branding project for their biggest competitor. He launched a multimedia presentation to show them their competitor’s branding overhaul, and by the end his prospects were asking themselves, “Why didn’t we think of that?” After the presentation, Ilicic revealed that he hadn’t actually made the campaign for the company’s competitor; it was for them.Stephanie Clifford, “Fasten Your Seatbelts,” Inc., February 1, 2007, www.inc.com/magazine/20070201/features-sales-performance-ilicic.html (accessed May 16, 2010).
So what techniques can you use to achieve these goals in your sales presentations? The tools you choose will depend on the situation and your presentation style. As Ilicic demonstrates, the possibilities are almost endless, but whatever tool you use, it is important to carefully consider your choice and how you can maximize its effectiveness.
PowerPoint Presentations
PowerPoint slides provide an easy way to organize your presentation and add helpful visuals. For many salespeople, PowerPoint is one of their go-to presentation tools. It can be an especially helpful tool for salespeople who are starting out and want the security of a clear framework from which to present. An added benefit is that it doesn’t take much technological know-how to put together a clean-looking PowerPoint demonstration.
On the other hand, not all presentation situations lend themselves to PowerPoint (e.g., conference rooms with no wall space on which to project or presentations given in the field), so if you plan to use this tool, make sure that you will be presenting in a space where you can make it work. Additionally, be aware of—and avoid—a number of common mistakes salespeople make when using PowerPoint that can ruin a presentation. As sales coach and author Anne Miller says, “Putting PowerPoint into the hands of some sales reps is like putting matches into the hands of some children.”Anne Miller, “Death by PowerPoint,” Sales and Sales Management Blog, February 22, 2008, http://salesandmanagementblog.com/2008/02/22/guest-article-death-by-powerpoint-by-anne-miller (accessed May 16, 2010). To maximize PowerPoint as a tool to successfully sell your story, use the tips in Figure \(5\).Anne Miller, “Death by PowerPoint,” Sales and Sales Management Blog, February 22, 2008, http://salesandmanagementblog.com/2008/02/22/guest-article-death-by-powerpoint-by-anne-miller (accessed May 16, 2010).
The following dos and don’ts can also be helpful as you are creating a PowerPoint presentation.
• Don’t turn down the lights. It takes the focus away from you, and it can put people to sleep.
• Don’t go overboard with technological gimmicks. Fancy fades and clever add-ons will only distract from you and from the content of your presentation.Jim Meisenheimer, “How to Use PowerPoint During Sales Presentations,” EvanCarmichael.com, www.evancarmichael.com/Sales/407/How-To-Use-PowerPoint-During-Sales-Presentations.html (accessed May 16, 2010).
• Don’t hide behind your computer screen when using PowerPoint; make sure you face your audience and make eye contact. This can be a temptation when the computer is set up on a podium close to eye level.
• Don’t fill your slides with words. Use bullet points, separate each point with white space, and cut out any unnecessary words you can.
• Don’t bore your audience with visual sameness. Slide after slide of bulleted lists gets monotonous; visuals and charts have a stronger impact.Anne Miller, “Death by PowerPoint,” Sales and Sales Management Blog, February 22, 2008, http://salesandmanagementblog.com/2008/02/22/guest-article-death-by-powerpoint-by-anne-miller (accessed May 16, 2010).
• Do make your slides easy to read. Avoid small fonts, visual clutter, and dark text against dark backgrounds.Jim Meisenheimer, “How to Use PowerPoint During Sales Presentations,” EvanCarmichael.com, www.evancarmichael.com/Sales/407/How-To-Use-PowerPoint-During-Sales-Presentations.html (accessed May 16, 2010).
• Do replace descriptive headlines with headlines that sell. No one cares about a headline that describes what’s already on the page.Anne Miller, “Death by PowerPoint,” Sales and Sales Management Blog, February 22, 2008, http://salesandmanagementblog.com/2008/02/22/guest-article-death-by-powerpoint-by-anne-miller (accessed May 16, 2010). For example, rather than writing “Our Statistics” at the top of the page, write “See Significant Savings in the First Year.”
• Do use the 10/20/30 rule: Make sure you limit your slides to 10 or fewer. Focus on the things you want people to remember, rather than overwhelming them with information. Give yourself 20 minutes to go through your 10 slides. Any more than this and you will reach the limit of your audience’s attention span. Finally, use only 30-point or larger font size so that your audience can clearly read what you’ve written.Jim Meisenheimer, “How to Use PowerPoint During Sales Presentations,” EvanCarmichael.com, www.evancarmichael.com/Sales/407/How-To-Use-PowerPoint-During-Sales-Presentations.html (accessed May 16, 2010).
10/20/30 Rule
Guy Kawaski, best-selling author, venture capitalist, and entrepreneur, created this rule and describes it in this video.
• Do remember that that PowerPoint is only an aid. “You are the star,” says communications consultant Ronnie Moore. “The media and visuals support you.”Geoff William, “The Perfect Presentation: Technology,” Entrepreneur, July 13, 2007, http://www.entrepreneur.com/marketing/marketingbasics/article181582.html (accessed May 16, 2010). Use dynamic speaking strategies, move around, keep your audience involved; don’t let your technology take over.
Brochures, Premiums, and Leave-Behinds
It is usually expected that you will have printed material to give your audience during a presentation. In addition to a printed supplement to your PowerPoint presentation (i.e., something that conveys the same information as your slides and on which your audience can take notes), you might decide to bring along brochures with information about your company, products, and services. What are the benefits of brochures? According to sales expert and author Geoffrey James, in some situations you need a brochure to make your firm look serious. However, James lists “I promise to read your brochure” as one of the top ten lies customers tell sales reps. His conclusion: the brochure might gain you credibility, but it probably won’t get read.Geoffrey James, “Top 10 Lies Customers Tell Sales Reps,” BNET, April 23, 2009, blogs.bnet.com/salesmachine/?p=2323&page=2 (accessed May 16, 2010). Don’t rely completely on brochures because they won’t be a focal point of your presentation.
Sometimes a brochure can work as a reminder about you and your company after you’ve left, but this is assuming your customer doesn’t throw the brochure away. When it comes to reminders, a better bet is leaving something functional that your customer will actually use regularly. These reminder objects—calendars, refrigerator magnets, pens, or mouse pads labeled with your company name—are called premium leave-behinds and are a proven method of reminding customers you exist.Brad Sugars, “Building Repeat Business from Day 1,” Entrepreneur, May 22, 2007, http://www.entrepreneur.com/startingabusiness/startupbasics/startupbasicscolumnistbradsugars/article178724.html (accessed May 16, 2010). Almost all salespeople bring some sort of brochure or premium leave-behind on their sales calls.
Samples and Demonstrations
There is almost no better way to make your story come to life for your customer than letting him experience it for himself. Think of television courtroom dramas: when the lawyer is making her final statement to the jury and she wants to pull out all the stops, what does she do? She doesn’t just give the jury the facts or tell them the version of the story she wants them to believe—she brings the story to life; she puts the gun in the defendant’s hand; she brings out the pictures of the stab wounds. Think about this when you plan your sales presentation. During the presentation, you can bring your story to life by offering product samples for your prospects to try or by running demonstrations that let them see for themselves what your product can do. When winemakers sell their products to large distributors, they don’t just bring in descriptions of their wines for the buyers to read; they offer tastings so buyers can experience the product. When caterers want to sell their services to someone who is planning a wedding, they bring in samples from their menus, so the customer can say, “Wow this pasta really is delicious!” Or think of Keith Waldon of Earth Preserv who didn’t just tell JCPenney, “We can make displays of our environmentally friendly products for your store windows;” instead, he set up a real shop window display so his prospects could see their place in his story.
Power Selling: Lessons in Selling from Successful Brands
Sell to Someone Unexpected
For the founders of Cranium, Inc., maker of the popular Cranium board game, playing is believing. When the company first launched in 1998, they knew that 50 percent of board games failed in their first year. Cranium’s strategy? Avoid the traditional board game buyers—toy stores—and sell to someone unexpected. Cranium’s founders managed to get an introduction to Howard Schultz, CEO of Starbucks, and they arrived at his office with a game board and challenged him to a match. After playing a few rounds, Schultz decided this was just the game Starbucks had been looking for—something that would support coffeehouse culture—and Cranium, Inc., had its first major sale.
Next on Cranium’s list? Barnes & Noble Booksellers. The company’s founders scheduled a meeting with Terese Profaci, the bookstore’s director of gift merchandising, whose boss told the sales reps on the way in, “I don’t know why you’re here. We don’t sell games.” Still, Profaci’s boss had her play a round of the game with some employees at corporate headquarters, and in the end, Barnes & Noble was won over.Julie Blick, “Inside the Smartest Little Company in America,” Inc., January 1, 2002, www.inc.com/magazine/20020101/23798.html (accessed May 16, 2010).
Besides bringing your story to life, there are a number of other good reasons to use demonstrations:
• To educate your prospect. If you are selling a complex product, such as a highly involved software program, the best way to help your customer understand how it works is to show her.
• To involve your prospect. Let him find the results for himself. Just as car shoppers get to take the wheel in a test drive—and this often makes the difference between a decision to buy or not to buy—customers who use your products for themselves are more likely to make a personal connection with it. A salesperson selling insulated windows, for instance, might place a piece of glass in front of a heat lamp and ask her customer to put out his hand and feel the heat. Then the salesperson might substitute the sheet of glass for a window sample. “Now put out your hand,” she will tell the customer. “Can you feel how this window is going to keep the elements out and save you money on your energy bills?”EDTM, Inc., “4 Steps to Close More Sales,” www.solarstop.net/edtm/sales_demonstration.htm (accessed May 16, 2010).
• To prove the performance of your product.EDTM, Inc., “4 Steps to Close More Sales,” www.solarstop.net/edtm/sales_demonstration.htm (accessed May 16, 2010). Of course, you can tell your prospect “our air purifiers are quieter than the leading model, and they take up less space in your home.” But if you bring your air purifier to the presentation and set it next to the leading model, and if you ask your prospect to turn both machines on, he can see for himself that your product is smaller, and he can hear for himself that it makes less noise.
Give Them the Numbers: Cost-Benefit Analysis and ROI
When you present your solution to the customer, especially in B2B sales, closing the sale usually depends on whether the cost of your solution is offset by the value it delivers.Gerald L. Manning and Barry L. Reece, Selling Today: Creating Customer Value, 9th ed. (Upper Saddle River, NJ: Prentice Hall, 2004), 256. If you can quantify your solution using cost-benefit analysis and ROI (return on investment) analysis, you can help your customer determine whether a project or purchase is worth funding.
A cost-benefit analysis asks the question “Will this purchase save more money in the long run than it costs?”David H. Miles, The 30 Second Encyclopedia of Learning and Performance (New York: AMACOM, 2003), 139–40. Imagine you are selling an energy-efficient commercial dishwasher to a pizza kitchen. The dishwasher costs \$3,000, but average cost savings per year are \$800 in energy bills and \$200 in water usage: a total of \$1,000.Energy Star, “Commercial Dishwashers for Consumers,” U.S. Environmental Protection Agency and U.S. Department of Energy, www.energystar.gov/index.cfm?c=comm_dishwashers.pr_comm_dishwashers (accessed May 16, 2010). Your dishwashers are guaranteed to last a long time; in fact, you offer a five-year warranty on any purchase. At a savings rate of \$1,000 each year, your customer will have saved \$5,000 in energy and water expenses by the time his warranty expires. Based on this information, you present this cost-benefit analysis to your prospect:
\$3,000 = cost (initial investment)cost savings – initial investment = benefit\$5,000 – \$3,000 = \$2,000
In this case, the cost savings is \$1,000 per year times five years for a total of \$5,000, minus the initial investment of \$3,000, means that there is a benefit of \$2,000.
In other words, the dishwasher has a three-to-two cost-benefit ratio over five years (\$3,000 in cost to \$2,000 in benefit). You can tell him, “This purchase will save you money in the long run. After you make back what you spent on the dishwasher in cost savings, you will continue to save \$1,000 each year.” Similarly, you can show your customer a return on investment (ROI) analysis. ROI shows the customer the return (profit or cost savings) compared to the investment he will make. In the case of the dishwasher, the ROI would be calculated by dividing the benefit (in this case \$2,000) by the cost of the product or initial investment (in this case \$3,000), then multiplying the result by 100, which would yield a 66 percent ROI after five years.
\$2000 (savings over five years) ÷ \$3,000 (initial investment) × 100 = 66% ROI
You can maximize ROI by cutting costs, increasing profits, or accelerating the rate at which profits are made.“Cost of Ownership, ROI, and Cost/Benefit Analysis: What’s the Difference?” Solution Matrix, www.solutionmatrix.com/tco-roi-cba-difference.html (accessed May 16, 2010). Some businesses have a minimum ROI that must be met before a purchase can be approved. While you might be able to learn this information in your preapproach, it is more likely that you will have to discuss minimum ROI with your customer during the sales presentation. You might present your solution and find out more about your customer’s specific needs (including budget constraints and minimum ROI) during the first sales presentation and then write up a proposal in response to your findings, which you deliver during a second presentation.
Key Takeaways
• Presenting to individuals requires a different set of skills and techniques than presenting to groups, so make sure you have a clear strategy for your presentation that takes the size of your audience into account.
• When presenting to an individual, keep your prospect’s personality in mind and adapt your approach accordingly. Take his position and responsibilities in the company into account in the way you present your solution.
• Selling to groups can be a more efficient presentation method, and sometimes it is required in your customer organization. When conducting a group presentation, take group dynamics into account, keeping in mind that people act differently in group situations than they do in one-on-one interactions.
• When you are delivering your presentation at your place of business or in a neutral location (like a rented space), treat the customer as you would treat a guest in your home. Set up refreshments and supplies well ahead of time so that you are well prepared when the prospect arrives.
• When you are presenting at your prospect’s place of business, try to find out about the presentation venue beforehand—but be prepared to adapt if your prospect doesn’t have the equipment or setup you were expecting. Arrive early so that you have time to set up.
• If your presentation is given as a Webinar or video conference, treat the presentation as you would treat an in-person interaction. Dress professionally and set up ahead of time. Make sure to minimize distractions.
• When delivering a PowerPoint presentation, keep your slides brief, uncluttered, and easy to read. Don’t let the technology overshadow you, the presenter.
• There is almost no better way to bring your product to life than by using samples or demonstrations to get your prospect involved.
• Your customer will expect you to bring a cost-benefit analysis or ROI analysis as a way to quantify your solution.
Exercise \(1\)
1. You are giving a presentation to a busy manager who initially tells you that she can only give you thirty minutes of her time. She seems brisk and businesslike at first, but when you are in her office, you notice a picture of her son in a soccer uniform and mention that your kids are involved in soccer. After this, she relaxes and begins discussing her children at length. Keeping in mind that (a) you have an agenda to get through but (b) establishing a connection is important to you, and you want to take your cue from your prospect, how do you respond, and why?
2. You are giving a presentation to a group and notice that one member of the group is more vocal than others and tends to dominate the conversation. What are some strategies you could use to make sure that other members of the group have a chance to participate and contribute their opinions?
3. Choose a product or service and prepare a short sales presentation that includes a demonstration. What other items do you need besides the product or service to perform the demonstration (e.g., Internet service for software; water for instant coffee; plates, silverware, and napkins for food products)? How is the product demonstration integrated into your presentation? How do you use the demonstration to engage the prospect with the product or service?
4. Assume you are selling environmental design consulting, and an important part of your sales presentation involves using your company Web site to demonstrate previous projects you have completed, interactive customer surveys, and your company’s brand image. However, when you arrive at your customer’s place of business to set up your presentation, you learn that the Internet has been down all morning and may not be back up until the next day. What could you have done to prepare for this sort of unforeseen problem in advance?
5. Find a PowerPoint presentation you have created for another class—or if this is unavailable, find a PowerPoint presentation online; Slideshare is a good resource: http://www.slideshare.net. Offer a critique of the presentation based on the information you learned in this chapter.
6. Assume you are a sales rep for an Internet advertising company. Your prospect, an online hardware retailer that specializes in compression pumps, is concerned about making the investment for Internet advertising. You want to incorporate the ROI into your presentation. If the prospect spends \$90,000 in advertising, it will generate 120,000 clicks to the company Web site. At a 2 percent conversion rate (2 percent of the customers who visit the site make a purchase), that is 2,400 orders. If each order is \$230, the sales generated from the online ad would be \$552,000. What is the prospect’s ROI (show your math)? How would you incorporate this ROI into your sales presentation?
7. Imagine that you are selling high-end electronic equipment. Your prospect has agreed to purchase a laptop for \$800. Now you tell him about the benefits of purchasing the service agreement, which includes free battery replacement and computer cleaning every year for three years for only \$120. A replacement battery costs \$200, and a computer cleaning costs \$85. How much will the customer save if he purchases the service agreement assuming he needs to replace the battery and have the laptop cleaned once a year? How would you incorporate this into your sales presentation? | textbooks/biz/Marketing/The_Power_of_Selling/10%3A_The_Presentation_-_The_Power_of_Solving_Problems/10.04%3A_Making_Your_Presentation_Work.txt |
Learning Objectives
• Understand SPIN and how to use it during the sales presentation.
In 1988, Neil Rackham and his company Huthwaite, Inc., researched more than 35,000 sales calls, observing successful and experienced sales professionals doing what they do best. In the process they disproved a number of popular myths about the selling process, and they developed a sales model of their own, which they called SPIN selling.Neil Rackham, The SPIN Selling Fieldbook (New York: McGraw-Hill, 1996), 8; Greg Woodley, “SPIN Selling Is Good,” SellingandPersuasionTechniques.com, http://www.sellingandpersuasiontechniques.com/SPIN-selling.html (accessed May 16, 2010). Today sales professionals around the world incorporate the SPIN selling model into their sales process with great success—and if you learn a few simple principles, you can too. The following section describes SPIN selling in a nutshell.
What Is SPIN Selling?
SPIN works from the theory that relationship selling is customer-centric. It requires you to adapt your selling process to your customer, and it delivers personal solutions. To make this work, you have to ask your buyer a lot of questions, let him do most of the talking, and give his responses your full attention.
In the SPIN model, there are four components of a sales call: opening, investigating, demonstrating capability, and obtaining commitment. SPIN gets its name from the four kinds of questions that take place during the investigation stage: Situation, Problem, Implication, and Need-payoff.
With smaller sales, these four components of the sale (opening, investigating, demonstrating capability, and obtaining commitment) often happen sequentially and in a short period of time; a customer might walk onto your car lot and commit to buying a car from you an hour later. But often in business-to-business (B2B) sales, especially complex ones, you will incorporate SPIN components into a number of the steps in your selling process. For instance, you will do some investigation during your preapproach, and you might make an early presentation in which you open, investigate, and demonstrate capability. Because larger sales take more time, you won’t close the sale at the end of your first presentation, but you might get a commitment from your customer to move the sale forward. SPIN selling is not a rigid, step-by-step model; rather it provides an effective, flexible framework for customer centered selling.Neil Rackham, SPIN Selling Fieldbook (New York: McGraw-Hill, 1996), 38.
Opening
According to Rackham, the opening of the sales call is not the most important part, but it does pave the way for the important steps that come after.Neil Rackham, SPIN Selling Fieldbook (New York: McGraw-Hill, 1996), 139. At the beginning of every call, you want to set the preliminaries and make any necessary introductions. (In larger B2B sales, you usually won’t spend very long on introductions because 95 percent of the time you will be meeting with an existing customer or a prospect you have already met.)Neil Rackham, SPIN Selling Fieldbook (New York: McGraw-Hill, 1996), 40. If you are following up on an earlier sales call, it’s important to recap the conclusions of your last discussion: “The last time we spoke, we talked about pricing and setting a timeline, and you agreed that you would like to move the sale forward if we could put together a proposal that matched your budget and would meet your deadlines.” Then, most important, you want to begin the conversation by getting your customer’s agreement to let you ask him some questions.Neil Rackham, SPIN Selling Fieldbook (New York: McGraw-Hill, 1996), 144. This builds rapport and establishes a buyer centered purpose for your call.“SPIN Selling,” review of SPIN Selling, ChangingMinds.org, http://changingminds.org/books/book_reviews/spin.htm (accessed May 16, 2010).
Investigation
Investigation—asking questions to uncover your buyer’s needs—is at the heart of SPIN selling. This is the stage during which you ask the types of questions that give SPIN its name: situation, problem, implication, and need-payoff. Here’s how each of these types of questions works during the sales presentation.
Situation Questions
Situation questions deal with the straightforward facts about the buyer’s existing situation and provide a starting place for understanding your buyer’s needs.Greg Woodley, “SPIN Selling Is Good,” SellingandPersuasionTechniques.com, http://www.sellingandpersuasiontechniques.com/SPIN-selling.html (accessed May 16, 2010). If you ask too many situation questions, you risk boring your prospect and damaging your credibility, so ask situation questions sparingly. If you do careful research before your sales call, you should find out most of the basic information about your customer’s current situation before your meeting so that the situation questions you ask are only the ones that will provide information you aren’t able to track down elsewhere.Neil Rackham, SPIN Selling Fieldbook (New York: McGraw-Hill, 1996), 76; Eric Wolfram, “How to Sell—SPIN Selling,” Wolfram, http://wolfram.org/writing/howto/sell/spin_selling.html (accessed May 16, 2010). For instance, if you are selling Internet connectivity, you might ask your buyer, “Which of your offices are currently using DSL?”
Customer: Our four branch campuses use DSL, but our main offices downtown use a cable service.
You: Oh, they use cable? Who is their provider?
Customer: Ajax Communications. We’ve been with them for about two years.
You: I understand Ajax sometimes offers their service on a contract basis. Do you currently have a contract with Ajax?
Customer: We had a contract, but that ended a couple of months ago.
Problem Questions
You already know that your prospect will only be motivated to buy if she recognizes she has a need. Asking problem questions helps customers understand their needs, and ultimately it paves the way for you to propose a solution that seems beneficial to your customer.Neil Rackham, SPIN Selling Fieldbook (New York: McGraw-Hill, 1996), 90. Problem questions are the most effective in small sales: “Was limited storage space ever an issue with your last computer? How much has the size and weight of your current laptop affected your ability to carry it with you?” But in B2B sales it is still important to ask a few problem questions so that you and your buyer share an understanding of the problem or need.Neil Rackham, SPIN Selling Fieldbook (New York: McGraw-Hill, 1996), 93. Sometimes it is tempting to jump right into presenting the benefits of your solution, but keep in mind that your prospect might not always see his problem right away, even if it is already evident to you.“SPIN Selling,” review of SPIN Selling, Changing Minds Book Reviews, http://changingminds.org/books/book_reviews/spin.htm (accessed May 16, 2010). Imagine you sell tractors. To understand the difficulties your prospect faces with his current machines, you could ask problem questions like “How much does it cost to maintain your current farm machinery?” “How often do your tractors break down?” and “Who is usually responsible for doing the maintenance work?”
Implication Questions
In larger sales, implication questions are closely linked to success because they increase a prospect’s motivation to seek change. Implication questions uncover the effects or consequences of a prospect’s problems. These questions are especially effective when your prospect is a decision maker whose success depends on understanding the underlying causes of a problem and its potential long-term consequences.Neil Rackham, SPIN Selling Fieldbook (New York: McGraw-Hill, 1996), 108. Say, for instance, your prospect has offices in five locations, but he only has IT staff at two of the locations. To help him understand the implications of this problem, you might ask questions like this:
You: If a computer crashes at one of your branch offices, who takes care of the problem?
Prospect: That depends. Our Bellevue and Redmond offices have their own IT people, but when we have a problem downtown or in North Seattle, we call someone from the east side offices to come fix it.
You: Wow, that must be a hassle for the IT people! How often do they have to drive out to another location for computer trouble?
Prospect: Usually not more than three or four times each week. If the problem isn’t an urgent one, the IT guys usually make a record of it so that they can fix it during their regular visits.
You: So your IT people have regularly scheduled maintenance visits that they make in addition to the occasional “emergency” trips?
Prospect: Yes. Someone from IT visits each of the three locations once a week to run maintenance and fix any issues that have come up since the last visit.
You: The travel time from Redmond to downtown is about half an hour each way, and it can take an hour during rush hour! Isn’t the commute from Redmond to your other locations even longer? In total, how much time and money would you guess your company invests in these maintenance trips each week?
Your buyer might have told you up front that the shortage of IT staff is a problem, but he might not yet realize all the implications of this problem (like higher costs, wasted time, and inefficiency). By asking this set of implication questions you have just asked, you are helping your prospect explicitly state a need (or needs) that you can solve for him.“SPIN Selling,” review of SPIN Selling, Changing Minds Book Reviews, http://changingminds.org/books/book_reviews/spin.htm (accessed May 16, 2010).
Need-Payoff Questions
Once you help your prospect uncover his specific needs, you can help him to discover a way out by asking how his problem could be resolved. These questions are called need-payoff questions. If you ask your prospect the right need-payoff questions, he will tell you how your solutions can help him; you won’t even need to spend much time talking about your product’s benefits because your prospect will have already convinced himself that your solution will be valuable to him.Neil Rackham, SPIN Selling Fieldbook (New York: McGraw-Hill, 1996), 128. For example, following the previous conversation about your customer’s IT problem you could ask “How would it help if the IT staff could fix at least half of your computer problems remotely?” or “How much time would you save if I could help you find a way to cut down on your IT support calls from the branch offices?”
Demonstrating Capability
When you present your solution, you can tell your customer about FAB, as discussed in Chapter 6.
• The product features, or what the product has: “This car has all-wheel drive, and the back seats fold down to expand the trunk.”
• Its advantages: “The all-wheel drive capability makes for better handling in ice and snow, and the ability to fold down the seats means you get a larger storage capacity than you would with other cars of its kind.”
• What the feature does and its benefits: “The all-wheel drive will give you peace of mind when your daughter drives the car in the winter, and the added storage capacity will be especially helpful for any odds and ends you need to transport during your upcoming move.” This includes what the features mean, or the ways in which your solution addresses your prospect’s acknowledged needs.Neil Rackham, SPIN Selling Fieldbook (New York: McGraw-Hill, 1996), 148.
All three methods demonstrate capability, but which method do you think moves you closer to a sale? If you guessed benefits, you’re right. SPIN selling is all about customization; when you are demonstrating capability, you want to show your prospect how your solution applies to the needs he has expressed. Listing a product’s advantages demonstrates how that product could be useful to anyone (a generic customer), but you don’t want to treat your buyer like a generic customer. OK, so the car you are selling has an excellent sound system that delivers a superior music-listening experience. But what if your prospect only ever listens to talk radio? If you go on at length about the advantages of the sound system, he won’t be impressed.
Rackham and his team concluded that salespeople who demonstrate capability by presenting benefits (rather than advantages) don’t have to deal with as many objections from their prospects. However, you can only demonstrate benefits successfully if you have asked the right questions to uncover your prospect’s specific needs. This is why the investigation stage is so important. Here are examples of some benefits you might share with your prospect:
Dr. Hogue, our software gives you the ability to organize large quantities of information (like those complicated medical records you mentioned) visually. If you use this software, it will be easy to identify relationships between patient’s medical histories so that you and your staff can save time whenever you have to perform a complicated diagnosis.
Ms. Lewis, you mentioned that you have a long commute to work each day, so I think the podcast versions of our training seminars will be a good solution for you. You can download them onto your iPod and listen to them on your way to work so that you can maximize your time and leave your evenings and weekends open to spend with your family.
Our custom engagement rings will allow you to choose an antique setting in the style you said your fiancée prefers and to pair this setting with a smaller stone that will fit your current budget.
Obtaining Commitment
In smaller sales, obtaining commitment is fairly straightforward: either your prospect decides to buy, or he tells you that he isn’t interested. In complex sales, on the other hand, fewer than 10 percent of calls have one of these two outcomes. It might take several years before your prospect agrees to purchase your solution, so a sales call that ends without a sale is in no way a failure. In between your first sales call and your prospect’s decision to buy, you will have a number of calls in which you either decide to move the process forward, terminate the process, or continue the process without an advance. Any time your prospect ends a call by agreeing on an action that moves you closer to the final sale, you have experienced a successful outcome.Neil Rackham, SPIN Selling Fieldbook (New York: McGraw-Hill, 1996), 42–43.
In the SPIN model there are three steps to obtaining commitment:
1. Check whether you have addressed key concerns.
2. Summarize the benefits you presented.
3. Propose a commitment that will move the sale forward.Neil Rackham, SPIN Selling Fieldbook (New York: McGraw-Hill, 1996), 44.
Consider this example:
You: So what I understand from our discussion is that you are concerned your image has become outdated, and you want your television advertising to appeal to a younger generation?
Prospect: Yes, that’s the biggest issue we’re facing right now.
You: As I mentioned earlier, Rockstar Marketing has successfully overhauled the brand image of a number of well-known retailers, and we think we could do the same for you by creating the youthful image you are looking for.
Prospect: That sounds like it might be a good fit, but I’d have to get the approval of our marketing committee before I could give you an answer.
You: Then what I’d like to do, if it’s OK with you, would be to write up a more specific proposal. If you could agree to arrange a meeting between our sales team and the members of your marketing committee in about two weeks from today, we could discuss the proposal options at the meeting.
Prospect: All right, that sounds like a good plan. I’ll have my secretary arrange the meeting and give you a call to confirm the day and time.
The commitment you propose at the end of the call will depend on your precall objectives. According to Rackham and his team, the most effective precall objectives are those that include actions on the part of the customer, such as “get the prospect to agree to call two of your past customers” or “get the prospect’s list of vendor selection criteria.”Neil Rackham, SPIN Selling Fieldbook (New York: McGraw-Hill, 1996), 45.
Why Use the SPIN Model?
In relationship selling, the idea of a sales “presentation” can be misleading. To deliver customized value to your prospect, you have to understand his needs and make sure that you are in agreement with him about a solution he could use. This means the sales presentation is a two-way communication. When you make the effort to listen to your prospect this way and when you work to understand his needs, not only will you close more sales, but you will also build stronger, lasting customer relationships. Your prospect will come to trust you and to rely on you as a problem-solving expert.
Key Takeaways
• SPIN selling is a four-step model that relies on the theory that successful selling is customer centered and offers customized solutions to your prospect’s problems.
• There are four steps to a SPIN sales call: opening, investigation, demonstrating capability, and obtaining commitment.
• The opening stage builds rapport and establishes a buyer centered purpose for your call.
• The investigation stage is at the heart of the SPIN model. The goal of this stage is to ask questions that will uncover your buyer’s needs.
• There are four types of investigation questions: Situation, Problem, Implication, and Need-payoff (SPIN).
• In demonstrating capability, you explain the benefits of your solution by showing your prospect how your product or service meets his explicit needs.
• In obtaining commitment, you get your prospect to agree to advance the sale, continue the sale without advancing, or make a purchase.
Exercise \(1\)
1. Assume you own a business that rents out retail space in a downtown area. You have found out from your prospect, the owner of a bagel shop, that his current store location is on a side street that doesn’t get much foot traffic. List at least one each of the following kinds of questions that may help uncover his unstated needs: situation, problem, implication, and need-payoff. Discuss how these questions would work during the investigation stage of your SPIN selling presentation?
2. Envision a selling situation between a travel agency that offers a variety of discount packages for business prospects and a consulting firm whose employees travel frequently for business. As a salesperson for the travel agency, what specific information would you need to know about your prospect’s current situation? How much of this information do you think you could find through research? What specific situation questions would you be likely to ask?
3. Your firm offers state-mandated alcohol handler’s training for restaurant employees, and you are making a sales call on a manager who has just opened three large restaurants and will be hiring a staff of over seventy-five servers and bartenders. Your training is more comprehensive than that of your competitors because it includes over five hours of training per employee. Many restaurant companies opt to leave the training up to the individual restaurants, which leads to inconsistency and lack of implementation. Companies may not realize how risky this is; the fine for improper implementation is \$10,000 per restaurant. Prepare a presentation that includes the four stages of SPIN selling (opening, investigation, demonstrating capability, and gaining commitment). Include the four types of questions during the investigation stage (situation, problem, implication, and need-payoff).
4. Assume you work for Apple in their B2B division and you are selling iPhones to a major medical supply company for their employees, including their five-hundred-person sales force. They are currently not using smartphones but realize they have a need for them for their employees to stay in touch throughout the day and to access the Internet while they are away from the office. The prospect is also considering other brands of smartphones. Create a short sales presentation for the iPhone using the four stages of SPIN selling (opening, investigation, demonstrating capability, and gaining commitment). Include the four types of questions during the investigation stage (situation, problem, implication, and need-payoff). | textbooks/biz/Marketing/The_Power_of_Selling/10%3A_The_Presentation_-_The_Power_of_Solving_Problems/10.05%3A_How_to_Use_SPIN_Selling_in_Your_Sales_Call.txt |
Learning Objectives
• Learn the five steps of a successful sales presentation.
As you have probably realized by now, there are many things to keep in mind when planning and executing a sales presentation. It can be enough to overwhelm even the most experienced sales professional.
Video Clip
Present Successfully
What are the most important ingredients to a successful presentation? Watch the following video to hear several experienced salespeople share their perspectives.
www.inc.com/inctv/2007/07/making-a-sale.html
While there is no one magic formula that will make your presentation come to life, successful presentations generally have a number of elements in common. Thinking of your presentation in terms of the following five steps will help you to plan and execute it with greater ease and success. Before the presentation, it’s a good idea to ask your prospect how much time is allotted for your presentation. That will help you tailor your presentation appropriately, keeping in mind your prospect’s time. It’s also a good idea to start the meeting by setting expectations in terms of time: “Just as a time check, I’ll spend thirty minutes on the presentation and allow fifteen minutes for discussion. We’ll plan to wrap up by 11 o’clock.”
Step 1: Build Rapport
In relationship selling, building rapport with your prospect lays the foundation for a selling partnership that could continue for many years. Especially if the sales presentation will be your first in-person interaction with your prospect, put effort into making a good impression. Offer your prospect a firm handshake and start with some small talk to break the ice. This isn’t difficult; you can establish a connection with a complete stranger over something as simple as the weather or a recent sporting event. Experienced salespeople use observation to their advantage, learning about the customer by noticing the environment of the prospect’s office. Are their photographs or artwork displayed on the walls? What items does the customer keep on her desk? As Bruce Harris, account manager with UPS, says, “A person’s office is a reflection of who they are and serves as an insight into his/her personality.”Bruce Harris, “The Eyes Have It,” American Salesman 54, no. 3 (2009): 17 You can make a personal connection and break the ice by questioning your prospect about a family photograph or a trophy he keeps on display. However, make sure not to go overboard on the small talk. Remember that your prospect is busy and has a limited amount of time to meet with you. If you spend too long on chit chat, you will eat up some of the time you need to get through your call objectives. Build rapport and then get down to business.Neil Rackham, SPIN Selling (New York: McGraw-Hill, 1996), 144. This is also a perfect time to confirm the time that is allocated to you for your presentation. Although you discussed it when you set up the appointment, it’s always a good idea to confirm since things change at the last minute. This will help you quickly make adjustments if need be.
In group presentations, it is harder to leave room for small talk because if everyone starts talking, the meeting could lose its focus quickly—and in very large sales presentations, small talk is impossible. Geoffrey James suggests building “group rapport” by opening your presentation with a memorable remark: something challenging or amusing.Geoffrey James, “How to Give a Killer Sales Presentation,” BNET, May 17, 2009, blogs.bnet.com/salesmachine/?p=2940 (accessed May 16, 2010). You could also open with a brief anecdote that establishes a common connection: “When I dropped my son off at school this morning, he told me to make sure not to give a boring presentation today.…” A comment like this might get a chuckle out of your audience and will build a connection because others in the audience probably have children as well and may have had common experiences.
Leading your presentation off with situation questions is another way to break the ice and get people talking. In group settings, people are often uncomfortable sharing their opinions right away, but if you ask questions that call for factual observation, rather than opinion (How many departments in your organization would be affected by this decision? What is the average turnaround time once an issue goes to press?), people can answer without feeling threatened.Gary M. Grikscheit, Harold C. Cash, and Clifford E. Young, The Handbook of Selling: Psychological, Managerial, and Marketing Dynamics, 2nd ed. (Hoboken, NJ: Wiley-Blackwell, 1993), 158. Once you get people talking, you can lead into problem and implication questions that require your audience members to voice an opinion.
Recall from the last section that it’s always a good idea to recap the findings of your last meeting in the opening of your call. This is another way to build rapport, remind your customer of your previous discussion, ensure that everyone is on the same page, and transition into your business topic.
Step 2: Make a General Benefit Statement
Keep in mind that to effectively demonstrate capability, you should sell benefits—solutions that address your prospect’s specific needs—rather than features or advantages. If your sales call is a follow-up on a previous call, you can make a benefit statement early on that will address issues you discussed in your last meeting:
In our previous discussion, you mentioned that you had a minimum ROI requirement of 20 percent per year, and you said that you would be interested in pursuing this sale further if we could propose a solution that would meet your requirements. I’ve created an ROI analysis here that shows how outsourcing your back office work through our firm will yield an annual ROI of 25 percent. Other businesses like yours have experienced these results with us and have been very satisfied with the transition. Here’s what we envision for your company. (Show a slide with a diagram or chart giving a visual representation of your prospect’s cost savings with this solution.) Is this something you would be interested in?
In this general benefit statement, the salesperson has
• recapped the findings of the previous conversation to provide context,
• explained the value in an idea that meets the customer’s needs, rather than trying to sell a service,
• helped the customer to see himself as part of the story,
• used a closed-ended question to lead into the rest of the presentation.
Closed-ended questions—questions that demand a yes or no response—can help to move your presentation forward, keep your customer involved throughout the presentation, and confirm your understanding. Closed-ended questions have a role during your sales presentation, as demonstrated above.
However, closed-ended questions should be balanced with open-ended questions that will help you probe further into the problem your product can solve. For instance, you might ask, “What are some of your biggest frustrations with your current back office operations?” It is virtually impossible to learn more from your prospect if you don’t use open-ended questions. If you are interested in learning more and engaging your customer in your presentation, be ready with open-ended questions. For example, the situation, problem, implication, and need-payback questions in SPIN selling are all examples of open-ended questions. Open-ended questions start with “who,” “what,” “when,” “where,” or “why.” Figure \(7\) provides examples of closed-ended and open-ended questions.
Asking the right questions is one of the skills required to be a successful salesperson. This is where your ability to ask the right questions really comes into play. It is the open-ended questions that you ask during this portion of the presentation that set the tone for the rest of your presentation. But don’t stop here. Ask open-ended questions throughout your presentation to engage the prospect and continue to gain valuable information.
Step 3: Make a Specific Benefit Statement
Once you have investigated to uncover your prospect’s needs, deliver a specific benefit statement: one that demonstrates in detail how you are going to solve his unique problem. It’s impossible to deliver a specific benefit statement at the opening of your sales call because there is no way you can understand your prospect’s needs and expectations without listening to him first.Michael T. Bosworth, Solution Selling: Creating Buyers in Difficult Selling Markets (New York: McGraw-Hill, 1995), 101. Sometimes, a prospect may ask you to solve a problem that sounds similar to one you just solved for another company or customer, but if you assume you already understand your prospect’s situation and treat her just like your old customer, you might lose the sale. Approach each new presentation as if it were your first.
In B2B sales, the specific benefit statement is generally something you prepare before your presentation (recall the discussion of this from Chapter 8). However, before launching into specific benefits, you can investigate to make sure you understand and have all the necessary information: “So let me make sure I understand. What you’re saying is that a 5 percent reduction in process time will reduce your costs by 20 percent?” In any selling situation, the information you get from your prospect is usually just the tip of the iceberg, and you won’t get an idea of the pressures she is facing unless you can get her talking.
After confirming that you and your prospect are on the same page, you can move forward with your presentation, adapting if you need to based on your prospect’s answers to the questions you asked. This is the part of your presentation where the solution really comes to life. Bring your customer into the story with videos, recordings, displays, or anything else that will allow him to experience the product for himself.
You: So let me just confirm—it sounds like your biggest priorities in purchasing this SUV are gas mileage, safety, and reliability and that you would sacrifice some luxury features if your vehicle met these other conditions?
Prospect: Yes, those are definitely the most important things. This is really going to be a family car, something I can use to drive the family around and take on camping trips. And our oldest child is going to college soon, so we want a car that will last for a while because we’ll need to save money to pay tuition.
You: OK Cindy, then I think you will be excited about the RAV4 we discussed earlier. It has the best fuel economy of almost any SUV on the market, and Toyotas are known for their reliability, so this car should last you well past the time your daughter graduates from college. You can also feel confident when you drive your kids around in this vehicle because the RAV4 received five stars in National Traffic Safety Administration crash tests.Joshua Rose, “New 2009 Toyota RAV4 Features and Prices,” Auto Broker Magic, www.auto-broker-magic.com/2009-toyota-rav4.html (accessed May 16, 2010). Does that sound like a good option to you?
Prospect: Yes, that sounds like just the kind of thing we’re looking for.
You: Great! Then why don’t we go for a test drive, and you can see for yourself how well this car handles on the road.
When you demonstrate your product for the prospect, make sure to draw attention to the features, advantages, and benefits that make it a good solution for her particular situation. In the example above, when your prospect takes the car out on the road, you could turn on the climate control settings that allow her to adjust for a different climate zone in the front and back seats and explain that this way she and her family can stay comfortable on long car rides. If she has her kids along, you could turn on the rear DVD player for them. These sort of extra, customer-specific benefits help your customer to make a personal connection to the product and to see her story aligning with your solution.
During this part of the presentation, make sure to ask open-ended questions that will help you learn more about the prospect’s needs and her perceptions about the product: “You said that you like to go on family camping trips; how well does your current vehicle meet your family’s needs on these trips? It sounds like your current vehicle gets poor gas mileage; how does this affect your frequency of use or the length of your road trips? How would the storage capacity of the RAV4 change the way you use your family vehicle? How important is the car’s sound system to you?” Not only will these questions help you to uncover your customer’s needs and expectations that are still below the surface, but they will also help you to anticipate potential objections as you transition to the next part of the selling process.
Step 4: Presentation
This is the reason you are here—to present your solution that will solve your prospect’s problem. Since you started your presentation by asking questions, your presentation is a perfect way to incorporate the things you just learned from your prospect and incorporate them into your presentation. Yes, this means you have to be quick on your feet. That’s another reason preparation is so important. It allows you to be comfortable with your presentation material, yet customize it on the spot to point out specific areas that address your prospect’s problem.
You are taking the prospect on a journey so make it interesting, compelling, and relevant. Here are a few tips:
• Keep your presentation pithy. A shorter presentation is better. It helps you get to the point more quickly and have more time for dialogue with your prospect.Kelley Robertson, “How to Create a Powerful Sales Presentation,” About.com, http://sbinfocanada.about.com/od/salesselling/a/presentationkr.htm (accessed January 4, 2010).
• Start with a quick review of the prospect’s objectives. This is a good technique to confirm that you were listening to your prospect throughout the process so far and confirms that you are on the same page. This also provides the ideal platform on which to present your solution and why it will help your prospect reach his objectives.Kevin Davis, “10 Tips for Winning Sales Presentations,” Business Know-How, http://www.businessknowhow.com/marketing/winslspres.htm (accessed January 4, 2010).
• Get a reaction from your prospect throughout your presentation. Use a combination of open-ended and closed-ended questions to confirm that your prospect is in agreement with the information you are presenting and to gain new insights into how your product or service can help him. “This time savings in your production cycle can help you save at least 10 percent over your current processing. Would you like to see how this would work?” is an example of a closed-ended question that helps keep your prospect engaged. “How do you think your team would like to submit invoices like this?” is an example of an open-ended question that helps the prospect think about the product or service in use in his organization.Kevin Davis, “10 Tips for Winning Sales Presentations,” Business Know-How, http://www.businessknowhow.com/marketing/winslspres.htm (accessed January 4, 2010).
• Use demonstrations whenever possible. Showing how a product or service will work is far more dramatic and memorable than simply talking about it. If it’s possible to demonstrate the product in person, do it. If not, have a demonstration video. Bring samples, mock-ups, or prototypes if the actual product is not yet available.Kelley Robertson, “How to Create a Powerful Sales Presentation,” About.com, http://sbinfocanada.about.com/od/salesselling/a/presentationkr.htm (accessed January 4, 2010).
• Have fun. When your passion and enthusiasm come through, it makes a difference to your prospect. A monotone or boring presentation is neither interesting nor compelling. Show your prospect you believe in your product or service with a powerful and personal presentation.Kelley Robertson, “How to Create a Powerful Sales Presentation,” About.com, http://sbinfocanada.about.com/od/salesselling/a/presentationkr.htm (accessed January 4, 2010).
Don’t Be Forgettable
(click to see video)
Learn why the worst sales presentation is one that is OK.
Step 5: Close
If you have successfully delivered value to your prospect in your presentation, it is time to think about closing the sales call. This is where you obtain your customer’s commitment, either to buy or to move the sales process forward. Especially if you are expecting your prospect to make a purchase at the end of the sales call, it is a good idea to use a trial closing technique to test his buying readiness. How likely is he to make a commitment now? By testing the waters with a trial close (e.g., “On a scale of one to ten, how important would this opportunity be to you?”), you can ask your prospect for an opinion rather than asking him for a commitment, so there is less pressure for both of you. A negative response to a trial close doesn’t mean that your prospect won’t buy or move forward with the sale; instead, it is a signal to change your strategy.Doug Dvorak, “How Trial Closing and Closing Techniques Can Save You Time and Help You Make More Sales,” EzineArticles, ezinearticles.com/?How-Trial-Closing-and-Closing-Techniques-Can-Save-You-Time-and-Help- You-to-Make-More-Sales&id=1019686 (accessed May 16, 2010). A trial close often leads to objections that you will need to overcome before your prospect feels prepared to make the purchase.Jim Holden, The Selling Fox (Hoboken, NJ: John Wiley and Sons, Inc., 2002), 25. Chapter 11 discusses overcoming objections in greater detail.
If your prospect responds positively to your trial close, it is time to close the sales call by asking for a commitment. There are a variety of closing techniques you might use that will be discussed in greater detail in Chapter 12. Whichever closing technique you choose will depend on the customer, the selling situation, and your goal for the end of the sales call.
Step 5: Recap
According to sales trainer and experienced salesman Tom Hopkins, all successful presentations and demonstrations have three steps: tell your audience what you’re going to tell them, tell them what you’re there to tell them, and tell them what you’ve just told them.Tom Hopkins, “Giving Champion Presentations,” Entrepreneur, February 7, 2005, http://www.entrepreneur.com/sales/presentations/article75918.html#ixzz0LqUNOCM3 (accessed May 16, 2010). After making your presentation and successfully closing, recap the important points of your meeting and the direction you and your customer have agreed to take from here: “I’ll touch base with you tomorrow once you’ve checked on that budget detail, and in the meantime, let me look up those part specifications for you.” This will reinforce the prospect’s decision and pave the way for the next steps, which may include anything from follow-up, to a next meeting, to a formal proposal, depending on the selling situation and the length of the sales cycle.
Sales Presentation Role-Play
(click to see video)
See how all the steps come together in this selling role-play. See if you can identify all the selling skills used by the “salesperson.”
Role of the Proposal in the Sales Presentation
In many B2B sales and some larger business-to-consumer (B2C) sales, once you have presented your solution, if your prospect is interested, she will ask for a proposal—a document that proposes the specific terms of the sale, including pricing, delivery time frame, and the scope of the products or services you are offering. In relationship selling there is no such thing as a standard proposal; the proposal should include the details of a customer-specific solution and should reflect the things your customer values most. Even in retail situations—like car buying—while you might have a basic template you use for your contracts, you will adapt and renegotiate the contract depending on your customer’s needs. The key is that the proposal, like your presentation, should be customized to the individual prospect.
Nitty Gritty: The Hows and Whys of a Proposal
While every proposal should be customized, there are a few common elements that good proposals share:
• an introduction
• a definition of the project or need
• a discussion of the solution and its benefits
• the costs associated with the projectEdward Lowe Foundation, “How to Write a Sales Proposal,” eSmallOffice, 2008, www.esmalloffice.com/SBR_template.cfm?DocNumber=PL12_4000.htm (accessed May 16, 2010).
• a time frame for completion of the project or project milestones
• a call to action that asks the prospect for a response
• evidence that you are qualified to perform the job.
In addition, a proposal should accomplish three things:
1. Educate the prospect about the specifics of his need and the pertinence of your solution. The proposal should showcase the value you are bringing to the individual prospect or organization; help your prospect to see why he can’t reach his objectives without the specific solution you offer.
2. Convince the prospect that you have the competence to deliver what she needs.Edward Lowe Foundation, “How to Write a Sales Proposal,” eSmallOffice, 2008, www.esmalloffice.com/SBR_template.cfm?DocNumber=PL12_4000.htm (accessed May 16, 2010). Show her how your expertise applies to her situation, by providing relevant information and presenting the proposal in a professional format. This is especially relevant in situations where the proposal will be reviewed by a committee that is unfamiliar with you or your company.Kimberly Kayler, “Send Me a Proposal! Proposals Are Often a Downfall of the Sales Cycle,” AllBusiness, August 1, 2005, www.AllBusiness.com/manufacturing/nonmetallic-mineral-product-manufacturing/521322-1.html (accessed May 16, 2010).
3. Provide justification for the prospect’s investment in clear terms.Edward Lowe Foundation, “How to Write a Sales Proposal,” eSmallOffice, 2008, www.esmalloffice.com/SBR_template.cfm?DocNumber=PL12_4000.htm (accessed May 16, 2010). The information in the proposal should be practical and should explain the problem and solution in terms that could be understood by someone outside of the industry.Kimberly Kayler, “Send Me a Proposal! Proposals Are Often a Downfall of the Sales Cycle,” Concrete Construction, August 1, 2005, AllBusiness, www.AllBusiness.com/manufacturing/nonmetallic-mineral-product-manufacturing/521322-1.html (accessed May 16, 2010). In addition, the proposal should include a cost-benefit and ROI analysis (discussed earlier in this chapter). This will give the prospect the financial information as it relates to cost and the expected return on investment.
In some B2B situations, your customer might submit a formal request for proposal (RFP), which sets out very specific guidelines for the format of the proposal and the information it should include. (See Chapter 6.) Organizations usually use RFPs when they are requesting proposals from a number of potential suppliers at once. By providing a proposal structure, RFPs simplify the process of assessing risks and benefits associated with the purchase and can help your prospect make a decision in complex buying situations.Glenn Wheaton, “Request for Proposal,” Epiq Technologies, November 20, 2008, http://www.epiqtech.com/request-for-proposal-rfp.htm (accessed May 16, 2010). If you receive an RFP, make sure that you stick closely to the requested formatting and respond to all the questions in the document. Plenty of qualified salespeople with strong solutions have lost prospects because they failed to respond to everything in an RFP document.Kimberly Kayler, “Send Me a Proposal! Proposals Are Often a Downfall of the Sales Cycle,” Concrete Construction, August 1, 2005, AllBusiness, www.AllBusiness.com/manufacturing/nonmetallic-mineral-product-manufacturing/521322-1.html (accessed May 16, 2010).
Whether or not you are responding to a formal RFP, here are a few things to keep in mind:
• Do make sure most of the document discusses your prospect and his objectives and how you and your company will meet them.
• Do keep the writing clear and concise. This will make it easy for your prospect to assess the proposal, and it demonstrates a respect for his time on your part. Select the most relevant information and present it in an efficient way.
• Do make sure you understand how the proposal will be reviewed, who will be reviewing the proposal, what the primary selection criteria will be, and when you can expect a response.
• Do use a straightforward approach to pricing that your customer can easily assess.
• Do pay attention to the visual presentation of the proposal. As Kimberly Kayler, president of Constructive Communication, puts it, “Prospective clients facing the task of wading through stacks of proposals filled with thousands of words usually welcome efforts designed to make their lives easier.”Kimberly Kayler, “Send Me a Proposal! Proposals Are Often a Downfall of the Sales Cycle,” Concrete Construction, August 1, 2005, AllBusiness, www.AllBusiness.com/manufacturing/nonmetallic-mineral-product-manufacturing/521322-1.html (accessed May 16, 2010). Graphics can add meaning and make the information more accessible.
• Do make it easy for the prospect to accept your services by attaching an agreement he can sign that outlines the terms of the contract.
• Don’t forget to check grammar and spelling. This is an important part of credibility and professionalism.
• Don’t overuse “we” or “us.” Your language should reflect a customer centered focus.
Figure 10.8 presents a sample of an outline to follow when preparing your proposal.Adapted from Cheryl Smith, “Writing Killer Proposals,” Edward Lowe Foundation, “How to Write a Sales Proposal,” eSmallOffice, 2008, www.esmalloffice.com/SBR_template.cfm?DocNumber=PL12_4000.htm (accessed May 16, 2010).
Timing: When to Deliver Your Proposal
Have you ever noticed that when you go into a high-end clothing boutique or a store that sells expensive jewelry and watches, the price tags are hidden? The thing you immediately see is the product itself, beautifully displayed. The goal is a psychological one: to get the buyer to make an emotional connection with the product before he considers the cost. As a buyer, if the cost were one of the first things you saw, you might never make that emotional connection with the product in the first place.
This is something to keep in mind in sales. Never present a proposal—or otherwise mention pricing—early on in the sales presentation, not until your prospect has fallen in love with your product. You want your prospect to pick out the color of the car before she asks about payment; if she picks out the color, she has already imagined herself owning the car, and you have probably made your sale.
Of course, in a situation like car sales, you generally present the proposal in the same day as you present the product. You discuss your prospect’s needs, show him the car, let him test drive it, and then tell him, “Let me go talk to my manager to see if we can work out the numbers.” The process is a relatively simple one. However, in complex B2B sales, your sales presentation will probably end with a request for a proposal, in which case you will agree to a future meeting when you can present your proposal to the customer. B2B proposals are generally more involved, and so they require careful planning and a greater investment of time. If your prospect says, “Just send us the proposal,” ask for a face-to-face meeting; you can always send them the proposal ahead of time, but following up with a meeting in-person will help you address objections, answer your prospect’s questions, and demonstrate your enthusiasm for the project.
Power Point: Lessons in Selling from the Customer’s Point of View
The Art of Bringing the Product to Life
Among the many accolades realtor Susie Stephens of Chico, California, receives for her work is that she knows how to make potential buyers fall in love with a house before they ever discuss offer details with her. As one homeowner put it, “As a seller, you want your house presented and marketed well.” According to her customers, Stephens has mastered the art of bringing the product to life: “The videos and photographs [she] produced of the properties we sold were so nice it almost made us want to buy them back from ourselves!” Another customer praised Smith’s customization, explaining that she considered “the applicability of the real estate transaction to our personal situation and objectives.”“Testimonials,” Chico Real Estate, Inc., http://www.chicorealestate.net/Testimonials.aspx (accessed May 16, 2010).
Delivering Value in Your Proposal
Until you understand the areas in which your customer places the greatest value, it is impossible to come up with a proposal. For instance, say your organization offers advertising services, and you find out from your prospect that her company especially values competitive pricing on individual projects. You decide the best way to deliver value is to drop your pricing below the competitor’s lowest price and to make up for the lower cost in your retainer fee—the fixed fee that your customer will pay in advance to secure your services. This way you can deliver value in the area that is most important to your customer while still generating the profits you need to run your business. In the end, you want a situation where everyone wins—but it takes some work to uncover the key to making this happen.
Sometimes your customer’s area of greatest value is determined by business needs, and other times the issues are emotional. For instance, if you are selling a car to a customer that wants a good value on his trade-in, recognize that he might have an emotional connection to his old car (in his mind it has a high value), so offering a low trade-in price, even if it is combined with competitive financing options, might be enough to drive your customer away. In fact, your customer might actually be willing to pay more for his new car if you can give him a good price for his trade-in.
Key Takeaways
• The most important ingredient of a successful sales presentation is you.
• While there is no single formula for a sales presentation, there are five basic steps: building rapport, making a general benefit statement, making a specific benefit statement, closing, and recapping.
• It’s best to ask questions throughout your presentation to learn as much information as possible from your prospect and to keep him engaged.
• Closed-ended questions help keep the prospect engaged and should be balanced with open-ended questions, which help you probe further into the problem your product can solve.
• The proposal is a written document that includes the specific terms of the sale and is usually prepared after the sales presentation.
• Some prospects submit a request for proposal (RFP), usually when they are evaluating proposals from a number of potential suppliers, which sets out specific guidelines for the format of the proposal and the information it should include.
Exercise \(1\)
1. Develop two examples each of closed-ended questions and open-ended questions. Ask both questions to at least five of your friends and document the responses. Which type of question was easier to control? Why? Which type of question provided more information? Why? How might you use both types of questions in a sales presentation?
2. Develop a five-minute sales presentation to sell your college to high school seniors using the five steps described in this section. Role-play your presentation. Is it difficult to stay within the time constraint? How should you adjust your selling presentation when you have a limited time frame to present?
3. Assume you are selling biodegradable bags to a major grocery store chain. The bags are 100 percent biodegradable and are priced comparably to nonbiodegradable bags. You are meeting the eight -person buying committee for the first time. Role-play how you would build rapport with the group before you begin your presentation. What questions would you ask to begin your presentation? What general benefit statement would you make?
4. Go to Best Buy or another electronics store and assume you are buying a new computer. What questions does the salesperson ask before he shows you a specific model? Which questions were closed-ended? Which questions were open-ended?
5. Imagine that you are selling children’s books to Borders and you arrive at the corporate office to make your presentation and your contact tells you that due to scheduling changes, he can only give you half as much time as he originally planned. How would you adjust your sales presentation?
6. Choose a product or service that can be demonstrated or sampled (e.g., a Web site, software, food, or a beverage). Create a five-minute sales presentation using the concepts in this section and incorporating the demonstration.
7. Assume your prospect is a restaurant on or near campus. Develop a new product or service that your prospect can offer to increase traffic during off-hours. Create a five-minute presentation using the concepts covered in this section.
8. Assume your prospect is one of your classmates. Create a five-minute sales presentation for an iPod Touch using the concepts covered in this section. Include a trial close when you present to your prospect.
9. Watch this scene from the AMC show, Mad Men. Evaluate the sales presentation based on the concepts covered in this section. Which elements of the presentation are effective? Why? Which elements are not effective? Why?
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Learning Objectives
• Understand how to prepare for a successful job interview.
In many ways, gearing up for a job interview is like gearing up for a sales presentation. You can’t control the outcome of the interview, but you can control the preparation that goes into the interview. Preparing beforehand, paying attention to logistics, and knowing what to expect will set you apart from your competitors and put you in the best possible position to let your personal brand shine. Here are ten steps that can guide you through preparation for and follow-up after every job interview.
Ten Steps to Successful Interviews
1. Be ready to show and sell.
2. Accept and confirm the interview.
3. Research the company.
4. Rehearse your elevator pitch.
5. Prepare your answers to popular interview questions.
6. Prepare your questions and answers.
7. Prepare for the logistics.
8. Prepare your wardrobe.
9. Make a personal connection during the interview.
10. Follow up, follow up, follow up.
Step 1: Be Ready to Show and Sell
In a sales presentation, you want to make your product come to life by showcasing it in a way that gets your audience involved. You want your prospect to “smell the leather in the car.” The same is true at a job interview; it’s not just about your résumé. Let your interviewer see examples of the work you’ve done and help her to envision the work you can do for her company. You can start preparing for this now, while you’re still a student. Bring your portfolio on every interview. If you need some tips, review the Selling U section in Chapter 6.
Step 2: Accept and Confirm the Interview
When one of your target companies calls or e-mails to offer you an interview, don’t leave anything to chance. Grab a pencil and paper or your personal digital assistant (PDA) and take down the information you’ll need to know on the day of the interview. Do you have the correct day and time written down? Do you know the name, title, and office location of the person with whom you’ll be interviewing? Do you have directions to the company’s location? Keep in mind that googling the company’s address on the day of your interview may not get you where you want to go. Sometimes companies have large campuses with a number of buildings, and Google won’t be able to tell you how to find the right entrance to the right building and how to find your contact person’s office once you get there. Take care of logistical details like this beforehand, so you won’t have anything to slow you down on the big day.
While you have a contact from the company on the phone, take the opportunity to ask whether there is a job description on the company Web site that you can review before the interview. It’s also a good idea to ask for the title of the job for which you will be interviewing and the names and titles of the people with whom you will be interviewing. You should also ask for your interviewer’s phone number and e-mail, and bring the phone number with you on the day of the interview in case you are unavoidably delayed. E-mail your interviewer several days in advance of the interview to confirm your appointment, or call her the day before. This demonstrates professionalism and ensures that everything will run smoothly.Kim Richmond, “10 Tips for Successful Interviews,” presentation in the How to Market Yourself as a Brand to Get the Job You Want Workshop Series, Upper Merion Township Library, King of Prussia, PA, June 1, 2009.
Step 3: Research the Company and Your Interviewer
Just as you would never go into a sales presentation without carefully researching your prospect’s company beforehand, you should never go to a job interview without the same kind of preparation. Begin by reviewing the job description on the company Web site if it’s available. Then spend some time on the Web site, researching the company’s mission statement and description. If you know which department you might be working in, pay careful attention to any specific details you can learn about this department on the Web site. Some of the basic facts employers will expect you to know include
• How many locations and employees does the company have?
• Does this company deal in B2B or B2C services?
• How long has the company been in business? Have they recently merged or been acquired?
• Is the company expanding globally?
• Does the company have any new products or services?
• Have the CEO or others in the company been recognized for any achievements or publications recently?Kim Richmond, Brand You, 3rd ed. (Upper Saddle River, NJ: Pearson Prentice Hall, 2008), 186.
Then go beyond Web site research; after all, your interviewer knows what’s on his company’s Web site, so don’t just repeat back the information you find there; show him your motivation and professionalism by coming prepared with your own research. Use the company’s product or service and talk to other people who use the product or service. Go online and read what customers have to say about the company. Go through the company’s purchasing process so you can understand the workings of the company from a customer’s point of view.Kim Richmond, “10 Tips for Successful Interviews,” presentation in the How to Market Yourself as a Brand to Get the Job You Want Workshop Series, Upper Merion Township Library, King of Prussia, PA, June 1, 2009. Read any recent press releases or press coverage about the company.
Don’t forget to research your interviewer. Chances are, he has a profile on LinkedIn so you can get some insight about him and even see what he looks like. Also do a Google search as you may learn about his personal hobbies and other pertinent information.
Step 4: Rehearse Your “Elevator Pitch”
Don’t be surprised if one of the first questions your interviewer asks is something along the lines of “tell me about yourself.” This is a common opening question, designed to put job candidates at ease, but it can be one of the hardest questions to answer. “As part of your job-search arsenal, having a good elevator speech is a critical tool,” says Alysin Foster, consultant and managing partner at the Centre for Strategic Management. “Sometimes all you get is 30 seconds to make a good impression.”Laura Raines, “Making Your Pitch,” The Atlanta Journal-Constitution, Jobs, 2007, www.ajc.com/hotjobs/content/hotjobs/careercenter/articles/2007_0225_elevatorsp.html (accessed May 16, 2010).
Review the Selling U section in Chapter 5 to be sure your elevator pitch is your strongest starting point. Then, rehearse, rehearse, rehearse so it sounds and feels natural as a response to that dreaded first question, “Tell me about yourself.”
Tell Me about Yourself
(click to see video)
This video provides insight into what the interviewer means when he says, “Tell me about yourself.”
Source: Collegegrad.com
Step 5: Prepare Your Answers to Popular Interview Questions
“Tell me about yourself” is only one of a number of popular interview questions for which you should prepare before going into the interview. While there’s no way to know which questions you’ll get for sure, you can be relatively certain that your interviewer will ask at least one or two of the common standbys. Preparing answers to popular interview questions beforehand will empower to respond with clarity and poise. “What traps a lot of people is they think and talk at the same time,” says Bill McGowan, founder of Clarity Media Group. “It’s better if you know your conversational path.”Sarah E. Needleman, “The New Trouble on the Line,” Wall Street Journal, June 2, 2009, http://online.wsj.com/article/SB124390348922474789.html (accessed May 16, 2010). The best way to have a powerful conversation is to review your brand positioning points from the Selling U section in Chapter 1 and your FAB in Chapter 6. These can be included in an answer to just about any interview question. Practice telling your stories out loud so they are concise and focused, yet sound natural.
The Best Interview Answers
(click to see video)
What’s the most important thing to remember when you are answering interview questions? Watch this video to find out.
Source: Collegegrad.com
Some common questions interviewers ask and a few pointers for coming up with a response are shown below.
Common Interview Questions
What Are Some of Your Greatest Strengths?
Most candidates will respond to this question in generalities like “I’m a strong self-starter” or “I’m highly organized.” You already have your personal brand positioning points and stories to go with them, so why not use them here? You will set yourself apart if you can illustrate your strengths with the anecdotes you have prepared ahead of time. For example, “My leadership skills are among my greatest strengths. As the shift leader at Olive Garden, I scheduled the wait staff and resolved customer service issues during my shift. The restaurant had the highest customer satisfaction ratings during the two years I worked there.”
What Are Some of Your Weaknesses?
The interviewer isn’t looking for any deep confessions when she asks you this question. According to CareerBuilder.com, “The secret to answering this question is using your weaknesses to your advantage.”CareerBuilder.com, “Answering 6 Common Interview Questions,” CNN.com, December 9, 2005, http://www.cnn.com/2005/US/Careers/12/09/six.questions/index.html (accessed May 16, 2010). For instance, if you say that you have trouble with organization, you can follow this up by saying that because organization doesn’t come naturally to you, you make a conscious effort at the beginning of a new project to plan out your goals. It’s never a good idea to simply name a weakness and finish off by telling the interviewer it’s something you are working on. On the other hand, it’s also important to be honest when you respond to the weakness question; don’t try to pretend that you are without faults because that won’t make you look good either.
This video clip addresses this challenging interview question.
(click to see video)
Source: Collegegrad.com
Have You Ever Had a Conflict with a Boss?
This is what is called a behavioral question. The interviewer is looking for how you behaved in a specific situation.
This video clip provides some tips for how to handle this question in an interview.
(click to see video)
Source: Collegegrad.com
What Can You Offer This Company, or How Do You See Yourself Fitting in at This Company?
This is one of those questions for which your research beforehand will pay off. This question is as much about your knowledge of the company as it is about your qualifications. Career strategist J. T. O’Donnell says “You can craft a better answer by asking [yourself] what the company wants and why.”Sarah E. Needleman, “The New Trouble on the Line,” Wall Street Journal, June 2, 2009, http://online.wsj.com/article/SB124390348922474789.html (accessed May 16, 2010). Then ask yourself how your story and the company’s story match up. This is a lot like presenting the customer-specific benefits of your product in a sales presentation. Prepare a story that can illustrate what you have to offer.
Why Do You Want to Work Here?
This question is another opportunity to showcase your company research. Consider what you know about any challenges or issues that the company faces and how your skills and experience will be beneficial.CareerBuilder.com, “Answering 6 Common Interview Questions,” CNN.com, December 9, 2005, http://www.cnn.com/2005/US/Careers/12/09/six.questions/index.html (accessed May 16, 2010). What community service or internship experiences might be relevant? For instance, “I know that your company is about to launch its first e-mail marketing campaign, and I would really like to be involved as this project gets off the ground. Last year I was in charge of writing the e-mail newsletters for the local food bank and expanding their list of subscribers, and I would look forward to putting that experience to work in a professional capacity.”
Why Should I Hire You?
This can be a difficult interview question, but not if you are prepared for the answer. Watch this video for some tips.
(click to see video)
Source: Collegegrad.com
What Is Your Favorite Ad Campaign (or Other Industry Specific Item)?
This is an example of an industry-specific interview question you might hear if you are interviewing for a marketing position. Whatever field you are going into, make sure you have done your research and understand the industry so you can respond to industry-specific questions. For example, if you are interviewing for a job in advertising, be familiar with the major advertising campaigns and be ready to discuss your favorite and why you think it works.
Where Do You See Yourself in Five Years?
Your interviewer won’t want to hear that your five-year goal is to be working in a different industry. Talk about your personal goals that relate to the job. This will demonstrate that you understand the company and are motivated to succeed there.CareerBuilder.com, “Answering 6 Common Interview Questions,” CNN.com, December 9, 2005, http://www.cnn.com/2005/US/Careers/12/09/six.questions/index.html (accessed May 16, 2010).
What Are Your Salary Expectations?
This is a problem you should avoid responding to directly if possible. A good response would be to deflect the question: “I would expect compensation that falls in the standard salary range for this industry.” It’s a good idea to research salary ranges for your industry so that you will be ready to negotiate when the topic of salary does come up, but let your employer put a figure on the table first.“Common Interview Questions,” USA Today, Careers and Workplace, January 29, 2001, www.usatoday.com/careers/resources/interviewcommon.htm (accessed May 16, 2010). If you feel that you have to respond to this question with a direct answer, just be warned that once you name a figure, you shouldn’t expect your employer to offer you more than that if you decide to take the job. It’s a good idea to do your research before any job interview by researching current salaries for the position for which you are interviewing at Web sites such as Salary.com, or use Web 2.0 techniques and ask an online community such as Salarymap.com. The following article includes several resources:
bizcovering.com/employment/10-extremely-useful-salary-and-job-websites
This video clip provides additional insights for how to answer this interview question.
(click to see video)
Source: Collegegrad.com
How Many Years of Experience Do You Have Using Excel (or Other Software Programs)?
You don’t want a question like this to cost you the position, especially considering that many software programs can be learned on the job. Don’t give false information, but you can try responding with your own question; try asking how much and what level of experience is required for the job. If you have a more specific idea of the answer the interviewer is looking for, you can provide a more convincing response as to why you should be considered for the job, even if your answer doesn’t match exactly what the interviewer is looking for.Sarah E. Needleman, “The New Trouble on the Line,” Wall Street Journal, June 2, 2009, http://online.wsj.com/article/SB124390348922474789.html (accessed May 16, 2010).
What Did You Like Least about Your Last Job?
Interviewers often ask this question to get you to reveal conflicts. Avoid going this route. In job hunting, you should never reveal anything negative about a former employer. Whatever you mention in your response, choose something that isn’t directly related to the job for which you are applying. And make sure to end your response on a positive note: “I’m ready for the challenges of my new job.”Kim Richmond, Brand You, 3rd ed. (Upper Saddle River, NJ: Pearson Prentice Hall, 2008), 196.
You might also find it helpful to review this video, which includes some frequently asked interview questions and some ways to answer them.
http://link.brightcove.com/services/player/bcpid26599544001?bclid=26964187001&bctid=31648299001
Step 6: Prepare Your Questions and Answers
Toward the end of the interview, every interviewer will ask you if you have any questions for him. So make sure you have three or four questions in mind. Preparing these ahead of time will show your interviewer that you have thought about the position and the company. Here are a few questions to consider asking:
• What opportunities will there be beyond this position in the company?
• What would the ideal person for this position look like?
• What are some challenges facing the department in the next three months? What role will the person in this position play in tackling these challenges?
• How would you describe your company culture?
• What are the next steps in the hiring process?
After the interviewer responds, be ready to follow up by restating your strengths.Kim Richmond, Brand You 3rd ed. (Upper Saddle River, NJ: Pearson Prentice Hall, 2008), 186. For instance, if you ask what qualities the ideal candidate for this job should have, your interviewer might mention something you hadn’t thought of mentioning earlier. You can respond by telling a relevant story about a specific time when you overcame an obstacle or helped a colleague solve a problem.
You can ask these questions even if you already know the answer. If you interview with multiple people in the organization, it is OK to ask the same question multiple times. It will help to get a variety of perspectives—and keep in mind that the questions you ask are also a way of showcasing your experience and your knowledge about the company.
You’ve Got the Power: Tips for Your Job Search
When Do I Ask about Salary?
Finally, even if you have questions about salary and benefits, don’t ask them now. Always delay a conversation about salary as long as possible. In a sales presentation, you wouldn’t pull out a pricing schedule before your customer had expressed a strong interest in buying the product; keep the same idea in mind going into a job interview. It’s best to let your interviewer bring up salary—and that might not be until after the second or third interview. Be patient; the longer your prospective employer has to get to know you, the more opportunities you have to point out why you would be a good addition to the company. If you sell yourself well throughout the interview process, you might even receive a higher offer.
This video provides some tips for how to handle the salary question during an interview.
(click to see video)
Source: Collegegrad.com
Step 7: Prepare for the Logistics
Before you interview, take care of the logistics just as you would for any sales presentation. Control the things that are in your power to control so that you can focus on your performance during the interview. Double check that you know where you’ll be going (including building, room, and/or suite number) and allow extra time for travel in case you get stuck in traffic. Make sure you know the title of the position for which you will be interviewing. Remember to assemble your materials the night before the interview: have your work samples ready to go in a portfolio and print at least four extra copies of your résumé on twenty-four-pound paper. Bring these extra résumés in your portfolio. Even though your interviewer will have already received your résumé, she may not have it on hand, and you should always be prepared in case you are asked to meet with anyone who was not on the original interview schedule.Kim Richmond, “10 Tips for Successful Interviews,” presentation in the How to Market Yourself as a Brand to Get the Job You Want Workshop Series, Upper Merion Township Library, King of Prussia, PA, June 1, 2009. Arrive early, fix any wardrobe malfunctions, and get ready to give a stellar presentation.
Step 8: Prepare Your Wardrobe
Your wardrobe is part of your personal branding, so make sure you dress like a professional when you go to your interview. This holds true even if you are interviewing in a more casual industry; you can always dress down after you get the job.
• Dress conservatively. Go for a suit or dress in dark or neutral colors (black, gray, or navy for the suit). Avoid hypertrendy clothes or clothes that otherwise make a bold statement. Women should avoid dresses with thin straps or low necklines.
• Make sure your suit or dress fits you well. If it needs to be tailored, have it done. You will use the suit in your new job, and a good fit will increase your confidence during the interview.
• Wear appropriate, professional shoes. Both men and women should wear conservative, close-toed shoes in a dark or neutral color to complement their wardrobe. Women should avoid stiletto, open-toe, or platform shoes. Men should avoid athletic shoes and make sure their shoes are polished.
• Wear appropriate accessories. Avoid flashy jewelry or watches. Carry a professional briefcase or handbag—no backpacks or messenger bags. Keep in mind that even your accessories are part of your personal brand.
• If you have tattoos or body piercings, make sure they are not visible during the interview. Make-up such as Conceal FX available at Sephora.com will camouflage tattoos. It is acceptable for women to wear conservative earrings—but avoid anything large or distracting.
• Select appropriate hosiery. If you are wearing a suit, your socks should match your pants or shoes. Women wearing skirts and dresses should always wear pantyhose (even if you think you have a great tan).
• Make sure your clothes are ironed. Do this the night before.Kim Richmond, Brand You, 3rd ed. (Upper Saddle River, NJ: Pearson Prentice Hall, 2008), 197. Lay our your clothes the night before so that you will have one less thing to worry about on the day of the interview. Wrinkled clothing and stains are considered among the biggest grooming red flags for job interviews, according to a recent survey of employers conducted by Gillette Career Advantage.“USA Today Snapshots,” USA Today, Money, December 30, 2009, 1B.
• Ensure impeccable grooming, including a conservative hairstyle and appropriately manicured fingernails. Don’t forget deodorant and a breath mint; body odor or bad breath can be a turn off in an interview. It’s a good idea to stop in the restroom right before you go to the interview for one last check in the mirror (it’s the perfect time to have a breath mint).
Step 9: Make a Personal Connection during the Interview
Make an effort to connect personally with your interviewer. People want to hire people they like. Smile, make eye contact, and greet him with a strong handshake. Allow yourself to relax and begin the conversation with some small talk. Notice the surroundings in your interviewer’s office. Does he have school memorabilia, family photographs, sports paraphernalia, or vacation photos? Try to discover commonalities that will allow you to make a connection. During the interview, remember to smile and maintain eye contact, and when the interview is wrapping up, make sure to close by telling the interviewer you want the job.Kim Richmond, “10 Tips for Successful Interviews,” presentation in the How to Market Yourself as a Brand to Get the Job You Want Workshop Series, Upper Merion Township Library, King of Prussia, PA, June 1, 2009. Most of all, relax, enjoy the conversation, and be yourself.
Step 10: Follow-Up, Follow-Up, Follow-Up
Don’t wait to do this! Get in contact while you are still fresh in your interviewer’s mind: write a thank-you e-mail the same day. Details on how to write a successful follow-up e-mail are covered in the Selling U section in Chapter 11. Besides the e-mail, send a hand-written thank-you note on a plain, white business note card. Mail this the same day, so that your interviewer will receive it the next day or the day after. Very few people send handwritten “thank-you’s” anymore, so this extra touch will make you stand out—and it only costs the price of postage, so why not do it?
During your interview you should ask the interviewer for a time frame so that you will know when to expect a response. If you haven’t heard back by the appointed date, follow up with a phone call. Asking your interviewer for a time frame is essential to follow-up: if she isn’t planning to make her hiring decision for another two weeks, calling her after one week will only be an annoyance. Be persistent, but keep in mind that there is a fine line between persistence and pestering. When you get voice mail, you can leave a message—once—but then keep calling back until you reach your contact. Following up by phone signals that you are still interested in the job and motivated enough to pursue it.Kim Richmond, Brand You, 3rd ed. (Upper Saddle River, NJ: Pearson Prentice Hall, 2008), 188. Sometimes hiring decisions get delayed because of issues that come up at the company, so not hearing back by the date you were expecting is not necessarily an indication that you weren’t selected for the position.
Lisa Peskin, Sales Trainer, Shares Her Insights, Experiences, and Tips for Successful Interviews
(click to see video)
Use your selling skills to prepare for and participate in successful interviews
Key Takeaways
• A job interview is like a sales presentation; a successful interview requires a lot of preparation.
• Always be ready for a job interview with a professional portfolio and interview suit including shoes and other accessories.
• When you receive a call for an interview, take the time to write down the date, time, location, title of the position for which you will be interviewing, and the people (names and titles) with whom you will be interviewing.
• Do your homework and thoroughly research the company, its products or services, customers, and competition. The company’s Web site is a good place to start, but if possible also use the product or service or call the company’s 800 number as though you were a prospective customer.
• Prepare for the most likely questions you will be asked including “tell me about yourself.” Review your brand positioning points and stories you want to tell in response to the most commonly asked interview questions.
• Be prepared with questions to ask during the interview.
• Delay the conversation about salary as long as possible; avoid the temptation to bring it up during an interview. It’s best to let the interviewer bring up the topic when she is closer to a final decision.
• Smile and be yourself during the interview. The best way to sell yourself is to be yourself.
• Follow up after the interview with a thank-you note.
Exercise \(1\)
1. Choose one company that is on your target company list, which you created in the Selling U section of Chapter 7. Research it thoroughly by visiting the company Web site and that of its competitors, using the product or service, contacting the company by e-mail or phone as a customer, reading about the company and its competitors in the news, and reading blogs and social networking Web sites to see what people are saying about the company and its key competitors. What did you learn about the company that you didn’t already know? Based on your findings, list three questions you would ask during an interview.
2. Conduct a role-play with one person acting as the interviewer and one acting as the interviewee using the commonly asked interview questions mentioned in this section.
3. Identify two ways to follow up from an interview and when each should be done. | textbooks/biz/Marketing/The_Power_of_Selling/10%3A_The_Presentation_-_The_Power_of_Solving_Problems/10.07%3A_Selling_U_-_Selling_Yourself_in_an_Interview.txt |
Power Wrap-Up
Now that you have read this chapter, you should be able to understand how to deliver value in a sales presentation.
• You can plan the steps you need to take to prepare for a sales presentation.
• You can describe how to dress for success at a sales presentation.
• You can explain how to deliver the message to your prospect.
• You can understand SPIN and how to use it during the sales presentation.
• You can list the five steps of a successful presentation.
• You can understand how to have a successful job interview.
TEST YOUR POWER KNOWLEDGE (AnswerS ARE BELOW)
1. Explain what it means to deliver value to your customer.
2. What is the best rule of thumb for dressing for a sales presentation?
3. List three dos and three don’ts for giving a PowerPoint presentation.
4. Explain the 10/20/30 rule for a PowerPoint presentation.
5. What are the benefits of using samples or demonstrations in your presentation?
6. List the four components of SPIN selling.
7. Give an example of a closed-ended question.
8. Give an example of an open-ended question.
9. What should you always do before making a specific benefit statement?
10. When should you deliver the proposal in a sales presentation and why?
POWER (ROLE) PLAY
Now it’s time to put what you’ve learned into practice. Following are two roles that are involved in the same selling situation—one role is the customer, and the other is the salesperson. This will give you the opportunity to think about this selling situation from the point of view of both the customer and the salesperson.
Read each role carefully along with the discussion questions. Then, be prepared to play either of the roles in class using the concepts covered in this chapter. You may be asked to discuss the roles and do a role-play in groups or individually.
In the Driver’s Seat
Role: Customer at a high-end car dealership
You are considering a new car. You want performance, but you still need some space for passengers. You want the latest and greatest, yet still be comfortable and be able to transport people and things easily. You are willing to pay for what you want, but given the current economic environment, you are concerned about paying too much.
• What type of presentation would you expect to get at the car dealership?
• What questions will you have for the salesperson about the car?
• What questions will you have for the salesperson about the financing for the car?
Role: Salesperson at car dealership
You want to be able to put this customer in the car he wants, but first you need to identify some things. If he has a family and needs space, you have just the car for him. And you have a price reduction for this week only since it’s the last week of the month (and you want to make your quota). But you’re not sure what is more important to him—luxury appointments, passenger space, gas mileage, status, or price. You want to use SPIN selling to understand exactly what he needs.
• How will you prepare for the presentation?
• How will you start the presentation?
• What questions will you ask in each of the areas of SPIN?
• How will you learn about the prospect’s objections (if there are any)?
• How will you trial close?
ACTIVITIES
1. Draft a list of the projects you have worked on for which you have samples that could showcase your work. Make a separate folder on your computer where you can save any of these files for use during your interview.
2. Use the list of popular interview questions and guidelines to generate answers that you can deliver during your interview. Write these answers down and save them somewhere where you will be able to review them before going to a job interview.
3. Visit your campus career center and learn about the opportunity to participate in mock interviews. Prepare for the mock interview and dress for success.
TEST YOUR POWER KNOWLEDGE AnswerS
1. Delivering value means asking questions, listening to your customer, and defining value in customer terms.
2. Dress one step above what you would wear if you worked at the organization. When in doubt, dress up.
3. Dos include the following:
• Do make your slides easy to read.
• Do replace descriptive headlines with headlines that sell.
• Do use the 10/20/30 rule.
• Do remember that that PowerPoint is only an aid.
Don’ts include the following:
• Don’t turn down the lights.
• Don’t go overboard with technological gimmicks.
• Don’t hide behind your computer screen when using PowerPoint.
• Don’t fill your slides with words.
• Don’t bore your audience with visual sameness.
4. Make sure you limit your slides to 10 or fewer. Give yourself 20 minutes to go through your 10 slides. And use only 30-point or larger font size so that your audience can clearly read what you’ve written.
5. Samples and demonstrations bring the product to life and help your prospect to see your solution as part of her story. Samples and demos also educate the prospect, prove the performance of your product, and get the prospect involved.
6. Opening, investigation, demonstrating capability, and obtaining commitment.
7. A closed-ended question requires a yes or no answer, such as “Do you currently use a recycling service?”
8. An open-ended question engages the customer in conversation, such as “How do you currently process invoices?”
9. Check for understanding.
10. Proposals should only come after your prospect has clearly made a connection to your product. Presenting specifics like pricing early on can create objections and prevent your prospect from making an emotional connection to the product. | textbooks/biz/Marketing/The_Power_of_Selling/10%3A_The_Presentation_-_The_Power_of_Solving_Problems/10.08%3A_Review_and_Practice.txt |
Handling Objections: The Power of Learning from Opportunities
Video Ride-Along with Paul Blake, Vice President of Sales at Greater Media Philadelphia
You heard Paul Blake talk about making a successful presentation in Chapter 10. Now hear his tips about handling objections. While this might sound like the most difficult part of the selling process, Paul shares his advice about how to make this the most productive part of the selling process.
If you think you have to memorize all kinds of responses to objections, you’ll be pleased to hear that handling objections is easy…when you use the skills you already learned.
(click to see video)
11.02: Objections Are Opportunities to Build Relationships
Learning Objectives
• Understand what a sales objection is.
• Learn how overcoming objections can strengthen a relationship.
• Understand when and why prospects raise objections.
You’ve been working really hard at school, and it’s paying off. You’re doing well this semester with a GPA right where you want it. Spring break is right around the corner, and you and your friends have been talking about going to Mexico. You even had an impromptu “fiesta party” at your place and even do some research about airfares and hotels; there are some great deals out there. You present your case to your parents and end with a strong “close” to seal the deal: “The timing is perfect since it’s my senior year. I can book the flights tonight online.” You thought you sold them on the trip when they say, “We’re worried about you going out of the country without a chaperone.” You are deflated, but you won’t take no for an answer so you wait for a moment, let it sink in, then deliver your response, just like you planned. You have just experienced the fine art of overcoming objections.
Since you are constantly selling in your everyday life, you have also undoubtedly encountered objections: your friend doesn’t want to see the same movie as you, your brother doesn’t want to share the car, your parents want you home earlier than you would like. When you attempt to convince someone or “sell” him on your point of view, you are not always successful. But each time you “sell” your idea, you usually have additional information or a fallback position so that you can get what you want while meeting the other person’s needs. You are probably more skilled in overcoming objections than you realize.
Occasionally in your sales career, you will encounter a situation in which you are able to close the sale directly after giving your sales presentation. Such a situation, however, is the exception not the rule. Objections are simply a natural outcome of the sales process. Each potential prospect has his own set of unique needs, and, though you may identify most of them during the preapproach stage of the selling process when you do your research, you will not be able to anticipate all of them. After all, you are not a mind reader. Besides, if all it took to excel in sales was to deliver a perfect script, anyone could do it. But that is not the case. The essence of sales is handling objections and truly understanding how you can help your prospect meet her needs. It is a demonstration of your skills as a salesperson to find the opportunity in these objections, listen to your prospect, and then respond. So an objection is simply a question from a prospect that indicates that she wants more information. If she weren’t interested, she wouldn’t be asking questions.John Boe, “Overcome Objections and Close the Sale,” Agency Sales, September 27, 2003, www.johnboe.com/articles/close_the_sale.html (accessed May 16, 2010).
The first myth to dispel is the assumption that objections are bad or an omen foreshadowing failure. On the contrary, resistance usually portends commitment. If a prospect is asking you questions, you can at least assume that he is interested in your product or service. In fact, in all likelihood, he already knows whether or not he needs or wants to make the purchase. Thus, the reason he is objecting isn’t necessarily because your presentation failed to communicate the features, advantages, and benefits of your offering. Rather, he is objecting because he is seeking reassurance; he is on the fence of indecision, and he wants you to provide him with the incentive that justifies an immediate purchase.R. T. Edwards, “Power Selling,” American Salesman 38, no. 3 (March 1993): 13. Supply your prospect with the right information, that is, show him why he wants to buy your product or service.
What Are Objections?
Objections, also called sales objections, are generally defined as prospect questions or hesitancies about either the product or company.William C. Moncrief and Greg W. Marshall, “The Evolution of the Seven Steps of Selling,” Industrial Marketing Management 34, no. 1 (January 2005): 13–22. While objection may sound like rejection, you should never assume that when a prospect asks a question or expresses a concern that you have failed to generate interest in your product or service. It is true sometimes that your prospect will object when he truly cannot or does not want to buy. Usually, though, objections mask—intentionally or unintentionally—a request for more information. They simply signal your prospect’s level of interest and alert you to what actions need to be taken to bring the sale to a close. If your prospect expresses objections, consider them invitations to continue to sell. Furthermore, leverage these objections into an opportunity to continue to build your relationship with your prospect so that you can continue to create a positive influence on the buyer’s decision. The fact is objections help you build your relationship and find the true reason for resistance. Think of objections as opportunities.
How Objections Build Relationships
As an analogy, consider asking someone out on a date for the first time. Even if you have hooked him in with a great pick-up line (approach) and dazzled him with your sparkling personality (presentation), he may still not be convinced that you are serious about him. Naturally, he might respond by playing hard-to-get. How you react will reveal to him your level of commitment. By allowing the relationship to grow slowly and organically, you demonstrate your patience, sensitivity, and sincerity. You establish a foundation of trust that eventually wins him over. On the other hand, if you respond by getting huffy and stomping off, he will probably be glad to see you go.
Objections as Opportunities
You might not keep track of objections in your everyday life (especially as they relate to dating). However, you may find it interesting to know that in sales, a prospect will say no an average of five times before he buys.John Boe, “Overcome Objections and Close the Sale,” Agency Sales, September 2003, www.johnboe.com/articles/close_the_sale.html (accessed May 16, 2010). That means that it’s more likely than not that you will experience a prospect who poses at least one objection: asking a question, requesting more information or time, or pushing back due to financial constraints. Without objections, you would have no way of knowing what a prospect is thinking, what concerns she has, or what barriers might be in the way of her saying, “Where do I sign?”
The fact is objections are an important part of the selling process. But thinking about overcoming objections might be the wrong frame of reference. The word “overcome” implies that you want to conquer, fight, or win (and, therefore, your prospect loses).Patty Morgan-Seager, “Handle Objections and Have Fun!” Multifamilypro, www.smmonline.com/Articles_handleobj.htm (accessed October 24, 2009). Instead, it’s best to think about objections as a perfect extension of the selling process. Think back to the steps of the selling process that you have covered so far: prospecting and qualifying, preapproach, approach, and presentation. Throughout each of these steps, your focus is on understanding your prospect’s needs and building a relationship based on trust. The same is true for this step: handling objections. This is all about learning more, finding common ground, and providing the solution that is best for your prospect. Objections and conversation help you better understand exactly what your prospect wants and needs. The bottom line is that you don’t want to avoid objections; you actually want to encourage objections and ask for them. According to the Selling Power Sales Management Newsletter, “Objections are not personal attacks; they’re gifts.”“Hug Your Objections,” Selling Power Sales Management Newsletter, August 15, 2007, http://www.sellingpower.com/content/newsletter/issue.php?pc=732 (accessed March 16, 2010).
Consider Objections before They Occur
If objections are such a positive part of the selling process, you might be wondering how to be prepared for them; how to think about them; how to consider them even before you get them. Here are some strategies for preparing for the objections portion of the selling process that will help you build your relationship.
• Understand your prospect and believe in your partnership. If you did your homework at every step of the process so far and put together a presentation and proposal that really makes sense for your prospect’s business, you should be confident in the fact that you are a true business partner to your prospect. Objections lead to sharing and learning and the ability for you to make adjustments in your proposal that will help your prospect manage her business.Janaé Rubin, “Overcoming Objections” Folio, November 2005, 80–81.
• Remember WII-FM. WII-FM (What’s In It For Me) is the radio station that everyone listens to. Never lose sight of your prospect’s buying motivations. If time is mission-critical to his success, know what you can deliver and by when. If national reach is important to your prospect, be sure you address it in detail in your proposal.
• Understand risk. Understand what your prospect considers a risk (e.g., time, money, changing suppliers). When risk outweighs reward in the mind of your prospect, chances are she will find a reason not to buy. Understand her risk factors and address them head on. This will allow you to employ a “risk-removal” strategy, rather than a selling strategy.Jeffrey Gitomer, Little Red Book of Selling: 12.5 Principles of Sales Greatness (Austin, TX: Bard Press, 2005), 153, 157.
• Anticipate objections. Think about every possible objection you might get—before you get it. That means making a list of every objection before you even make your presentation and building in the response into the presentation.Paul Karasik and James Benson, 22 Keys to Sales Success (New York: Bloomberg Press, 2004), 119. Your success as a salesperson will largely be determined by your ability to anticipate and handle objections.Felice Philip Verrecchia, “How to Identify and Overcome Objections,” Edward Lowe Peerspectives, August 11, 2004, www.bankseta.org.za/downloads/faisII/benefits/objections.pdf (accessed October 24, 2009). Write down all the possible objections and go back and incorporate them into your presentation. Then, give your presentation to a friend or colleague and see if they can find any additional objections. Although you can’t make your presentation “objection proof,” you can anticipate and be prepared for most objections that will be raised.Felice Philip Verrecchia, “How to Identify and Overcome Objections,” Edward Lowe Peerspectives, August 11, 2004, www.bankseta.org.za/downloads/faisII/benefits/objections.pdf (accessed October 24, 2009). Anticipating objections helps you be responsive, rather than reactive.Keith Rosen, “Respond to your Prospect’s Objections,” AllBusiness, www.AllBusiness.com/sales/selling-techniques-active-listening/4019422-1.html (accessed May 16, 2010).
• Raise objections first. Since you have done so much preparation and you understand and have a good relationship with your prospect, be proactive and be prepared to raise objections first. When you raise an objection, you actually turn it into a discussion point rather than an objection. It shows your prospect that you are thinking about the sale from her perspective and helps you build the relationship.Janaé Rubin, “Overcoming Objections” Folio, November 2005, 80–81.
Objections should not intimidate you or dissuade you from continuing the selling process. Rather, you should consider objections opportunities to learn more about your prospect’s needs. The more you understand about your prospect’s needs, the greater your ability to determine how your product or service can satisfy them or how your product or service can be improved to satisfy them. Remember, selling is about solving problems. The solution that you offer will demonstrate to your prospect whether or not you truly understand his needs and whether or not you have his best interests at heart. By embracing your prospect’s objections and handling them effectively, you will inspire his trust, confidence, and most important, loyalty. As a result, both you and the prospect benefit.
Power Selling: Lessons in Selling from Successful Brands
Handling Objections: All in a Day’s Work
At iCore Networks, a leading VoIP (voice over Internet protocol) provider, handling objections is an everyday learning experience. Sales reps gather at 8:00 a.m. sharp every day to discuss successes and failures from the previous day and role-play overcoming objections and then put what they learn to work in the field all day. The commitment to coaching and being in front of customers works for the company and its sales force: the average compensation for a first-year sales rep is \$92,000.Mike Hofman and April Joyner, “A Salesforce Built around Cold Calling,” Inc., September 1, 2009, www.inc.com/magazine/20090901/a-sales-force-built-around-cold-calling.html (accessed November 22, 2009).
Learn more about how iCore sells in the following article.
www.inc.com/magazine/20090901/a-sales-force-built-around-cold-calling.html
Why Prospects Object
While prospects may voice their objections in different ways, just about every objection comes down to one of four reasons: no or not enough money, no perceived need, no sense of urgency, and no trust.John Boe, “Overcome Objections and Close the Sale,” Agency Sales, September 2003, www.johnboe.com/articles/close_the_sale.html (accessed May 16, 2010). As a selling professional, you have control over each one of these objections. But it’s too late if you address it only when the prospect objects. In other words, you are actually handling objections at every step of the selling process. For example, you can avoid the price objection with thorough qualification during your first step of the selling process.Joan Leotta, “Overcoming Doubts: The Road to a Sale Is Blocked by the Prospect’s Doubts,” Selling Power 20, no. 2, http://www.sellingpower.com/content/article.php?a=5351 (accessed March 16, 2010).
If a prospect does not have a perceived need or high sense of urgency to buy your product or service, your challenge is to understand the drivers of his business. Every business has challenges, and your role from the time you qualify the prospect is to understand your prospect’s “pain points,” those issues that cause problems for him and his company and present barriers to growth. If you truly understand your prospect’s business, it is much easier to present a solution that addresses the perceived need and reasons to buy it now. “There is no reason for buyers to buy today unless we build in that sense of urgency and give them a reason to buy today,” says Dana Forest, director of sales at Simons Homes.William F. Kendy, “An Uncertain Situation: How to Kick-Start the Hesitant Buyer,” Selling Power 27, no. 9, http://www.sellingpower.com/content/article.php?a=7658 (accessed March 16, 2010).
Many objections are raised because the relationship between the prospect and the salesperson is not fully developed. business-to-business (B2B) selling is dependent on trust. If the trust is not there, or the relationship is not yet fully developed, it can be difficult for a prospect to make a change or finalize the purchase. If this is the case, prospects will frequently delay or stall before making a decision, which can be an attempt to derail the sale.
When Prospects Object
While you may not be able to predict your prospect’s every objection, you can at least predict that he will object. Knowing when to expect objections is the first step to handling them: you will eliminate the chance of appearing caught off guard or unprepared to discuss the product or service that you are selling.
Of course, it is possible that the prospect may object at any time during your sales call—from introduction to close. Still, there are specific points in time during the sales process where these objections are more likely to occur: when you are first trying to make contact, when you are making a sales presentation, and when you are attempting to close the sale, or make a trial close. As you learned in Chapter 10, a trial close includes any attempt to close the sale but usually focuses on asking the prospect’s opinion: “What do you think about the turnaround time?” A trial close may occur at any point during the selling process. In other words, if the prospect indicates that she may be interested in making the purchase, it is an opportunity to make a trial close.
Objections are likely to occur at several points during the selling process, including the trial close. It’s best to be prepared for objections at every step in the selling process, including the qualifying stage. Know your prospect and be ready to incorporate objections into your sales presentation.Jeffrey Gitomer, “Objection Prevention & Objection Cure,” video, May 18, 2009, http://www.youtube.com/watch?v=CgfmcuE_06w (accessed October 24, 2009).
Setting Up the Appointment
Imagine that you are in the middle of a cold call and you are attempting to set up an appointment to meet your prospect. You have barely uttered your name when your prospect exasperatedly grunts, “Don’t waste your breath. I’m not buying anything you’re selling.” How do you respond?
This scenario is meant to illustrate the fact that you may meet resistance as soon as you try to establish contact with your qualified prospect. Hopefully, you will have reduced the rate of this problem occurring by properly qualifying your prospect beforehand and preparing for the most common objections. Nonetheless, anticipate resistance from the beginning.
Using the questioning technique is a good way to engage your prospect in conversation and learn more about what can help her run her business.“Telemarketing Tips about Overcoming Objections,” September 25, 2009, http://www.mindtools.com/pages/article/newTMC_88.htm (accessed October 25, 2009).
Prospect: No thanks, I’m satisfied with my current supplier.
You: May I ask you who you are currently using?
Prospect: We work with Advanced, and they have been doing a good job.
You: Advanced is very good at what they do. Did you know that Symone offers a money-back guarantee? In other words, if you are not completely satisfied with the conversion or the service, we will completely refund your money. It would be worth thirty minutes of your time to learn more about it. How does Tuesday at 8 o’clock look?
When you are giving a sales presentation, very often the prospect will ask you questions as you go. It is unlikely that your prospect will wait until you have finished your presentation before asking you questions. However, the experienced salesperson will actually encourage questions throughout her presentation since she knows that responding to them supplies her with precious time that she can use to further demonstrate how her offering can solve her prospect’s problem. As a rule, you will want to acknowledge objections as they arise. If you feel that the objection will be addressed at a later point during the presentation, you may postpone your response, but you will need to communicate this information to your prospect. For example, you might say something like the following:
Prospect: I’m a little concerned about the financing.
You: I’m glad you brought that up. I’m going to address that in the next slide, which I think will provide you with the information you are looking for.
During the Presentation
Otherwise, he may think that you are avoiding the question and that you are trying to hide something, are unprepared and do not know how to respond, or are simply not listening—all kinds of impressions that you do not want give.
During the Trial Close
Recall from Chapter 10 that you can test your buyer’s readiness after your sales presentation by employing a trial close. If your prospect hasn’t expressed any opinions at this point of the selling process, then the trial close is your opportunity to seek them out. If your prospect responds positively to it, then congratulations! This response indicates that you have skillfully executed each step of the selling process: creating rapport, gaining the prospect’s trust, listening, identifying his problem, and presenting products and services that will provide him with solutions and value.“Telemarketing Tips about Overcoming Objections,” September 25, 2009, http://www.articlesbase.com/sales-articles/telemarketing-tips-about-overcoming-objections-457823.html (accessed October 25, 2009). From this point, you can move to the next step of the process, the close.
If, on the other hand, an objection is raised, then you will use this time to respond to it. Always remember that an unacknowledged concern lessens the opportunity for a sale. Responding means fully listening to your prospect’s concerns and objections, asking clarifying questions to determine whether or not you understand them, identifying the types of objections they are, and meeting them. To be clear, “meeting” an objection does not mean saying what you think the prospect wants to hear; you should never make a promise about a product or service that you cannot deliver. How you meet an objection will depend on the type of objection you are dealing with. Simply put, meeting the objection means returning to the presentation stage, elaborating on your product’s capabilities, and emphasizing in what ways they benefit your prospect. For example, assume you are making a sales presentation for a software product to a B2B client and she presents an objection about the timing of the installation.
Prospect: This is really an interesting option for us to pursue, but we are planning on launching our service much sooner than your timeline suggests. I’m not sure your implementation timing will work for us.
You: When are your planning on launching your new service?
Prospect: We want to have everything in place and tested in less than forty-five days.
You: So it sounds like the biggest challenge is the installation date. I can talk to our head engineer and see if we can change the installation date. If we can guarantee installation within thirty days, will you commit to the two-year agreement?
In this example, it’s important to note that the objection led to the prospect sharing information that was not previously known: the date of the launch. This is valuable information that the salesperson can use to potentially overcome other objections and provide service that will help the prospect meet his goals.
After you think you have responded to and have overcome all your prospect’s objections, you can proceed with another trial close. If you determine that your prospect has new objections, then you will want to repeat the response process. You may have to use a trial close several times before moving to a close. Keep in mind that the sales process is not perfectly linear; rather, it is iterative. Depending on the prospect and the product, it is perfectly appropriate to repeat steps.
When you are certain that you have addressed all your prospect’s objections and that he has no further objections, move to the close. Don’t be shy! You have earned this right and, besides, your customer expects you to!
In the same way, you should never allow yourself to become defensive or antagonistic when a prospect makes an objection. Since your goal is to build and sustain an enduring customer relationship, you will want to handle your prospect’s objections with as much delicacy as possible. For example, avoid responding to objections with statements beginning with “but”: “But our company is better” or “But we offer more value for your money.”Keith Rosen, “Respond to your Prospect’s Objections,” AllBusiness, www.AllBusiness.com/sales/selling-techniques-active-listening/4019422-1.html (accessed May 16, 2010). It’s better to respond in a positive way, such as “We are the only company that offers a guarantee on our product. If you’re not satisfied for any reason, we’ll refund your money. Our goal is for you to be more than satisfied—we want you to be delighted.”
Key Takeaways
• Objections are a normal part of the selling process and are not a personal reflection on you but rather an opportunity to learn more about how the customer is evaluating the potential purchase.
• Objections actually help build relationships because they give you the opportunity to clarify communication and revisit your relationship with the prospect.
• The best way to handle objections is to be thorough in every part of the selling process from qualifying through the preapproach, approach, and presentation.
• It’s a good idea to anticipate objections by reviewing your presentation, writing down every possible objection, and building it into your presentation.
• It’s especially important to understand risk from your prospect’s perspective so you can create a risk-removal strategy.
• Prospects object for four reasons: money, no perceived need, no sense of urgency, and no trust.
• Prospects may pose objections at any time, but especially while setting up the appointment, during the presentation, and during the trial close.
Exercise \(1\)
1. Go to a local health club and go through the sales presentation as if you were going to join. What objections would you have for the salesperson? Which objections did the salesperson address to your satisfaction? Which objections did the salesperson not address to your satisfaction? Why?
2. Try to sell your professor on conducting class as a study period next week. How would you prepare for the “presentation” to make your case? What are some objections you might receive? How might you handle the objections?
3. Identify the three most common points at which objections occur in a sales presentation. Provide an example of each one in your everyday life.
4. Assume you are selling real estate and you are calling a prospect to set up an appointment. How would you handle an objection that she doesn’t have the time to meet with you?
5. Assume you are a financial services salesperson. You have presented an investment strategy to your prospect, and he has objected because he is concerned about the state of the market. How would you handle this objection by making him feel more comfortable with the risk?
6. Contact a salesperson for a local business and ask him how he handles objections. Share your findings with the class. | textbooks/biz/Marketing/The_Power_of_Selling/11%3A_Handling_Objections_-_The_Power_of_Learning_from_Opportunities/11.01%3A_Introduction.txt |
Learning Objectives
• Learn strategies to handle objections.
• Understand the different types of objections and how to handle them.
Learning how to handle objections is key, especially when many of the same ones occur regularly. There are six strategies that can help you handle virtually any objection.
1. View the objection as a question. Many times salespeople hear an objection as a personal attack. Instead, an objection such as “Why are your prices so high?” should be considered a question. That allows a more positive conversation rather than a defensive one.Pam Lontos, “10 Strategies for Dealing with Objections,” FrogPond, www.frogpond.com/articles.cfm?articleid=plontos12 (accessed October 24, 2009).
2. Respond to the objection with a question. As in every step of the selling process, asking the right questions is critical, and handling objections is no exception. Questions, such as “Can you share you concerns in this area?” or “Is there another way to look at this to make it work for you?” are good ways to engage prospects in dialogue that will help you better solve their problems.Keith Rosen, “Respond to your Prospect’s Objections,” AllBusiness, www.AllBusiness.com/sales/selling-techniques-active-listening/4019422-1.html (accessed May 16, 2010).
3. Restate the objection before answering the objection. It’s a good idea to check for understanding and demonstrate that you are listening by restating your prospect’s objection. For example, “So what you’re saying is you’re concerned about the capacity during peak periods” is a good way not only to acknowledge the objection but also to give you time to formulate your response.Pam Lontos, “10 Strategies for Dealing with Objections,” FrogPond, www.frogpond.com/articles.cfm?articleid=plontos12 (accessed October 24, 2009).
4. Take a pause before responding. Many times salespeople “oversell” when they are answering an objection. When a prospect raises an objection, stop, listen, and pause for a few seconds. This shows the prospect that you are legitimately listening to her objection, not just trying to sell.Felice Philip Verrecchia, “How to Identify and Overcome Objections,” Edward Lowe Peerspectives, August 11, 2004, www.bankseta.org.za/downloads/faisII/benefits/objections.pdf (accessed October 24, 2009).
5. Use testimonials and past experiences. Don’t avoid answering any part of an objection. In fact, objections are the perfect time to share testimonials. For example, “I have another customer who was concerned about the turnaround time. He found that not only were we able to deliver on time, we were slightly under budget.”Felice Philip Verrecchia, “How to Identify and Overcome Objections,” Edward Lowe Peerspectives, August 11, 2004, www.bankseta.org.za/downloads/faisII/benefits/objections.pdf (accessed October 24, 2009).
Testimonials can be very powerful at any point in your sales presentation but especially when a prospect presents an objection.
Prospect: I’m not sure this is the right database management tool for us. Technology is not our strong suit, and I’m concerned that we would be buying a product that has more horsepower than we need.
You: I have several other clients with businesses that are about the size of yours, and they felt that way initially, too. In fact, John Jackson at Premier Services felt the same way, but he said that the product is so easy to use that it took very little time to train his people. He was able to increase his sales by 3 percent and reduce his sales and marketing costs by 5 percent when using our database management tool. Chris Ling at IBS was worried about the same issue. He increased his sales over 5 percent with an 8 percent reduction in selling and marketing costs. Let’s take a look at the demo again.
You can also simply respond to an objection by letting your customers speak for you.Bob Bly, “Overcoming Objections,” http://bly.com/blog/general/overcoming-objections (accessed January 6, 2010).
Prospect: We’ve tried other cleaning products, but they didn’t really work for us.
You: Here’s what my customers say…
6. Never argue with the prospect. “The customer is always right” is always true when it comes to handling objections. It’s never a good idea to disagree or argue with the customer, even when he is wrong. Relationships are built on trust, so it’s best to use an objection to build the trust, not break it.Felice Philip Verrecchia, “How to Identify and Overcome Objections,” Edward Lowe Peerspectives, August 11, 2004, www.bankseta.org.za/downloads/faisII/benefits/objections.pdf (accessed October 24, 2009).
Dos and Don’ts of Handling Objections
The following are things you should concentrate on doing when you are handling objections:
• Do maintain a positive attitude and be enthusiastic.
• Do remember that objections are a natural part of the sales process and should not be considered as a personal affront.
• Do maintain good eye contact, even when under fire.
• Do listen closely to an objection.
• Do acknowledge the objection and then give your point of view.
• Do prepare to prove your position with testimonials, references, and documentation.
The following are things you should avoid doing when you are handling objections:
• Don’t knock the competition. That takes the focus off you and your company, and you never want to do that.
• Don’t say anything negative about your company.
• Don’t say anything negative about your product or service.
• Don’t tell the customer that they are wrong.
• Don’t tell the customer, “You don’t understand.”
• Don’t argue with a customer.
• Don’t lie to a customer. Long-term relationships are built on trust and honesty. It is far better to say, “I don’t know, but I’ll find out and get right back to you.”
• Don’t be defensive. That’s not a positive approach to an objection.
• Don’t lose your cool with the customer.
• Don’t let an objection go by without an answer.
Reprinted with permission from Edward Lowe Peerspectives.Felice Philip Verrecchia, “How to Identify and Overcome Objections,” Edward Lowe Peerspectives, August 11, 2004, www.edwardlowe.org/index.elf?page=sserc&storyid=6407&function=story (accessed October 24, 2009).
Types of Objections
Prospects may object for any reason, but there are six major categories into which most objections fall. When you are prepared for all these types of objections, you will be able to successfully handle them.
• Product objection
• Source objection
• Price objection
• Money objection
• “I’m already satisfied” objection
• “I have to think about it” objection
Product Objection
Sometimes prospects voice an objection as it relates to the product, called a product objection. Comments such as “This isn’t as good as your competitor’s product” or “I don’t want to take that kind of risk” are a reflection of a concern about the performance of the product. For complex purchases, prospects may not fully understand all the functions of the product due to lack of familiarity. Listening is an important skill to use, especially when a prospect voices a product objection. It’s a good idea to handle product objections by describing warranties, using testimonials, getting the prospect engaged in a product demonstration, or presenting industry or third-party research to support your claims.Charles M. Futrell, Fundamentals of Selling: Customers for Life through Service, 10th ed. (New York: McGraw-Hill Irwin, 2008), 385. For example, consider the following:
Prospect: I’m not sure your product stacks up to your competition.
You: So what you’re saying is you are not convinced that this product will perform as well as others on the market? I’m glad you brought that up. I have customers who felt the same way when I began talking with them. Now they actually speak for the product themselves. Let’s take a look at these three short videos from some of our current customers talking about the product performance and how much better it is than that of the competitors.
Power Player: Lessons in Selling from Successful Salespeople
The Edge That Works
How do you compete with the big players in a crowded business-to-business (B2B) industry? Bob Ladner, founder and president of a market research firm in Florida, wanted to compete with the big players but couldn’t get any prospects to give him a chance. Finally, in the middle of a sales presentation when he was overcoming objection after objection, he asked the prospect, “What do you want? A guarantee?” While it’s almost impossible to offer a guarantee in the market research business, Ladner ultimately designed one that works. His successful firm now boasts major clients thanks to the guarantee. “The guarantee is a method of generating confidence,” says Ladner.Leslie M. Schultz, “Guaranteed Advantage,” Inc., June 1, 1984, www.inc.com/magazine/19840601/7042.html (accessed October 24, 2009).
Source Objection
Some prospects voice objections about the company or about doing business with you as a salesperson. This is called a source objection. While this type of objection doesn’t happen often, it does happen so it’s important to know how to handle it.
Source objections as they relate to the company may be voiced with comments about the stability or financial health of the company or about how the company does business. But this is an opportunity for you to help your prospect understand your company’s strengths. Consider the following example:
Prospect: Your company hasn’t been around for very long. How can I trust that your company will be here in three years to support the warranty?
You: I’m glad you brought that up. I can see why that might be a concern for you, but let me give you some information about the company that I think will put your mind at ease. Our company is backed by some of the largest investors in the industry. The reason they invested in the company is because they see the vision of how we can bring more solutions to companies like yours. They have made a commitment to support all customer warranties for the next ten years. Talk about putting your money where your mouth is. The bottom line is that we are trying to reduce your risk.
When a prospect has a source objection as it relates to you as a salesperson, it might not be as obvious to overcome. As with other objections, the best way to handle it is to get it out in the open:Charles M. Futrell, Fundamentals of Selling: Customers for Life through Service, 10th ed. (New York: McGraw-Hill Irwin, 2008), 386
Prospect: I don’t think we would make this purchase from you.
You: I can respect that. May I ask you why?
Price Objection
One of the most common objections is the price objection. It is important to ask probing questions to really understand the nature of this objection. Many prospects use the price objection as a negotiating ploy to determine how much flexibility there is in the pricing, while others use it as a way to object due to budget constraints. It’s best to always be prepared for the price objection. The bottom line on the price objection is that people buy when they see the value. Cost (or price) is what the customer actually pays for the product or service. Value is the benefit that the customer receives from the product or service. It is value that customers assign to a product or service that determines the price. For example, value is what dictates that a shack on the beach in Monterey, California, is worth more than a similar home in Omaha, Nebraska. Or in another example, value is what causes customers to pay more for an iPod than a comparable MP3 player. Customers perceive that the design and function of an iPod delivers more value, even at a higher price, than comparable products made by other manufacturers. This is the essence of value.
“The customer is typically going to throw the price objection out there just out of habit, out of rote,” according to sales trainer Chuck Reeves. When salespeople really listen to customers, Reeves says that they actually hear customers saying, “I don’t see the value, and if you can convince me there is value, there is return, then I just might pay.”Rick Weber, “How to Overcome the Price Objection,” Trailer/Body Builders, January 1, 2003, http://trailer-bodybuilders.com/mag/trucks_overcome_price_objection (accessed November 7, 2009). Even when budgets are tight, companies make adjustments to purchase the products or services that they find compelling and can help them profitably grow their business. If you think about it, the same is probably true for your personal purchasing; when you want something bad enough, you are able to somehow find the money for it.
Many salespeople believe that price is the barrier standing in the way of making a sale. That is, they think that cutting the price will help them get the sale. Many times salespeople are willing to cut the price or a product or service when a prospect objects because they feel that if the product or service is priced lower, they will get the sale. This situation is sometimes compounded if the salesperson rationalizes cutting the prices because she believes the margins are high enough, or even too high. This “sense of fairness” approach never recognizes the value that the product or service brings to the prospect. If simply reducing the price were the answer, selling would be easy—and probably wouldn’t require your skills and intuition.
The truth is that price is not the driving factor in most purchasing decisions. More important, pricing shouldn’t be determined based on your product cost. To be successful, you need to understand more about the value your product or service is delivering to the customer. It’s the value that should determine the price, not product cost, or even prospect objections.Tom Reilly, “What Is a Fair Price?” Tom Reilly Training, www.tomreillytraining.com/CPO%20article%205.htm (accessed November 11, 2009).
So be prepared for the price objection. Preparation will make you look at the product or service through the eyes of the prospect and will help you establish the value. The price objection might be handled in the following way:
Prospect: Your prices are much higher than anyone else I’ve looked at.
You: So what you’re saying is you think that our prices are higher than others? Certainly, price is part of the equation, but it’s also important to look at the value for the price. You mentioned that real-time inventory information was an important strategic issue for your business. Ours is the only product on the market that provides real-time inventory information without any integration costs. Our system is a true plug-and-play application so you can begin getting real-time inventory the day we sign the deal. In fact, one of my customers was concerned about the same thing, and now we provide his entire backend logistics.
Handling the Price Objection
(click to see video)
This video, featuring best-selling author and sales expert Jeffrey Gitomer, discusses how to handle the price objection.
Timing Is Everything
Timing is everything when it comes to objections. While a prospect may raise an objection at any time during the selling process, it’s best to keep the pricing discussion until the end of your sales presentation rather than discussing it early on. (In fact, the same is true about salary when you are on a job interview—always postpone the conversation about salary until an offer is made.) The reason for this is simple: it gives you the opportunity to talk about value rather than price.
Think about the process of buying a new car. First, you go into the showroom and talk to a salesperson, then you go for a test drive and really fall in love with the car—how it handles, the smooth ride, the sound system, the GPS system, the smell of the leather seats. While you probably looked at the sticker price before you got into the car, you don’t really start talking about price until after you determined that this car has what you want. At this point, the value has been established, which makes it easier for the salesperson to sell on value than to simply sell on price.Lance Baird, “Overcoming the Price Objection,” B2B Insights Blog, October 1, 2009, http://www.godfrey.com/blog/post/2009/10/01/276 (accessed November 7, 2009).
Money Objection
An objection that is related to the price objection is the money objection, sometimes called the budget objection, which relates to the prospect’s financial ability to make the purchase. While some budget objections are true, when the prospect really doesn’t have the means to purchase the product or service, it’s important to avoid these types of objections with proper qualifying.
Even if you do your homework before you begin the selling process, there is still a good chance that a prospect may present a money objection. In some cases, the prospect’s budget may not be large enough to accommodate the cost of your product or service. If this is true, you may determine that this is a prospect for the future when his business is large enough to afford your offering. However, it is worth probing to determine if the objection is price or budget related. Like the price objection, this objection is also related to value. When prospects can’t see the value for the price, they object by saying either the price is too high or they can’t afford it. The best way to handle it is to anticipate it and be prepared:
Prospect: I really can’t afford this right now.
You: You mentioned that you are already paying \$5,000 per month on your current plan. This plan even gives you a broader service at a lower cost per transaction cost. If you continue with your current plan, you will actually be paying a higher cost per customer. The fact is you really can’t afford not to switch. Let’s try this service for thirty days, and I can prove to you that your cost per transaction will be lower.
In this example, the broader service, which results in a lower cost per transaction, is what establishes the value in this example. It’s the value that allows the salesperson to handle the money objection and make a trial close.
Another approach to this objection is to help the prospect see how they can afford your product or service. Consider the following example:
Prospect: We really can’t afford this in our budget right now.
You: It sounds like this can really help you increase your sales. If I can show you how this product can pay for itself, would you be interested?
Power Point: Lessons in Selling from the Customer’s Point of View
Just Ask
Want to be able to handle objections with ease? Deliver value. When prospects object with price or money objections, differentiate your product with a value-added service. If you want to know which service will make a difference—and help make the sale—just ask your customers. You’ll be surprised what you learn when you just ask.Jack Carroll, “Your Price is too High—Not!” Inc., December 7, 1998, www.inc.com/articles/1998/12/14304.html (accessed November 22, 2009). This article by Jack Carroll from Inc. will help you think differently about handling the price or money objection.
www.inc.com/articles/1998/12/14304.html
“I’m Already Satisfied” Objection
Many times prospects will object with what is called the “I’m already satisfied” objection (also called the need objection). This can be a more challenging objection than price because it might include a hidden objection, an objection that is not openly stated by the prospect but is an obstacle in the way of making the sale. In this situation, a prospect doesn’t state his concern about making the purchase. Instead, he might ask trivial questions to avoid the issue or he might not ask any questions at all and simply state that he does not have a need for the product or service.Charles M. Futrell, Fundamentals of Selling: Customers for Life through Service, 10th ed. (New York: McGraw-Hill Irwin, 2008), 378. The best way to handle hidden objections is to bring them to the surface. In other words, ask questions to get your prospect to talk openly about her objections. If she says no simply continue to ask questions until you are able to identify the true objection.Pam Lontos, “10 Strategies for Dealing with Objections,” FrogPond, www.frogpond.com/articles.cfm?articleid=plontos12 (accessed October 24, 2009).
Anticipation is best to avoid the “I’m already satisfied” objection. According to sales maven Jeffrey Gitomer, engaging the prospect is key. He preaches that there is a huge difference between customers being satisfied and being ecstatic and profitable. The secret is in engaging the prospect and talking about the value that other customers have received. According to him, when a prospect is satisfied with their current supplier, it’s the perfect time to make a sale.
Is Being Satisfied Good Enough?
(click to see video)
That’s the question to ask prospects if they use the “I’m already satisfied” objection, according to this video featuring Jeffrey Gitomer.
“I Have to Think about It” Objection
While the “I have to think about it” objection might sound like an objection, it is actually a stall. This “objection” usually occurs when a prospect isn’t completely comfortable with you and your product or service. This is the classic stall tactic and is a signal for you to build your relationship. Prospects usually use this objection when they are trying to mask some fear or risk that they have about committing to the sale. Your challenge is to uncover the risk that the prospect sees and build your relationship with him to build a deeper trust.Jeffrey Gitomer, “I’d Like to Think about It—and Other Sales Stalls,” video, June 22, 2009, http://www.youtube.com/watch?v=cCyf8af78A8&feature=related (accessed October 24, 2009). Just as with other objections, asking questions is important to understand why the prospect is stalling and what kind of information will help him feel more comfortable. In reality, this objection is one that is a signal for you to work on improving your relationship with the prospect:
Prospect: I need some time to think about it.
You: I want to give you the time you need to think about it. But let’s talk specifically about your reasons for buying now versus later.
This type of approach will help you engage the prospect in conversation so you can understand more specifically what the barriers are to the sale.
Ultimate Stall
(click to see video)
This video, featuring Jeffrey Gitomer, highlights how to deal with the “I have to think about it” objection.
Key Takeaways
• There are six strategies that will help you handle any objection: view the objection as a question, respond to the objection with a question, restate the objection before answering the objection, take a pause before responding, use testimonials and past experiences, and never argue with the customer.
• There are six major types of objections: product, source, price, money, need, and thinking about it (which is actually a stall).
Exercise \(1\)
1. Assume you are a sales rep for an interactive advertising company. Your prospect is learning about how social networking works and has responded to your presentation with the following comment: “I’m not sure this is really for us.” What type of objection is this? How would you respond?
2. Imagine that you are a sales rep for a commercial landscaping company. You have just finished a presentation that includes a five-year landscaping plan for your client’s property. She responded by saying that she doesn’t think there’s enough money in the budget for the plan. What kind of objection is this? How would you respond to her?
3. Assume you just presented your ideas to help your prospect increase traffic to his store by adding a sign on the side of the building. The customer was polite and listened to the presentation but said that he’s not sure he really needs the additional sign since there is already one in front of the store. What type of objection is this? How would you respond?
4. Choose a type of car that you might like to own. Review the company’s Web site along with Edmonds.com to identify the elements that create value for the car. How does the value relate to the price?
5. Assume you work for the school you are attending and are responsible for selling sponsorships of campus events to local companies such as restaurants, gyms, and retail stores. If your prospects say the price is too high, how would you overcome this objection?
6. Visit a retail store that engages in personal selling. Assume you are a customer for the product and present an objection to the salesperson. Record how she responds to it. Is it an effective handling of your objection? If so, why? If not, what you would suggest to make it more effective?
7. Read the objection outlined in this article: blogs.bnet.com/salesmachine/?p=5207&tag=content;col1. Then, take the quiz to identify the correct answer. | textbooks/biz/Marketing/The_Power_of_Selling/11%3A_Handling_Objections_-_The_Power_of_Learning_from_Opportunities/11.03%3A_Types_of_Objections_and_How_to_Handle_Them.txt |
Learning Objectives
• Learn about common objections you may hear in a job interview and the best way to respond.
• Understand how follow-up to a job interview can help “overcome objections.”
It’s exciting to get a call to go on a job interview. During your preparation (described in detail in the Selling U section of Chapter 10), you will, of course, research the company and learn everything you can about how it does business. You’ll identify some questions that you want to ask because you realize that a job interview is a two-sided exchange—the company wants to learn about you, and you want to learn about the company. You’ll plan your wardrobe, transportation, and other details well in advance of the big day. But one thing you may not think about is how to overcome objections during the job interview.
Common Interview “Objections”
Be prepared to answer the most common objections that may be voiced during your interview. Focus on the positive and keep your answers professional. In fact, you should practice your answers to these questions out loud so that your answers are crisp and conversational. When an interviewer presents an objection, take a breath before you answer the question. Restate the objection and then answer it. It’s best not to dwell on an objection and talk too much, simply handle them and move on.Randall S. Hansen and Katharine Hansen, “Closing the Sale and Overcoming Objections in the Job Interview,” Quintessential Careers, www.quintcareers.com/printable/interview_objections_closing.html (accessed October 24, 2009). Here are some common objections and suggested ways to handle them.
Objection 1: You Don’t Have Enough Experience
The best way to anticipate and even avoid this objection is to review your portfolio during the interview (see the Selling U section of Chapter 6 for more details about preparing your portfolio).Randall S. Hansen and Katharine Hansen, “Closing the Sale and Overcoming Objections in the Job Interview,” Quintessential Careers, www.quintcareers.com/printable/interview_objections_closing.html (accessed October 24, 2009). A portfolio is a visual way to demonstrate your skills and experience. It’s one thing to talk about what you’ve done, it’s quite another to bring it alive to your interviewer. It’s especially important to show your work from internships, major class projects, volunteer projects, and other examples of your work.
Objection 2: I’m Not Sure You Will Fit In with the Team
This is another opportunity to refer to your portfolio by talking about projects that you work on with other people. Chances are you’ve worked on teams for class projects, internships, volunteer projects, and other areas. Be prepared with specific examples about how you have worked in collaboration with a team or taken on the leadership role within a team.Randall S. Hansen and Katharine Hansen, “Closing the Sale and Overcoming Objections in the Job Interview,” Quintessential Careers, www.quintcareers.com/printable/interview_objections_closing.html (accessed October 24, 2009).
Objection 3: The Position Doesn’t Pay as Much as You Are Looking For
Your response to this objection should be something like “Salary is only one part of compensation. I’m looking for the right opportunity, and I’m willing to look at other areas of the total compensation program, including benefits, advancement, exposure, and other elements of my personal and professional growth.” It’s best not to take this conversation into a salary discussion. Wait to have the salary conversation until the company has extended an offer. It’s a good idea to have a salary range in mind before you go into an interview. Do your research on Web sites such as Salary.com so that you are prepared if your interviewer asks how much you are expecting as a starting salary.Mary Moss, “Tips for Overcoming Objections during a Job Interview,” Associated Content, August 13, 2007, www.associatedcontent.com/article/337859/tips_for_overcoming_objections_during.html?singlepage=true&cat=31 (accessed October 24, 2009).
Objection 4: You’re Too Experienced for This Position
When you are starting out, it will be rare to hear that you have too much experience for a particular position. However, if you do hear it, be ready with the right answer. It’s always best to seek a job you really want. But starting at a level that might be below your expectations is a good strategy, especially in this economy.
When interviewers say this, they are worried that when the job you want comes along, you will leave. Answer this objection by pointing out that you are willing and excited about learning about the business from the ground up. Based on your research of the company, give your interviewer a specific reason about why you want to work for that particular company. People are more willing to give you a chance if you are really interested in working for the company.
“Hidden Objections” during Job Interviews
Although there are some common objections you may hear in a job interview, chances are you will rarely hear an objection on a job interview. This is one major difference between a sales call and an interview. Most managers and recruiters respond during an interview in a more neutral way so as not to imply that the job is going to one candidate over another.Kim Richmond, Brand You, 3rd ed. (Upper Saddle River, NJ: Pearson Prentice Hall, 2008), 188. Prospective employers prefer to interview all the candidates and then make their hiring decision. Therefore, their objections are often more like hidden objections, those that are not openly stated during the interview. Unlike the sales call, it is not appropriate to keep probing to identify the objection. The best way to overcome objections, hidden or stated, is to be prepared to sell yourself in the most compelling way possible.
The concept of value, described earlier in this chapter, can be a successful way to overcome objections in a job interview whether the objections are stated or hidden. Prepare for the interview, understand the company’s needs, and demonstrate how you can meet the needs. Simple. Effective. Powerful.
Follow Up after Job Interviews: Set Yourself Apart
After you’ve shaken hands and finished your interview, keep in mind that your ability to stand out is not over. Follow-up is the currency of sales; those who follow up significantly increase their chances of getting the sale (or getting the job). Here are some ways to follow up and make yourself memorable.
Thank-You E-mail after a Job Interview
Prospective employers want people who want to work for the company. A thank-you note can set you apart from other candidates and show your interviewer that you really want the job (it’s easy for every candidate to say she wants the job, but not every candidate writes a thank-you note).
You have the opportunity to say thank you more than once. It’s also a good idea to take advantage of every opportunity to demonstrate your interest and enthusiasm for the company. Start with a thank-you e-mail that you send the day of the interview. It’s important to use e-mail to thank your interviewer for his time, and it is also is the perfect way to deliver value. Take a minute and recap some of the topics you discussed with each interviewer (if there was more than one). Jot down a list and go online and look for an article, video, or interesting blog that would be worth sharing. Send a personal thank-you e-mail to everyone with whom you interviewed (no group e-mails here). Also, be sure to send a thank-you e-mail to the recruiter, if you worked with one to get the interview. It’s important to remember that a thank-you e-mail should be as formal and professional as a handwritten thank-you note.
Now, it’s time to write your thank-you e-mail. There are three major parts to a thank-you e-mail. It can be short, but effective.
• First, thank your interviewer for her time.
• Mention something specific that you discussed. Include the link in the e-mail.
• Close your e-mail with a note about next steps.
See Figure \(5\) for a sample thank you e-mail. Additional sample thank-you e-mail notes can also be found at http://jobsearch.about.com/od/thankyouletters/a/thankyouemail.htm.
Handwritten Thank-You Note
Sending a thank-you e-mail is good etiquette, and it reminds your interviewer that you can deliver value to the organization. But don’t stop there. As soon as you send your thank-you e-mail, write a handwritten thank-you note to each person with whom you interviewed. You might think that it is unusual to send two thank-you notes, but it is the perfect way to communicate your interest and value to your interviewer in two ways: the thank-you e-mail demonstrates immediacy and helps you deliver value with a link to a relevant article, video, or blog, and the handwritten thank-you note provides a personal touch that few candidates take the time to do. As with the thank-you e-mail, timing is important for the handwritten note. It’s best to write and mail it the same day so your interviewer receives it within a day or two of the interview. It’s the perfect way to reinforce the fact that you go the extra mile to make an impression and build a relationship.
Thank-You Note
(click to see video)
This video highlights some key elements of a handwritten thank-you note.
See Figure \(7\) for a sample handwritten thank-you note. Additional sample thank-you notes can also be found at http://jobsearch.about.com/od/thankyouletters/a/samplethankyou.htm.
You’ve Got the Power: Tips for Your Job Search
Dos and Don’ts of Thank-You Notes
Here are some tips for writing effective thank-you e-mails and notes:
• Do ask for a business card at the end of each interview so that you have the correct spelling and title for each person with whom you interviewed.Randall S. Hansen, “Job Interview Follow-Up Do’s and Don’ts,” Quintessential Careers, www.quintcareers.com/interview_follow-up-dos-donts.html (accessed November 8, 2009).
• Do write individual thank-you notes to each person with whom you interviewed. If a recruiter arranged the interview, send a thank-you e-mail or note to her, too.Randall S. Hansen, “Job Interview Follow-Up Do’s and Don’ts,” Quintessential Careers, www.quintcareers.com/interview_follow-up-dos-donts.html (accessed November 8, 2009).
• Do write a thank-you e-mail or note even if you are not interested in the job. It’s always a good idea to say thank you to someone for his time.Alison Doyle, “Writing Thank You Letters,” About.com, http://jobsearch.about.com/od/thankyouletters/a/thankyouletters.htm (accessed November 8, 2009).
• Do send a thank-you e-mail or note within twenty-four hours.
• Do proof your thank-you e-mail or note before you send it, including the spelling of the person’s name.
Here are some things to avoid when sending thank-you e-mails and notes:
• Don’t stop job hunting even if you had a good interview. The job isn’t yours until you get an offer.Randall S. Hansen, “Job Interview Follow-Up Do’s and Don’ts,” Quintessential Careers, www.quintcareers.com/interview_follow-up-dos-donts.html (accessed November 8, 2009).
• Don’t bother the employer and follow up in a way that becomes annoying.Randall S. Hansen, “Job Interview Follow-Up Do’s and Don’ts,” Quintessential Careers, www.quintcareers.com/interview_follow-up-dos-donts.html (accessed November 8, 2009).
• Don’t follow up sooner than the interviewer or recruiter indicates is appropriate.
What If You Don’t Hear Back?
At the end of a job interview, it’s a good idea to ask about next steps. Usually interviewers or recruiters will tell you the expected time frame in which they will make a decision. This is valuable information because it will help you determine how and when you should follow up.
If you don’t hear back from the employer or recruiter within the specified time frame, it’s recommended that you call and follow up. Companies frequently have good intentions of making a decision quickly, but other business issues take priority. Following up with a phone call helps remind your prospective employer that you are interested in the position. While it is appropriate to follow up by e-mail, it is more effective to follow up by phone. It’s easier to have a conversation with the interviewer or recruiter and get some insight about the timing as well as reinforcing why you are a good choice for the position. Continue to do research on the company so that when you follow up, you can discuss company news. For example, you might say something like “I noticed that you were recently awarded the ACON business. It sounds like this is an exciting time at the agency and one that will need some motivated salespeople. I wanted to follow up on our conversation last week to see where you stand with filling the position.”
Follow-Up Tip
Set up a Google News Alert (http://www.google.com/alerts) using keywords for every company in which you are interested in working. The news alerts will be delivered to your e-mail (or other source you specify), and you will know all the latest news about the company—as it happens. It’s a good idea to send an e-mail to your contact about the news as a follow-up and a way to keep in touch.
Follow-Up after Sending Résumés
You can see that follow-up is critical after an interview. It helps overcome objections even after the interview is over. The same principle of follow-up applies to every contact you make during your job search.
When you use the tools described in the Selling U sections of Chapter 7 and Chapter 8 to get the word out about your personal brand, follow-up will be especially important. Your list of twenty-five target companies and the appropriate people to contact at each that you created in the Selling U section of Chapter 7 should include a phone number and e-mail address for each person on your follow-up list. Within one week of sending a cover letter and résumé, a phone call to each person (or at least the top twenty people) on your mailing list will help reinforce your cover letter and résumé and give you the opportunity to sell yourself on the phone.
Follow-Up after Networking
You learned about the power of networking in Chapter 3. But like other forms of contact, networking requires follow-ups. Make it a point to follow up by e-mail or phone with each person on your networking list every four to six weeks.
It’s especially important to follow up quickly with those people with whom you connected about a possible job or contact to someone at a company. It’s appropriate to follow up within a week, unless the person told you otherwise.
Key Takeaways
• Unlike a sales call, a job interview usually doesn’t include stated objections.
• The secret to overcoming hidden objections such as experience or salary is to be prepared and establish the value you can bring to the company during the interview.
• Follow up after a job interview is a powerful way to make yourself memorable even after the interview is over.
• Thank-you notes (both e-mail and handwritten) should be sent within twenty-four hours of an interview to each person with whom you met. It’s also a good idea to send one to the recruiter who arranged the interview.
• Thank-you notes are a reflection of your personal brand. Correct spelling and grammar are required, including each person’s name and title.
• Follow-up, which may include a phone call or e-mail, is also important for each stage of your job search.
Exercise \(1\)
1. Assume you went on an interview for a job you want. Write a thank-you e-mail and handwritten thank-you note to the person with whom you interviewed.
2. Imagine that you are networking with someone who said his company may have an opening and asked you for your résumé. It’s been a week since you sent your résumé to him. When would you follow up? How would you follow up?
3. Assume that you are on a job interview and the interviewer says, “You have an interesting background, but I’m not sure you have the experience we need for this position.” How would you respond? | textbooks/biz/Marketing/The_Power_of_Selling/11%3A_Handling_Objections_-_The_Power_of_Learning_from_Opportunities/11.04%3A_Selling_U_-_How_to_Overcome_Objections_in_a_Job_Interview.txt |
Power Wrap-Up
Now that you have read this chapter, you should be able to understand how to handle objections.
• You understand objections are a normal part of the selling process and are not a personal reflection on you.
• You learn that objections are opportunities to build a relationship.
• You recognize that anticipating objections is the best way to handle them.
• You understand the role that risk plays in your prospect’s decision and how to help him minimize the risk.
• You can list the six strategies for handling objections.
• You can discuss the five types of objections and how to handle them.
• You learn how to handle objections in job interviews.
• You understand how to use a follow-up, including thank-you notes, to set yourself apart and overcome objections even after the interview.
TEST YOUR POWER KNOWLEDGE (AnswerS ARE BELOW)
1. What is an objection?
2. What is the best way to anticipate objections?
3. At what point in the selling process might the prospect or customer object?
4. Name the six strategies to handle an objection.
5. Name the five types of objections.
6. What is value?
7. What is a hidden objection?
8. How can you overcome objections after a job interview?
POWER (ROLE) PLAY
Now it’s time to put what you’ve learned into practice. Following are two roles that are involved in the same selling situation—one role is the customer, and the other is the salesperson. This will give you the opportunity to think about this selling situation from the point of view of both the customer and the salesperson.
Read each role carefully along with the discussion questions. Then, be prepared to play either of the roles in class using the concepts covered in this chapter. You may be asked to discuss the roles and do a role-play in groups or individually.
Meeting Objection
Role: Meeting planner at Capstone Industries, a distribution company
You are responsible for planning the annual meeting for the company. It is the only time that all five hundred employees are in one place. The three-day conference is usually quite a lavish affair; however, this year the budget is much smaller. Your objective is to book a five-star venue despite the budget reduction. You have just taken a tour of the lavish Premier Hotel, and you are impressed. However, the price you received in the proposal is still too high considering the fact that you would be booking five hundred rooms for three nights and three meals per day plus snacks, not to mention the additional business the lounge will realize from your attendees.
• Now that the salesperson has made his presentation, what will you say to tell her that the price is too high?
• What are the points you want to make in your objection?
Role: Event sales rep, Premier Hotel
You are responsible for booking the events at this spectacular five-star hotel. The convention facilities are state-of-the-art and ideal for large corporate meetings. The accommodations are suites, not rooms, so two people can stay comfortably in each one, which helps reduce the overall cost of rooms. The service is impeccable and has ratings above the Ritz Carlton and Four Seasons. In fact, Premier Hotel has received the J. D. Power and Associates Award for the best service in the hospitality industry.
You have done your presentation along with a pricing proposal and presented it to the prospect. This is an important meeting for the hotel, and it’s important that you close the sale. However, first you will need to handle some objections.
• What is the value that Premier Hotel offers to Capstone Industries for this meeting?
• What objections are you most likely to get?
• How you will prepare for each one?
• You are not willing to lower the price, so if you get the price objection, how will you handle it?
ACTIVITIES
1. Assume you are on a job interview and the interviewer has indicated that you might be overqualified for the position. How would you prepare for a question like this? How would you respond?
2. Visit your campus career center and meet with a career counselor to discuss common objections that may come up in job interviews. How would you handle each one?
3. Meet with your advisor or one of your professors or other professional. Share your career aspirations with them. Ask each of them about objections he may have if he were interviewing you. How would you handle each objection?
TEST YOUR POWER KNOWLEDGE AnswerS
1. Questions or hesitancies on the part of the prospect or customer.
2. Review your presentation with someone, write down all the possible objections, and incorporate them into your presentation.
3. A prospect may object at any time, especially when you are setting up the appointment, during the presentation, and during the trial close.
4. View the objection as a question, respond to the objection with a question, restate the objection before answering the objection, take a pause before responding, use testimonials and past experiences, and never argue with the customer.
5. Product objection, source objection, price objection, money objection, “I’m already satisfied” objection, and “I have to think about it” objection.
6. Value is the worth that a product or service provides to a customer. It is not based on cost but on perceived benefit.
7. An objection that is not openly stated by the prospect but is an obstacle in the way of making the sale.
8. Send a personal thank-you e-mail and handwritten thank-you note within twenty-four hours of the interview. | textbooks/biz/Marketing/The_Power_of_Selling/11%3A_Handling_Objections_-_The_Power_of_Learning_from_Opportunities/11.05%3A_Review_and_Practice.txt |
Closing the Sale: The Power of Negotiating to Win
Video Ride-Along with Lisa Peskin, Sales Trainer at Business Development University
Lisa Peskin shared her tips for successful prospecting in Chapter 7. During this ride-along, she is going to give you insights about closing the sale. The term close implies that this step comes at the end of the process. You may be surprised to learn when you start closing the sale.
Ride along with Lisa and learn about how she successfully closes sales.
(click to see video)
12.02: Closing Starts at the Beginning
Learning Objectives
• Discuss how to successfully close a sale.
“Show me the money.”
It’s this line from the classic 1996 movie Jerry Maguire that says it all about negotiating and closing the deal. In the movie, Jerry Maguire (Tom Cruise) is a sports agent who has second thoughts about the way business is conducted, and when he voices his concerns, he loses his job and all his clients except one. Maguire’s passionate plea to his sole client, NFL player Rod Tidwell (Cuba Gooding, Jr.), has become a dramatic metaphor for negotiations and deal making ever since.Jean-Marc Rocher, “Plot Summary for Jerry Maguire,” IMDb, www.imdb.com/title/tt0116695/plotsummary (accessed November 19, 2009).
While the movie is fictional, Maguire’s character was based on real-life sports agent Leigh Steinberg, whose firm has negotiated and closed more than one hundred multimillion-dollar deals for high-profile clients in every professional sport. Steinberg’s philosophy on negotiations and closing deals is based on the fact that life is filled with negotiations and deals—from deciding where to eat to buying houses and cars—and each should be handled with “a clear focus and principled philosophy.”Alan M. Webber, “How to Get Them to Show You the Money,” December 18, 2007, Fast Company, www.fastcompany.com/magazine/19/showmoney.html (accessed November 19, 2009). There’s nothing better than closing a big deal…the right way.
Whether it’s a major professional sports deal, business deal, or a major purchase, it’s easy to visualize what the “desired state” is in any kind of deal. You can actually see the athlete in your team’s uniform, imagine two companies merging together as one, or see yourself in the car you want to buy. In fact, you negotiate every day. You negotiate with everyone from your roommate about how to arrange the furniture to your siblings about who will use the car. You might even negotiate with your professor about when you can hand in an assignment that is late.
The step in the selling process that moves the conversation to a sale (or the desired resolution) is the close. Many people believe that the close takes place at the end of the selling process because that’s when the prospect agrees to buy the product or service. But nothing could be farther from the truth. Closing the sale, or getting the order, starts at the beginning of the selling process, long before you even come in contact with the prospect.
Start Strong
“What it takes to win a championship is to have your preparation meet the opportunities, whether it’s out on the racetrack or behind the scenes,” according to NASCAR driver Kurt Busch.Joe Guertin, “When Did ‘Closing’ Become a Bad Word?” Agency Sales, March 2006, www.allbusiness.com/sales/1064380-1.html (accessed March 16, 2010). This is true in sports and in selling. Winning in selling—delivering value to customers and to your company—requires good solid preparation and hard work. Sure, there are some sales that fall into your lap. Those are the ones that make it feel like selling is easy. But most sales don’t happen that way. In fact, in many industries closing the sale may take weeks, months, or even years.
Despite the term “close,” which implies the end, closing the sale starts with the first step in the selling process—qualifying. Sometimes salespeople want to fill their sales funnel (or pipeline) with lots of leads so they don’t take the time or ask the right questions when they are qualifying. While it’s true that you want to “go out and get as many nos as you can,” you’ll get a lot more yeses when you pitch to the right prospects.Joe Guertin, “When Did ‘Closing’ Become a Bad Word?” Agency Sales, March 2006, www.allbusiness.com/sales/1064380-1.html (accessed March 16, 2010). In fact, the selling process is analogous to building a house; if the foundation is poured right, everything else will easily come together. The same is true in selling—prospecting is the foundation of the entire process.Tim Connor, “The Myth of Closing Sales,” Roderick Martin, roderickmartin.com/the-myth-of-closing-sales (accessed November 17, 2009).
Not only does closing start at the first point in the selling process, but it also is far from the end of the selling process. In fact, closing is a lot like graduation—it is actually the beginning, not the end. Just like graduation is not the end of your education but rather the beginning or commencement of the rest of your life, the closing in sales is the beginning of the relationship with the customer, not the end of the selling process.
Closing Time
The close sounds like it might be a definitive part of the selling process. It’s actually not a single statement, question, or event. Rather, the close is an ongoing series of events that occurs throughout the selling process, according to Mary Delaney, vice president of sales for CareerBuilder.com.“Closing the Deal,” Selling Power Sales Management eNewsletter, May 17, 2004, http://www.sellingpower.com/content/newsletter/issue.php?pc=368 (accessed March 16, 2010). Qualifying is the key; it’s virtually impossible to close a sale with the wrong prospect. But the preparation doesn’t stop there. The preapproach, approach, presentation, and overcoming objections all play a role in the closing the sale. According to author Ray Silverstein, the close is made in the first thirty seconds of the sales presentation. He says that’s when a customer has an emotional response to you and your product or service story. Silverstein points to research that was conducted by William Brooks and Thomas Travisano that concludes that people want to buy from people they like and trust.Ray Silverstein, “How to Close a Sale in the First 30 Seconds,” Entrepreneur, http://www.entrepreneur.com/management/leadership/leadershipcolumnistraysilverstein/article178590.html (accessed November 17, 2009). If this sounds familiar, it should be. The concept of building a relationship based on first impressions was covered in detail in Chapter 3. And understanding the difference between needs, which are rational, and wants, which are emotional, makes a difference in how your prospect perceives you and the message you are delivering.
To demonstrate that the close takes place at virtually every point in the selling process, Daniel Sheridan from Extensis Group LLC, a sales training consultancy, says it best: “If you’re waiting for a proposal to close, it’s too late.” He goes on to say that the most important meeting is the first one because that’s when trust and rapport are established.Simona Covel, “Finding the Right People to Make the Sale,” Wall Street Journal, May 29, 2008, http://online.wsj.com/article/SB121199448885726503.html (accessed November 17, 2009).
The close builds on everything that has already taken place throughout the selling process—rapport, trust, information sharing. It’s also important to know what the close is not. The close is not a high-pressure exchange between seller and buyer. It’s not a time when the salesperson resorts to trickery, manipulation, or other unsavory tactics just to get a sale.Geoffrey James, “Close More Sales: Train Your Sales Team,” Selling Power 23, no. 8, http://www.sellingpower.com/content/article.php?a=6389 (accessed March 16, 2010). While sales are the ultimate financial goal of the selling process, relationships, trust, and understanding a customer’s business and providing cost-effective solutions are driving factors behind making the sale.Ram Charan, “What Your Customer Isn’t Saying about Your Sales Pitch,” Wall Street Journal, May 29, 2008, http://online.wsj.com/article/SB121182439378120865.html (accessed November 17, 2009). The same principles that guide the rest of the selling process also guide the close.
If closing is not a specific event that happens during the selling process, you might be wondering how you effectively get the order. You learned about the trial close in Chapter 10. The trial close can take place during any part of the selling process. The trial close gives you the opportunity to get specific feedback from the customer as it relates to her likelihood to make the purchase at any point during the process. While the trial close is most likely to come during the presentation, it could come even earlier in the process depending on the prospect and the product or service being purchased. A trial close asks for an opinion (“What is most important to you about this product or service?”), whereas a closing question asks for a decision (“Shall we complete the paperwork?”).D. Forbes Ley, “Trial Closing Questions Tell You When to Ask for a Decision,” The Bachman Company, www.bachmanco.com/pretz/PDF/Trial%20Closing%20Questions.pdf (accessed November 18, 2009). The trial close gives you the opportunity to learn what the prospect is thinking and will give you some insight as to when to make the close. In some cases, the trial close may result in a close, but if it doesn’t, the prospect’s response provides valuable insight. The trial close should be done early and often throughout the selling process. Getting the prospect’s opinion at various points throughout the process helps you determine your path and how and when you should make your close.D. Forbes Ley, “Trial Closing Questions Tell You When to Ask for a Decision,” The Bachman Company, www.bachmanco.com/pretz/PDF/Trial%20Closing%20Questions.pdf (accessed November 18, 2009).
ABC or ABO?
There is an old adage in selling that says, “Always Be Closing” (ABC). This means that a salesperson should never miss the opportunity to close a sale, no matter where it occurs in the selling process.Michelle Nichols, “The Two-by-Four Closing Question,” BusinessWeek, April 19, 2007, http://www.businessweek.com/smallbiz/content/apr2007/sb20070419_586407.htm (accessed November 17, 2009). But in today’s collaborative environment, it’s better to approach closing more like “Always Be Opening” (ABO).Joe Takash, “Connect with the Buyer,” www.joetakash.com/media-resource/wp-content/uploads/2009/03/independent-agent.pdf (accessed May 16, 2010). In other words, the best strategy is to always be helping your customer identify and solve his problems, just like you do when you are opening the selling process. Focus on asking the right questions and learning about how you can suggest solutions (in some cases, the solution might not even be your product or service). When you deliver value to your prospect, they will look to you for advice and counsel. “You become much more than a salesperson, you become their marketing expert, a resource, an ally,” according to Mario Russo, general sales manager at radio station WBEN-FM in Philadelphia. “That’s when you are successful in selling.”Mario Russo, Executive Panel in Marketing 2335—Public Relations and Publicity, Saint Joseph’s University, Philadelphia, PA, November 18, 2009.
It’s true that asking for the order is critical for success in selling. But if you close too soon, you might run the risk that the customer thinks that the process is over and mentally moves on to something else.Mark Hunter, “Close Too Quick and You Lose Profit,” Fast Company, November 4, 2009, www.fastcompany.com/blog/mark-hunter/sales-hunter/close-too-quick-and-you-lose-profit (accessed November 17, 2009). That’s why it is a good idea to ask exploratory questions: open-ended, nonthreatening questions that encourage your prospect to discuss her business needs. This helps supplement the information you gathered during the preapproach, enabling you to understand what the customer needs and how to meet those needs. For example, if you are selling accounting software, you might ask the following exploratory questions: “What are the top three activities that consume your people’s time daily?” “What is the ideal way you would like your people to spend their time?” “What are the types of activities that you think can be automated?”“Closing the Deal,” Selling Power, May 17, 2004, http://www.sellingpower.com/content/newsletter/issue.php?pc=368 (accessed November 17, 2009). None of these is a hard-sell question. Rather, each question allows you to listen and gather information so that you can identify how you can help the prospect solve his problem.
While you always have your eye on the prize of closing a sale, the focus is to extend your relationship with your prospect beyond selling to servicing and being a business partner. That’s what ABO is all about. When you focus your selling efforts in this way, it makes it easier to sell additional products and services to existing customers because you are constantly learning about ways in which you and your company can add value.
Always Be Opening
(click to see video)
This video featuring sales guru, Jeffrey Gitomer, highlights the shift from ABC to ABO.
Ask for the Order
When you focus on delivering value to your prospects and customers, you have earned the right to close or ask for the sale. It might seem obvious, but sometimes salespeople get caught up in the selling process and lose track of the fact that it is a buying process for the prospect. Sometimes, simple questions like “Will delivery on Tuesday work for you?” or “Should we start your service the week of the twenty-first?” help you and the customer focus on moving from the sales presentation to the delivery of the product or service. The specific closing questions will most likely differ based on the product or service you are selling. For example, in pharmaceutical sales, industry sales expert Jane Williams adds, “Never end a successful close without adding the proper patient dosing.” She says, “It is very important that your physician prescribe your product properly.”“Closing Arguments,” Selling Power Pharmaceuticals eNewsletter, September 11, 2007, http://www.sellingpower.com/content/newsletter/issue.php?pc=648 (accessed March 16, 2010).
Sometimes salespeople don’t feel comfortable asking for the order. Earn the right to ask for the order. Be confident: believe in yourself and your product or service.Laura Lorber, “Three Tips for Closing a Sale,” Wall Street Journal, http://online.wsj.com/article/SB121198785761226199.html#printMode (accessed November 17, 2009). The trust you establish from the beginning will translate into how you can close the sale. Closing the sale is all about presenting solutions for the biggest problems that your prospect faces. “If you can’t help them with their biggest challenge, they won’t have time for you,” says Mary Delaney from CareerBuilder.“Closing the Deal,” Selling Power Sales Management eNewsletter, May 17, 2004, http://www.sellingpower.com/content/newsletter/issue.php?pc=368 (accessed March 16, 2010). Author Barry Farber includes the element of confidence in the closing equation by saying, “The important factor that contributes to your success at closing (or knowing when to move on) is the leverage you have going in and the confidence you have to back it up.”Barry Farber, “Wrap It Up,” Entrepreneur, April 2008, http://www.entrepreneur.com/magazine/entrepreneur/2008/april/191580.html (accessed November 17, 2009).
Not every contact results in a sale. Typically, 80 percent of prospects say no to a sales offer, and that percentage may be as high as 90 percent during these challenging economic times.“Sales Closing—Closing Throughout the Sales Cycle Process Using Different Types of Closes,” Money Instructor, http://www.moneyinstructor.com/art/saleclose.asp (accessed November 17, 2009). This underscores the fact that it usually takes several closes to actually close the sale. In some cases, it will take at least three tries. In other cases, it can take as many as five or more attempts. It’s best to view closing as an ongoing part of the process, not a single event in which a prospect can say no. Confidence and the right mental attitude can make all the difference in being able to take all the nos on the way to yeses.Joe Takash, “Connect with the Buyer,” www.joetakash.com/media-resource/wp-content/uploads/2009/03/independent-agent.pdf (accessed May 16, 2010).
When to Close
It’s rare that a prospect will say, “I’m ready to close this deal.” That step in the process usually belongs to the salesperson to actively close the sale. The best way to know when to close is to listen and watch. There are verbal and nonverbal cues that prospects provide that help you understand when she is ready for you to close. Here are some of the signals that the prospect is ready to buy:
• When the prospect displays positive body language and interaction. The prospect is engaged, interested, asks questions, reviews literature, and provides insights about his business.
• When the prospect asks questions. It is a good time to close after answering a question. Questions like “How long will delivery take?” or “How would that integrate into our current system?” are good cues that the prospect is close to buying.“Tips for Closing a Sale,” AllBusiness, www.AllBusiness.com/sales/selling-techniques-closing-sales/450-1.html (accessed November 17, 2009).
• After you handle an objection. This can be the perfect time to close, as you have just provided some insight that will help the customer make her decision.
Tips for Closing with a Committee
It’s one thing to close a deal with an individual buyer. It’s another thing to close with a buying committee. Here are four steps to close with a committee:
1. Have a specific, measurable, actionable, realistic, and time-bound (SMART) objective.
2. Know each committee member’s name and role in the decision.
3. Identify your champion on the committee.
4. Leverage your champion to help “sell” the committee for you.Steve Atlas, “Closing: How to Use the Right Techniques to Close a Committee,” Selling Power 20, no. 8, http://www.sellingpower.com/content/article.php?a=5547 (accessed March 16, 2010).
Types of Closes
There is not a single surefire way to close every sale. You should be prepared with several different types of closes and use them as appropriate for each situation. Some situations may require a combination of closes.
Direct Request Close
Direct request close means that you simply ask for the order. This is the most straightforward approach to a close. The fact is customers expect salespeople to ask for the order. This is a simple but effective way to close the sale.Gerald L. Manning, Barry L. Reece, and Michael Ahearne, Selling Today: Creating Customer Value, 11th ed. (Upper Saddle River, NJ: Pearson Prentice Hall, 2010), 310.
You: Can I write up the order as we discussed?
Prospect: I think we have covered everything. Yes, let’s wrap it up.
Benefit Summary Close
The benefit summary close is a natural extension of the selling process. It simply summarizes the benefits of everything you have discussed throughout the process. This approach is especially effective when you are able to integrate and present benefits from the prospect’s point of view that you have discussed over the course of several meetings. This is an opportunity to focus on how you can help her solve the largest problem that she faces.Barton A. Weitz, Setphen B. Castleberry, and John F. Tanner, Jr., Selling: Building Partnerships, 7th ed. (New York: McGraw-Hill, 2008), 319.
You: We’ve talked about the fact that speed is extremely important to you and your company. We can deliver your complete order to your twenty-seven construction sites within forty-eight hours of your commitment. In addition, you’ll never be at risk for product performance because we guarantee the product 100 percent. If you ever have a problem, you just call us, and we’ll replace it, no questions asked. Will you be willing to commit to an initial order of fifty?
Prospect: Yes, we are looking for a partner who will not only provide the highest quality product but also be able to deliver it on time to all our locations. It sounds like you have your bases covered. If you can deliver what you say, we have a deal.
Assumptive Close
The assumptive close asks a question that when the prospect replies, she is committing to the sale.Geoffrey James, “Sales Reps’ Frequently Asked Questions on Closing,” Selling Power, http://www.sellingpower.com/content/article.php?a=6389 (accessed November 17, 2009). In other words, you are assuming that the customer is going to make the purchase. This close can be effective if you have done your job of developing trust and rapport with your prospect.
You: Shall we set you up on automatic billing?
Prospect: Automatic billing definitely works best for us.
Alternative-Choice Close
The alternative-choice close gives the prospect a choice between two options rather than a choice between buying and not buying.“Alternative Close,” ChangingMinds.org, http://changingminds.org/disciplines/sales/closing/alternative_close.htm (accessed November 18, 2009). This close is related to the assumptive close but gives your prospect the option of which product or service they will buy.Charles M. Futrell, Fundamentals of Selling: Customers for Life through Service, 10th ed. (New York: McGraw-Hill Irwin, 2008), 417.
You: Would you prefer the white or blue?
Prospect: White is a more neutral color.
Types of Closes
(click to see video)
Hear Lisa Peskin, sales trainer at Business Development University, discuss the assumptive close and the alternative-choice close.
Compliment or Vanity Close
The compliment (or vanity) close helps you relate the purchase to the person and appeal to his or her sense of identity. You are making a positive connection between the purchase decision and the judgment of the buyer. When you use this approach to closing, you are confirming their role as a subject matter expert. You are, in fact, paying them a compliment.
You: One of the reasons I like calling on you is because you and your team really understand your business and your customer. You make it easy for your customer to buy from you and you offer them the product at a fair price. No games, no coupons just good, honest value. I think that our product can expand your offering to your customers with a company that shares your values about putting the customer first. I suggest you start by adding this item to your line and let’s gauge the customer response.
Prospect: I’m glad to hear that you feel that way. We do take our commitment to our customers very seriously and we only like to do business with people who feel the same way. I think it would be a good idea to start out with this one product and get some customer feedback. If they like it, we can talk about expanding to more products.
Combination Close
It’s best to have several types of closes ready to deliver. In some cases, it’s a combination of closes that helps you ultimately gain agreement with the prospect. Virtually any of the different closes can be used together.
You: The horsepower on this model is the highest in the industry. And the model is so efficient that it will lower your cost per unit in all your factories starting on day one. Can we wrap this up?
Prospect: It looks like this is going to be a good short-term and long-term investment for us. Yes, let’s get the paperwork ready.
Keep It Brief
Whatever close you use, it’s best to keep it focused and brief. Salespeople have a habit of talking too much, especially when they’re ready to close. According to Michelle Nichols, contributor for BusinessWeek, author, and sales trainer, “Ask yourself what aspect of your offering would customers want so badly that they would miss lunch or cross a very busy street to get it?”Michelle Nichols, “The Two-by-Four Closing Question,” BusinessWeek, April 19, 2007, http://www.businessweek.com/smallbiz/content/apr2007/sb20070419_586407.htm (accessed November 17, 2009). That should be the focus of your close.
What Works?
Closing is part of the selling process. A process is a systematic approach, which, by its very nature, can be measured. You won’t be able to be successful closing every sale. After all, even professional baseball players only hit the ball three times out of every ten pitches to be considered above average. While hitting the ball 100 percent of the time would be considered unrealistic, every professional hitter takes batting practice to help increase his batting average. His batting coach gives him tips as to how to stand, swing, and ultimately increase his percentage of hitting the ball. The same can be done in closing. Record the information about your closings—what works and what doesn’t.Raymund Flandez, “Sales Outsourcing First Teaches Workers to Ask A Lot of Questions,” Wall Street Journal, September 12, 2008, http://online.wsj.com/article/SB121199452139126417.html (accessed November 17, 2009).
You don’t have to wait until the close to be able to track your progress. Sales veteran and author Barry Farber suggests managing accounts and the sales process with a simple visual tool. Post your prospects in the different stages of the sales cycle on a corkboard. While there are several software programs that perform this function, there’s nothing more powerful than seeing it play out on the wall in front of you every day.Barry Farber, “Wrap It Up,” Entrepreneur, April 2008, http://www.entrepreneur.com/magazine/entrepreneur/2008/april/191580.html (accessed November 17, 2009).
Closing Complex Sales
A complex sale is a term that usually refers to high-value purchases (usually \$100,000 and higher). Products and services such as enterprise systems, health care providers, commercial real estate, manufacturing equipment, logistics services, and other major business-to-business (B2B) purchases are considered complex sales. These types of sales have a long selling cycle because there is a lot at stake for such a major purchase and there are multiple people involved in the decision-making process. In fact, it may take as long as six months to three years to close the sale.Steve Kayser, “Shooting the Donkey in the Complex Sales Process…Hollywood Style,” http://scottymiller.wordpress.com/category/tips-on-navigating-the-complex-sale (accessed January 7, 2010). The product or service commitment is usually a long-term commitment with a contract as long as three, five, or even ten years or longer.
While the selling skills discussed throughout this book apply to complex sales, there are some differences. According to Jeff Thull, author of Mastering the Complex Sale, there are four phases to a complex sale.
1. Discover. As with any other sale, research about the prospect and his needs is critical to success. During the discover phase, you set the stage for the ongoing relationship or engagement. This stage includes your detailed research about the company and its current provider including several meetings and phone calls with the prospect. It is at this stage that the prospect determines whether the engagement has potential.
2. Diagnose. In a complex sale, the decision is likely to be centered on what should be changed, such as the location of a warehouse, and includes a collaborative effort between the salesperson and the customer to determine if the change is feasible and desirable. This stage also includes extensive financial analysis to determine the impact of the decision on the company. The role of the salesperson is to be a true business partner and help the prospect understand the trade-offs and benefits of making a major change in the operation.
3. Design. Complex sales usually involve products and services that are customized for each customer. For example, an ad campaign, software, retail fixtures, or other major purchases are adapted, adjusted, or designed exclusively for that customer. At this stage, the sales rep works closely with key people in the customer’s organization to design the best solution to fit the customer’s needs.
4. Deliver. If the first three phases are implemented correctly, this final stage should logically follow. At this point, the key people at the customer’s organization have been involved in the design and financial justification of the product or service so the presentation of the formal proposal should lead to acceptance. Then the efforts are focused on the delivery of the product or service and implementation. For products such as software and other major purchases, there may even be training, troubleshooting, and other transition issues that are handled by the salesperson.“Winning Strategies to Succeed in Complex Sales,” Prime Resource Group, March 2010, www.masteringthecomplexsale.com/sales-training-book-press-release.htm (accessed January 7, 2010).
During each of these phases, it’s important to identify all the decision makers and their positions in the process. As with every stage in the selling process, this is about asking the right questions. “How will your organization go about making this decision?” and “Who else do I need to talk to?” are good questions to ask during the discover phase so that you can get input and feedback from all involved at the beginning of the process. Once you identify all the people involved in the decision-making process, you’ll want to identify the decision makers. Again, the right questions will help you focus your efforts appropriately. Knowing to whom the expense will be charged helps you identify the ultimate authority. The person who controls the budget is most likely different from the person who will be evaluating the technical aspects of the product or service. For example, while the chief information officer may make the budget decision, the systems implementation manager may be evaluating the technical aspects of the software. Finally, you want to identify the “power broker,” the person who will ultimately make the final decision. This is usually the person, a subject-matter expert, who is the right hand of the person who controls the budget.“Ten Keys to Winning Complex High Dollar Sales,” Best-Coaching-Training.org, May 9, 2009, www.best-coaching-training.org/2009/05/29/ten-keys-to-winning-complex-high-dollar-sales (accessed January 7, 2010). In other words, you want to identify with whom you will be negotiating and ultimately closing the sale.
Key Takeaways
• Closing is not an event but an ongoing part of the selling process that starts with prospecting and qualifying.
• Closing is all about helping the customer solve the single largest challenge he faces.
• Salespeople should always ask for the sale and make it easy for the customer to go from the conversation or sales presentation to the sale.
• The prospect provides verbal and nonverbal cues that make it easier to know when to close.
• There are several different types of closes. Each can effectively be used alone or in combination with other closes.
• Complex sales have a longer selling cycle, have many people involved in the decision making, and require a modified selling process.
Exercise \(1\)
1. Assume you are selling coffee to a chain of restaurants. The buyer is very concerned about changing the brand of coffee that the restaurant uses because coffee is the last experience that the customer has with the restaurant and she doesn’t want to change anything about the current experience. You have sampled your coffee in a blind taste test with her and in several of her restaurants, and all who have tasted it have chosen it as the better-tasting coffee. Now that she is convinced that this change would be a good one, which type of close would you use and why?
2. Assume you are selling paper to a major high-volume printer. Your firm has just introduced a new type of recycled paper that is less expensive than previous options. The buyer is someone whom you know and respect. You have learned a lot of what you know about the industry from him. You feel like you are bringing him a new product that can bring benefit to his company. You are preparing a compliment close. What would you include in the close?
3. Think about a high-ticket product or service that you recently purchased from a salesperson. How did the salesperson approach the close? Which approach to closing did she use? Was it effective? Why or why not?
4. Name the type of close that is used in each of the following examples:
• “Would you like the pay-as-you-go or the family plan?”
• “Shall we formalize the deal with your signature?”
• “I really enjoy working with you and your team, and the way you are growing the company so fast. That’s why I’d like to suggest this service plan.”
• “With the extra capacity, you’ll be able to expand your service as you need it, yet it won’t cost you any additional monthly fees. You can sign right here, and we can start your service on Monday.”
5. Create a closing for each of the following situations and identify the type of close you are using:
• You are a real estate agent, and you just finished showing a house to a newly married couple.
• You are a fine jewelry salesperson, and you are showing a diamond engagement ring to a young man.
• You are selling high-end electronics, and you are demonstrating a home theater system to a couple who just bought a new house (and it’s the week before the Super Bowl). You are able to have it delivered and installed before Super Bowl Sunday.
• You are selling memberships to a health club, and you just took a couple on a tour. They recently moved to the area and are not familiar with all the competitors.
• You are selling accounting software, and you just finished a demonstration of the product for a group of lawyers in a firm. | textbooks/biz/Marketing/The_Power_of_Selling/12%3A_Closing_the_Sale-_The_Power_of_Negotiating_to_Win/12.01%3A_Introduction.txt |
Learning Objectives
• Learn how to negotiate so that all parties win.
Now that you have learned about the role of closing in the selling process and techniques to close the sale, it’s time to dig a bit deeper into the process of negotiating. Depending on the product, service, or prospect, some sales might be straightforward like, for example, buying a computer (“I’ll take the MacBook Pro with the fifteen-inch screen”). The price is posted and there is no room for negotiation. However in many situations, especially in business-to-business (B2B) selling, the pricing, length of contract, terms, options, delivery dates, services, and other aspects of the sale can all be negotiated. Negotiation, like selling, is a process. Following the process helps improve your chances of getting what you want.
The Art of Negotiation
Simply put, “negotiating is the act of discussing an issue between two or more parties with competing interests with the aim of coming to an agreement.”“Negotiation,” Entrepreneur, http://www.entrepreneur.com/encyclopedia/term/82556.html (accessed November 20, 2009). While that might sound like an impossible task, it is not as difficult as you might think. Even people with differing positions or points of view share a common interest, which becomes the basis for finding common ground. It’s these common interests—security, economic health, personal recognition, control—that motivate people. If you take the time to understand your prospect’s interests in a negotiation, you can successfully collaborate and find a solution that supports the interests of all parties.Stephanie Mojica, “The Art of Sale Negotiation Skills,” Associated Content, December 26, 2008, www.associatedcontent.com/article/1313361/the_art_of_sale_negotiation_skills.html?cat=35 (accessed November 19, 2009).
It is negotiating that provides profit for organizations. The collaboration between parties is what provides companies the opportunity to exchange goods and services for money.
Link
Sales Negotiations
This series provides insights about how to negotiate in B2B selling.
Why negotiate:
http://www.sellingpower.com/content/video/?date=9/7/2007
How to negotiate using value:
http://www.sellingpower.com/content/video/?date=9/10/2007
What makes a good negotiation:
http://www.sellingpower.com/content/video/?date=9/11/2007
It might be helpful to think about a negotiation like an iceberg. Although you can see the tip of the iceberg, it can be deceiving because it does not tell the entire story. The same is true when you are negotiating; your prospect may say something that appears to be obvious but really wants to achieve other things that are hidden below the surface. Using the process of negotiation to learn more about your prospect’s motivations and interests, you can understand what is below the tip of the iceberg. It’s usually the part of the iceberg that you can’t see that is more substantial and has more impact that the portion that is visible. When you come prepared, listen, and probe during the negotiation process, you can learn a lot about what lies below the tip of the iceberg and use this information to collaborate and eventually reach a common ground on the issues. For example, assume you are selling advertising space for a men’s magazine to the hottest new beer company. Your contact at the beer company wants to get the word out about this new brand but has a very small budget, so he doesn’t want to pay the full published rate for the ads. You don’t want to sell at less than the published rate because that will lower the value of your ad space. The tip of the iceberg shows that this is a price negotiation. However, if you ask the right questions and listen more, you will learn that his ultimate objective is to get people to taste the beer because that is the best way to get new customers. If he can get a major sampling opportunity, then he can use it to go to other media partners to get other sampling campaigns. Now you have gotten below the surface of the iceberg and understand his motivations. With this additional information that wasn’t readily visible on the surface, you can offer him an advertising package that includes ads in the magazine in addition to sampling opportunities at three upcoming national events that the magazine sponsors. Now the negotiation is focused on all parties winning by getting something they want, rather than simply negotiating on price. Getting below the surface provides valuable information and insights for negotiating.
Definition of Negotiating
Understand that negotiation takes place only before you agree to anything: “If you ask for something before a contract is signed, it’s called ‘negotiating.’ If you ask for something after a contract is signed, it’s called ‘begging.’ It’s better to be a good negotiator than an expert beggar.”RCM Staff Report, “27 Principles of Negotiating with a Meeting Facility,” MeetingsNet, February 1, 2003, meetingsnet.com/ar/meetings_principles_negotiating (accessed November 19, 2009).
Negotiate to Win-Win-Win
A successful negotiation can be measured by its ability to deliver a mutually beneficial solution to all parties. Some people believe that negotiation is an act that yields a “win” for one side and therefore a “lose” for the other side. The win-lose approach usually ends up in a lose-lose deal that doesn’t work for anyone.Robert J. McGarvey, “Covering the Bases,” Entrepreneur, June 1997, http://www.entrepreneur.com/magazine/entrepreneur/1997/june/14260.html (accessed November 20, 2009). This philosophy of negotiating is selfish and short term. In addition, this approach implies that negotiation includes some kind of confrontation or manipulation to “trick” one side into doing something that it doesn’t want to do. This is an unethical approach to negotiating which doesn’t have a place in the business world.Daniel Roach, “5 Simple Rules for Unbeatable Sales Negotiation,” Associated Content, September 29, 2008, www.associatedcontent.com/article/1047808/5_simple_rules_for_unbeatable_sales.html?cat=3 (accessed November 19, 2009).
In selling, negotiating and closing go hand-in-hand. Just as closing is not a one-time event, negotiating is a process that has both short-term and long-term impacts.“Negotiating to Win-Win,” Selling Power Sales Management eNewsletter, January 6, 2003, http://www.sellingpower.com/content/newsletter/issue.php?pc=248 (accessed March 16, 2010). The best negotiations are collaborative in nature and focus on delivering mutual satisfaction. According to Leigh Steinberg, lawyer and sports agent, “The goal is not to destroy the other side. The goal is to find the most profitable way to complete a deal that works for both sides.”Alan M. Webber, “How to Get Them to Show You the Money,” Fast Company, October 31, 1998, www.fastcompany.com/magazine/19/showmoney.html (accessed November 19, 2009). Effective negotiating is based on respect and is seeded with open communication. Collaborative negotiating is dependent on the following three elements:Herb Cohen, You Can Negotiate Anything (New York: McGraw-Hill, 1980), 163.
1. Building trust.Herb Cohen, You Can Negotiate Anything (New York: McGraw-Hill, 1980), 163. You’ve already learned in Chapter 3 that establishing and building trust is key to relationship building. Negotiating is the ultimate extension of a relationship because you and your customer are agreeing to concede on some points to make the relationship go even farther. If your prospect signs a contract with your company for products or services, you are now even more dependent on each other to make the relationship work. It is the true win-win-win relationship. But if your prospect doesn’t trust you, or you don’t trust her, it will be difficult to enter into a negotiation that will work for both of you and both of your companies. Building trust is the precursor to all business transactions, especially negotiating and closing.
The best way to build trust during the negotiation process is to gain trust before the formal negotiation. And then, during the formal negotiation, focus on the ends rather than the means.Herb Cohen, You Can Negotiate Anything (New York: McGraw-Hill, 1980), 163. In other words, instead of focusing on going head to head on each issue to be negotiated, concentrate on keeping the end goal in mind. Take the time to listen and understand exactly what is motivating your prospect so you can deliver what is important to her. “Negotiation is needs based,” according to the online Selling Power Sales Management Newsletter. Understanding what is important to you and to your prospect drives your negotiation.“Negotiating to Win-Win,” Selling Power Sales Management eNewsletter, January 6, 2003, http://www.sellingpower.com/content/newsletter/issue.php?pc=248 (accessed March 16, 2010).
Power Player: Lessons in Selling from Successful Salespeople
Honesty: The Best Negotiating Tool
Marty Rodriguez, one of the top real estate brokers worldwide for Century 21, has a simple formula for successful negotiations. She feels strongly that the real estate business isn’t just about closing the deal—it’s about providing honest information to help customers make the decision that’s right for them. She tells prospects everything from the fact that there is structural damage on a property to whether she thinks a deal is out of their price range. “When you treat people that way they’re not only happy to give you a commission—they become raving fans,” according to Rodriguez.Polly LaBarre, “Saleswoman for the 21st Century,” Fast Company, December 18, 2007, www.fastcompany.com/node/36271/print (accessed November 18, 2009).
2. Gaining commitment.Herb Cohen, You Can Negotiate Anything (New York: McGraw-Hill, 1980), 163. Part of the process of closing is gaining commitment on every specific element of the sale. To do that effectively, strive to gain commitment long before you begin the formal negotiation. That means using every touch point you have at the company to help you. While you might think it is impossible to enlist others in your prospect’s company to help you sell, consider the creativity of Art Fry, the creator of 3M Post-it notes. Fry stumbled upon the semisticky adhesive years before the product was introduced after creating the first version of the product as a way to mark hymns in his hymnal at church, he started giving his new invention to secretaries and coworkers at 3M. Soon secretaries were taking the pilgrimage between buildings on the 3M corporate campus just to get more of the sticky note pads. It was the demand from the people who used the product that ultimately generated interest in marketing the product to consumers. Fry successfully gained commitment from others in the company as a way to “sell” his new invention as a marketable product.Greg Beato, “Twenty-Five Years of Post-it Notes,” March 24, 2005, archives.secretsofthecity.com/magazine/reporting/features/twenty-five-years-post-it-notes-0 (accessed November 20, 2009).
3. Managing opposition.Herb Cohen, You Can Negotiate Anything (New York: McGraw-Hill, 1980), 183. It’s true that although a negotiation is a collaborative effort, it is inherently a situation that addresses opposing views. The best way to manage this is to be prepared and know what’s important to you and your prospect.
Power Selling: Lessons in Selling from Successful Companies
Searching for Common Ground
Microsoft wanted to be more dominant in the Internet search business and saw the acquisition of Yahoo! and the development of a new search engine named Bing as the way to gain market share quickly. Although Microsoft made a bid to buy Yahoo! in early 2008, it wasn’t until July 2009 that a deal was closed. The original \$45 billion takeover bid was shunned by Yahoo! much to the dismay and dissatisfaction of the shareholders because senior management wanted the company to remain a separate company.Peter Burrows and Robert D. Hof, “Yahoo Gives in to Microsoft, Gives Up on Search,” BusinessWeek, July 29, 2009, http://www.businessweek.com/technology/content/jul2009/tc20090728_826397.htm (accessed January 7, 2010). Then, newly appointed Yahoo! CEO Carol Bartz saw an opportunity for common ground and negotiated a deal that was a win for everyone. Under the ten-year agreement, Microsoft’s Bing will be used to power Yahoo! searches. Yahoo! will receive 88 percent of the revenue from all searches done on Yahoo! Web sites. Customers and advertisers now have a viable alternative to Google. Negotiating a solution that lets everyone win, including the customer, takes creativity and time.David Goldman, “Microsoft and Yahoo: Search Partners,” CNNMoney.com, July 29, 2009, http://money.cnn.com/2009/07/29/technology/microsoft_yahoo/index.htm (accessed January 7, 2010).
The Three Elements of Negotiation
Every negotiation, whether it is in business, politics, or your personal life, includes three critical elements. Understanding the role of these elements can help make you a better negotiator.
1. Information. When you do your homework, research, and ask questions about what is important to your prospect, you may be able to avoid negotiating on price all together. If you have information, and share information at the appropriate time, you can make a negotiation a huge win for everyone.Herb Cohen, You Can Negotiate Anything (New York: McGraw-Hill, 1980), 19.
2. Power. According to Herb Cohen, known as the world’s best negotiator, power is based on perception. If you perceive you have the power to influence your situation, you do (conversely, if you don’t believe you have the power, you don’t).Herb Cohen, You Can Negotiate Anything (New York: McGraw-Hill, 1980), 20.
3. Time. Time is the great negotiator. Ninety percent of all negotiating occurs during the last 10 percent of the set time frame. Deadlines force decisions to be made and negotiations to come to fruition. Use time to your advantage by never revealing your deadline. Don’t negotiate when you’re in a hurry; chances are you won’t get the result you want.RCM Staff Report, “27 Principles of Negotiating with a Meeting Facility,” MeetingsNet, February 1, 2003, meetingsnet.com/ar/meetings_principles_negotiating (accessed November 19, 2009).
Everything Is Negotiable
Many salespeople are afraid of negotiating. They are worried that they won’t be up to the challenge to persuade someone to do what they want or to pay their price. Confidence and preparation go a long way to achieving a satisfactory result on both sides. Negotiating and closing are ways of gaining agreement. The old saying goes, “Everything is negotiable,” and it’s true. Your prospect believes the same thing so be prepared to negotiate about virtually every aspect of the sale. For less complex sales, the close might come as a result of a simple question at the end of the presentation. However, for more complex sales, there are various elements of the sale that must be agreed upon to close the sale. Elements such as price, length of contract, service, terms, and options are common points to be negotiated as part of the close.
One for All
Negotiations in B2B selling usually require multiple parties to be involved from both companies. You may find yourself negotiating one-on-one with a prospect or being a member of a negotiating team that works with a prospect team to negotiate a deal. Either way, the same principles of negotiating apply.
Many salespeople are concerned about negotiating price. They think that lowering the price will make the sale. In fact, price is rarely the motivating factor behind any purchase. That’s not to say that price isn’t important, but customers buy value, not price. If price were always the determining factor in purchases, premium brand such as Porsche, Apple, and Neiman Marcus would not exist. If you’ve ever shopped at Nordstrom, Banana Republic, or Abercrombie & Fitch, you decided that those retailers offered more value than Old Navy, eBay, or Wal-Mart for the item you bought. Price is a part of the value equation but not all of it. According to author Kelley Robertson, “Everything you say and do from the first contact with a prospect affects the value of your product or service in their mind.”Kelley Robertson, “Let’s Make a Deal: Negotiating Techniques,” The EyesOnSales Blog, January 18, 2008, www.eyesonsales.com/content/article/lets_make_a_deal_negotiating_techniques (accessed November 19, 2009). That means establishing value with your presentation, demonstration, testimonials, follow-up, and everything that comes before the actual negotiation. How is your product or service different? What advantage does it offer? What is the most important problem it will solve for your prospect?Kelley Robertson, “Let’s Make a Deal: Negotiating Techniques,” The EyesOnSales Blog, January 18, 2008, www.eyesonsales.com/content/article/lets_make_a_deal_negotiating_techniques (accessed November 19, 2009).
Holding Firm
Forty percent of customers ask for a price concession not because they want it to close the sale but because “they had to ask.”
Fifty percent of salespeople give price concessions on the first request.
The best salespeople negotiate on value, not price, and use creative negotiating to find common ground.Colleen Francis, “Negotiation Quick Hits,” The EyesOnSales Blog, November 13, 2008, www.eyesonsales.com/content/article/negotiation_quick_hits (accessed November 19, 2009).
If your prospect wants to negotiate on price, use your creative problem solving skills to get to the end that will work for all parties. Use concessions, something that you are willing to compromise, to create value during the negotiation. For example, use length of the contract, payment terms, service, delivery date, training, or other elements to demonstrate to your prospect that you are willing to work with him and give him something that has value to him.
You: I’m not able to meet that price, but I can offer you three months of training worth \$3,000 at no charge.
Prospect: How many employees would be included in the training?
The following is another example:
You: That pricing is only available if you carry the entire product line. If you add all ten of the products into all your stores, I can meet that pricing.
Prospect: We can take a look at that.
The bottom line is that it’s best not to make a concession without getting a concession. In these examples, the salesperson always used another part of the deal to give something and get something in return. This win-win-win approach helps reach common ground and close the sale faster.RCM Staff Report, “27 Principles of Negotiating with a Meeting Facility,” MeetingsNet, February 1, 2003, meetingsnet.com/ar/meetings_principles_negotiating (accessed November 19, 2009).
Steps of the Negotiation Process
While negotiation has some elements of being an art, there are three specific steps that can be followed to help ensure success with each negotiation.
Three Steps of the Negotiation Process
Steps
Activities
1. Prenegotiation
• Get in the right frame of mind; be confident about the value of your product.
• Do your homework; know who’s sitting on the other side of the table and what’s important to him.
• Set prenegotiation goals; identify the minimum that you will accept for the deal and be ready to walk away if you can’t get it.
• Identify an offer that is higher than your prenegotiation goals to allow some room for negotiating.
2. Negotiation
• Make your initial offer and hold firm.
• Identify other “currencies” with which to negotiate to reach common ground.
• Be specific and identify every element of the deal in detail; put it on paper to avoid surprises later.
• If you encounter a deadlock, put the issue aside and come back to it at a later time in the negotiation.
• Avoid getting emotionally involved; be ready to walk away if you can’t make a deal that is mutually beneficial.
3. Postnegotiation
• Celebrate with all appropriate people; consider dinner, cocktails, or another get-together.
• Use the negotiation to build your relationship.
• Record what you’ve learned.
• Be ready for the next negotiation.
Step 1: Prenegotiation
Start off in the right frame of mind. Be confident by knowing that you are one of the finalists for your prospect’s business. If you are confident that you have the best product and represent the best value for the price, you already have the beginning of a good negotiation. On the other hand, if you’re not confident or don’t believe in the value of your product, chances are you will not negotiate well.Colleen Francis, “Negotiation Quick Hits,” The EyesOnSales Blog, November 13, 2008, www.eyesonsales.com/content/article/negotiation_quick_hits (accessed November 19, 2009).
Once you believe you are presenting the best option with the best value to your prospect, dig below the surface in research and conversation to learn what’s really important to your prospect. Ask a lot of questions; negotiators report that they often have to ask five levels of “why” to get to the “root cause” or true motivator of the person with whom they are negotiating.John Hoult, “Negotiation 101,” Fast Company, September 30, 2000, www.fastcompany.com/articles/2000/10/act_podziba.html?page=0%2C0 (accessed November 17, 2009).
Identify your prenegotiation goal, the minimum that you will accept during the negotiation. This is critical to your success as a negotiator so that you don’t give away more than you want in order to make the deal. Prenegotiation goals should be realistic based on what you want to get out of the negotiation and what your prospect wants or needs to get out of the negotiation. This is where you have the opportunity to explore creative solutions that may address different aspects of the sale. (Are you willing to provide additional services rather than provide a price concession? Will shorter payment terms help your prospect be able to sign on the dotted line?)
It’s a good idea to realize that your prenegotiation goals should not be the same offer you put on the table. Always allow some negotiating room as the first offer is rarely, if ever, accepted. Your prospect wants to feel as if she was able to get you to move from your original position. When you identify your prenegotiation goals, you know where you may end up, and also give yourself some room to negotiate.John Hoult, “Negotiation 101,” Fast Company, September 30, 2000, www.fastcompany.com/articles/2000/10/act_podziba.html?page=0%2C0 (accessed November 17, 2009).
Step 2: Negotiation
This is where it all comes together—your preparation, prenegotiation goals, strategy, and understanding of your prospect’s needs. Although you have done your homework and set your prenegotiation goals, hold firm on your initial offer. This allows you to learn more about what your prospect thinks is important and why. If you give in too early in the process, your prospect will feel like the negotiation was too easy and may have an expectation of getting even more concessions than you are willing to give. The general rule of negotiating is not to accept the first offer. That means you will need to reiterate the value you deliver and hold firm to your initial offer.Anthony Tjan, “Four Rules of Effective Negotiations,” Harvard Business Review, July 28, 2009, http://blogs.harvardbusiness.org/tjan/2009/07/four-rules-for-effective-negot.html (accessed November 17, 2009).
As the negotiation progresses, consider offering a concession to move toward common ground. But for every concession you give, get one in return. For example, “I’ll be able to look at pricing like this if we were able to be your exclusive distributor in the Northeast.” This is an example of using other “currencies” to make the negotiation work. In this case, the currency of exclusivity is used in exchange for a price concession.Anthony Tjan, “Four Rules of Effective Negotiations,” Harvard Business Review, July 28, 2009, http://blogs.harvardbusiness.org/tjan/2009/07/four-rules-for-effective-negot.html (accessed November 17, 2009).
Specificity is key in negotiating and closing, because once an issue is negotiated, it will be difficult to revisit it. Define each negotiated point in specific terms such as the number of days until delivery, specific payment terms, options that are clearly spelled out, and any other information that will clearly define your agreement. In most cases, all these elements are included in the contract that is signed as a result of the negotiation. It’s always best to clarify each point during the negotiation and put it on paper to avoid misconceptions, bad memories, or surprises down the road. If there is no contract, it’s a good idea to follow up the negotiation with a written summary of the agreed upon points.“Customers’ Negotiating Tactics,” Selling Power Sales Management eNewsletter, July 1, 2002, http://www.sellingpower.com/content/newsletter/issue.php?pc=212 (accessed March 16, 2010).
If you encounter an issue during the negotiation that causes a deadlock, or a stop in the discussion, set the issue aside and revisit it after other elements have been negotiated. You may find a way to include the thorny issue in a concession for a different negotiating point. It’s not worth getting held up on points during the negotiation; simply set them aside and revisit them at a later point in the negotiation. When you leave the most difficult issues until the end, other issues have already been resolved and both parties are motivated to find a resolution.RCM Staff Report, “27 Principles of Negotiating with a Meeting Facility,” MeetingsNet, February 1, 2003, meetingsnet.com/ar/meetings_principles_negotiating (accessed November 19, 2009).
Negotiator and author Herb Cohen says, “Negotiation is just a game. You care about the outcome, but not that much.” You have to avoid getting emotionally involved in the negotiation because the more emotionally attached to the outcome you become, the more you push to get what you want.Danielle Kennedy, “Let’s Make a Deal,” Entrepreneur, October 1996, www.entrepreneur.com/article/printthis/13404.html (accessed November 20, 2009). Getting emotionally involved in a negotiation makes it extremely difficult to walk away from it. That’s why many professional people such as actors and actresses, professional athletes, writers, and others have agents negotiate their contracts.Christina Novicki, “Secrets of a Superagent,” Fast Company, October 31, 1996, www.fastcompany.com/magazine/05/superagent.html (accessed November 19, 2009). It’s a good idea to remember that it’s not personal, it’s business.Alan M. Webber, “How to Get Them to Show You the Money,” Fast Company, October 31, 1998, www.fastcompany.com/magazine/19/showmoney.html (accessed November 19, 2009). Not all negotiations end in a deal. Based on your prenegotiation goals, you may need to walk away from a deal if it isn’t mutually beneficial. Keep in mind that your ability to negotiate is directly linked to your ability to walk away from the deal. If you don’t have any other options, you have given up any power you might have. It’s a good idea to always keep your options open.RCM Staff Report, “27 Principles of Negotiating with a Meeting Facility,” MeetingsNet, February 1, 2003, meetingsnet.com/ar/meetings_principles_negotiating (accessed November 19, 2009).
Learn from the Master
(click to see video)
This video features an excerpt of a speech by Herb Cohen. Cohen is an entertaining and thought-provoking speaker who underscores the concept of “care, but not too much” in negotiating.
Step 3: Postnegotiation
At this point, every element of the deal has been negotiated, agreed to, and documented on paper. It’s a good idea to take some time to celebrate a successful negotiation including all appropriate people at dinner, cocktails, or another get-together. This is a good way to recognize everyone’s contribution to making the negotiation a success and to look forward to enjoying the benefits of the partnership.Alan M. Webber, “How to Get Them to Show You the Money,” Fast Company, October 31, 1998, www.fastcompany.com/magazine/19/showmoney.html (accessed November 19, 2009). There’s one thing that’s true about every negotiation—it will surely lead to other negotiations in the future.Christina Novicki, “Secrets of a Superagent,” Fast Company, October 31, 1996, www.fastcompany.com/magazine/05/superagent.html (accessed November 19, 2009).
Key Takeaways
• Many times closing includes negotiating, the act of discussing an issue between two or more parties with competing interests with the aim of coming to an agreement.
• A successful negotiation is one that focuses on open, honest communication and yields a win-win resolution.
• Negotiations require building trust, gaining commitment, and managing opposition.
• Every negotiation includes three elements—information, power, and time.
• Negotiating starts long before the formal exchange; it begins with your first communication with the prospect and includes every contact you have had with her. Those communications establish the value of your product or service.
• While price is a common negotiating point, it is rarely the deal breaker that most salespeople perceive it is.
• Every negotiation includes three parts—prenegotiation, negotiation, and postnegotiation.
• Avoid getting emotionally involved in a negotiation as it makes it easier to walk away, if need be.
Exercise \(1\)
1. Assume you are buying a used car from someone. If your prenegotiation goal is \$10,000 and he is holding firm at \$12,000, how would you find common ground for a successful negotiation?
2. Assume you are buying a house from someone. She has indicated that the chandelier in the dining room has sentimental value. You think that the chandelier makes the dining room, and you want it included in the sale of the house. You are willing to increase your offer to reflect the inclusion of the chandelier. How would you approach this negotiation?
3. Assume you are selling medical supplies to a doctor’s office and the doctor says, “I won’t pay anything over \$3,000 for the machine, take it or leave it.” How would you respond?
4. Imagine that you are a sales rep for a paint manufacturer and you are selling to Home Depot. The buyer provided positive responses in all your previous meetings and is ready to narrow down his choices for paint suppliers.
• Identify three ways you could prepare for your negotiation to make it as productive as possible.
• How would you go about identifying your prenegotiation goals?
5. You are trying to sell accounting software to a regional grocery store chain, but negotiations have stalled. How can you get back on track?
6. Think about a negotiation in which you have been involved that yielded a win-win-win resolution. How did you get to the win-win-win solution? Think about a negotiation in which you have been involved that didn’t result in an agreement. Why do you think the negotiation wasn’t successful? What would have made it more successful?
7. How would you handle a situation in which a prospect wanted a guarantee that your company will not raise the price of the product he was buying for the next five years? Would you agree to hold the price to get the sale?
8. Contact a local law firm or company that specializes in negotiating. Invite a person from the firm to come to class and share tips and techniques that she uses in successful negotiations. | textbooks/biz/Marketing/The_Power_of_Selling/12%3A_Closing_the_Sale-_The_Power_of_Negotiating_to_Win/12.03%3A_Collaborate_to_Negotiate.txt |
Learning Objectives
• Understand how to negotiate and accept the right job offer.
So you’ve completed all your interviews and it’s the moment of truth…you are on the verge or receiving an internship or job offer. Congratulations!
As difficult as it has been to get to this point, you’re not quite there yet. This is the stage of the job searching process that really tests your mettle to get what you want. Just like negotiating and closing (outlined in the previous sections of this chapter), the quality of the job offer starts long before you actually receive the offer.
Know What You’re Worth
Before you even begin thinking about looking for an internship or job, your first step should be to determine your value based on the marketplace. As with every step of the selling process, doing your homework is key. If you don’t do your research to find out competitive compensation packages for the position and city in which you are seeking an internship or job, you might be disappointed with the job offers you receive.
There are several Web sites—Salary.com, JobStar.com, and SalaryExpert.com are just a few—that include compensation ranges for hundreds of different positions in areas across the country. Visit the sites listed in Table 12.1 to gather compensation information before you go on any interviews.
Table \(1\): Web Sites for Researching Compensation
Riley Guide rileyguide.com/salguides.html
Salary.com http://swz.salary.com
JobStar.com http://jobstar.org/tools/salary/sal-prof.php
SalaryExpert.com http://www.salaryexpert.com/index.cfm?fuseaction=Main.Home_Personal
CareerBuilder.com www.cbsalary.com/?siteid=cbcrcnav
Bureau of Labor Statistics www.bls.gov/NCS
It’s a good idea to use these tools as a guide as there are many assumptions that are made when these numbers are prepared. However, this information can be extremely helpful to understand the range of compensation being paid for a specific role in a specific city.“Evaluating the Salary Information You’ve Found,” The Riley Guide, www.rileyguide.com/saleval.html (accessed November 21, 2009). You will be able to negotiate more effectively if you walk into every job interview knowing how much you are worth.Kim Richmond, Brand You, 3rd ed. (Upper Saddle River, NJ: Pearson Prentice Hall, 2008), 201.
Establish Your Value Early
Just as in the selling process, establishing your value begins with your first contact with your prospective employer. Many times you have the opportunity to meet at least one or two people at the company, usually someone in human resources as well as your hiring manager. In some cases, you may meet with several different people with whom you will be working. In some cases, you may also talk with or meet with a recruiter. At any rate, you have the opportunity to establish your value with as many people as you meet. Everything you say and do has an impact on how people perceive you and your value. Are your résumé and cover letter professional? Did you do your research before you contacted the employer? How did you make contact? When you went in for an interview, did you dress appropriately and professionally? Were you prepared for the interview? Did you bring samples of your work to demonstrate your skills? Did you follow up with a thank-you e-mail and handwritten thank-you note within twenty-four hours? All these elements help establish your value long before an offer is extended. When it comes to making an impression on a prospective employer, everything matters.
Just as in the selling process, if you do your homework and establish your value early in the process, you will be more likely to get the offer you want.
Compensation versus Salary
One thing to know before you walk into any interview: compensation is different from salary. Compensation is the total amount of money and benefits that you are paid for a particular position. Compensation can include salary, insurance, vacation or sick leave, stock options, signing bonus, car allowance, 401(k), child care or elder care assistance, and any other type of payment received in exchange for your services to the company. Salary, a fixed amount of money that is paid regularly in exchange for services provided, is only one element of compensation. When you are considering a job offer, it’s best to keep in mind that salary is not the only element of compensation. This will allow you to be creative in your negotiating as there are several elements other than salary that can be included in your total compensation package.
Starting Out at a Start-Up
Working for a start-up company can be exciting and lucrative—with the right compensation agreement. The high-risk environment of a start-up might provide exactly the right place for you to start your career. But enter the business with your eyes wide open. “There’s no shortage of start-ups to work for, but most are going to fail,” says Greg Carney of Carney-Neuhaus.Dona DeZube, “Negotiating Compensation for a Job at a Startup,” Monster.com, career-advice.monster.com/salary-benefits/negotiation-tips/negotiating-compensation-startup/article.aspx (accessed November 22, 2009). This article provides insight about how to structure compensation with a start-up company.
career-advice.monster.com/salary-benefits/negotiation-tips/negotiating-compensation-startup/article.aspx.
Timing Is Everything
Although you may want to discuss compensation on your first interview, it’s a good idea to postpone discussing the topic as long as possible.Kim Richmond, Brand You, 3rd ed. (Upper Saddle River, NJ: Pearson Prentice Hall, 2008), 202. Just as in the selling process, you want to put focus on establishing your value and learning about what’s important to your prospective employer before you begin discussing compensation. A word of caution: you should be prepared to give your desired salary range on an interview as many employers want to understand your salary expectations as soon as possible. If you can avoid the topic, do so until you receive a job offer. Here’s an example of how you can deflect the conversation.
Interviewer: What are your salary expectations?
You: I’m sure your total compensation package is competitive for the position. What’s most important to me is the opportunity to learn and be a part of an organization like yours.
If you are forced to give a salary range, be sure you can live with the lowest number you give. Once you say a number, it will be extremely difficult to negotiate above that salary.Kim Richmond, Brand You, 3rd ed. (Upper Saddle River, NJ: Pearson Prentice Hall, 2008), 202.
Receiving the Offer
Most job offers are extended over the phone, although some may be extended in a letter and still others may be presented in person. Since companies usually interview multiple candidates for each position, chances are you won’t receive a job offer on an interview. However, you should be prepared to respond to a job offer if one is presented during an interview.
Responding to an Offer in an Interview
(click to see video)
The following video provides suggestions about how to handle this situation.
When you receive an offer, it will most likely come over the phone. When you get a phone call about an offer, write down every element of the offer (it’s OK to ask the person to hold while you get a pencil and paper). Thank the person who made the offer and tell her how pleased you are to receive the offer. Even if you think you want to accept the offer, don’t accept it right away.
You: I’m very excited about this offer. Thank you so much for extending it to me. It’s a very big decision, and I’d like to have a few days to think about it. Can I get back to you on Thursday? What time is good for you to talk?
Employer: I’m glad to hear that you are happy about the offer. We are all very excited about the prospect of you joining our team. I’m happy to answer any questions you might have about the company or the offer. Don’t hesitate to give me a call. In fact, let me give you my cell phone number so you can call me at any time. Then let’s touch base on Thursday at 10:00 a.m.
You: I just want to repeat the elements of the offer so that I have it correct. The base salary is \$45,000 with the opportunity to earn a bonus of 5 percent based on meeting performance objectives. There is a car allowance of \$3,000 a month. I’ll be eligible for medical insurance after thirty days of employment, and I’ll receive one week of vacation after working for twelve months. Is that right?
It’s worth noting the time element of this negotiation. You should take as much time as you need to evaluate the offer, but you should be reasonable and state the time frame you need. Recall from earlier in this chapter that time is one of the elements that is always present in a negotiation. Sometimes a prospective employer or recruiter will try to create a deadline to force you to make a decision by a specific date. Use time to your advantage and negotiate for more time so that you don’t feel as if you are under pressure to make this important decision.
Evaluating the Offer
Congratulations on your offer! Although it’s a difficult economy, don’t feel pressured to take the first offer you get. Take the first offer you get for a job you really want. This is a special moment; it is the time when the power shifts from the prospective employer to you (remember that power is one of the elements that is always present in a negotiation). Your prospective employer has now indicated that you are their choice for the position. You have the opportunity to take the offer, counteroffer (your response to the job offer), or walk away. Whatever your choice, you have the power. As soon as you make your choice, the power shifts back to your prospective employer. That’s why it’s a good idea to take your time and completely evaluate your offer before you respond.
Now that you have all the elements of the offer, you can begin to evaluate it. Just as in a negotiation in selling, identify what is important to you. Consider making a list that includes both elements of compensation as well as other elements such as culture, opportunity, environment, commuting distance, and so on. Figure \(6\) and Figure \(7\) can provide some ideas to help you create your list.Adapted from Kim Richmond, Brand You, 3rd ed. (Upper Saddle River, NJ: Prentice Hall, 2008), 204; and Paul W. Barada, “Job Offer Evaluation Checklist,” Monster.com, career-advice.monster.com/salary-benefits/Negotiation-Tips/Job-Offer-Evaluation-Checklist/article.aspx (accessed November 21, 2009).,Adapted from Kim Richmond, Brand You, 3rd ed. (Upper Saddle River, NJ: Prentice Hall, 2008), 206; and Paul W. Barada, “Job Offer Evaluation Checklist,” Monster.com, career-advice.monster.com/salary-benefits/Negotiation-Tips/Job-Offer-Evaluation-Checklist/article.aspx (accessed November 21, 2009).
Only you can determine if a job offer is right for you. The following are some additional resources that you may want to review to help you evaluate a job offer:
• http://careerplanning.about.com/cs/joboffers/a/evaluate_offer.htm
• rileyguide.com/salguides.html
• www.ehow.com/how_2068763_evaluate-job-offer.html
• www.black-collegian.com/career/joboffer-199702.shtml
• www.salary.com/careers/layouthtmls/crel_display_nocat_Ser16_Par41.html
• career-advice.monster.com/salary-benefits/Negotiation-Tips/Job-Offer-Evaluation-Checklist/article.aspx
Negotiating the Offer
After you have had the time to evaluate the job offer, it’s time to identify any gaps that there might be between what you think is important and the offer. If you think the offer is perfect as is, then accept it as is. Keep in mind that many employers expect candidates to negotiate by presenting a counteroffer, a candidate’s response to a job offer. Since the economy is challenging, candidates don’t have as much bargaining power as when the economy is healthy. However, this is the time you have the most negotiating power with your prospective employer so it’s a good idea to take advantage of the opportunity.
Should You Negotiate a Job Offer?
(click to see video)
Hear why Tonya Murphy, general sales manager at WBEN-FM, thinks candidates should negotiate a job offer.
This is the time when you should identify your prenegotiation goals. Remember that your prenegotiation goals are the minimum that you will accept. Your counteroffer will be above your prenegotiation goals to allow room for negotiation. You won’t be able to negotiate every element of the offer. Choose one or two key areas and focus your negotiation on those. Keep in mind the things that are important to you and to your prospective employer so that you can easily find common ground. For example, if speed and availability are important to your prospective employer, you may want to use that fact to negotiate a more flexible work arrangement. While it may be difficult to negotiate a higher base salary in this economy, you may be able to negotiate on another area such as getting additional vacation time.Kim Richmond, Brand You, 3rd ed. (Upper Saddle River, NJ: Pearson Prentice Hall, 2008), 208.
When you have identified the areas you wish to negotiate along with your counteroffer for each, contact the recruiter or prospective employer to begin the negotiation. As with any negotiation, approach it with a confident, collaborative attitude. It’s important to note that you should not accept the offer until you negotiate the offer. Once you accept the offer, you have lost your power to negotiate.
You: I wanted to follow up and thank you again for the offer to join the company. I’m really excited about it. Based on the interviews, I believe I can bring value to your company. I wanted to talk about one area of the offer.
Employer: Great. What questions can I answer for you?
You: The base salary is lower than I expected. [Important note here: say this point and wait for a response. Many people feel obligated to talk more, but less is more in a negotiation.]
Employer: We have made the base salary as high as we could. There’s really nothing we can do to make it any higher.
You: One of the things that could make the offer more attractive is additional vacation time.
Employer: We might be able to take a look at that. Let me touch base with Casey. I can’t make you any promises, but I can talk to him and let you know.
You: That would be great. I really appreciate it.
Just as in a selling negotiation, you have to be ready to accept the offer as is or be ready to walk away. That’s a lot harder to do when you are negotiating on behalf of yourself since you are emotionally involved with the decision.
Negotiating a Job Offer
(click to see video)
The following video outlines these key points about negotiating a job offer.
Video Clip
Negotiating Tips
Read about how to negotiate your best compensation package.
career-advice.monster.com/salary-benefits/negotiation-tips/salary-negotiation-guide/article.aspx
You’ve Got the Power: Tips for Your Job Search
Negotiate before You Accept
Use your power when you receive a job offer. Thank the employer for the offer, evaluate it, and negotiate the offer before you accept it. Once you accept the offer, you have lost any power to negotiate.
The Offer Letter
Once you agree on the final elements of the offer, you should ask for an offer letter, a formal letter from the company (on company letterhead) that outlines the terms of the offer. All companies should provide an offer letter as a matter of course for an internship (paid or unpaid) or a job offer. If you received the original offer in the form of a letter, you already have the offer letter; however, you should request an updated letter to reflect the final offer on which you agreed. If you find any discrepancies in your offer letter, contact the person at the company as soon as possible to have a new offer letter issued.
A offer letter simply reiterates the terms of employment that you have negotiated and may be conditional based on requirements such as a background check or drug test or may make reference to company documents such as the benefits summary or employee handbook. While some information in offer letters may vary depending on the company, some key information should be included in the offer letter:
• Title
• Salary
• Bonus
• All other elements of compensation (e.g., stock options, benefits)
• Start date
• Any conditions of employment
Some companies request that you sign a copy of the offer letter and return it to the company. If this is the case, sign the letter and make a copy for your files before returning it to the company. The offer letter is your documentation of the compensation the company has agreed to give you.Kim Richmond, Brand You, 3rd ed. (Upper Saddle River, NJ: Pearson Prentice Hall, 2008), 212.,John Steven Nisnick, “Job Offer Letter Sample,” About.com, http://jobsearchtech.about.com/od/jobofferletters/a/jobofferletter.htm (accessed November 21, 2009). See sample offer letters in Figure \(8\) and Figure \(9\).
Key Takeaways
• Before you begin your job search, do your research and know what you are worth in the marketplace.
• Salary is only one element of total compensation. Use all elements of compensation to creatively negotiate to get what you want.
• Avoid discussing compensation as long as possible; don’t bring it up unless the interviewer brings it up. Your goal on every interview is to establish your value so that your offer reflects what you are worth.
• Before you begin negotiating a job offer, be sure you understand all the elements of the offer.
• Carefully evaluate an internship or job offer based on what is important to you including the offer as well as other aspects of the job and company.
• Identify one or two elements of a job offer that you want to negotiate. Determine your prenegotiation goals for each and approach your prospective employer to discuss each element. Focus on what is important to the company as you negotiate each point.
• The final offer that you accept should be documented in an offer letter. Whether you are being offered an internship (paid or unpaid) or a full-time job, the company should provide an offer letter within a few days of your acceptance of the offer.
Exercise \(1\)
1. Visit Salary.com, or one of the other Web sites mentioned in this section, and determine the total compensation for at least three different positions in which you are interested in pursuing. Is the compensation higher or lower than you expected for each position?
2. Identify three ways that you can establish your value in the eyes of your prospective employer during the interview process.
3. Assume you received a job offer with a base salary of \$35,000 and commission of 10 percent. How would you plan to approach your prospective employer to increase your overall compensation?
4. Have you ever received an offer letter? If so, what position was it for? What information did it include?
5. When you are negotiating your job offer, is it ever appropriate to exaggerate your accomplishments a little bit to get an offer that you think you deserve? Why or why not? | textbooks/biz/Marketing/The_Power_of_Selling/12%3A_Closing_the_Sale-_The_Power_of_Negotiating_to_Win/12.04%3A_Selling_U_-_Negotiating_to_Win_for_Your_Job_Offer.txt |
Power Wrap-Up
Now that you have read this chapter, you should be able to understand how closing and negotiation work in the selling process.
• You learn the role of the close in the selling process.
• You understand how the close is an opportunity to build a relationship.
• You recognize that closing is a natural part of the selling process.
• You can list the different types of closes.
• You understand how to negotiate so that all parties win.
• You realize that a job offer can be negotiated.
TEST YOUR POWER KNOWLEDGE (AnswerS ARE BELOW)
1. Explain the statement “The close, or getting the order, starts at the beginning of the selling process, long before you even come in contact with the prospect.”
2. What is a trial close?
3. Describe three times during a sales call that are good times to close.
4. Assume you are selling a video game. Give an example of an alternative-choice close.
5. Describe the role that trust plays in negotiating.
6. What are the three elements that are always present in a negotiation?
7. Why do salespeople think they need to lower the price to have a successful negotiation?
8. Describe what a concession is in a negotiation.
9. Name the three steps in the negotiation process.
10. What is a prenegotiation goal?
11. Is the following statement true or false? You can get more as a result of a negotiation in which you are emotionally involved.
12. How do you know if you received a good job offer?
13. What is the difference between compensation and salary? Why is it important to know this when negotiating a job offer?
POWER (ROLE) PLAY
Now it’s time to put what you’ve learned into practice. The following are two roles that are involved in the same selling situation—one role is the customer, and the other is the salesperson. This will give you the opportunity to think about this selling situation from the point of view of both the customer and the salesperson.
Read each role carefully along with the discussion questions and be prepared to play either of the roles in class using the concepts covered in this chapter. You may be asked to discuss the roles and do a role-play in groups or individually.
Sweet Success
Role: Purchasing manager at ProFood, the food service supplier for campus cafeterias and restaurants
You are responsible for purchasing the products to be offered in college cafeterias and restaurants. You try to include new products that reflect the eating trends of the students. One of the trends is for more natural and organic food choices. The challenge is that, in order to offer new menu options, some of the existing options need to be eliminated. Any new products must be able to generate more revenue than existing items at a lower cost. You are especially interested in increasing sales at the snack bars with impulse items like individually wrapped cookies and cakes. The Organic Delight Desserts option is exactly what you are looking for, but the price is too high, and you’re not sure you want all the flavors that come packed together in one case. The price from the sales rep is \$20 per case. There are four flavors packed in a case—chocolate, strawberry, lemon, and mocha. At this rate, you might only put this in your top ten schools. If you can get a better price with the option to order individual flavors by the case, you might consider putting the line in all three hundred colleges and universities.
• Are you interested in negotiating to get what you want from the sales rep, or will you just take a pass and wait for another product?
• If you want to negotiate, what are your prenegotiation goals?
• What will you ask for during the negotiation? Is this different from your prenegotiation goals? Why or why not?
Role: Territory manager, Organic Delight Desserts
You are selling a new line of 100 percent organic desserts. These cookies and mini cakes are individually wrapped and are an excellent impulse item, or ideal for cafeterias. Since this is a new product line, it would be ideal to get placement with ProFood because it could lead to distribution at hundreds of schools. You just need to sell the purchasing manager on the line. You have sampled the products, and she likes the taste and thinks the packaging is perfect for her schools. Now you are down to negotiating on price and packaging. You have quoted \$20 a case for a case that includes all four flavors—chocolate, strawberry, lemon, and mocha. You might have some flexibility to have a custom cases made up in each flavor so she can order only the flavors she wants. However, it will cost additional handling to do that.
• Are you going to make this a “take it or leave it” proposal?
• If you are going to negotiate, what are your prenegotiation goals?
• How will you find common ground to make this a win-win-win situation?
• What will you ask for during the negotiation? Is this different from your prenegotiation goals? Why or why not?
ACTIVITIES
1. Visit the campus career center and ask about salary information that is available for positions that you are interested in pursuing. Compare this information to similar information you have gathered from Web sites mentioned in this section that include salary information. What information is consistent? What information is different? Ask a career counselor to help you understand the differences.
2. Talk to a campus career center counselor, advisor, or other professor or professional (and use this information in this section) and create a list of elements that might be included in a job offer. Identify those elements that are most important to you. What are your prenegotiation goals as it relates to this list?
TEST YOUR POWER KNOWLEDGE AnswerS
1. If you do your prospecting and qualifying correctly, you can significantly improve the number of times you are able to close a sale.
2. A trial close is when you ask your prospect their opinion. A close is when you ask for a decision.
3. When the prospect is demonstrating positive body language, when the prospect asks questions, and after you handle an objection.
4. “Would you like to preorder Guitar Hero Van Halen or take Guitar Hero Metallica with you now?”
5. Negotiating is based on trust. If your prospect doesn’t trust you, chances are she will be unwilling to compromise and find common ground during the negotiation.
6. Information, power, and time.
7. Forty percent of prospects ask for a lower price. Salespeople should work to get below the surface and understand the prospect’s true needs. Prospects are looking for value, not necessarily price. Salespeople should demonstrate the value of their product or service and negotiate on other elements rather than price. Reducing the price decreases profit, commission, and value of the product or service in the mind of the prospect.
8. A concession is something on which you are willing to compromise such as price, service, terms, options, or other elements of the deal. It’s best to get a concession when you give a concession.
9. Prenegotiation, negotiation, and postnegotiation.
10. Goals that you identify before the beginning of a negotiation that establish the minimum that you are willing to accept to make the deal happen.
11. False.
12. Do research before beginning your job search by visiting Web sites that include salary information.
13. Salary is only one portion of total compensation, payment for services provided to an employer. There are several elements of compensation, including salary, vacation, insurance, hours, travel, relocation, and others that can be used to increase the total value of your job offer. | textbooks/biz/Marketing/The_Power_of_Selling/12%3A_Closing_the_Sale-_The_Power_of_Negotiating_to_Win/12.05%3A_Review_and_Practice.txt |
Follow-Up: The Power of Providing Service That Sells
Video Ride-Along with Rachel Gordon, Account Manager at WMGK Radio
You met Rachel Gordon in Chapter 6 when she shared her tips for finding the decision maker. Now she talks about the importance of follow-up and provides some valuable tips about how she follows up with prospects and customers.
Ride along with Rachel and hear about the importance of handwritten thank-you notes and other elements that are important to making the sale again and again.
(click to see video)
13.02: Follow-Up - The Lasting Impression
Learning Objectives
• Understand what follow-up entails and why it is so important.
• Discuss the ongoing nature of follow-up.
You have spending power, and lots of it. Millennials (or Gen Y, if you prefer) are estimated to have over \$1.3 trillion in direct spending for apparel, food, music, entertainment, and other products and services. That number is understated due to the influence you have on parents and other older people who seek your tech-savvy advice on all types of products from computers to cars.Sarah Littman, “Welcome to the New Millenials,” Response Magazine, May 1, 2008, www.responsemagazine.com/response-magazine/welcome-new-millenials-1192 (accessed November 25, 2009). You are one of the most sought-after consumer groups around. More sales and marketing efforts are aimed at you than at any other generation. You determine where and when you will spend your money. You have the power.
So what is it that makes you decide to choose Nintendo over Xbox, Mini Cooper over Chrysler, or Apple over Toshiba? Of course, the product has a lot to do with your choice. Price is certainly a consideration, but you don’t always buy the lowest-priced product or service. Think about it. It’s the ongoing relationship you have with the brand that makes a difference. It’s the fact that the company continues to serve up exactly the new products and services you need (how do they do that?). It’s how the company keeps in touch on Facebook and other ways that keeps you engaged in the conversation. And it’s the fact that you feel appreciated as a customer. When a company makes you feel like they forgot about you, it’s time to move on and spend your money elsewhere.
What Is Follow-Up?
Follow-up entails everything that takes place after the sale is closed from getting signatures on all contracts and paperwork to scheduling delivery. It also includes your ongoing relationship with your customer. Relationship is the key word here. If you were involved in transactional selling, only focused on making the short-term sale, you would not be worried about follow-up because someone else in your company would take care of it. You would move on to the next customer. In many retail selling environments, this may be the case. You would not expect to receive a thank-you note from the checker at the grocery store or the cashier at a fast-food restaurant. However, you would expect to hear from a real estate agent who sold you a new home, or from a financial services consultant who is managing your money.
It’s the attention to detail to be sure that your transaction goes smoothly that you rely on your salesperson to do. Think about how you feel when your salesperson adds value to your new investment with additional information and insights. That makes you feel like a valuable customer. Chances are, when you need something else (another house or more money to invest), the first person you will call will be the salesperson who continues to follow up with you. When one of your friends wants to buy a house or invest some money, you will be very likely to go out of your way to recommend your salesperson.
While the specific follow-up activities may vary from company to company and even customer to customer, Figure \(1\) provides a summary of some of the most common follow-up actions that are expected. Many companies have a checklist or best practices that are used as guidelines to ensure that all details are covered. In the case of complex sales, follow-up may include a transition team with members from both the company and the customer. The transition team may work closely together, including weekly or in some cases daily status calls, to ensure that the transition to the new product or service goes smoothly. For example, the implementation of a new logistics system or software program may require that the old system runs parallel with the new system until all aspects are completely set up and appropriate training is conducted. This is especially true for products or services like these that have a direct impact on the operation of the customer’s business.
Why Follow Up?
No matter what product or service you are selling, the sales process can be challenging. The selling process starts with prospecting and qualifying (that was six chapters ago!). Depending on the complexity and buying cycle of the product or service, it could takes weeks, months, or even years until you close the sale. In fact, 81 percent of all sales happen on or after the fifth sales call, according to study conducted by the Association of Sales Executives.David Frey, “Follow-up Marketing: How to Win More Sales with Less Effort,” Marketing Best Practices, www.marketingbestpractices.com/Articles/FollowUpMarketing.htm (accessed November 22, 2009). It takes time, energy, and commitment to get to the point where the deal is done. Some salespeople spend all their time and effort to research the prospect, get the appointment, make the presentation, handle objections, and close the sale—and then expect to collect their commission check. They seem to literally disappear after the sale is completed.Jeff Schmitt, “The Personal Touch: Make the Sale…after the Sale,” Sales & Marketing Management, September 9, 2009, www.salesandmarketing.com/article/personal-touch-making-salex2026after-sale (accessed November 23, 2009).
Relationship selling doesn’t work that way. The relationship really begins with the close of the sale; follow-up is what makes a relationship grow and prosper. Follow-up is how most customers evaluate the performance of the product or service they just bought. As you may recall from Chapter 1, you are the brand to the customer. How you proactively handle follow-ups will make all the difference in your relationships and your sales. In other words, the best way to make the sale is by the way you handle things after the sale.
Here’s the not-so-subtle point here. Even though the sale is closed, you should never assume the sale is closed. Jeff Schmitt, “The Personal Touch: Make the Sale…after the Sale,” Sales & Marketing Management, September 9, 2009, www.salesandmarketing.com/article/personal-touch-making-salex2026after-sale (accessed November 23, 2009). This is especially important when there is a gap in time between the closing of the sale and the delivery of the product or service (as in the delivery of a major software package, installation of new equipment, or bringing on board a new product or service vendor). A customer can have second thoughts, sometimes called buyer’s remorse or cognitive dissonance (covered in detail in Chapter 6). This is when a customer may think that the decision she made is not the right one. She may be in contact with a competitor, receive additional information, or be concerned that she made the wrong decision, paid too much, or didn’t consider some alternatives properly. You can help avoid letting your customers be vulnerable to alternatives.Jeff Schmitt, “The Personal Touch: Make the Sale…after the Sale,” Sales & Marketing Management, September 9, 2009, www.salesandmarketing.com/article/personal-touch-making-salex2026after-sale (accessed November 23, 2009).,Joan Leotta, “When Buyers Change, Grin and Sell It,” Selling Power 21, no. 5, http://www.sellingpower.com/content/article.php?a=5769 (accessed March 16, 2010). Increase your return on your time investment and your customer’s return on her financial investment and put your follow-up plan into place immediately.
Plan Your Follow-Up
Put together your follow-up plan even before you begin your prospecting efforts. While follow-up is the last step in the selling process, it is the step that can have the most impact on your customer. You worked hard to establish trust with your customer during the selling process. After the sale is the time to put that trust to work and continue to earn it every day. Lip service, saying that you’ll do something but not really putting in the effort to do it, doesn’t go very far in sales. And just going through the motions will put you farther behind. It may seem more exciting to be working on a new proposal rather than doing follow-up for a sale that has already closed.
Think about your follow-up plan with the following five elements in mind:
1. Demonstrate your personal commitment and connection to the customer. Start by saying thank you to your customer for her business. “Customers want to know you care about them, their business, their challenges, and them as individuals,” according to author and professional speaker George Hedley. “The number one reason customers stop doing business with a company is an attitude of indifference,” he says.George Hedley, “Customer Care = Cash,” American Salesman, March 2009, www.hardhatpresentations.com/CustomerCareCash.htm (accessed March 16, 2010). How you follow up after the sale is a good indication of how you will respond throughout the relationship.
Start off on the right foot by sending a thank-you letter. Everyone likes to feel appreciated, especially right after they have made a commitment to spend money. Your letter should be professional, yet personal, and sincere. This is the perfect opportunity to reinforce to the customer that she has made a wise decision; this is a perfect opportunity to reiterate the product or service benefits with a focus on the information you learned about the customer’s business during the selling process.Dana Ray, “Phenomenal Follow-up,” Selling Power 19, no. 6, http://www.sellingpower.com/content/article.php?a=5081 (accessed March 16, 2010).,Joan Leotta, “When Buyers Change, Grin and Sell It,” Selling Power 21, no. 5, http://www.sellingpower.com/content/article.php?a=5769 (accessed March 16, 2010).
Besides demonstrating good business etiquette, a personal thank-you letter also serves some operational objectives. It should include your contact information, phone numbers, e-mail address, Web sites for customer contact (in addition to your contact information), receipt or order confirmation, and a list of next steps.Joan Leotta, “When Buyers Change, Grin and Sell It,” Selling Power 21, no. 5, http://www.sellingpower.com/content/article.php?a=5769 (accessed March 16, 2010).
Don’t just say thank you after you close the sale. Be ready to follow up with three to five “selling points” timed after the sale. For example, after a salesperson sells a car, she follows up with an article about a safety award that the brand was awarded. She also sends a birthday card to the customer with a note to indicate the value of the car has increased based on current market conditions.Sean McPheat, “Post Sales Follow Up,” Master of the Sales Force Blog, http://www.mtdsalestraining.com/mtdblog/post-sales-follow-up.html (accessed November 23, 2009).
Most of all show your customers that you appreciate them and their business regularly with a handwritten thank-you note, an unexpected visit, or small gift like a box of candy. Little gestures go a long way; they are like “one-a-day vitamins” for your business.George Hedley, “Customer Care = Cash,” American Salesman, March 2009, www.hardhatpresentations.com/CustomerCareCash.htm (accessed March 16, 2010).
Follow-Up Letter
(click to see video)
Looking for tips about how to write a sales follow-up letter? This video includes some great tips.
What If the Answer Is No?
So what if you didn’t get the sale? Send a thank-you note anyway. It’s a professional way to set yourself apart and keep the door open for future conversations. A personal thank-you note or letter really stands out in today’s fast-paced world. You might be surprised where a thank-you note or letter can lead. See a sample thank-you letter.
www.bestsampleletters.com/sales-and-marketing/appointment-setting-and-follow-up/post-sale-follow-up/follow-up-letter-to-lost-sale-letter.html
2. Deliver as promised. While you are the person on the front line with the customer, you have a team of people who are responsible for delivering the product or service as specified. “Don’t just check the box,” says executive coach and author Marshall Goldsmith.Marshall Goldsmith, “Don’t Just Check the Box,” Fast Company, February 1, 2005, www.fastcompany.com/magazine/91/mgoldsmith.html (accessed November 23, 2009). Take the time to follow up internally to be sure all the i’s are dotted and t’s are crossed so that your customer’s delivery is flawless. That means taking the time to share details and insights about the customer’s business and preferences with your entire team (whether your team is large or small). When salespeople just fill out the forms to get things moving internally, there’s a high likelihood that some nuances can fall between the cracks. Keep in mind that your customer made the purchase because you can deliver consistently for her, but you can’t deliver the product or service alone. There are most likely internal processes for communication and delivery, contracts to be signed, schedules to be communicated, and other operational activities that require the entire team to be working in harmony. Follow the internal processes and go a step farther. Make your coworkers care as much about delivering consistently for the customer as you do; take the time to share information about the customer that goes above and beyond your internal forms. You’ll also be surprised to see that everyone involved will add value when each has a connection to the customer. And don’t forget to say thank you to your team. You couldn’t do it without them; share the positive feedback from your customer with the team.Jeff Schmitt, “The Personal Touch: Make the Sale…after the Sale,” Sales & Marketing Management, September 9, 2009, www.salesandmarketing.com/article/personal-touch-making-salex2026after-sale (accessed November 23, 2009).
Call the customer to be sure the delivery was made as promised and everything is to the customer’s liking.Kelley Robertson, “The Power of Follow Up,” About.com, http://entrepreneurs.about.com/od/salesmarketing/a/poweroffollowup.htm (accessed November 23, 2009).
3. Add value to your customer’s business. Follow-up isn’t a one-time event. Rather, it is an ongoing process that takes place after the sale is closed. Just like when you researched, asked questions, and listened to your customer to learn as much as possible about you might solve his business challenges before he made the commitment to buy, you want to continue to do the same thing as part of your ongoing follow-up.
Build your credibility by creating a systematic follow-up system so that your customer knows he can count on hearing from you regularly. You might touch base in person or by phone, e-mail, text, or a combination of these contact methods. The key is to communicate regularly in the manner or manners in which your customer prefers. It’s a good idea to get into a routine to get and give status updates.Dana Ray, “Phenomenal Follow-up,” Selling Power 19, no. 6, http://www.sellingpower.com/content/article.php?a=5081 (accessed March 16, 2010).,Jeff Schmitt, “The Personal Touch: Make the Sale…after the Sale,” Sales & Marketing Management, September 9, 2009, www.salesandmarketing.com/article/personal-touch-making-salex2026after-sale (accessed November 23, 2009). Believe it or not, some salespeople actually forget to follow up.Kelley Robertson, “The Power of Follow Up,” About.com, http://entrepreneurs.about.com/od/salesmarketing/a/poweroffollowup.htm (accessed November 23, 2009). They get so busy with making new proposals and putting out fires that they lose track of time and details.
What’s important to the customer should be important to you so make yourself easily accessible and respond to his inquiries in a timely manner.Dana Ray, “Phenomenal Follow-up,” Selling Power 19, no. 6, http://www.sellingpower.com/content/article.php?a=5081 (accessed March 16, 2010). Deliver the same energy, enthusiasm, and level of service you did before you closed the sale. And just as you did when you were working to close the business, be honest about timing and resolution of issues. In other words, set expectations and then overdeliver on them.Jeff Schmitt, “The Personal Touch: Make the Sale…after the Sale,” Sales & Marketing Management, September 9, 2009, www.salesandmarketing.com/article/personal-touch-making-salex2026after-sale (accessed November 23, 2009).
Adding value goes beyond the typical “I’m just checking in.” Every time you contact your customer, offer some insight, news, or expertise to help him and his business. Make yourself the trusted advisor and key collaborator. Provide insights from industry events, forward copies of relevant white papers, make introductions to subject matter experts in your company, and send company (or your own) newsletters. You can complement your personal follow-up with the Internet to provide valuable updates and networking connections through a blog, Twitter updates, LinkedIn discussions, and other social networking tools. All these types of communications help add value to your customer’s business so that when she has a problem (any problem), you deliver so much value that she calls you first to help her solve it. This is how you earn your seat at the table as a true business partner, not a salesperson.Jeff Schmitt, “The Personal Touch: Make the Sale…after the Sale,” Sales & Marketing Management, September 9, 2009, www.salesandmarketing.com/article/personal-touch-making-salex2026after-sale (accessed November 23, 2009).
4. Get feedback. It’s not enough to talk to your customers; it’s also important to listen.Tamara Monosoff, “Focus on Core Customers,” Entrepreneur, October 21, 2009, http://www.entrepreneur.com/article/printthis/203774.html (accessed November 23, 2009). Ask for their input, insight, and ideas about everything from things you can do better to new products and services. Customers, especially those with whom you have good relationships, can provide invaluable guidance to you and your company. One-on-one planning meetings, product development meetings, and other forward-looking events are ideal ways of gaining firsthand feedback and getting buy-in from the start. There’s nothing that your customer would rather talk about than his business. Be genuine and ask him about it, then listen and use the information to help his business (and yours) grow.George Hedley, “Customer Care = Cash,” American Salesman, March 2009, www.hardhatpresentations.com/CustomerCareCash.htm (accessed March 16, 2010).
Customer Feedback Meets Social Networking
IdeaStorm (www.ideastorm.com) is a Web site created by Dell that literally turns customer feedback into a social network. You can post, vote, promote, or demote ideas for Dell. What makes this Web site so unique is that you can actually see the ideas that have been put into action. Talk about showing customers you care about what they think, Dell puts customer feedback to good use.Alister Cameron, “Dell IdeaStorm—Ultimate Customer Feedback Example,” WebProNews, February 27, 2007, www.webpronews.com/blogtalk/2007/02/27/dell-ideastorm-the-ultimate-customer-feedback-example (accessed November 23, 2009).
Starbucks has incorporated MyStarbucksIdea into its Web site as a place for customers to share their ideas, vote on their favorites, discuss the pros and cons, and see the actions that have been taken as a result.Starbucks, mystarbucksidea.force.com (accessed November 24, 2009). Suggest an idea at mystarbucksidea.force.com.
5. Make your customers into fans. Focusing on your customers’ businesses as if they were yours, adding value, and showing your customers that you appreciate their business makes them more than customers—it makes them fans. Fans share stories of their great experiences. Your customers can help you sell with testimonials, referrals, and references. One of the most effective ways to handle objections from prospects is to call on excited and energized customers who are more than satisfied with your product and service. There are no more powerful words to win over a new prospect than those of a more-than-satisfied customer.Jeffrey Gitomer, “Objection Prevention & Objection Cure,” video, May 18, 2009, http://www.youtube.com/watch?v=CgfmcuE_06w (accessed November 24, 2009). Use customer testimonials as part of your selling presentation, on your company’s Web site, and on your professional Web site and social networking pages. In fact, it’s a good idea to ask customers to write a recommendation for you on LinkedIn.
Link
Referrals Build Sales
See how testimonials are used by Atlanta REMAX real estate agent Ellen Crawford on her professional Web site.
http://www.atlantabesthomes.com/testimonials.htm
Reward your best customers with special offers and added value such as additional training, additional advertising space or time, or other additional service.Jeff Schmitt, “The Personal Touch: Make the Sale…after the Sale,” Sales & Marketing Management, September 9, 2009, www.salesandmarketing.com/article/personal-touch-making-salex2026after-sale (accessed November 23, 2009). While you may extend a special pricing offer, focus on delivering value and giving your best customers the opportunity to experience the other services you have to offer. This lets your best customers know you appreciate their business and gives you an opportunity to move your relationship to the next level by becoming an even more important business partner to them.
It is these loyal customers who build your business in two ways. First, they buy more from you because they feel that you are bringing them value in more ways than simply selling a product. Second, when they are loyal customers, they become fans or advocates of your product or service, and they tell their friends about you.
Power Point: Lessons in Selling from the Customer’s Point of View
The Making of a Fan—Yahoo!-Style
Blogger Michael Eisenberg went from a detractor to a promoter of Yahoo! with one e-mail. Eisenberg made a “not-so-flattering post” about the functionality of what was then the new MyYahoo! in March 2007. Within twenty-four hours he received an e-mail from the manager of Yahoo!’s Front Doors Group that said, “I would love to find out what you would like to see and which features you are most concerned about losing. We want to be sure that our heavy users remain satisfied. If you have a few minutes to e-mail me, I’d very much appreciate it.” Eisenberg promptly posted the response from the Yahoo! manager on his blog along with his fanatic endorsement of the company that can be summed up in one word: “Kudos!”Michael Eisenberg, “Yahoo!—Great Customer Feedback Loop,” March 13, 2007, http://sixkidsandafulltimejob.blogspot.com/2007/03/yahoo-great-customer-feedback-loop.html (accessed November 19, 2009).
Heroic Recovery: How a Service Failure Can Be a Good Thing
No matter how good you and your company are at taking care of customers, there will be a time when something doesn’t go as planned or as your customer expected. When you experience a setback, your mettle is put to the test. “Errors are inevitable, dissatisfied customers are not.”Chia-Chi Chang, “When Service Fails: The Role of the Salesperson and the Customer,” Psychology & Marketing 23, no. 3 (March 2006): 204. It’s not about the fact that the problem occurred; it’s how you respond that matters. When a salesperson responds quickly to a service failure and delights the customer with the outcome, it is called heroic recovery. The salesperson has the opportunity to perform a “heroic” action to save the customer’s business. For example, when a food service distributor sales rep personally delivers a case of ground beef that was missing from the truck earlier in the morning to a restaurant before lunch, he goes above and beyond to demonstrate service and help the customer avoid missed lunch sales.
In some cases, heroic recovery can improve a customer’s perception of the quality of service provided by a salesperson. Some customers actually rate companies higher when there has been a service failure and it has been corrected quickly than if there was no service failure at all. In addition, service failures can ultimately help identify service issues that are important to the customer. For example, an industrial packing company had an internal service standard of shipping 95 percent of all orders complete. This had a negative impact on the company’s ability to make deliveries within seventy-two hours, which is the industry average. After conducting focus groups, the company learned that customers valued complete shipments more than the seventy-two-hour delivery window. The company has since changed its policies and has created a competitive advantage based on service that is important to the customer.Gabriel R. Gonzalez, K. Douglas Hoffman, and Thomas N. Ingram, “Improving Relationship Selling Through Failure Analysis and Recovery Efforts: A Framework and Call to Action,” Journal of Personal Selling & Sales Management 25, no. 1 (Winter 2005): 58.
This is not to imply that a constant state of heroic recovery is acceptable to a customer. In fact, providing excellent service begins with understanding what the customer values and then having internal operations in place to be able to consistently deliver that level of service. Recall from Chapter 1 that consistency is one of the elements of a brand. If you as a salesperson, or your company, can consistently deliver on a service promise, then heroic recovery is not efficient or effective in servicing the customer or creating a loyal customer.
Part of heroic recovery includes taking care of the customer—whatever it takes to make the impact of the service failure right for the customer. In addition, it includes internal analysis to identify where and why the service failure occurred, what it takes to correct the problem, and how to prevent it from happening again. As a salesperson, you want to be able to recover from a service failure with confidence so that you know the root cause of the problem has been fixed.
Power Player: Lessons in Selling from Successful Salespeople
Inspiration from Air Conditioning
Said Hilal, CEO of Applied Medical Resources, owned one of the early Mercedes S series and was happy with the performance of the car. After one year, Mercedes notified him that the air conditioner was appropriate for Europe but was underpowered for the United States and offered to replace the air conditioner. Hilal was so impressed with how Mercedes proactively handled the issue that he decided to use the same approach to his business. “We ask our customers what they want to see in our future product—what problems they have that we can help resolve,” says Hilal. “We consistently remind ourselves to listen to what the customer needs, not what we need.”Ilan Mochari, “What You Learn on the Other Side,” Inc., November 1, 2002, www.inc.com/magazine/20021101/24833.html (accessed November 23, 2009).
The bottom line is that companies and salespeople should view heroic recovery efforts as an investment in customer service perceptions, rather than as a cost. If handled properly, service failures can improve a relationship with a customer even more so than excellent service.Gabriel R. Gonzalez, K. Douglas Hoffman, and Thomas N. Ingram, “Improving Relationship Selling through Failure Analysis and Recovery Efforts: A Framework and Call to Action,” Journal of Personal Selling & Sales Management 25, no. 1 (Winter 2005): 58.
Key Takeaways
• Follow-up is what builds a relationship after the sale. You should never assume the sale is closed.
• Follow-up should take place regularly so your customer knows he can count on hearing from you.
• A personal thank-you note or letter is appropriate after the close of the sale. The letter can also include some operational information such as contact information and receipts.
• Follow up to be sure everything is delivered as promised. Do your follow-up inside the company and touch base with the customer to be sure everything is to her satisfaction.
• Add value to your customer’s business with industry information, white papers, blogs, and newsletters. These bring value to your customer and keep your name in front of him.
• Feedback is an important part of follow-up.
• Customers can become your best-selling tool with testimonials and referrals.
• Heroic recovery can be a way to delight your customer (only if a service failure occurs infrequently and it is handled in a satisfactory manner).
Exercise \(1\)
1. Identify a company with whom you have a relationship (you purchase its products or services on an ongoing basis). What makes the relationship work? What role does follow-up play in the relationship?
2. Identify a company from which you have purchased products or services that doesn’t follow up with you. Why do you continue to purchase the products or services? If another alternative comes along, will you be open to trying the new alternative? Why or why not?
3. Assume you work for a video game manufacturer and you sell video games to bricks-and-mortar and online retailers. Identify three things you would do as part of your follow-up plan after you close the sale to Best Buy.
4. Assume you are selling security systems to businesses, how would you use a news article about recent security issues as part of your follow-up with your customers?
5. Assume you sell landscaping to businesses. Once you have arranged for the landscaping to be installed, are there any other opportunities for follow-up?
• If so, what would you do to follow up during the spring and summer?
• What would you do to follow up during the fall and winter?
6. Imagine that you are a sales rep for a major insurance company. How can you gather customer feedback to improve your service? How can you use customer feedback that you receive about products and services for which you are not responsible? | textbooks/biz/Marketing/The_Power_of_Selling/13%3A_Follow-Up_-_The_Power_of_Providing_Service_That_Sells/13.01%3A_Introduction.txt |
Learning Objectives
• Understand how customer satisfaction relates to customer loyalty.
Customer loyalty and retention are the holy grail in sales—and in all areas of business, for that matter. Loyal customers are how successful businesses are built. Not only is it easier to sell more to existing customers, it is financially prudent to do so. Some companies have increased their profit by as much as 100 percent by focusing on retaining an additional 5 percent of customers. Since it costs about five times more to acquire a new customer than to retain an existing customer, companies are well served to focus on retaining existing customers and making them into advocates for their brand.Chia-Chi Chang, “When Service Fails: The Role of the Salesperson and the Customer,” Psychology & Marketing 23, no. 3 (March 2006): 204. In other words, “Customer acquisition is an investment, but customer retention delivers profitability.”Guy Maser, “How to Earn Your Customers’ Loyalty,” CRMBuyer.com, July 16, 2009, http://www.crmbuyer.com/story/67608.html (accessed November 23, 2009).
Follow-Up, Feedback, and Fans
Earlier in this chapter, the five elements of follow-up were discussed including getting feedback from customers. This concept is so important, it’s worth drilling a little deeper into it. It is loyal customers who buy more from you in the form of more products and services more often. Companies that focus on creating customer loyalty usually invest in developing an effective customer feedback loop, a formal process for gathering, synthesizing, and acting on customer feedback. The most successful customer feedback loops are simple, focus on understanding what is important to customers, and empower front-line employees (i.e., those who interact with customers on a day-to-day basis, such as salespeople). For example, Charles Schwab, an online investment services company, has a process whereby managers review customer feedback daily from comments on the company Web site, transactions, and other communications with the company. Managers and sales reps respond personally to negative customer comments. Cheryl Pasquale, a branch manager, says she looks forward to customer calls to follow up on complaints or less-than-positive comments. She feels she has an opportunity to turn “critics into fans.”Rob Markey, Fred Reichheld, and Andreas Dullweber, “Closing the Customer Feedback Loop,” Harvard Business Review, hbr.harvardbusiness.org/2009/12/closing-the-customer-feedback-loop/ar/pr (accessed November 23, 2009).
There are several different types of customer feedback loops that companies use such as mystery shopper programs, customer satisfaction surveys, and other measurement tools. Some of these methods are expensive, require elaborate reporting, and take a long time to compile and act on the data.Rob Markey, Fred Reichheld, and Andreas Dullweber, “Closing the Customer Feedback Loop,” Harvard Business Review, hbr.harvardbusiness.org/2009/12/closing-the-customer-feedback-loop/ar/pr (accessed November 23, 2009). Simply asking customers what they think can defeat the purpose if companies don’t act quickly on the feedback. It raises customer expectations that action is going to be taken.Dr. Laura Brooks, “Closing the Loop on Customer Feedback,”Sales & Marketing Management, April 23, 2009, www.salesandmarketing.com/article/closing-loop-customer-feedback (accessed November 23, 2009).
Power Selling: Lessons in Selling from Successful Brands
Follow-Up Is Just a Tweet Away
Personal follow-up meets technology with more than half of Fortune 100 companies using Twitter as one of the tools in their arsenal to respond to customer service issues. Comcast is a leader in this area. The company believes that Twitter has provided more transparency and improved communication with customers in multiple channels.Jon Swartz, “Twitter Helps Customer Service,” USA Today, November 18, 2009, 3B. Comcast uses Twitter to address follow-up issues such as a service call that didn’t happen on time, service that isn’t operating properly, and even billing issues. According to Frank Eliason, director of digital care at Comcast, Twitter is not a replacement for phone and e-mail follow-up. However, he says, “It gives immediacy to interactions.” He finds that customers are surprised—and pleased—to hear from him so quickly on Twitter.Rebecca Resisner, “Comcast’s Twitter Man,” BusinessWeek, January 13, 2009, http://www.businessweek.com/managing/content/jan2009/ca20090113_373506.htm (accessed January 8, 2010). The bottom line is to take care of the customer, no matter what method you use for follow-up.
One Simple Question
Successful companies have found that customers can be more than customers; they can be advocates, supporters, promoters, and fans. It’s these passionate fans that not only spend their money with these companies but also tell their friends and ultimately their friends’ friends to patronize the company. The mutual admiration of brand and customer starts with the culture of the company. Those companies that not only listen to their customers but also engage them in communities, new product development, and other improvements are the ones that have a maniacal focus on the customer. They get it. For example, watch this video of a Southwest Airlines flight attendant that was posted to YouTube by a passenger. Talk about being a fan of the brand—it’s hard not to be after you watch this video.
Southwest Gets It
(click to see video)
See how an ordinary activity can create an extraordinary customer experience.
In another example, it’s no surprise that Zappos, the dominant online shoe and apparel retailer, has a maniacal focus on the customer when you listen to CEO Tony Hsieh talk about his philosophy of customer service. Zappos has grown to be a billion-dollar business in just ten years. Although shoes have a notoriously high return rate due to fit problems, Zappos offers free shipping both ways to encourage purchases. Hsieh’s vision for the ultimate experience in customer service is clear throughout the company (try calling their 800 number for customer service and experience Zappos’ unique telephone greeting).
Zappos Gets It
(click to see video)
Hear CEO Tony Hsieh talk about why Zappos is a fan favorite.
Many companies have found that Net Promoter Score (NPS) is the ideal customer feedback tool because it is simple, keeps the customer at the forefront, allows frontline employees to act, thereby closing the customer feedback loop.Rob Markey, Fred Reichheld, and Andreas Dullweber, “Closing the Customer Feedback Loop,” Harvard Business Review, hbr.harvardbusiness.org/2009/12/closing-the-customer-feedback-loop/ar/pr (accessed November 23, 2009). Net Promoter Score is based on asking customers the ultimate question: “How likely are you to recommend this product or company to a colleague or friend?” The response is based on a ten-point scale and categorizes responses as follows:
• Promoters (customers who answer with a 9 or 10). These are customers who are advocates or loyal fans who will willingly tell their friends to do business with the company.
• Passives (customers who answer with a 7 or 8). These are customers who might be categorized as satisfied, but do not enthusiastically support the company. They are vulnerable to competitive offerings.
• Detractors (customers who answer with a 0 to 6). These are customers who are not happy and are likely to pass along stories about their bad experiences to their friends via word of mouth or social networking.
A company’s Net Promoter Score is determined by taking the percentage of promoters (scores of 9 or 10) and subtracting the percentage of detractors (scores of 0 to 6).Net Promoter, “How to Calculate Your Score,” www.netpromoter.com/np/calculate.jsp (accessed November 25, 2009). For example, assume that Widgets, Inc., received the following ratings:
Promoters (score of 9 or 10) = 60%
Passives (score of 7 or 8) = 30%
Detractors (score of 0 to 6) = 10%
The Net Promoter Score for Widgets, Inc., is calculated as follows:
60% − 10% = 50%
Promoters − Detractors = NPS
The premise of Net Promoter Score is simple and elegant. The answer to one question says it all. Customers are then asked why they would be likely or unlikely to recommend the company.Rob Markey, Fred Reichheld, and Andreas Dullweber, “Closing the Customer Feedback Loop,” Harvard Business Review, hbr.harvardbusiness.org/2009/12/closing-the-customer-feedback-loop/ar/pr (accessed November 23, 2009). If the customer is not a promoter after their experience with the brand, they are at risk either to try another brand or become a detractor of the brand. As you can see from the formula, customers that are passives (scores of 7 to 8), reflect poorly on the brand’s NPS. Being satisfied isn’t enough; a brand’s goal is to have promoters or fans. This process quickly lets front-line managers and employees identify where problems exist and allow them to act quickly to respond and fix them.Net Promoter, “How to Calculate Your Score,” www.netpromoter.com/np/calculate.jsp (accessed November 25, 2009).
Net Promoter Scores vary by industry. The Net Promoter Score Web site includes a comparison by industry here: www.netpromoter.com/np/compare.jsp. Some companies that use Net Promoter Score are American Express, Southwest Airlines, FedEx, eBay, Harley-Davidson, and Dell. See additional companies listed on this Web site: http://www.theultimatequestion.com/theultimatequestion/good_profits.asp?groupCode=2.
While Net Promoter Score is a simple concept, it does require a complete operational commitment on the part of every level of management of the company to make it work effectively. Listen to Fred Reichheld, author of The Ultimate Question: Driving Good Profits and True Growth, and Brad Smith, CEO of Intuit, Inc., talk about how Net Promoter Score works.
How Net Promoter Score Works
(click to see video)
Hear about how Net Promoter Score changes the way companies do business.
Key Takeaways
• Customer loyalty pays. It costs five times more to acquire a new customer than to keep an existing customer.
• A customer feedback loop is a formal process for gathering, synthesizing, and acting on customer feedback. Customer feedback loops are most effective when front-line employees have the power to respond to customer feedback to turn “critics into fans.”
• Net Promoter Score (NPS) is a closed customer feedback loop that is based on the theory that a loyal customer is one that will recommend the brand to their friends.
• NPS is determined based on a brand’s percentage of promoters minus the percentage of detractors.
Learning Objectives
• Describe why Net Promoter Score is a closed customer feedback loop.
• Assume you worked as a financial planner. How would you use Net Promoter Score with your customers? How would you respond to promoters? How would you respond to passives? How would you respond to detractors?
• Imagine that you are a sales rep for a medical supply company and you have just received your Net Promoter Score for the past month, which is as follows: Promoters: 63 percentPassives: 28 percentDetractors: 9 percent
Calculate your overall Net Promoter Score. What steps would you take to communicate with the customers in each of the categories?
• Research one of the companies that use Net Promoter Score and identify at least one way it impacts how the company does business.
• Research Net Promoter Score online and find some articles that discuss the drawbacks of using it as the customer feedback loop. What do you think? Is Net Promoter Score something you think you might find helpful in sales?
• Imagine that you are a salesperson for a software company and a portion of your compensation is based on your Net Promoter Score. Is it ethical for you to tell your customers that you need their positive comments to earn your salary? Why or why not? | textbooks/biz/Marketing/The_Power_of_Selling/13%3A_Follow-Up_-_The_Power_of_Providing_Service_That_Sells/13.03%3A_Customer_Satisfaction_Isnt_Enough.txt |
Learning Objectives
• Learn how to follow up after accepting a job offer.
• Understand how to adapt to your new job.
So you’ve got your offer letter, and you’re excited about starting your new job in a few weeks. Time to take it easy? Maybe a little. But don’t kick back completely. There’s follow-up work to be done.
From Classroom to the Corporate World
Just as you should never assume the sale is closed, the same is true about your job. Even though you have your offer, it’s really the beginning of proving yourself in your new career. Whether you decide to work for a large corporation, a small company, or start your own business, it all starts right here.
The first thing to realize is that the corporate world is very different from the classroom. For starters, everything will not be mapped out for you in a syllabus with predetermined reading, homework, and final exams. If you think you’re busy now while you are in school, wait until you start working! At work, everything is due “yesterday,” so it’s up to you to prioritize what you need to get done.Alexandra Levit, The Don’t Teach Corporate in College: A Twenty-Something’s Guide to the Business World (Franklin Lakes, NJ: Career Press, 2009), 134. There are no tests, but you are being tested everyday. You don’t get a report card or grades; you get a performance review that provides a platform for feedback and self-improvement as well as a record of your performance for the company.Dawn Rosenberg McKay, “From College Campus to Corporate Climate: How to Make the Transition to Your First Job after College Graduation,” About.com, http://careerplanning.about.com/cs/firstjob/a/post_grad.htm (accessed November 23, 2009). And even if you’ve had a job while you were in school, there’s more expected of you as a full-time employee than as an intern or part-time employee.Dawn Rosenberg McKay, “Your First Job: Making a Good Impression,” About.com, http://careerplanning.about.com/cs/firstjob/a/first_job.htm (accessed November 24, 2009).After all, it’s no longer about you; it’s about how your performance impacts the company’s results.Dawn Rosenberg McKay, “From College Campus to Corporate Climate: How to Make the Transition to Your First Job after College Graduation,” About.com, http://careerplanning.about.com/cs/firstjob/a/post_grad.htm (accessed November 23, 2009). Welcome to the “real world.”
Before You Start
Starting strong is important in any job. The first ninety days can make the difference in how well you do at your job, so do your follow-up from your job interviews before you even start working. It will not only give you a head start; it can make the difference about how well you do at the company.Andy Wang, “The First 90 Days,” Forbes, September 7, 2006, http://www.forbes.com/2006/09/06/leadership-pink-careers-cx_ag_0906ninetydays.html (accessed November 25, 2009).
Here are five things you should do before you start your new job.
• Say thank you. Drop a handwritten note to your new boss, the human resources person, and any other people with whom you interviewed. Although you already sent thank-you notes to each of these people after your interviews, it’s a good idea to send each one a personal note to thank them for their support and tell each how much you are looking forward to working with him. This is a great way to set yourself apart even before you begin your new job.
• Continue to do your research on the company. Just because you have a job offer doesn’t mean you should stop researching the company. In fact, you should do just the opposite. Visit the company’s stores, Web site, talk to customers, read press releases, and talk to current employees. Do everything you can to learn even more about the company you will work for.“Brave New World: What to Do before You Start a New Job,” Workplace911, February 22, 2008, http://careerplanning.about.com/cs/firstjob/a/new_job.htm (accessed November 25, 2009).
• Dress for success. Plan what you are going to wear on your first day, even your first week of work. It’s best to dress more conservatively during your first days until you can begin to really understand the company culture. Even if the company is very casual, dress up on your first day. According to Alexandra Levit, author of They Don’t Teach Corporate in College: A Twenty-Something’s Guide to the Business World, “You might be overdressed, but I guarantee no one will criticize you for it. Rather, your colleagues will respect that you mean business, and your boss will be proud to introduce you around the company.”Alexandra Levit, They Don’t Teach Corporate in College: A Twenty-Something’s Guide to the Business World (Franklin Lakes, NJ: Career Press, 2009), 51. Try on your clothes, take items to the dry cleaner, or have them tailored as needed. You want to avoid any last-minute fashion emergencies on your first day of work.Dawn Rosenberg McKay, “Starting a New Job: What You Can Do before Your First Day,” About.com, http://careerplanning.about.com/cs/firstjob/a/new_job.htm (accessed November 24, 2009).
• Plan your route. Even though you probably know your way to the office, it’s a good idea to take a test run during actual conditions during rush hour. You want to avoid being late for any reason so that includes knowing the public transportation schedule, traffic, or parking situation, depending on how you will get to work. Have an alternate route in mind just in case there is a traffic problem on your first day. Allow extra time on your first day. It’s better to be early than to be late.Dawn Rosenberg McKay, “Starting a New Job: What You Can Do before Your First Day,” About.com, http://careerplanning.about.com/cs/firstjob/a/new_job.htm (accessed November 24, 2009).
• Walk in with a smile. While you will most likely be nervous with anticipation on your first day of work, follow the process similar to what you did for your job interview. Arrive a bit early, use the restroom, take one last look at yourself, use a breath mint, and smile. People will be helpful, so just relax and enjoy your first day on the job. A smile goes a long way on your first day and every day.Dawn Rosenberg McKay, “Starting a New Job: Fitting In,” About.com, http://careerplanning.about.com/od/newjobfirstjob/New_Job_First_Job.htm (accessed November 24, 2009).
You’ve Got the Power: Tips for Your Job Search
Impress Your New Boss
You already sent your thank-you note to the people with whom you interviewed and have just accepted your offer. What’s next? It’s a good idea to send a handwritten note to your new boss and tell her how much you are looking forward to working with her. It’s the perfect way to make a good first impression before you even start your new job.
After You Start
Your first few weeks on the job will be a whirlwind. You will meet lots of people, and it will be difficult to remember anyone’s name, title, or function. It takes a while to adjust and fit in at any company. Remember how it felt when you were a freshman? By the time you became a sophomore, you knew a lot of people, and you knew the ropes. The same thing happens at a job. There’s no magic time frame to adjust to a new job; everyone is different. It’s good to know that you’re not alone and that adjusting to your new job just takes time—and commitment.Dawn Rosenberg McKay, “Starting a New Job: Fitting In,” About.com, http://careerplanning.about.com/od/newjobfirstjob/New_Job_First_Job.htm (accessed November 24, 2009). Here are five tips to help you get your feet on the ground at your new job.
• Listen, observe, and ask questions. This is the best way to learn the ropes and the company culture. There is no stupid question, so take advantage of the fact that you are new to ask as many questions as possible. When you watch and listen to other people, it’s easier to understand the culture or the unwritten rules of the company.Dawn Rosenberg McKay, “Your First Job: Making a Good Impression,” About.com, http://careerplanning.about.com/cs/firstjob/a/first_job.htm (accessed November 24, 2009).
• Avoid office gossip. It might sound obvious, but engaging in office gossip can only hurt you. You never know to whom you are speaking so it’s better to heed your mother’s words: “If you can’t say something nice about someone, don’t say anything at all.” But do pay attention to the office grapevine. This will help you understand the informal rules, who’s who in the office, and how people perceive what’s going on in the company.Dawn Rosenberg McKay, “Your First Job: Etiquette and Gossip,” About.com, http://careerplanning.about.com/cs/firstjob/a/first_job_2.htm (accessed November 24, 2009). On similar note, it’s never appropriate to use company time and resources to check or update your status on social networking sites. Even if other employees do it, avoid the temptation to participate in social networking at work.
• Find a mentor. A mentor is someone who has experience in the area you wish to pursue and who exhibits a “generosity of spirit,” a natural gift to go out of her way to help others.Alexandra Levit, They Don’t Teach Corporate in College: A Twenty-Something’s Guide to the Business World (Franklin Lakes, NJ: Career Press, 2009), 106. A mentor is a person with whom you develop a personal relationship: someone whom you trust and are comfortable asking questions to and getting feedback from to take your career to the next level. Some companies offer formal mentoring programs, but at most companies finding a mentor is usually a less formal process. Go out of your way to get to know people whom you think might be a good mentor and take the time to get to know them. You should consider having several mentors throughout your career.
• Stand out. Perception is reality so be the person who stands out.Alexandra Levit, They Don’t Teach Corporate in College: A Twenty-Something’s Guide to the Business World (Franklin Lakes, NJ: Career Press, 2009), 75. Volunteer to work on projects, especially those that others don’t want to do, come in early, stay late, and deliver high-quality work on time.Dawn Rosenberg McKay, “Starting a New Job: Fitting In,” About.com, http://careerplanning.about.com/od/newjobfirstjob/New_Job_First_Job.htm (accessed November 24, 2009). Going the extra mile pays off.
• Fine-tune your writing and speaking skills. Now that you are working, you have to develop and communicate your ideas and point of view to your boss, your colleagues, and even your clients. Be a good listener and a confident communicator. It will make a difference in how people perceive you and your work.Alexandra Levit, They Don’t Teach Corporate in College: A Twenty-Something’s Guide to the Business World (Franklin Lakes, NJ: Career Press, 2009), 161.
Now, it’s time to relax, enjoy, and start this next chapter in your life.
Key Takeaways
• Even though you receive a job offer, there are still a lot of things you can do to follow up after your interview and before you start your new job.
• The corporate world is different from the classroom with a different environment and expectations. Your performance is no longer just about you; it’s about how you help the company achieve its goals.
• It takes time to adapt to a new job.
Exercise \(1\)
1. Assume you just accepted a job offer to become a sales rep at a national food manufacturer. Write a personal note to your new boss to tell him how you are looking forward to starting your new job. Who are some other people in the company to whom you might also write a note?
2. What is a mentor? Identify someone who is currently a mentor to you. What makes him a good mentor? How might you be able to find additional mentors when you begin working?
3. Identify two resources that would be helpful to fine-tune your writing and speaking skills. How can you use these resources to help prepare you for your career? | textbooks/biz/Marketing/The_Power_of_Selling/13%3A_Follow-Up_-_The_Power_of_Providing_Service_That_Sells/13.04%3A_Selling_U_-_What_Happens_after_You_Accept_the_Offer.txt |
Power Wrap-Up
Now that you have read this chapter, you should be able to understand the importance of follow-up on your relationships and sales.
• You can understand the role of follow-up in the selling process.
• You can identify how to plan your follow-up even before you begin prospecting.
• You can learn that follow-up is a personal commitment and has a reflection on you as a brand.
• You can identify ways to add value to your customers’ businesses as part of follow-up.
• You can describe how follow-up can build your business with additional sales from your existing customers, testimonials, and referrals.
• You can define heroic recovery and the impact it can have on how customers perceive you.
• You can understand how the customer feedback loop works.
• You can describe how Net Promoter Score works to improve follow-up and customer service.
• You can list things you can do after you accept a job offer.
TEST YOUR POWER KNOWLEDGE (AnswerS ARE BELOW)
1. How many calls does it take on average to close a sale?
2. True or false: After the sale is closed, the role of the salesperson is finished.
3. What does this statement mean: “Even though the sale is closed, you should never assume the sale is closed”?
4. Name three areas that require follow-up on the part of the salesperson.
5. Identify three ways that you can add value to your customers’ businesses during the follow-up process.
6. Name three benefits of having a loyal customer.
7. Describe how heroic recovery can have a positive impact on your relationship with your customer.
8. What is a customer feedback loop?
9. Describe Net Promoter Score?
10. What is the formula to calculate NPS?
11. Identify at least one thing you can do after you receive your job offer but before you start your job.
POWER (ROLE) PLAY
Now it’s time to put what you’ve learned into practice. The following are two roles that are involved in the same selling situation—one role is the customer, and the other is the salesperson. This will give you the opportunity to think about this selling situation from the point of view of both the customer and the salesperson.
Read each role carefully along with the discussion questions. Be prepared to play either of the roles in class using the concepts covered in this chapter. You may be asked to discuss the roles and do a role-play in groups or individually.
Let It Snow
Role: Facilities manager at the Tri-County Office Complex
You are responsible for the overall maintenance at the largest office complex in the area. There are ten office buildings in the complex, which provides office space for thirty companies. You oversee the exterior maintenance, which includes everything from trash and snow removal to lawn care and window washing. You have just signed a contract with All Weather Maintenance Co., two days ago. It’s 5:00 a.m., and a major snowstorm just hit, so you are on your way to inspect the property to be sure that the walkways are shoveled and parking lot is plowed.
• What role do you expect the salesperson to play now that the contract has been signed?
• Who will you call if the snow removal is not completed to your satisfaction?
• How will this experience impact your expectations of All Weather Maintenance Co., for other snowstorms and situations that require maintenance, especially time-sensitive maintenance?
Role: Sales rep, All Weather Maintenance Co.
You recently signed your largest client, Tri-County Office Complex. You have a very good relationship with the facilities manager based on the selling process. You have communicated the maintenance requirements to your company’s operations department. Now the job is up to them to conduct year-round maintenance. Your normal hours are 8:00 a.m. to 5:00 p.m., but you were concerned about the weather report last night, so you set your alarm early. You wake up at 5:00 a.m. to see a blanket of snow and ice and immediately wonder if the maintenance crew made it to the Tri-County Office Complex.
• What action, if any, do you take?
• What kind of follow-up will you do with the customer?
• When will you do follow-up?
• What will you say to the customer?
• What will you do to ensure that time-sensitive maintenance is completed as expected?
ACTIVITIES
1. Visit your career center and ask them for information about mentors. Learn how you can get a mentor even before you start your job.
2. Identify someone who already works at the company from which you received an offer. Set up a meeting with her before you start your new job to learn more about the company, company culture, and other things that will be important to know for your new job.
TEST YOUR POWER KNOWLEDGE AnswerS
1. Five.
2. False.
3. Good salespeople help avoid buyer’s remorse by following up quickly after the sale is closed and reinforcing the fact that the buyer made a good decision.
4. Contracts to be signed, delivery to be scheduled, customer shipping and billing information to be added to CRM system, credit checks, addition of customers to all appropriate correspondence, invoice to be generated, welcome package to be sent to customer, introductions to be make to all appropriate internal people on the team, and status calls to be scheduled.
5. Phone and in-person regular status updates, newsletters, white papers, industry information, networking, asking questions, spending time in the business.
6. Additional sales from the loyal customer, testimonials to be used in presentations for prospects, and referrals to new customers.
7. If a service failure is handled quickly and meets or exceeds the expectations of the customer, it can have an even more positive impact on how the customer perceives the service from the sales rep and the company.
8. A formal process for gathering, synthesizing, and acting upon customer feedback.
9. NPS is a closed loop customer feedback system that relies on the answer from customers to one key question: “How likely would you be to recommend this product or service to your friends or colleagues?”
10. NPS = Promoters – Detractors.
11. Say thank you with a personal note to your new boss, continue to do research on the company, dress for success, plan your route, and walk in with a smile. | textbooks/biz/Marketing/The_Power_of_Selling/13%3A_Follow-Up_-_The_Power_of_Providing_Service_That_Sells/13.05%3A_Review_and_Practice.txt |
The Power of Learning the Ropes
Video Ride-Along with Priya Masih, Sales Representative at Lupin Pharmaceuticals
Meet Priya Masih. Priya has been in sales for five years with experience in the telecommunications, insurance, and pharmaceutical industries. She is currently a sales representative at Lupin Pharmaceuticals. She has learned how to manage herself, her time, and her results for a successful career in sales. But it’s not always easy. One of the biggest challenges of being successful in sales is to stay motivated, even when you don’t make the sale.
Priya shares how she stays motivated to achieve new heights every day.
(click to see video)
14.02: Managing Yourself Your Income and Your Results
Learning Objectives
• Understand how to manage yourself as a selling professional.
• Learn the keys to managing your time.
• Identify the elements that drive your results and income.
So imagine that you landed your dream job in sales, you’ve been to the corporate office for training and orientation, you’ve set up your home office, and you’ve picked up your company car—now what?
Sales is a challenging, exhilarating, demanding, and rewarding profession. You want to be successful and enjoy what you do, but you really haven’t had a chance to focus on the actual job between graduation and job interviews. Here’s your chance to look ahead to how you learn the day-to-day activities that go on in the profession of selling, identify the resources to help you be a partner to your customers, and bring success to yourself and your company. It sounds like a tall order, but it’s easy when you have people to guide and support you.
Be an A-Player
No matter what job you have or what company you sell for, you can and should be an “A-Player.” That means being the best at what you do. You don’t have to be a celebrity or a person who went to Harvard, according to blogger Auren Hoffman in her April 2009 post “The A-Player Janitor”: “An ‘A-Player’ by definition is incredibly productive and smart and has that ‘it,’ that rockstar-esque factor that makes everyone want to work with her.”Valeria Maltoni, “How Do You Become an A-Player?” Social Media Today, April 17, 2009, www.socialmediatoday.com/SMC/85675 (accessed September 4, 2009). Her point is that hiring managers want to hire the best person for every job. So you don’t have to be an A-Player in everything, just be an A-Player in the one thing you do best.Auren Hoffman, “The A-Player Janitor,” Summation Blog, April 9, 2009, blog.summation.net/2009/04/the-aplayer-janitor.html (accessed September 4, 2009). Find your sweet spot and focus on it. In sales, being an A-Player means connecting with customers. You might be surprised to learn what makes someone an A-Player in sales according to this video.
Video Clip
What Makes an A-Player?
http://www.sellingpower.com/content/video/?date=8/29/2009
Managing Yourself: Making the Most of Your Resources
“The best part of a career in sales is that it is undefined,” according to Ann Devine in a recent article on The Black Collegian Online.Ann Devine, “Is a Career in Sales Right for You?” The Black Collegian, www.black-collegian.com/career/career-reports/sales-grad05.shtml (accessed August 19, 2009). Every day is completely different; some days you will be researching leads, and other days you might be making a presentation to a prospective customer. This exciting, unstructured, and sometimes unpredictable environment rarely gets boring. But it’s this lack of structure that can present a challenge in choosing priorities and accomplishing goals. Those who are successful realize how to manage themselves and their time and use the resources that are available to them from their company, their colleagues, and their community.
You might be wondering what managing yourself means. When you are in sales, one of the most important jobs you have is being sure that you have clear direction about what you want to accomplish and what you need to do to get there. Even though you are used to managing yourself and your time at school, it can be a daunting task to be responsible for calling on customers and generating sales, especially if you are based in a location remote from the company office such as your home office. So first things first—identify your resources. Even though you’re traveling solo, you are not alone.
Manage Yourself for Success
A great salesperson starts with great habits. Here are a few tips from Richard E. Goldman, author of Luck by Design: Certain Success in an Uncertain World.
• Learn by doing. Take the initiative to seek out information and teach yourself how to do things; the power of learning is by doing.
• Make your own choices. You might not have all the information you need at the time, but the best decision is the one you make. Don’t let someone else make your decisions for you.
• Believe in yourself. You got this job because you are smart and talented. Don’t ever believe you can’t succeed.Richard E. Goldman, “Managing Yourself First,” Focus, July 8, 2009, www.cuckleburr.com/book-excerpt-managing-yourself-first-from-luck-by-design (accessed August 18, 2009).
Ride-Alongs
One of the best ways to learn the ropes and get the inside track is to go on ride-alongs (also referred to as shadowing) with colleagues, traveling with an experienced sales rep or sales manager to make sales calls. The video ride-alongs at the start of each chapter are a virtual way for you to get some powerful insights from experienced sales professionals. Sometimes a ride-along is included in the interviewing process; it’s an opportunity for you to experience firsthand exactly what the job entails and for the company to see how you react in the selling environment before you get a job offer. Other times a ride-along is an training opportunity that takes place after you’ve been hired. Either way, always take advantage of as many opportunities as you can to ride-along with experienced salespeople. There are some tips that will help maximize your ride-along experience.
• Always be professional. It is likely that you will be traveling with a salesperson or sales manager at least for a day and sometimes for a week or longer. Even though you will get to know each other, always remember your role on the ride-along.
• Avoid highly personal or inappropriate topics. While it’s always appropriate to tell the truth, it’s best to avoid controversial topics, especially as they relate to the company.
• Mind your manners and avoid alcohol if you go out to lunch or dinner. Since you will ultimately be in the role of entertaining clients, the person with whom you are riding will undoubtedly be watching your social behavior.
• Above all, be yourself. You won’t be able to learn if your focus is on acting in a way that isn’t natural.“What Do I Do on a Ride Along?” PharmBoard.com, pharmboard.com/what-do-i-do-on-a-ride-along (accessed August 29, 2009).
Use Your Sales Manager
Many salespeople don’t realize that their sales manager (i.e., the person to whom they report) is ultimately responsible for delivering the company’s sales goals. As such, the sales manager wants to do everything he can to help his salespeople be successful. Even before you start your job, it’s a good idea to touch base with your sales manager. Chances are you interviewed with him, so you probably have his contact information. A good way to get off to a great start is to send him a handwritten thank-you note after you’ve accepted the position. What better way to start a new relationship than with a personal note.
Your sales manager can be your most important source of company information as well as customer insights. He had a lot of experience selling before he became a sales manager, and he would likely share his insights to help you be successful. Not only can he make your job (and your life) easier, he can teach you a lot about selling. It’s always a good idea to keep your sales manager updated with the status of your customers and prospects. He will appreciate your proactive and regular updates about the standing of each lead and customer in addition to your regular one-on-one meetings, staff meetings, or conference calls.
Sometimes new salespeople are nervous about asking questions of their sales managers, which is natural. It’s best to remember that your sales manager doesn’t expect you to know everything. Your questions show him that you are interested in learning more about the business from him and help him identify what areas would be most beneficial for coaching. Your sales manager can be a part of your success story. Ask questions, ask his opinion, keep him in the loop, help make him look good, and you will have a relationship that works and grows.
Just as communication is important with customers, it is critical to building your relationship with your sales manager. He probably has a span of control, or the number of people reporting to him, ranging from a two to twenty or more people. It’s important to understand the organizational structure of a sales department. While each company is different, the basic structure of a selling organization is shown in Figure \(2\). In some companies, salespeople may be responsible for a city or cities, region, or other geographic area. This is called territory management. In this case, salespeople, usually called territory managers, are responsible for the customers in their specific geographic area. This organizational strategy helps minimize the amount of travel time between customers.
In other companies, salespeople may be responsible for specific brands, products, or product categories. In the case of food manufacturers, these categories might be noncarbonated beverages, prepared meals, or dairy products. In the professional services arena, the organization might be vertical, such as retail sales, financial services, or health care. This product or category approach may require salespeople to travel to customers in various parts of the country based on the needs of the customers. It requires the salesperson to develop expertise in a specific product or discipline. These sales positions may have titles such as account manager, product manager, or sales rep. The different types of sales positions are discussed in more detail in Chapter 2.
Resources and Resourcefulness
The company you work for, whether it is large or small, has resources. A laptop, the customer relationship management (CRM) system, your expense account, the company owner, the human resources department, accounts receivable department, and others are all resources that can help you do your job. Take the time to explore all the resources when you start with the company. In larger companies, you will most likely participate in an orientation session or process frequently referred to as onboarding to learn about how the company operates and how your can take advantage of resources to help you do your job. In a small company, the process is less formal and requires you to be more proactive about understanding what’s available. Either way, it’s your responsibility to explore and understand your resources. Remember that all the skills you use when you are communicating with customers are the same when you are communicating inside your company: build lasting relationships that are mutually beneficial. While every company is different, here are some internal resources that are available in most companies.
• Human resources department. Whether you work for a large company or a start-up, it’s a good idea to know the key people in HR. Chances are, you interviewed with someone in the department, but don’t stop there. Continue your relationship by learning who handles employee relations (for questions about the company policies or an ethical dilemma) and who handles benefits (for questions about medical, dental, other insurance, 401(k) plan, and other company benefits).
• Finance department. You’ll want to get to know the people who handle accounts receivable. Since most salespeople are responsible for collections, you will most likely be working closely with people in finance, accounting, or accounts payable. They can provide helpful information about company processes and policies for payment of invoices. You’re not the first person to be challenged by customer payment issues, so take advantage of their knowledge and experience.
• Procurement or product development department. Whether you are selling a product or a service, you will want to know those who make the decisions about exactly what will be available for sale. Customers may have specific questions about the performance of the product or service that you may need some additional information to answer. In addition, building a relationship with people in this department will help give you insight into what will be available in the future. More important, it will help you provide input and feedback based on the customer’s perspective.
• Marketing department. The people who are responsible for getting the word out about your company’s brand are important to know. You can get insights about advertising, promotions, and other communication activities. You can also get important information about future plans and help shape the marketing plan for the future based on your experience with customers.
• Information technology department. Everything from your laptop to your reports is supported in the IT department. It’s especially important to get to know the people who man the help desk. Chances are, you will have a technology emergency at some point in time so it’s best to build a strong relationship from the start.
• Other salespeople. Create relationships with the best-performing salespeople so you can learn the best practices. Go on ride-alongs and learn what makes them successful.
• Other resources. Explore the CRM system and company intranet, especially the online communities. This is an excellent way to learn about how sales were won, see examples of successful proposals, and learn about best practices of the top performers.
Power Selling: Lessons in Selling from Successful Brands
School of Hard Rocks
Imagine going to employee orientation and getting the employee handbook that looks more like a comic book than a manual. That’s how Hard Rock Café onboards its mostly millennial sales force of wait staff and other support roles during its one-day orientation. Jim Knight, senior director of training and development, completely revamped the company’s School of Hard Rocks corporate university. Knight used comic books as his inspiration and got employees involved in telling the Hard Rock Café story; all the illustrations and photos in the handbook were done by Hard Rock employees. The results are impressive: employee turnover rate is now fifty-five points lower than that of the industry.
Besides using company resources, it’s also important for you to stay on top of changes in technology, not only to be effective but also to redefine practices. In fact, Helen Hast, a professor at the Harvard Graduate School of Business, has identified managing technological change as of the five core competencies for the twenty-first century. According to a recent article on BNET, she said, “When we have a new tool, we first use it for what we are already doing, just doing it a bit better. But gradually, the new tool changes the way we do things.”Sean Silverthorne, “5 Personal Core Competencies for the 21st Century,” BNET, August 13, 2009, blogs.bnet.com/harvard/?p=3332&tag=nl.e713 (accessed August 19, 2009).
While resources are important for you to be effective in sales, it’s resourcefulness that will make you successful.Tony Robbins, “Tony Robbins: Why We Do What We Do and How We Can Do It Better,” video, January 16, 2007, http://www.youtube.com/watch?v=Cpc-t-Uwv1I (accessed September 5, 2009). Think about it: Evan Williams, Biz Stone, and Jack Dorsey figured out a way to make Twitter—the microblogging site they founded in March 2006—one of the most popular Web sites in the world without the use of traditional advertising to spread the word.Ashton Kutcher, “The Twitter Guys: The 2009 Time 100,” Time, www.time.com/time/specials/packages/article/0,28804,1894410_1893837_1894156,00.html (accessed September 5, 2009).
It would be hard to argue that Williams, Stone, and Dorsey had all the resources they needed to launch this hugely successful Web site; they had no money for advertising, or anything else for that matter. But they were resourceful about getting people to try their new service, use it, and engage with it. While you might not invent the next Twitter, you can certainly sell the next big idea by using your resources and being resourceful.
Managing Your Time: Organizing and Prioritizing
Depending on the type of business you are in and the company you work for, you might have as few as one customer and as many as a hundred or more. You might be wondering how you determine which customers to call on each day, how much time should be spent on prospecting versus calling on existing customers, how much time should be devoted to nonselling activities such as travel, paperwork, and internal meetings. While there is no hard-and-fast answer to these questions, your goal should be to spend as much time as possible with customers or prospects. It’s impossible to sell if you are not in front of a customer.
Consider this: Salespeople spend approximately fourteen hours a week engaged in face-to-face selling. That means that 70 percent of the time, in an average forty-six-hour workweek, salespeople are doing something other than face-to-face selling.Gerald L. Manning, Barry L. Reece, and Michael Ahearne, Selling Today: Creating Customer Value (Upper Saddle River, NJ: Pearson Prentice Hall, 2010), 31. See Figure \(3\) for a complete breakdown of activities.
Since your objective is to spend as much time as possible with customers, you’ll need to balance where you physically spend your time and with which customers you spend it. This is where territory management strategies come into play. Based on the call cycle, the frequency at which you call on each of your customers, and where each is located, you’ll develop a plan to call on your existing customers and allow time for prospecting. In other words, you will need to have a plan to invest your time wisely to meet your goals.
To plan your sales calls, you’ll need a map (Google maps or MapQuest) and sales and potential sales information by customer (your company CRM system should include some, if not all, of this information), and your call cycle. Identify the location of each of your customers with a red dot or push pin. Then, divide your territory into sections by geography (designated as one, two, three, etc.), this can become the basis of your territory management plan. Review your customer data, including current sales and potential sales, to organize and prioritize your customers and calls. Figure \(4\) includes an example of a territory management worksheet.
Based on this, you would plan your route so that you are making calls in one section of your territory on a given day, then covering another section on another day. This will ensure that you regularly visit your best customers and those with the most potential for growth, minimizing your travel time. While this might seem like a lot of work to do, it will save you time in the long run and help you increase your sales…and your income.
Time Management
I am definitely going to take a course on time management…just as soon as I can work it into my schedule.
Louis E. Boone
If you’ve ever felt this way, it’s time to focus on time management. Salespeople get paid on results, not on the number of hours worked. As a salesperson, there are so many demands on your time: client needs, internal meetings, follow-ups, proposals, phone calls, e-mails, text messages, and the emergency du jour. All these can be time thieves, or activities that literally steal your time away from selling. You can easily fill your days with demanding tasks like these that really do not bring value to customers or ultimately close sales. Keep in mind that according to renowned sales consultant and motivational speaker Zig Ziglar, “Nothing happens until someone sells something.”Ann Devine, “Is a Career in Sales Right for You?” The Black Collegian, www.black-collegian.com/career/career-reports/sales-grad05.shtml (accessed August 19, 2009). To understand how to avoid getting caught up in the daily sea of details, it’s a good idea to realize why these interruptions and administrative demands consume your day. Here are three key reasons that time can get away from you:
• Poor planning. Avoid getting caught up in the moment and make a plan every day of selling activities—not time-fillers—that you want to accomplish. True selling activities include things like identifying six new prospects, setting up three appointments for the coming week, or closing at least one sale. “Write your top three outcomes at the top of your plan” is good advice.John Hacking, “Time Management for Sales People,” Buzzle.com, October 15, 2007, www.buzzle.com/articles/time-management-for-sales-people.html (accessed September 5, 2009).
• Procrastination. Fear of rejection causes many salespeople to stay involved in meaningless tasks. It’s hard to get an appointment with a customer, as they don’t always have time to give to salespeople. Customers want true solutions, not a sales pitch. It takes time, research, and creativity to really understand a customers’ business.“Procrastination Costing Your Sales Team,” ArticlesBase, April 29, 2009, http://www.articlesbase.com/education-articles/procrastination-costing-your-sales-team-893170.html (accessed September 5, 2009).
• Making tasks too big. Thinking about how long it takes to go from identifying a prospect to actually closing a sale can sometimes make the job seem overwhelming. Sales success comes from a series of wins, not one home run. It’s best to set short-term goals to make steady progress toward the larger, longer-term goal.John Hacking, “Time Management for Sales People,” Buzzle.com, October 15, 2007, www.buzzle.com/articles/time-management-for-sales-people.html (accessed September 5, 2009).
Mastering Time Management
While there are many theories on the best way to manage yourself and your time, one of the best resources is The Seven Habits of Highly Effective People by best-selling author and management expert Stephen R. Covey. The book is based on seven principles that appear to be simple, but provide a framework to make you more efficient, effective, and successful.
• Habit 1: Be proactive. Take ownership and control your environment.
• Habit 2: Begin with the end in mind. Develop personal leadership that helps you stay focused on your goals.
• Habit 3: Put first things first. Avoid distractions and time wasters with personal management; this is the essence of time management.
• Habit 4: Think win-win. Build success through cooperation with others, not on a win-lose attitude.
• Habit 5: Seek first to understand, then to be understood. Develop strong relationships by listening and understanding.
• Habit 6: Synergize. See and appreciate what others have to contribute.
• Habit 7: Sharpen the saw. Focus on self-renewal in four areas: spiritual, mental, physical, and social/emotional.“The Seven Habits of Highly Effective People,” Businessballs.com, http://www.businessballs.com/sevenhabitsstevencovey.htm (accessed January 2, 2010).
Covey’s philosophy has been embraced by so many that his consulting firm, FranklinCovey, advises thousands of people and companies around the world. His time management and personal planning tools are very popular with a loyal following. You can learn more about Stephen Covey and his philosophy at www.franklincovey.com/tc.
Choosing Success
(click to see video)
Hear Stephen Covey talk about choosing success.
Power Point: Lessons in Selling from the Customer’s Point of View
Don’t Waste My Time
If you think your time is valuable, think about how valuable your customer’s time is. When your customer thinks that doing business with you helps her save time, it can be a reason she won’t do business with anyone else. Ask Marcia F. Borello, who sings the praises of BankAtlantic in Tampa, Florida: “I do my banking exclusively at BankAtlantic because I save so much time. At so many other banks, I waste my precious free time in my lunch hour waiting in long lines hurrying to make my banking transactions before the bank closes at 4 pm. BankAtlantic’s long hours and seven day service make it convenient for me to do my banking when I choose to.”Marcia F. Borello comment, Bank Atlantic, www.bankatlantic.com/Customerfeedback/default.html (accessed September 13, 2009).
The moral of the story is that when you save time and save your customer’s time, you get more business.
Top Three Time-Wasters for Salespeople
Selling is all about making things happen. According to Ray Silverstein, “When you’re selling, time is your most valuable asset.”Ray Silverstein, “Time Management for Sales Pros,” Entrepreneur, March 20, 2007, http://www.entrepreneur.com/management/leadership/leadershipcolumnistraysilverstein/article176034.html (accessed September 5, 2009). But sometimes salespeople can get sidetracked doing tasks that don’t really generate sales. Here are the top three time-wasters:
1. Focusing on the urgent. E-mails, phone calls, paperwork, and even meetings can be unnecessary tasks that appear to be urgent but take time and focus away from selling.
2. Being too comfortable. Habit, routine, and being comfortable can be barriers to breaking through to sell the next big idea.
3. Lacking trust in other people. Salespeople can miss a huge opportunity for teamwork and sharing the workload when they think that no one else can do it as well as they can.Dave Kahle, “Biggest Time Wasters for Sales People,” Business Know-How, http://www.businessknowhow.com/growth/timewast.htm (accessed September 5, 2009).
Work Smarter, Not Harder
Being successful in sales doesn’t require working longer hours; it requires taking control.Tom Metcalf, “3 Steps to Better Time Management for Sales Reps,” Sales Reps, Increase Your Productivity! Blog, January 3, 2007, telenotes.blogspot.com/2007/01/3-steps-to-better-time-management-for.html (accessed May 16, 2010). Time management is all about taking control of your time, your life, and your results. Here are six road-tested tips for effective time management that you can use for school and in sales:
1. Get organized.Doug Dvorak, “How to Use Time Management to Become a More Successful Sales Professional!” EzineArticles, ezinearticles.com/?How-to-Use- Time-Management-to-Become-a-More-Successful-Sales-Professional!&id=1081316 (accessed September 6, 2009). Get all the right tools to do your work efficiently and effectively. Be sure your work space is adequate with appropriate light, get file folders for each subject, organize your electronic files by folder, and choose a naming convention (e.g., customer name_topic_date) so it will be easier to find files that may have been saved to the wrong folder.Margot Carmichael Lester, “5 Ways to Get and Stay Organized” SalesHQ.com, www.saleshq.com/training/articles/1353-5-ways-to-get-and-stay-organized (accessed September 6, 2009).
Consider using a time management product to help you stay organized. Franklin Covey offers a world-renowned planning system at shopping.franklinplanner.com/shopping/index.jsp?. Day-Timer also offers paper and electronic options (including iPhone apps) for planning at http://www.daytimer.com. In addition, Microsoft Outlook and other e-mail programs offer excellent tools to help you organize and plan your time.
2. Set goals for the day, week, month, and year. If you don’t know what you expect to accomplish, you’ll never know if you get there. Write down the goals you want to accomplish every day in a to-do list; it’s a good idea to write down your goals at the end of the day for the next day.Doug Dvorak, “How to Use Time Management to Become a More Successful Sales Professional!” EzineArticles, ezinearticles.com/?How-to-Use- Time-Management-to-Become-a-More-Successful-Sales-Professional!&id=1081316 (accessed September 6, 2009). Invest fifteen minutes at the end of every day to plan for the next day.Jim Meisenheimer, “25 Ways to Get Motivated to Start Selling More,” EvanCarmichael.com, www.evancarmichael.com/Sales/407/25-Ways-To-Get-Motivated-To-Start-Selling-More.html (accessed August 19, 2009). Take the time to write down your goals for the coming week, Sunday night is a good time to do this. Be clear and realistic about what you want to accomplish and by when.
Time Is Money
(click to see video)
Hear how Andrew Sykes, pharmaceutical sales specialist whom you met in the video ride-along in Chapter 5, manages his time.
3. Prioritize your activities. Now that you have created your action plan, or to-do list, review it and reorder it to put the most important things first. Focus your time on the most important activities.“How Good Is Your Time Management?” Mind Tools, http://www.mindtools.com/pages/article/newHTE_88.htm (accessed September 6, 2009). Lee Iacocca, the former CEO of Chrysler, said it best: “If you want to make good use of your time, you’ve got to know what’s most important and then give it all you’ve got.Donald Latumhahina, “Time Quotes: 66 Best Time Management Quotes,” Life Optimizer Blog, March 8, 2007, http://www.lifeoptimizer.org/2007/03/08/66-best-quotes-on-time-management (accessed August 19, 2009). In other words, do important and challenging things first.
Sometimes people think it’s best to do a lot of small things first so that you can scratch them off your list. But it’s best to take on more challenging things when you are fresh and leave the smaller things for later or when you have a few minutes in your day.
4. Create a schedule. Using your to-do list as a guide, put times to your activities so that you can identify the amount of time it will take to accomplish each one. Also, during the day this schedule will serve as a guide and help keep you on track. And “manage minutes” effectively; use travel time, waiting time, and other downtime to return phone calls and e-mails or to think about solutions for customer problems.Margot Carmichael Lester, “5 Ways to Get and Stay Organized” SalesHQ.com, www.saleshq.com/training/articles/1353-5-ways-to-get-and-stay-organized (accessed September 6, 2009). Include addresses, phone numbers, and e-mail addresses for each person you need to contact if they are not already in your address book. This will help save time and prevent distractions.Jim Meisenheimer, “25 Ways to Get Motivated to Start Selling More,” EvanCarmichael.com, www.evancarmichael.com/Sales/407/25-Ways-To-Get-Motivated-To-Start-Selling-More.html (accessed August 19, 2009).
5. Delegate work to others. Although you might be working independently, chances are there are other people in the company that can help with certain activities. Clerical and administrative tasks should be delegated to your assistant or other support person. Always thank someone (subordinate, colleague, family member) who helps you get your work done.Doug Dvorak, “How to Use Time Management to Become a More Successful Sales Professional!” EzineArticles, ezinearticles.com/?How-to-Use- Time-Management-to-Become-a-More-Successful-Sales-Professional!&id=1081316 (accessed September 6, 2009). Delegation is the true secret to success: a survey conducted by Watson Wyatt shows that high-performing salespeople spend 30 percent less time on administrative tasks than the low performers.Doug Dvorak, “How to Use Time Management to Become a More Successful Sales Professional!” EzineArticles, ezinearticles.com/?How-to-Use- Time-Management-to-Become-a-More-Successful-Sales-Professional!&id=1081316 (accessed September 6, 2009).
6. Maximize selling time. Your objective should be to spend as much time as possible in front of customers; it’s practically impossible to sell anything if you’re not talking to and learning about your customers. Shane Gibson, “How to Operationalize Your Selling Strategy,” SalesHQ, www.saleshq.com/training/articles/1990-how-to-operationalize-your-selling-strategy (accessed September 6, 2009). That means that you will have to manage interruptions.
Sales Traffic School
(click to see video)
See how all these time management tips come together to maximize selling time in this video featuring George Ludwig speaking about the “sales traffic school.”
Link
How Good Are Your Time Management Skills?
Take the quiz by clicking on the link and learn about the areas of your time management that might need some focus.
http://www.mindtools.com/pages/article/newHTE_88.htm
Managing Your Results: Set Goals and Determine Your Income
Patricia Schneider, a former beauty queen, aspiring actress, and law firm clerk, started selling Mary Kay Cosmetics because she heard that the company awards a pink Cadillac for meeting certain goals. With 150,000 miles on her Toyota Celica, she decided that she could sell makeup. She got her first pink Cadillac in 2003; in 2009 she earned her fourth one.Lori Basheda, “May Kay Consultant Scores Fourth Pink Cadillac,” Orange County Register, August 18, 2009, www.bta.org/ (accessed September 6, 2009). Patricia learned that if you want to earn cars and money, go into sales.
• salary. This is a regular payment from your employer in exchange for your services. Salary is a set amount and is usually the same amount for every pay period, or interval of time for which you are paid. A pay period may be weekly, biweekly, monthly, or quarterly depending on the company and the position. Most companies have biweekly pay periods. In most sales positions, if salary is included as one of the components of the compensation plan, it is usually a small portion of the compensation. This allows the company to provide incentive to the salespeople with a greater opportunity to earn more money based on the amount of sales (or gross profit) generated but still provides some regular guaranteed income to the salesperson. In other words, salary doesn’t necessarily provide incentive for a salesperson to sell more since it is paid no matter what sales are generated.Jim Kahrs, “Sales Compensation: Creating a Plan that Works for Your Dealership,” Prosperity Plus Management Consulting Inc., www.prosperityplus.biz/ArticleSalesComp.html (accessed May 16, 2010). If a company pays salary, the salary usually makes up 15 percent to 40 percent of total compensation.“Sales Compensation Plan Components,” Online Business Advisor, www.onlinebusadv.com/?PAGE=178 (accessed August 19, 2009). The percentage of salary will be higher for new salespeople; whereas more experienced salespeople will earn a higher percentage of their compensation from commissions. For example, if total compensation is \$50,000, salary might be \$20,000 for a new salesperson (approximately 40 percent of total compensation); whereas an experienced salesperson might earn \$24,000 in salary but earn a total of \$120,000 including commission (20 percent of total compensation).
• commission. This is income that is based on the percentage of sales or gross profit generated. Commission is usually the largest portion of salesperson compensation. It is designed to be an incentive to the salesperson to sell more. This is one of the ways that salespeople have virtually unlimited income. Most sales jobs include some kind of commission element; others pay straight commission, which means that the salesperson makes only a percentage of what she sells without any guaranteed salary.“Sales Compensation Plan Components,” Online Business Advisor, www.onlinebusadv.com/?PAGE=178 (accessed August 19, 2009). Depending on the company, commission might be paid on sales dollars, on gross profit dollars, or as a percentage. Gross profit is the difference between sales generated and the cost of the product or service. Gross profit may be expressed as dollars or a percentage. Gross profit dollars are calculated by multiplying the gross profit percentage times the sales.Jim Kahrs, “Sales Compensation: Creating a Plan that Works for Your Dealership,” Prosperity Plus Management Consulting Inc., www.prosperityplus.biz/ArticleSalesComp.html (accessed May 16, 2010). When salespeople have control over pricing, commission plans are usually based on gross profit to ensure that the company makes a profit on each sale.“Sales Compensation Plan Components,” Online Business Advisor, www.onlinebusadv.com/?PAGE=178 (accessed August 19, 2009). For example, if a 15 percent commission is paid on sales of \$1 million, the income for the salesperson is \$150,000 (\$1,000,000 × 0.15 = \$150,000). If a 25 percent commission is paid on gross profit (the difference between the selling price and the profit) based on a 35 percent gross profit and \$1 million in sales, the commission would be \$87,500 (\$1,000,000 × 0.35) × 0.25 = \$87,500. This calculation is also shown in Figure \(6\).
While most sales positions include commission, some positions pay a combination of salary plus commission. This helps provide some steady income for a salesperson, especially during businesses that have peaks and valleys.
Do you want to earn enough money to drive a pink Cadillac, a BMW, or Mercedes? Or is your goal to buy a condo? Maybe you want to be able to travel to the islands during the winter or experience exotic locations around the globe. All these can be possible in sales because how much money you earn every year is usually up to you. It’s not too good to be true; it’s the reality of sales. The lifeblood of every company is its sales force, those people who connect to customers and generate sales for the company. That’s why most selling jobs provide at least some portion of compensation—money and benefits received in exchange for providing services to a company that is based on performance. Compensation may include one element such as salary or several components including salary, commission, bonus, benefits, and more. To understand how compensation works in sales, it’s important to know the terms.
Generally, it takes time for a new person to build up a customer base and begin earning higher commissions. For more detail about how commissions are calculated, follow this link:
http://compforce.typepad.com/compensation_force/2009/01/sales-commission-calculation-basics.html
• draw. This is an advance against future commissions or bonuses. Earning a draw provides steady income to the salesperson, especially if commissions are paid on a monthly or quarterly basis.Jim Kahrs, “Sales Compensation: Creating a Plan that Works for Your Dealership,” Prosperity Plus Management Consulting Inc., www.prosperityplus.biz/ArticleSalesComp.html (accessed May 16, 2010). For example, a salesperson might be guaranteed a draw of \$2,000 per month; if the salesperson earns more than \$2,000 in commissions she makes whatever she earns. If she earns \$1,500, she is paid \$2,000 for the month. In some cases, the \$500 shortfall would be deducted from future commission earnings; this is called recoverable draw. On the other hand, a nonrecoverable draw means that a shortfall in commissions earned would not be owed to the company. Generally, a draw is designed to provide an income to a salesperson while he is building his customer base.“Sales Compensation Plan Components,” Online Business Advisor, www.onlinebusadv.com/?PAGE=178 (accessed August 19, 2009).
• bonus. This is an incentive paid to sell a particular product or service or to reach a specific sales goal. Bonuses are paid in addition to salary and commission. They are usually paid quarterly but may be paid monthly.“Sales Compensation Plan Components,” Online Business Advisor,www.onlinebusadv.com/?PAGE=178 (accessed August 19, 2009). Bonuses can be a significant portion of total compensation, depending on the industry and company. For example, in primary care pharmaceutical sales, a bonus can be between \$20,000 and \$25,000 and as high as \$50,000.Cory Nahman, “Frequently Asked Questions Regarding the Profession of Pharmaceutical Sales Representative,” http://www.coreynahman.com/how_many_products.html (accessed August 19, 2009).
Video Clip
Who Wants to Be a Millionaire?
Imagine making \$1 million a year. Irina Haydon, executive director of sales and service at Heartland Payment Systems, shares what it takes to make \$1 million a year.
http://www.sellingpower.com/content/video/index.php?mid=150
Plan to Earn
Now you can see why managing yourself, managing your time, and understanding compensation plans are so important to success in sales. All these elements are linked to the company’s goals, which ultimately determine your sales goals. For example, if the company is planning a 6 percent sales increase for the year, each salesperson is responsible for delivering a certain portion of that increase. Since some salespeople are new, their goals will undoubtedly be less than those salespeople who have been at the company long enough to develop customer relationships and steady income streams.
The most important aspect of sales is to understand your sales goals: exactly what is expected and by when. Most companies establish annual sales goals or quotas, expectations of sales for a specific time frame, by salesperson, and then break down the goals by month and sometimes by week. Establishing specific, measurable, actionable, realistic, and time-bound (SMART) sales goals (covered in Chapter 8) provides a clear set of expectations for the salesperson and the company. For example, a SMART sales goal is “to increase dollar sales of accounting software with current customers by 8 percent by December 31, 2011.”
When this goal is broken down by month and by week, it provides a way to measure progress regularly. More important, the SMART goal provides a method by which to have a regular conversation with your sales manager to discuss how to remove barriers or gain access to additional resources to achieve the goal. SMART goals become the basis of sales quotas.
Since many business-to-business (B2B) sales have a long sales cycle, many companies use key performance indicators (KPIs) to help gauge the productivity of each salesperson. KPIs might be compared to miles per gallon; they are a measure of efficiency and effectiveness. So while sales or gross profit might be included in a SMART goal, KPIs provide insights into performance; they can act as a way to diagnose problems in the selling process. KPIs are used to evaluate performance and compensation. Results are how salespeople are evaluated and paid. If a salesperson is not generating the desired results, chances are he won’t last long in his position at that company.
KPIs may be organized by type of goal—performance KPIs or conversion KPIs. Performance KPIs are those that include outcomes such as sales, new accounts, units sold, or gross profit percent. Conversion KPIs are used as a measure of a salesperson’s productivity or efficiency but do not have outcomes. Examples of conversion KPIs are sales per customer or closing ratio.Baron A. Weitz, Stephen B. Castleberry, and John F. Tanner, Selling: Building Partnerships, 5th ed. (New York: McGraw-Hill, 2003), 397. The following are some performance KPIs and conversion KPIs that are commonly used to measure the effectiveness of salespeople.
Performance KPIs
• Sales quota (sales goal). Expected sales volume to be generated in a specific time frame; salespeople are usually given quotas by day, week, month, quarter, and year, which may be used as the basis for compensation and sales incentives.
• Sales versus quota. Sales generated compared to the sales goal or quota by the salesperson during the designated time frame; when a salesperson falls short of his sales goal, it is an opportunity for improvement.
• Gross profit. Difference between the cost of the product and the selling price.
• Number of new accounts. Number of customers who were not doing business with the company during the prior period.
Conversion KPIs
• Sales per customer. Total sales generated by the salesperson divided by the number of customers; high sales per customer indicates sales rep productivity.
• Sales per employee. Total sales generated divided by the number of employees at the company; high sales per employee indicate a productive sales force.
• Customer penetration. The percentage of a customer’s business (in total dollars and across product lines) that is being done with the salesperson; high penetration usually indicates a productive salesperson (and usually a good relationship with the customer).
• Cost per sale. The cost of generating the sale (cost of sales rep compensation, travel and entertainment, marketing materials, promotional discounts, and other expenses); low cost per sale usually indicates a productive salesperson who is able to close the sale quickly and at a higher gross profit (and, therefore, lower cost).
• First appointment-to-proposal ratio. The number of days it takes after a first appointment with a prospect until a proposal is made; a low number of days usually indicates a salesperson who moves quickly on an opportunity.
• Closing ratio. The percentage of times that a salesperson coverts a prospect to a customer by making a sale; a high closing ratio usually indicates a productive salesperson.Jeff Hardesty, “Setting and Exceeding Sales Goals through Key Performance Indicators (KPIs),” UnArchived Articles, June 14, 2006, articles.webraydian.com/article312-Setting_and_Exceeding_Sales_Goals_through_Key_Performance_ Indicators_KPI.html (accessed August 19, 2009).
• Call cycle. The frequency at which a salesperson calls on a customer (e.g., once every twenty days); call cycle will vary depending on the size and potential of the customer; a shorter call cycle indicates that there is more contact with the customer.
• Call-to-sale ratio. The percentage of calls that result in a sale; a low call-to-sale ratio usually indicates a productive salesperson.
Your sales manager will undoubtedly set quotas for you for many KPIs based on the goals of the company. Sales goals or quotas are used by companies “to align sales force performance to the business plan.”Renee Houston Zemanski, “Tough Truth about Quotas,” Selling Power 22, no. 6, http://www.sellingpower.com/content/article.php?a=5998 (accessed March 16, 2010). In many instances, sales quotas are used as the basis of incentives, such as additional commission, cash, and other incentives. You can use KPIs to set your goals for your annual income and see what it will take to make your earnings goal a reality. See Table 14.1 for this example.
Assume you wanted to make \$45,000 in a year and you are paid a \$500 commission on every sale. What will it take to earn your target income? Do the math below.Baron A. Weitz, Stephen B. Castleberry, and John F. Tanner, Selling: Building Partnerships, 5th ed. (New York: McGraw-Hill, 2003), 397.
Table \(1\): Goal Setting
KPI Name Calculation KPI Goal
Annual earnings \$45,000
Commission per sale
Number of sales
Earnings ÷ commission per sale
\$45,000 ÷ \$800
57
Closing ratio 10%
Number of prospects
Sales × number of prospects per sale
50 × 10
570
Number of prospect calls
Number of prospects × number of calls per prospect
570 × 2.5
1,425
Average number of sales calls per month
Number of prospect calls divided by 12
1,425 ÷ 12
119
Average number of sales calls per week
Number of monthly prospect calls divided by 4
119 ÷ 4
30
Average number of sales calls per day
Number of weekly prospect calls divided by 5
30 ÷ 5
6
Set Your Goals
It might seem a little overwhelming to think about achieving a specific sales goal. But it’s easier than you think when you use these tips of the trade to help you plan:
• Write down your goals. Believe it or not, you actually increase your chances for success when you put your goals in writing. Whether you are setting goals for your career, for the year, or for the day ahead, write them down and prioritize them.
What Can You Do in Twenty-Four Hours?
(click to see video)
Listen to author and selling expert Brian Tracy talk about what you can accomplish when you write down your goals and commit to achieving them.
• Understand what it takes to achieve your goal. If your goal is to generate a 10 percent increase in sales over last month’s sales, do the math and determine what that means in dollar sales, then determine how many sales calls you will have to make to achieve your goal. See Table 14.1 for an example.
It’s All in the Numbers
(click to see video)
Watch this video to see why life is a numbers game.
• Schedule success. Once you determine how many sales calls you will need to make to achieve your goal, plan your schedule so you plan the time it takes.
• Track your progress. Track your daily progress against your goal and make adjustments where necessary.
• Stay focused. It’s easy to lose focus, especially if things aren’t going according to plan. Review your plan and see where you have opportunities and start each day with determination to reach your goal.“How to Exceed Monthly Sales Targets,” eHow, www.ehow.com/how_2252974_exceed-monthly-sales-targets.html (accessed August 19, 2009).
Video Clip
How Measuring and Metrics Drive Success
Learn more about how metrics and measurements can help you achieve your goals in this video.
http://www.sellingpower.com/content/video/?date=7/30/2009
Key Takeaways
• Companies want to hire A-players for their sales positions, people who can connect with the customer and help the company achieve its goals.
• Resources such as ride-alongs, your sales manager, CRM, and other technology tools can help you learn more about the company, especially during the onboarding period.
• Territory management is the practice of managing your customers in a geographic area or territory; you determine whom you call on and when you call on them to minimize your travel time and maximize your selling time.
• Time management is the practice of organizing and prioritizing your activities to ensure that you can achieve your goals. This is especially important in sales because your goals can only be achieved by maximizing your selling time.
• Compensation can include many elements such as salary, commission, draw, bonus, and more. Commission is designed to provide incentive to the salesperson to increase her income by achieving and exceeding the sales goal.
• Key performance indicators (KPIs) provide a roadmap to improve performance and achieve sales goals.
Exercise \(1\)
1. Identify three resources that are available to you through your school. How do these resources help you succeed? Do you use these resources? Why or why not?
2. Go on the “virtual ride-along” with a medical device salesperson by reading this article: www.e-shadow.com/an-interview-with-a-medical-device-salesman. Discuss three things that you learned about the role of a medical device salesperson. Does this position appeal to you? Why or why not?
3. Think about a situation in which you have gone through an onboarding process. What information or resources were available to you to help you become familiar with your new environment? How did you learn about these resources? Did you use these resources after you learned your way around?
4. Visit your campus Student Services Center (or similar office) and ask about the availability of a time management seminar (most schools offer them to students at various times of the year at no cost). Attend the seminar and identify at least two new habits that you will implement into your personal time management process.
5. Using the concept of the “Sales Traffic School” discussed in the video clip in Section 1.5, which of the following activities would you classify as “red,” “yellow,” or “green”? Indicate your choices in the chart below.
Activity Color (Red, Yellow, Green)
Prospecting
Responding to customer e-mails
Attending internal meetings
Customer follow-up
Writing a proposal
Meeting a friend for lunch
Precall research
6. Assume you are a financial advisor and you want to earn \$7,000 a month. Based on earning \$1,000 per sale in commission and having a 10 percent closing ratio and an average of 2.5 calls per prospect, use the following form to determine how many sales, prospects, and calls you will need to make each month to meet your goal. Why did you choose the priority of each of your activities?
KPI Name Calculation KPI Goal
Monthly earnings
Commission per sale
Number of sales
Closing ratio
Number of prospects
Number of prospect calls
Average number of sales calls per month
Average number of sales calls per day
7. Assume you are a salesperson earning 10 percent commission and you have sold \$540,000 in products this year. What are your commission earnings for the year (show your math). Based on this, if you were on a draw of \$50,000, would you earn your draw or commission?
8. Assume you are territory manager for a health care insurance company. The activities listed in the table below need to be completed tomorrow. The time it takes to complete each activity is also included. Using the “Day Planner” below, plan your day by entering the activity in the time of day that you would use to get that activity completed. You may not have enough time to complete all activities so you will need to decide what activities will not get done (don’t forget to allow time for lunch).
Activity Comments Time to complete activity
Travel to and from sales call Prospect sales call 1 hour
Return call to boss Boss sent an e-mail and asked you to call him as soon as possible 15 minutes
Check e-mails and voice mails and respond as needed Check at least three times daily 15 minutes each time
Travel to and from customer call Key customer call 1 hour and 15 minutes
Travel to and from customer call Customer with low sales but high potential 1 hour
Paperwork Complete once daily 30 minutes
Customer follow-up Complete at least twice daily 15 minutes each
Urgent phone call Call comes in at 10:15 a.m. 30 minutes
Internal follow-up and meetings One meeting during the day 1 hour
Prospecting and qualifying Allow time at least once during the day 1 hour
Precall preparation for upcoming prospect call Prospect call is in one day 30 minutes
Write a proposal Proposal is due in two days 1 hour
Finish up proposal Proposal can go out as soon as it is finished 15 minutes
Day Planner
Time of day Activity Time of day Activity
8:00–8:15 12:00–12:15
8:15–8:30 12:15–12:30
8:30–8:45 12:30–12:45
8:45–9:00 12:45–1:00
9:00–9:15 1:00–1:15
9:15–9:30 1:15–1:30
9:30–9:45 1:30–1:45
9:45–10:00 1:45–2:00
10:00–10:15 2:00–2:15
10:15–10:30 2:15–3:00
10:30–10:45 3:00–3:15
10:45–11:00 3:15–3:45
11:00–11:15 3:45–4:00
11:15–11:30 4:00–4:15
11:30–11:45 4:15–4:30
11:45–12:00 4:30–4:45
4:45–5:00 | textbooks/biz/Marketing/The_Power_of_Selling/14%3A_The_Power_of_Learning_the_Ropes/14.01%3A_Introduction.txt |
Learning Objectives
• Learn how to stay motivated and expand your knowledge every day.
• Discuss how to stay positive with a healthy mind and body.
Never Give Up
It was Super Bowl Sunday in 2005, and the New England Patriots were playing the Philadelphia Eagles. Everything was perfect for New England Patriots linebacker Tedy Bruschi as he waited with excited anticipation before the game, visualizing its outcome. Talk about pressure—the Patriots had a chance to win their third Super Bowl. He played with his young sons, both under the age of five at the time, on the field hours before game time; it was great that his wife Heidi came to the Alltel Stadium with the boys early so that he had some time with them before the game. He was motivated by more than simply winning this historic game; he wanted to show his sons that you can do anything you set out to do.
The game, like the day, could not have been more perfect for Bruschi. He had a sack, seven tackles, and an interception. He was only one story in a team filled with winners. The final score was 24–21, Patriots. They had done it!
It was almost impossible to go to sleep that night. In fact, Bruschi didn’t get to bed until 4:00 a.m. and then was up at the crack of dawn (literally) to appear on Good Morning America. The next few days were nonstop celebrations, interviews, and photographs. Life was indescribably good.
On February 16, 2005, just ten days later, Bruschi awoke at 4:00 a.m. with a headache and numbness in his body that was so severe that he had to crawl to the bathroom. Seven hours later things went from bad to worse when his vision blurred and he could no longer move his arm or leg. At the hospital he learned that he had had a stroke.Tedy Bruschi with Michael Holley, Never Give Up: My Stroke, My Recovery & My Return to the NFL (Hoboken, NJ: Wiley & Sons, Inc., 2007).
After an agonizing recovery, eight months later Bruschi returned to the game he loved. He battled back to the field and played for four more seasons before retiring in August 2009. His teammate Larry Izzo said of Bruschi, “To come back from his stroke in ’05 and play four more seasons at such a high level, was nothing less than amazing. When I think of Tedy Bruschi, I think of his toughness, his courage, the passion and desire he played with, and his production. He made plays. He was a true warrior. The heart and soul of our team.”Karen Guregian, “Tedy Bruschi a Role Model to Pats, Fans,” Boston Herald, September 1, 2009, www.bostonherald.com/sports/football/patriots/view/20090901tedy_bruschi_a_role_model_to_pats_fans (accessed September 7, 2009).
What motivated Bruschi? It would have been enough just to survive a stroke, but to come back and play football at a professional level is almost unthinkable. Most players don’t ever make the cut to play in the NFL, but Bruschi survived a life-threatening stroke and came back to play at the top of his game. His passion, drive, and will to survive and win outpaced even his physical challenges.
Be a Rock Star
(click to see video)
Are you a rock star…or a top-performer wannabe? Listen to what it takes to be a rock star in this video featuring sales guru and best-selling author, Jeffrey Gitomer.
You Can Do It!
“Don’t Stop Believin’,” the number one hit from the rock band Journey in 1981, became the anthem for the Chicago White Sox throughout the 2005 season in which the team won the their first World Series championship after eighty-eight years.Mark Newman, “Soxabration: Reliving 2005,” White Sox, March 27, 2006, http://chicago.whitesox.mlb.com/cws/history/championship05.jsp (accessed January 3, 2010). The song was more than background music for the team; it became the promise to their fans…and themselves throughout the season.
Is being successful in sales as easy as having a great song or a catchy slogan? Not really. In fact, the White Sox were far from being the favorites at the beginning of the season. After all, eighty-eight years is a long time to go without a championship. No one really believed they could do it. But even when others might not see your vision, you have to believe in yourself even when things don’t go your way. Successful sales professionals will tell you that’s what it takes to make it in sales: an unwavering belief in yourself that you can achieve the goals you set. Hard work? Absolutely. Setbacks? Just about every day. Believing in yourself? As they say, priceless.Priceless.com, http://www.priceless.com/us/personal/en/index.html (accessed September 7, 2009).
Motivation is especially important in sales because you will hear no more than you will hear yes. Your motivation, goals, and drive to succeed will make you successful.
Believe and Achieve
(click to see video)
Listen to Lisa Peskin, sales trainer at Business Development University, talk about the need to believe in yourself, stay focused, and stay motivated in sales.
You can do it, but only if you believe you can. What is the difference between the salesperson who makes \$1 million a year and one that makes \$50,000? It is the belief in himself that he can achieve her goals. It starts with a positive mental attitude every day. That means making the most of every day and taking control of your plan to accomplish your goals. “Motivation is an inside job. It’s up to you,” according to sales expert and author Jim Meisenheimer.Jim Meisenheimer, “25 Ways to Get Motivated to Start Selling More,” EvanCarmichael.com, www.evancarmichael.com/Sales/407/25-Ways-To-Get-Motivated-To-Start-Selling-More.html (accessed August 19, 2009). Here are a few of his tips from his article “25 Ways to Get Motivated to Start Selling More”:
• Take pictures of your top ten customers and top ten prospects. Put the pictures in clear view (your computer wallpaper, on your cell phone, on your refrigerator) along with your SMART goal for each one. This visual reminder will help you stay focused.
• Tell your family how you are going to celebrate together when you become the number one salesperson at your company. You will make your goal real by telling someone close to you, and you will have their support to get through the challenging days.
• Invest fifteen minutes every day to read articles, books, blogs, listen to podcasts, or view videos about your industry or the selling profession. Sellingpower.com, Salesandmarketing.com, bnet.com, SalesHQ.com, and SalesVantage.com are all excellent sources of information about selling.
• Before the year ends, write yourself a check for December 31 of the next year for the amount you want to earn. Make three copies and have each laminated. Put one in your brief case, one on your car console, and one in your office. When you look at it every day, ask yourself, “What can I do today to get closer to this goal?”Jim Meisenheimer, “25 Ways to Get Motivated to Start Selling More,” EvanCarmichael.com, www.evancarmichael.com/Sales/407/25-Ways-To-Get-Motivated-To-Start-Selling-More.html (accessed August 19, 2009).
Link
Twenty-Five Ways to Get Motivated to Start Selling More
You can read the entire article by clicking on the following link:
www.evancarmichael.com/Sales/407/25-Ways-To-Get-Motivated-To-Start-Selling-More.html
Act Like You Run the Place
David C. Novak, chairman, CEO, and president of Yum Brands, whose chains include KFC, Pizza Hut, Taco Bell, and Long John Silver’s shares his advice for young people: “I tell people that once you get a job you should act like you run the place. Not in terms of ego, but in how you think about the business.”Adam Bryant, “You Win a Floppy Chicken,” New York Times, July 12, 2009, business, 2. In other words, if you think about your sales territory or product line as if it is your own business, you’ll make decisions that will be in the best interest of growth.
Fail...to Succeed
It may seem counterintuitive, but the best way to succeed is to fail. The fact is, failures can be a positive experience because they can help you avoid repeating mistakes.Stacy Blackman, “Want to Succeed? Learn How to Fail,” BNET, July 21, 2009, blogs.bnet.com/mba/?p=962 (accessed September 7, 2009). Since failures are much more painful than the sweet taste of success, we tend to remember our failures more vividly.Dave Kahle, “Learning from Failure,” American Salesman, February 2009, http://www.davekahle.com/article/learningfromfailure.html (accessed May 16, 2010). But as important as the actual failure is what you do as a result of the experience. “You don’t have control over what happens to you in life,” says Lisa Peskin, sales trainer at Business Development University, “but you absolutely have control over how you choose to handle it.”Lisa Peskin, “Top 10 Secrets of Selling in a Recession” Philadelphia Business Journal Workshop, Philadelphia, PA, July 29, 2009. Peskin has over twenty years of experience in business-to-business (B2B) selling. To overcome the feeling of failure especially on daily sales calls, she suggests the “rocking chair test”: will you remember that someone said no to you today when you are sitting in a rocking chair fifty years from now? “Don’t get upset over the small stuff” is her advice to salespeople. “If you want something you never had, you must do something you’ve never done, and that may result in some failures, but a lot of successes.”Lisa Peskin, “Top 10 Secrets of Selling in a Recession,” Philadelphia Business Journal Workshop, Philadelphia, PA, July 29, 2009.
It might be hard to imagine that successful people ever had failures. But Shantanu Narayen, CEO of software maker Adobe Systems, says, “You know, there is no such thing as failure. You’re always learning.” He goes on to share his personal experiences: “I have looked back at aspects of my career where somebody might look at it and say, you know, that start-up was not successful, and I look at it and I say, ‘I learned how to build a team, how to raise money, how to sell a vision, how to create a product.’ It was a great steppingstone for me.”Adam Bryant, “Connecting the Dots Isn’t Enough,” New York Times, July 19, 2009, business, 2.
Failure is a fact of life. Although the White Sox were eventually named World Champions again in 2005, each member of the team missed more balls than they hit. In baseball, a 0.333 batting average is considered outstanding (Ty Cobb’s average, the highest in baseball is 0.366), which means that the batter misses almost seven times out of every ten at bats. Similarly, an average of 70 percent of people who walk into a retail store don’t buy anything, and 99 percent of people who visit a company’s Web site don’t make a purchase.Baseball Almanac, “Career Leaders for Batting Average,” http://www.baseball-almanac.com/hitting/hibavg1.shtml (accessed September 7, 2009).,Amanda Ferrante, “Retailers Counting on Conversion to Drive Store Metrics,” Retail Store Ops Blog, March 17, 2008, http://retailstoreops.blogspot.com/2008/03/retailers-counting-on-conversion.html (accessed September 7, 2009).,The Conversion Chronicles, www.conversionchronicles.com/, September 7, 2009 (accessed May 16, 2010). So, it is inevitable that you will have to fail in order to succeed. But that doesn’t mean that failure should become a way of life. With failure comes personal responsibility, acknowledging and accepting that you are accountable for the choices you make with your prospects and customers, in your career, and in life. Someone who is personally responsible doesn’t rationalize why a failure occurs, doesn’t blame others, and doesn’t feel sorry for himself. Here are four simple steps that can help you turn failure into growth:
1. Objectively analyze your role in the failure. What did you do that may have caused an outcome other than the one your preferred?
2. Imagine if you had done something different. What impact would it have had on the outcome?
3. Determine what prompted you to take the actions you took.
4. Decide that when the situation occurs again, you will do something differently.Dave Kahle, “Learning from Failure,” American Salesman, February 2009, http://www.davekahle.com/article/learningfromfailure.html (accessed May 16, 2010).
Failure is about learning and taking personal responsibility, which can be the key to your personal success. “The price of greatness is responsibility,” said Winston Churchill.Wayne Mansfield, “Seven Tips for Handling Stress in Challenging Times,” Article Dashboard, www.articledashboard.com/Article/7-Tips-for-Handling-Stress-in-Challenging-Times/612133 (accessed September 8, 2009).
Link
Rate Your Personal Responsibility
You can rate yourself on a personal responsibility scale to identify if you have areas in which you need to develop personal responsibility.James Messina, “Accepting Personal Responsibility,” LIVESTRONG.COM, November 18, 2009, http://www.livestrong.com/article/14698-accepting-personal-responsibility (accessed September 9, 2009).
http://www.livestrong.com/article/14698-accepting-personal-responsibility
Unfortunately, you are going to hear no more often in sales than you hear yes. In fact, no is part of the game of sales. But don’t take it personally. “Don’t get dejected when you’ve been rejected—just get your skills perfected,” is advice from selling expert and author Harvey Mackay.Harvey Mackay, “8 Tips for Handling Rejection,” WMAR-ABC2, July 5, 2009, www.abc2news.com/content/financialsurvival/yourjob/story/8-tips-for-handling-rejection/dM-Sg9DHiEaJcmqMgDp44w.cspx (accessed September 9, 2009). No is what helps you hear yes. Of course, you wouldn’t expect every prospect you contact to buy your product or service. Think about it: do you buy everything that is pitched to you? So it is hearing no that helps you fine-tune your sales presentation to ultimately hear yes.Hal Becker, “Become a Pro at Dealing with Rejection, and You’ll Win More often at the Sales Game,” Kansas City Business Journal, March 4, 2005, http://kansascity.bizjournals.com/kansascity/stories/2005/03/07/smallb6.html (accessed May 16, 2010). There should be no fear in no.
Positive Reinforcement
(click to see video)
Want to stay motivated to keep going even when you hear no? Watch this video about being positive and being creative featuring sales guru Jeffrey Gitomer:
Power Player: Lessons in Selling from Successful Salespeople
The Eyes Have It
“Sell with your eyes” is the advice that Jessica Sciarabba, AT&T retail sales consultant, gives to all sales reps. “The best piece of advice about how to have a successful career in sales came from my first boss. He taught me how to make a personal connection with customers by looking at them and showing my interest in them with my eyes. It makes a difference and it really works.”Conversation with Jessica Sciarabba at AT&T store in King of Prussia, PA, August 26, 2009.
Positive Energy from a Healthy Mind and Body
It’s virtually impossible to be successful in sales and in life if you don’t take care of yourself. Conflicting priorities; lack of time; demands of work, family, and friends; the negativity of some people; and even the state of the economy can take a toll on you. Stress is a real part of everyday life; unfortunately there is no magic formula to avoid it. But you can learn to balance work and your personal life for a better balance and potentially less stress.
Make Time for Yourself
(click to see video)
Listen to how Rachel Gordon, account manager at WMGK, balances work and her personal life to be a better salesperson.
Take Good Care of Yourself
Have you ever been on an airplane and listened to the directions from the flight attendant about safety? She says that in case of an emergency, put on your oxygen mask first and then help those around you to put theirs on. The theory is that you can’t really help anyone else until you are taken care of. That same theory applies to mental, emotional, and physical health. You won’t be able to provide support and ideas to your customers unless you are healthy in mind and body. To start off every day with the energy and enthusiasm to conquer the world, take the time to take care of yourself with the following tips:
• Get a good night’s sleep. Avoid caffeine in the late afternoon and evening, and go to sleep early. It’s best not to assume that your current sleep time is enough; experiment with what is your optimum night’s sleep.Donald Latumahina, “How to Get Your Morning Off to a Great Start,” Life Optimizer Blog, July 28, 2009, http://www.lifeoptimizer.org/2009/07/28/how-to-get-your-morning-off-to-a-great-start (accessed August 19, 2009).
• Eat a healthy breakfast. You might think you don’t have time, but it’s best to make time to eat breakfast. It provides fuel to start your day. A healthy breakfast can be fast and easy, especially if you plan it the night before. Oatmeal with almonds, cold cereal with fruit, a smoothie with fruit, low-fat yogurt and wheat germ, and even cold veggie pizza can be a healthy breakfast. You owe it to yourself to start right. Use some tips from this article from the Mayo Clinic for some healthy options: www.mayoclinic.com/health/food-and-nutrition/NU00197.Mayo Clinic Staff, “Healthy Breakfast: Quick, Flexible Options to Grab at Home,” MayoClinic.com, www.mayoclinic.com/health/food-and-nutrition/NU00197 (accessed September 8, 2009).
• Exercise. It’s a good idea to do some kind of regular physical exercise such as working out or walking. This helps reduce stress and helps you manage weight.
• Don’t procrastinate; get right into your day. Jump in with both feet and start your day off right. If you stop to watch television, text your friends, or do other activities, your morning will be half over before you know it with nothing to show for it.Donald Latumahina, “How to Get Your Morning Off to a Great Start,” Life Optimizer Blog, July 28, 2009, http://www.lifeoptimizer.org/2009/07/28/how-to-get-your-morning-off-to-a-great-start (accessed August 19, 2009).
• Start your day with the most important task. Use your prioritized to-do list and conquer the most important item first; leave everything else for later. You might be tempted to do the small things first so you can scratch them off you list. But when you take on the more important and biggest challenge first, you have the most amount of energy and drive. And you will have a huge sense of accomplishment with time to spare for your smaller to-dos.Jonathan Figaro, “A Simple Tip to Be More Productive,” Life Optimizer Blog, July 25, 2009, http://www.lifeoptimizer.org/2009/07/18/be-more-productive (accessed August 19, 2009).
• Smile and enjoy your day. Your attitude sets the tone for your customers. Always wear a warm, genuine smile; people like to do business with people who enjoy what they do. A smile can break tension in a meeting and put everyone at ease.Diane Gray, “Some Sales Tips from the Best Salespersons,” Associated Content, March 14, 2007, www.associatedcontent.com/article/166636/some_sales_tips_from_the_best_salespersons.html?cat=35 (accessed August 19, 2009).
• Don’t worry about what you can’t control. Worry causes stress, and stress breeds doubt. It’s a good idea to make a list of all the things that are causing stress in your life. Review them and identify which ones you have control over and develop an action plan to take control over that element. For those over which you don’t have control, don’t stress. It’s simply a waste of time to worry about things you can’t control.Tim W. Knox, “Can You Handle the Stress of Running a Business?” Entrepreneur, October 6, 2003, http://www.entrepreneur.com/startingabusiness/startupbasics/askourideasandinspirationexpert/article64824.html (accessed September 8, 2009).
• Take time for yourself. Chances are you have multiple responsibilities and demands including work, family, school, community, and others. Even though you want to do it all, it’s a good idea to schedule some time for yourself regularly. Do something that you enjoy whether it is reading a book, going to the mall, or taking a bike ride. This “me time” can go a long way to rejuvenate yourself and refresh you to take on the next big challenge.
Be Inspired
(click to see video)
The bottom line is you can be successful in your career. It’s up to you. This video will inspire you.
Key Takeaways
• You have to believe in yourself so that others will believe in you.
• Expand your knowledge every day by reading an article, part of a book, or blog about selling or the industry you are selling in.
• Failure can be one of the best ways to succeed; it teaches you how to avoid the same mistakes twice.
• Personal responsibility is a key element in success.
• You can create positive energy by taking care of yourself in mind and body.
Exercise \(1\)
1. Think about a situation in your life that seemed almost impossible. What did you do to overcome it? How did you determine your course of action? How long did it take? How did you stay motivated?
2. Identify at least one goal that you would like to accomplish in your life. How will you achieve it? What are the steps you plan to take?
3. Assume you are a designer and you want to sell your new designs for home furnishings called “The Ultimate Dorm Room” to Target. You met with the buyer once, and she made some suggestions and said she would consider it again in six months. What would you do to continue to stay motivated until you meet with her again?
4. Identify at least three resources for information about the selling profession (hint: several are mentioned in the chapter). Read, watch, or participate in them at least once a week for the next three weeks. What did you learn? Will you continue to use these as resources? Why or why not?
5. Contact a B2B salesperson and ask him what he does to stay motivated. Which pointers might you find helpful? Why? Which pointers don’t you find helpful? Why not?
6. Discuss a group situation in which you were involved and someone did not take personal responsibility. What impact did the lack of personal responsibility have on the team in achieving its goal? What might the outcome had been if the person had taken personal responsibility?
7. Watch this video and discuss three things that can make a failure into a learning experience: http://www.sellingpower.com/content/video/index.php?mid=309. | textbooks/biz/Marketing/The_Power_of_Selling/14%3A_The_Power_of_Learning_the_Ropes/14.03%3A_Motivation_Learning_Enjoyment_Success.txt |
Learning Objectives
• Understand how to leverage internships and professional organizations to get the job you want.
When Jay Leno was young, he saw a Mercedes/Rolls-Royce dealership in his hometown in Boston and thought it would be a great place to work given his passion for cars. When he applied for a job the manager responded with the usual: “We’re not hiring right now.” Leno was undaunted; the next Monday morning he returned and went to the car wash bay. He told them he was the new guy and started washing cars. A few days later, the manager saw him and said, “What’s he doing here?” The head of the car wash team said he was a hard worker; Leno said that he figured he would work there until he got hired. Needless to say, he got the job.Jay Leno, “Jay Leno Says: Persistence Pays Off,” Parade, September 6, 2009, www.parade.com/celebrity/2009/09/jay-leno-persistance-pays-off.html (accessed May 16, 2010).
This same “can-do” positive attitude and willingness to work can help you get the job you want. Even in this difficult economy, there are opportunities to demonstrate your passion and skills and set yourself apart, just like Jay Leno did. Finding the right job requires focus, time management, and motivation…and sometimes even working for free. You have to keep a positive mental attitude throughout your search and manage your time to gain experience while you are going to school. Here are two key things that you can do every day to help you get the job you want: internships and professional organizations.
Build Your Résumé with Internships
“One hundred percent of the students I hire have had internships,” says Michelle Goza, a campus recruiter for Gap. “It’s foolish not to pursue the opportunity.”“Make the Most of Your Internship,” WetFeet, www.wetfeet.com/Undergrad/Internships/Articles/Make-the-Most-of-Your-Internship.aspx (accessed September 8, 2009). Paid or unpaid, internships can make a difference in whether you are considered for another internship or the job you want.
An internship, or more than one internship over the course of your academic career, can provide insight into an industry or a specific company or position. What better way to learn about something you might want to do during your career…or not want to do. (Some internships teach you what you don’t want to do during your career, which is as valuable as learning what you want to do.) Internships are almost the norm today, and many employers expect to hire recent grads for entry-level positions who have had some kind of internship. “There is no such thing as too much experience, just not enough,” says Craig Bollig, a journalism major at the University of Wisconsin, Oshkosh.Anya Kamenetz, “Take This Internship and Shove It,” New York Times, May 30, 2006, www.nytimes.com/2006/05/30/opinion/30kamenetz.html (accessed September 11, 2009).
Internships aren’t just for undergraduates any more. Due to the challenging economy, recent graduates are finding internships to be an excellent way to build their experience while they continue to look for the full-time job of their dreams. “The need for experience is always growing and one internship may not cut it like it did before,” adds Bollig.Craig Bollig, “Maybe One Is Not Enough,” Internships for Dummies Newsletter, Spring 2009, www.uwosh.edu/journalism/docs/LowRes09.pdf (accessed September 8, 2009).
Get Experience
(click to see video)
Andrew Sykes, pharmaceutical sales specialist at AstraZeneca, shares his advice on getting experience.
If internships are so important to building your résumé and your experience, you might be wondering how you go about getting the right internship. First, stop by your campus career center. The people there are skilled at helping you understand the options that are available and can provide insight as to how to find the right internship to help you meet your career objectives. And most campuses include internship postings on the campus Web site. It’s a good idea to take the time to learn all about your internship options.
Consider an internship the same way you would consider a job: Is this the right fit for my experience and skills? Are the company values in line with my personal values? What will I be doing? Who will I report to? Will I be paid for the internship? If so, how much? How will I be evaluated during the internship? What is the possibility of getting a full-time position after graduation? While the interview process is usually more abbreviated for an internship than for a full-time job, take the time to ask questions so you understand the expectations of the role.
You’ve Got the Power: Tips for Your Job Search
Internship Offer Letter
Before accepting any internship (or full-time job), it’s best to get an offer letter.Kim Richmond, Brand You, 3rd ed. (Upper Saddle River, NJ: Pearson Prentice Hall, 2008), 210. Even if you are accepting an unpaid internship, an offer letter outlines your areas of responsibility, dates of employment, and other specific information that you should agree upon with your employer before you begin. See the Selling U section of Chapter 12 for more specifics about offer letters.
The Right Internship for You
Internships come in all shapes and sizes; some are formal, structured programs while others are created by the student. Either way, here are some considerations when you are pursuing an internship:
Paid versus unpaid. Some, but not all, internships include a paycheck. Many large corporations have structured internship programs that include paid internships. However, many industries such as advertising, entertainment, and public relations offer unpaid internships as a way for you to gain experience. So you want to make some money and gain some experience in the field of your choice? You might be able to do both since approximately half of all internships are paid. But in today’s challenging job market, you might find it worth your while to accept an unpaid internship. While that might be a tough pill to swallow, consider Lindsey Roberts’s point of view after she graduated with an MBA from a top business school, “I could either sit at home all day and drive myself nuts going from company websites, to Indeed.com and back to Gmail and Facebook, or I could get out there, put my education and experience to work while I continued my job search.”Lindsey Roberts, “A Millennial’s View on Cost/Benefits of an Unpaid Internship Post-MBA,” Millennial Marketing, August 10, 2009, http://millennialmarketing.com/2009/08/a-millennial’s-view-on-costbenefits-of-an-unpaid-internship-post-mba (accessed September 11, 2009). It’s not only graduate students who are accepting unpaid internships. According to the Wall Street Journal, the recession has tightened the internship market; experienced workers who have been laid off are now successfully participating in unpaid internships for the same reasons students do: to build their résumés and increase their chances for full-time work.Sarah H. Needleman, “Starting Fresh with an Unpaid Internship,” Wall Street Journal, July 16, 2009, http://online.wsj.com/article/SB10001424052970203577304574280201046918712.html (accessed September 11, 2009). So you might think twice about holding out for a paid internship. Before you make a decision about a paid versus an unpaid internship, put your internship consideration to the test. The right internship, paid or unpaid, allows you the opportunity to get experience, a chance to network, and the opportunity to test drive a job to see if it’s something you like to do.Rich DeMatteo, “3 Reasons to Take an Unpaid Internship,” Corn on the Job Blog, July 23, 2009, http://cornonthejob.com/2009/07/23/3-reasons-to-take-an-unpaid-internship (accessed September 11, 2009). Those three things can be well worth temporarily foregoing a paycheck.
Credit versus noncredit. In many cases, you may be able to earn credit hours for your internship. Begin by visiting your faculty advisor and learn about the requirements to earn credit for internships. Internships usually require a faculty mentor or other university liaison. Besides working the specified number of hours each week, students are usually required to write a paper. In addition, an employer evaluation is usually included in the grade. It’s best to understand the requirements for credit long before the internship is finished so that you can be sure that all the details are in order between your employer and the school. Be sure to fill out all the appropriate paperwork before the beginning of the internship. Just a note, although the internship does not usually require a textbook or classroom learning, most schools include the standard course fee for a credit internship.
It doesn’t matter if your internship is paid or unpaid, for credit or noncredit, the payoff can be significant. Internships are like an extended job interview; the company gets to know you and your work, which could result in a full-time job offer after graduation. “Internships are so powerful,” affirms Wendy Washington, a senior vice president at Universal Records, whose assistant is a former intern for the company. “We get our employees from our intern pool. They know the system. They know how things work, and you can’t get a better character reference. Interns who work for our company have a better shot of becoming employed here than someone who just sends in an application.”“Getting the Big Break with the Right Internship: You Can Beat the Odds and Become a Success in the Entertainment Industry,” TheFreeLibrary.com, http://www.thefreelibrary.com/Getting+the+big+break:+with+the+right+internship,+you+can+beat+the...-a094672526 (accessed September 11, 2009). That’s why it’s especially important to stand out in everything you do. And don’t forget to keep copies of the projects you work on; they are excellent examples of your work to include in your portfolio and serve as a demonstration of your on-the-job experience.“Make the Most of Your Internship,” WetFeet, www.wetfeet.com/Undergrad/Internships/Articles/Make-the-Most-of-Your-Internship.aspx (accessed September 8, 2009).
Internship “How To”
(click to see video)
This video provides an overview of how to get an internship.
The Best Places to Look for an Internship
Your campus career center and faculy advisor can be the best resources for getting an internship. In addition, there are several Web sites that can give you access to internships by industry or by geography. Table \(2\) provides some places for you to start your search.
Table \(2\): Recommended Web Sites for Internships
Web Site Comments
Buddingup.com
buddingup.com
Internships, entry-level jobs, co-ops
GoAbroad.com
http://www.internabroad.com/search.cfm
International internships
InternWeb.com
http://www.internweb.com
Internships and entry-level jobs in technology industries
InternSHARE
http://internshare.com
Internship listings with ratings and reviews
Internshipprograms.com
http://internshipprograms.com
Internship listings
Internshipratings.com
http://www.internshipratings.com
Internship ratings including best and worst internships
Vault
www.vault.com/wps/portal/usa
Internship listings
TheFreeLibrary.com
http://www.thefreelibrary.com/Getting+the+big+break:+with+the+right+internship,+you+can+beat+the...-a094672526
Monster.com
www.monster.com/internships-entry-level-college-jobs.aspx
Internship listings
Summerinternships.com
http://www.summerinternships.com
Internship listings
Idealist.org
www.idealist.org/if/as/Internship
Nonprofit internship listings
Internzoo.com
www.internzoo.com
Internship listings
Link
Companies That Hire the Most Interns
Don’t forget to apply directly to companies for internship opportunities. Enterprise Rent-A-Car, Walgreens, and General Electric are among companies on the Forbes list of “U.S. Companies That Hire the Most Interns.”Joyce Lee, “U.S. Companies that Hire the Most Interns,” Forbes, July 7, 2009, http://www.forbes.com/2009/07/07/biggest-intern-companies-leadership-careers-jobs.html (accessed September 11, 2009). This is an excellent source of possible internships in addition to local companies.
http://www.forbes.com/2009/07/07/biggest-intern-companies-leadership-careers-jobs_slide_9.html?thisSpeed=15000
Professional Organizations: Your Key to Growth
You’re probably really pressed for time this semester. A full course load, your job, your community service work, family, friends—it seems like you can’t possibly fit in another thing to do. You already have twenty-eight hours for every twenty-four-hour day. Just when it seems like you can’t do another thing, there is one thing you should consider: it’s worth making time to join a professional organization while you’re in school, then continue as a member after you graduate. It can build your experience and enhance your résumé.
There are most likely several professional organizations on your college campus. organizations such as the American Marketing Association, American Association of Advertising Agencies, American Society of Women Accountants, Sales & Marketing Executives International are just a few that may have a chapter on your campus called a collegiate chapter. If you’re not sure about what professional organizations are available on campus, you might consider visiting your campus student services center or career center. The people who work there will likely have information about the purpose of each group; date, time, and place of the next meeting; and more. In addition, many professional organizations may not have a collegiate chapter, but they offer student membership into their organization at reduced rates. It depends on the organization whether they offer a collegiate chapter or a student membership. Collegiate chapters are usually extensions of national organizations such as the American Marketing Association, and usually have student members on campus and hold regular events and activities on campus while leveraging and participating in national conferences, competitions, and best practices. Professional organizations that do not have a collegiate chapter but offer a student membership rate usually have events and activities with professionals in the local area but not on campus.
Being a member of a professional organization in school helps build your professional network, and hone your skills. It’s also a great résumé-builder because it signals to your prospective employer that you are willing to take the time to get involved in a business function during your spare time (of which you probably have none). Participation in a professional organization can help make you stand out as a candidate or help you meet the right people. In fact, some professional organizations, such as the Philly Ad Club, offer a formal mentoring program. This is an excellent way to meet an executive in the industry of your choice.Philly Ad Club, “Philly Ad Club Mentor Program,” May 5, 2009, http://www.phillyadclub.com/news_article.php?id=1973 (accessed September 12, 2009). The networking aspects of professional organizations are well documented and are covered in Chapter 3. The reason for the existence of most professional organizations is to promote the health and advancement of the industry and bring people together for networking purposes. In addition, most professional organizations offer a newsletter that includes information about the industry and companies that can be very helpful for job leads and interviewing research.Sally Kearsley, “It Pays to Join a Professional Association,” www.jobscareers.com/articles/developingyourcareer.html (accessed September 12, 2009).
You might think that simply joining a professional organization is enough. However, what will help you stand out within the organization is to get involved. Keep in mind that all professional organizations are volunteer organizations, so it is easy to get onto one of the committees or even take on a leadership role of a committee. This allows you to demonstrate your skills, work ethic, and commitment to people who are usually more senior than you are (in the case of a campus professional organization, you stand out to your professors, which is a good strategy). It’s a great way to build your leadership, teamwork, and networking skills.
There are several professional organizations off campus that invite students to join, usually at a reduced rate. For example, the PRSA (Public Relations Society of America) costs \$290 for an annual membership while the fee for a student in the PRSSA (Public Relations Student Society of America) is \$50.Public Relation Society of America (PRSA), “Receive a Free PRSA Chapter Membership,” http://www.prsa.org/membership/documents/PRSA%20Individual%20Membership%20Application%20with%20Chapter%20Pricing.pdf (accessed September 12, 2009).,Public Relations Student Society of America (PRSSA), “Join PRSSA,” http://www.prssa.org/about/join.aspx (accessed September 12, 2009). It’s a good idea to talk to your faculty advisor, as she will most likely know what local organizations have a student rate. These off-campus professional organizations are excellent for networking to find the right people to whom you should be speaking to get the job you want. Professional organizations will serve you well throughout your career. Even after you graduate, you will be a student of the business. Professional organizations provide a platform for ongoing learning about industry trends, case studies, and best practices. Successful people stay involved in one or more professional organizations even after they have established careers.
There are professional organizations for virtually every profession and industry. Table \(3\) includes a list that can help you see some of what’s available.
Table \(3\): Professional Organizations
Professional Organization Information and Web Site Comments
Sales & Marketing Executives International
http://www.smei.org
Web site of SMEI
United Professional Sales Association
www.salestrainingdrivers.org/upsa
Web site of UPSA
Marketing Trade Organizations and Industry Groups
www.knowthis.com/marketing-lists/marketing-groups-and-meetings/trade-and-professional-marketing-associations-and-groups/
List of resources and associations
Sales and Marketing Organizations
dir.yahoo.com/Business_and_Economy/Marketing_and_Advertising/Organizations/
List of sales and marketing organizations
San Francisco State University
cob.sfsu.edu/cob/undergrad/pro-orgs.cfm
List of professional organizations
Wikipedia
http://en.Wikipedia.org/wiki/List_of_professional_organizations
List of professional organizations
American Marketing Association
www.marketingpower.com/Pages/default.aspx
Web site of the American Marketing Association
PRSSA (Public Relations Student Society of America)
http://www.prssa.org/about/join.aspx
Web site of PRSSA
10 Top Professional Networks for Women in Finance
http://www.theglasshammer.com/news/2009/07/23/top-10-professional-networks-for-women-in-finance
Professional organizations for women in finance
Women’s Career Networking and Professional Organizations
www.quintcareers.com/womens_networking_organizations.html
Women’s networking organizations
Gale’s Encyclopedia of Associations
www.apl.org/quick/associations.html
List of all types of professional associations
Key Takeaways
• The best way to get the job you want when you graduate is to work on it right now with internships and by joining professional organizations. Both are expected by prospective employers of new hires for entry-level jobs.
• Internships provide an opportunity to gain experience, test drive a job, and network; many result in full-time job offers.
• Some internships are paid while others are unpaid; some internships qualify for college credit while others do not. It’s best to consult your faculty advisor before you accept an internship.
• Professional organizations provide exposure to executives, industry news, and best practices and enhance your résumé. The best way to take advantage of your professional organization membership is to get involved one of the organization’s committees.
Exercise \(1\)
1. Visit your campus career center and meet with one of the counselors. Learn about internships that are offered and how to apply for them. Discuss three things that you learned during this meeting.
2. Identify a student who has had an internship in your target industry. Set up a meeting to discuss how she landed her internship and what advice she can give you about finding the right internship.
3. Review your campus Web site and identify at least two professional organizations that may be of interest to you. Attend at least one of their meetings and determine which might be the right organization for you to join. | textbooks/biz/Marketing/The_Power_of_Selling/14%3A_The_Power_of_Learning_the_Ropes/14.04%3A_Selling_U_-_Its_Your_CareerOwn_It.txt |
Power Wrap-Up
Now that you have read this chapter, you should be able to understand the key elements of how to manage your time and resources to be successful in sales.
• You understand that although you might be working alone, you have resources available to help you be successful every day.
• You can discuss how to manage your time to accomplish your goals.
• You can recognize all the key elements of compensation and how you can leverage each to earn the income you want.
• You can describe how to set SMART goals and use key performance indicators to help measure progress.
• You can appreciate how important believing in yourself is to being successful in sales and in life.
• You can understand that failure is a part of selling, but how you react to failure is what can make you successful.
• You can recognize that having a healthy mind and body contributes to your chances of success.
• You can understand that you can prepare now for the full-time job you want by having internships and getting involved in professional organizations.
TEST YOUR POWER KNOWLEDGE (AnswerS ARE BELOW)
1. What is a ride-along?
2. If you are going through the onboarding process, what are you doing?
3. Name three reasons time can get away from you.
4. List three things you can do to improve your time management.
5. What is the difference between a commission and a draw?
6. If you are earning 12 percent of sales as a commission, how much would you earn on annual sales of \$1,100,000?
7. Is it possible for an employer to offer a salary plus commission plus bonus as part of a compensation plan?
8. What is a KPI, and how is it used in sales?
9. Name three ways to get motivated to sell more.
10. How does personal responsibility relate to failure?
11. Name three things you can do to ensure you have a healthy mind and body?
12. Why is it important to have at least one internship?
13. Why is it important to join a professional organization?
POWER (ROLE) PLAY
Now it’s time to put what you’ve learned into practice. The following are two roles that are involved in the same selling situation—one role is the customer, and the other is the salesperson. This will give you the opportunity to think about this selling situation from the point of view of both the customer and the salesperson.
Read each role carefully along with the discussion questions. Be prepared to play either of the roles in class using the concepts covered in this chapter. You may be asked to discuss the roles and do a role-play in groups or individually.
Time Is on My Side: Managing Customers and Your Time
Role: Food and beverage manager at a major luxury hotel chain
You are a significant customer, one that all your liquor distributors want to have because you have a four-star brand name. All the sales reps want to add your hotel chain to their client list. That’s why you split the business between several reps. You feel like that provides your business the best service and keeps all the reps on their toes.
You like it when sales reps spend time talking with you and demonstrate that they care about the business. In fact, you are always impressed when one goes out of her way to personally deliver a case of something that didn’t get delivered on the truck (it seems like that happens often). When a rep doesn’t respond quickly to a delivery error, you take away some of the business from them.
• You believe that when sales reps spend more time at your account, they care more about your business. How do you tell the sales rep that you expect more face time?
• When something is missing on a delivery, you expect the sales rep to bring the missing part of the shipment to the hotel that day. Do you think that is unreasonable?
• You call the sales rep often to see if he will do special things for the hotel, including tastings and other events. You want the sales rep to host a sampling event for a convention next week. You realize it’s not very much advance notice, but you want to ask anyway. What will you say to the sales rep?
Role: Liquor distributor sales rep
You have a very small part of this luxury hotel’s business; you really want to get more because the hotel has the potential to be your largest customer. However, the food and beverage manager is very demanding, and it takes a lot of your time; in fact, too much time. It seems that a sales call always takes at least two hours. It also seems that you are regularly doing hand deliveries because the food and beverage manager forgets to order things and expects a delivery within a few hours of his call. This account has been a time management challenge. You have to determine if you can get a commitment for more of the hotel’s business and reduce the amount of time you spend servicing the account.
• Although this customer has the potential to give you more business, he hasn’t yet. He asks for a lot of your time with in-person meetings, personal follow-up on missed deliveries and short notice for special tastings and other events. Should you always say yes to his requests in hopes of getting more business?
• If the customer is always right, what is the best way to balance your time and get additional business to offset your time investment?
• You are extremely busy with another customer on the date that this customer has requested a sampling at his convention, and you have a personal commitment that night. What will you say to this customer about his request?
ACTIVITIES
1. Use your professional social networking skills by going to LinkedIn (see the Selling U section in Chapter 3) and join your school’s alumni group. Use the Q & A feature to identify people who are in your target industry. Ask for their advice about what to look for when choosing an internship. If you haven’t already done so, join The Power of Selling group on LinkedIn and start a discussion or use the questions and answers feature to get input about what to look for when choosing an internship. This group is there to help you.
2. Visit your campus career center and review the Web sites mentioned in the Selling U section in this chapter. Identify at least two professional organizations in which you might be interested. Visit their Web sites and review the mission, events, news, membership benefits, and cost. Attend one meeting for each of the organizations to help determine if one might be a good organization to join.
TEST YOUR POWER KNOWLEDGE AnswerS
1. Traveling with an experienced sales rep or sales manager to make sales calls.
2. Being exposed to an employee orientation process or method for a new employee to learn about company practices, policies, and procedures.
3. Poor planning, procrastination, and making tasks too big.
4. Get organized, set goals, prioritize activities, create a schedule, delegate work to others, and maximize selling time.
5. Commission is the income that is based on the percentage of sales (or gross profit) generated; a draw is an advance against future commissions or bonuses.
6. \$1,100,000 × 0.12 = \$144,000.
7. Yes. Compensation may include as many elements as the employer chooses to offer. Usually commissions and bonuses are higher than salary to provide incentive for salespeople to sell more.
8. KPI stands for key performance indicator: a measure of productivity that relates to achieving goals. KPIs are used to measure progress against SMART goals and are often used to determine compensation and incentives for salespeople.
9. Take pictures of your top ten customers and top ten prospects; tell your family how you are going to celebrate when you achieve your goal; invest at least fifteen minutes every day to read articles, books, or blogs about your industry or profession; and write yourself a check for the amount you want to earn and keep copies with you.
10. Although failure is a part of selling (and of life), personal responsibility means acknowledging and accepting that you are accountable for your choices, learning from failures, and not making the same mistake again.
11. Get a good night’s sleep, eat a healthy breakfast, exercise, get right into your day, start your day with an important task, smile, don’t worry about what you can’t control, and take time for yourself.
12. An internship provides practical work experience and gives you insight about what you might want to do (or not want to do) after graduation. Also, it’s a great way to network. In addition, many internships result in full-time job offers.
13. Professional organizations offer students and professionals unique opportunities to network and learn about trends and best practices in the industry; they also provide a chance to stand out and be noticed by getting involved in a committee. | textbooks/biz/Marketing/The_Power_of_Selling/14%3A_The_Power_of_Learning_the_Ropes/14.05%3A_Review_and_Practice.txt |
Entrepreneurial Selling: The Power of Running Your Own Business
Video Ride-Along with David Fox, Founder and CEO of Brave Spirits
Meet David Fox. He is an entrepreneur who founded Brave Spirits, a company that makes vodka, gin, whiskey, and rum. What makes Brave Spirits different from other spirits makers is that \$2 of every bottle sold is donated to charities that support the men and women of America’s military, fire departments, and police departments. The company is dedicated to celebrating the bravery of those who serve our country.
Ride along with David and learn about the role selling plays in being an entrepreneur.
(click to see video)
15.02: The Power of Entrepreneurship
Learning Objectives
• Understand the entrepreneurial spirit and what it takes to be an entrepreneur.
• Discuss the role of entrepreneurial businesses in the economy.
Her owners couldn’t imagine putting their beloved Zoe, a seventeen-year-old Jack Russell terrier, in the dark belly of the cargo hold of a jetliner when they moved from California’s Bay Area to Delray Beach, Florida, in 2005. Although most commercial airlines have been working on policies and procedures to make pets more comfortable when traveling, and give their owners more peace of mind, husband-and-wife team Alysa Binder and Dan Wiesel just didn’t feel that options like Delta’s Pet First and Continental’s PetSafe really filled the bill. So on July 14, 2009, they launched Pet Airways, the first airline dedicated exclusively to transporting animals in a safe and comfortable environment. The airline includes nineteen turboprop Beech 1900 planes that have been converted to comfortably carry up to fifty live animal crates and one certified pet attendant. Pet “passengers” can fly one way for \$149 and round trip for \$250 to or from any of five major airport locations.Dan Reed, “For Passengers of New Airline, When the Fur Flies, It’s in Style,” USA Today, June 19, 2009, B1. Who else but entrepreneurs would conceive a business idea like this?
It All Starts with an Idea
Entrepreneurs have started all different kinds of businesses from overnight shipping to electronics, music, skin-care products, to retail stores, because each saw an unmet need in the market. Fred Smith started FedEx in 1971 based on a paper he wrote for a Yale University economics class and used his \$4 million to start the company.Funding Universe, http://www.fundinguniverse.com/company-histories/FedEx-Corporation-Company-History.html (accessed September 21, 2009). Sam Walton founded Wal-Mart in 1962 because he was convinced that Americans wanted a new type of store, a discount store, so he and his wife put up 95 percent of the money to build the first Wal-Mart store in Rodgers, Arizona.Wal-Mart Stores, Inc., About Us, http://walmartstores.com/AboutUs/297.aspx (accessed September 21, 2009). In 1930, Colonel Harland Sanders started cooking for weary travelers who stopped by his gas station; they ate at his own dining table because he didn’t have a restaurant.“Colonel Harland Sanders: From Young Cook to KFC’s Famous Colonel,” KFC.com, www.kfc.com/about/colonel.asp (accessed January 3, 2010). William Hewlett and David Packard decided to start a business and “make a run for it themselves” in 1939; their first product was an audio oscillator, an electronic instrument used to test sound equipment. They decided the name of their company on the toss of a coin.HP, “HP Timeline—1930s,” www.hp.com/hpinfo/abouthp/histnfacts/timeline/hist_30s.html (accessed September 21, 2009). Jay-Z saw hip-hop music as a way to get out of the projects in Brooklyn and parlayed his passion and business prowess into a net worth of over \$350 million. He is CEO of Def Jam Recordings and Roc-a-Fella Records, part-owner of the New Jersey Nets, and co-owner of the 40/40 Club, among other things.Black Entrepreneur Profile, “Shawn ‘Jay Z’ Carter,” www.blackentrepreneurprofile.com/profile-full/article/shawn-jay-z-carter/ (accessed September 21, 2009). Estée Lauder started selling creams that were created by her uncle and founded Estée Lauder in 1946 “with four products and the belief that every woman can be beautiful.”“Famous Women Entrepreneurs,” About.com, http://entrepreneurs.about.com/ (accessed September 21, 2009).,Estée Lauder, “About Estée Lauder,” www.esteelauder.com/about/index.tmpl (accessed September 21, 2009). Walt Disney, a cartoonist, saw the opportunity to entertain as his driving force when he founded Walt Disney Company in 1923.The Walt Disney Company, “Corporate Information,” http://corporate.disney.go.com/corporate/overview.html (accessed September 21, 2009). His passion for his craft led to the creation of the company when he said, “I am interested in entertaining people, in bringing pleasure, particularly laughter, to others, rather than being concerned with ‘expressing’ myself with obscure creative impressions.”“Quotes,” JustDisney.com, http://www.justdisney.com/walt_disney/quotes/quotes01.html (accessed September 21, 2009).
Entrepreneurship is about helping people see the benefit of a new way of doing things; it’s about having an idea and having the passion and perseverance to make it come alive. What sets an entrepreneur apart from any other businessperson is that fact that she is willing to assume risk to make a profit.“What Is an Entrepreneur?” ZeroMillion.com, www.zeromillion.com/business/starting/entrepreneur.html (accessed September 21, 2009). An entrepreneur is not only open to new ways of thinking and of doing things, but he also has the vision, drive, energy, and optimism to bring an idea from concept to reality. Generally, an entrepreneur is someone who says, “There’s a better way, and I will find it.” Entrepreneurs are willing to take risks to make things better.Federal Reserve Bank of Dallas, “Everyday Economics,” www.dallasfed.org/educate/everyday/ev3.html (accessed September 19, 2009).
Why Become an Entrepreneur?
(click to see video)
David Fox, founder and CEO of Brave Spirits, talks about why someone should consider being an entrepreneur. It might surprise you to hear what he says about being your own boss.
Guy Kawasaki, famous entrepreneur, venture capitalist, speaker, and author of several books on entrepreneurialism including The Art of the Start: The Time-Tested, Battle-Hardened Guide for Anyone Starting Anything, provides a holistic way of viewing entrepreneurship.
Making Meaning
(click to see video)
The idea of starting, restarting, or growing something—whether a business or a nonprofit organization—is grounded in “making meaning,” which is the cornerstone of Guy Kawasaki’s concept of entrepreneurship as shown in this video.Guy Kawasaki, “The Art of the Start,” video, April 29, 2006, http://www.youtube.com/watch?v=L3xaeVXTSBg (accessed September 28, 2009).
Entrepreneurialism and the Economy
Imagine that in 1899, the director of the United States Patent Office made the proclamation that everything that could be invented had already been invented. He certainly underestimated the power of the American entrepreneur. Everything from airplanes to telephones, computers to the Internet, not to mention the iPod and even skateboards, have all been products developed and marketed by entrepreneurs over the past 110 years.Guy Kawasaki, “The Art of the Start,” video, April 29, 2006, http://www.youtube.com/watch?v=L3xaeVXTSBg (accessed September 28, 2009).
Today, entrepreneurs literally power the country. There are twenty-seven million businesses in the United States; 99.7 percent of them are considered small businesses, according to David C. Dickson, district director for the Small Business Administration. That means that only 80,000 businesses are considered large businesses; the other 26.9 million are small businesses.District Director David C. Dickson, Philadelphia, Small Business Association, SCORE Open House, September 22, 2009, Valley Forge, PA. Small businesses provide approximately 75 percent of the new jobs added to the American economy every year.
Besides providing jobs, entrepreneurial businesses are more likely to provide specialty and custom goods and services to consumers and businesses. In fact, small businesses produce nearly thirteen times more patents (rights of exclusivity to make and market the product or service granted by the United States government) per employee than large firms.United States Patent and Trademark Office, “What Is a Patent?” www.uspto.gov/go/pac/doc/general/#patent (accessed September 28, 2009).,U.S. Department of State’s Bureau of International Information Programs, “Entrepreneurship Aids the Economy: Most Economists Agree that Entrepreneurship Is Essential to Any Economy,” May 12, 2008, www.america.gov/st/business-english/2008/May/20080603233010eaifas0.8230554.html (accessed September 19, 2009). Small businesses represent one-third of all companies that have fifteen or more patents.U.S. Department of State’s Bureau of International Information Programs, “Entrepreneurship Aids the Economy: Most Economists Agree that Entrepreneurship Is Essential to Any Economy,” May 12, 2008, www.america.gov/st/business-english/2008/May/20080603233010eaifas0.8230554.html (accessed September 19, 2009).
Entrepreneurialism is critical not only for the growth of the economy in the United States but also globally. The Kauffman Foundation, in conjunction with researchers from Babson College and the London School of Business, found that the disparity in entrepreneurial activity in some countries is contributing to a gap in economic growth. “The Global Entrepreneurship Monitor (GEM) report provides conclusive evidence that promoting entrepreneurship and enhancing the entrepreneurial dynamic of a country should be an integral element of any government’s commitment to boosting economic well-being,” according to Paul Reynolds, GEM project coordinator at both Babson College and the London Business School.Small Business, “Entrepreneurs Add Vitality to the Economy,” http://www.smallbusinessnotes.com/aboutsb/vitality.html (accessed September 19, 2009). According to the study, Canada, Israel, and the United States are those countries that are experiencing the highest level of activity, while Denmark, Finland, France, Germany, and Japan have the lowest levels of activity. The GEM constructed a framework for countries to work within to encourage entrepreneurial activity that includes raising the participation level to those outside the core age group of twenty-five to forty-four and increasing the participation of women in the entrepreneurial process.Small Business, “Entrepreneurs Add Vitality to the Economy,” http://www.smallbusinessnotes.com/aboutsb/vitality.html (accessed September 19, 2009).
Entrepreneurship is rewarded in the United States because the economy is based on a free market system—one in which an individual’s success is dictated by demand on the part of the consumer, not by the government.Federal Reserve Bank of Dallas, “Everyday Economics,” www.dallasfed.org/educate/everyday/ev3.html (accessed September 19, 2009). Because entrepreneurial businesses are so important to the economy, free market governments not only support them, but they often encourage entrepreneurial business ventures. According to Harvard Business School professor Josh Lerner, the United States government played a key role in the early development of Silicon Valley. The government has also provided support for the growth of innovation in places such as Tel Aviv and Singapore.Sean Silverthorne, “Government’s Positive Role in Kick-Starting Entrepreneurship,” HBS Working Knowledge, December 7, 2009, http://hbswk.hbs.edu/item/6318.html (accessed January 3, 2010).
While statistics are no guarantee of success for the future, especially in today’s tough economy, you might find it interesting to know that small businesses employ a little more than half of all private sector workers, pay 44 percent of all private sector payroll, and have generated 64 percent of new jobs over the past fifteen years. In 2008, approximately 530,000 new businesses were created every month, which was a slight increase over 2007.Laura Petrecca, “Tough Times Drive Start-Ups,” USA Today, September 14, 2009, B2.
These facts reflect the importance of creativity, ideas, passion, drive, independence, and risk taking in the economy. Entrepreneurialism isn’t limited to a specific type of organization or business; it’s a state of mind, a way of thinking and behaving, a way of pushing for something better than the status quo. Entrepreneurs can be found in large multinational corporations as well as nonprofit organizations, and new business start-ups. Every large business or organization had to start small but think differently.
Power Selling: Lessons in Selling from Successful Companies
An Entrepreneur Searching for Entrepreneurs
Imagine auditioning your product idea in front of a panel of judges. The reward for the winner? A television commercial for your product sponsored by Telebrands, the company that advertises “As Seen on TV,” quirky, practical products on television and in their retail stores. CEO A. J. Khubani goes on the hunt regularly for new and unique products. Would-be entrepreneurs demonstrate and sell their products in front of a panel that includes Khubani, his wife, and other executives of the company. He looks for products with “mass appeal, inexpensive production costs, and good demonstrability in ads.”Theresa Howard, “Investors Seek Pot of Gold on TV,” USA Today, June 25, 2009, B1, 2.
Getting Started
Being an entrepreneur can be exciting and invigorating. Having the vision to create products and services that can meet and exceed customers’ needs and imaginations and the passion to bring them to market can be exhilarating. Imagine having a hobby in college that turns into a multibillion-dollar business. That’s how Mark Zuckerberg created Facebook and Michael Dell created Dell. Both had a unique idea and vision and were willing to take the risk to make it a reality.
Power Player: Lessons in Selling from Successful Salespeople
Meet the Chef
It started when formally trained chef Kimberly Davis Cuthbert’s infant son refused to eat the sweet potato puree she made for him. Although her bakery, Sweet Jazmines, used the puree for some recipes, she was determined never to waste sweet potato puree again. Her famous Sweet Potato Muffins were born and have been a signature treat at her successful bakery ever since. Read more about Chef Kim’s passion for baking, commitment to baking from scratch, and her sweet taste of success at sweetjazmines.com/index.html.
Where does it all start? The entrepreneurial drive starts early for most who have it. Think about when you were young. If you were the one in the neighborhood who set up a lemonade stand, went door-to-door selling products, or set up a lawn-mowing service in the neighborhood (and hired your friends), you may have the makings of an entrepreneur. While you and many of your friends (and perhaps your siblings) participated in activities like this when you were young, not everyone will grow up to be an entrepreneur. A recent study, which focused on the behaviors of identical twins, has found evidence that entrepreneurialism may be based on genetics, not environment.Nicos Nicolaou, Scott Shane, Lyn Cherkas, Janice Hunkin, and Tim D. Spector, “Is the Tendency to Engage in Entrepreneurship Genetic?” Management Science 54, no. 1 (January 2008): 167–79, mansci.journal.informs.org/cgi/content/abstract/54/1/167 (accessed September 21, 2009). In fact, the study found that a connection between different genes causing someone to be extroverted is important to salesmanship, which is also a strong trait in many entrepreneurs.Jim Hopkins, “Starting a Business: What It Takes,” USA Today, October 25, 2006, www.usatoday.com/money/smAllBusiness/2006-07-30-starting-your-business_x.htm (accessed September 19, 2009).
Video Clip
A Virtual Idea
Listen to Philip Rosedale, the CEO of the virtual world Second Life, talk about how he conceived the concept when he was young and how he created and has grown the business.
cmypitch.com/entrepreneur-tv/show/95/philip_rosedale_created_the_virtual_world_known_as_second_life
Genetics or not, the current state of the economy has forced many who are out of work to consider owning their own business. “These are people that two years ago did not aspire to own a business, but circumstances have dictated that they look at freelance opportunities,” says Ken Yancey, CEO of nonprofit entrepreneur-mentoring group Service Corps of Retired Executives (SCORE). In fact, Internal Revenue Service (IRS) data confirm that the number of nonemployer firms (those that have paid employees) is up 8.1 percent from 2007 to 2008.Laura Petrecca, “Tough Times Drive Start-Ups,” USA Today, September 14, 2009, B1. These “accidental entrepreneurs” have come of age due to a variety of circumstances. Some, like Maureen Rothman, president of Rothman Associates, started her business when her employer went out of business. Although she had over fifteen years of selling and sales management experience, when people encouraged her to start her own business her first thought was, “I’m not an entrepreneur.” After developing a business plan and getting input and direction from resources such as SCORE, she said that everyone told her to “just do it.” “On 09.06.05 [September 6, 2005],” as she likes to say, “I did it!” and launched Rothman Associates, a manufacturers’ representative for hospitality seating products.Maureen Rothman, SCORE Open House, September 22, 2009, Valley Forge, PA.
Entrepreneurial at Any Age
Whether it’s the challenging economy or the idea of controlling your destiny, entrepreneurs begin their journey at different ages.
Tech-savvy teenagers are starting online businesses since the Internet has lowered the barriers to entry and provided anonymity from their age. Challenges in finding part-time work also drive teens to start traditional businesses such as yard work, party planning, and tutoring.Eilene Zimmerman, “Teenagers Are Building Their Own Job Engine,” New York Times, June 28, 2009, B10.
At the other end of the spectrum, many baby boomers who are reaching retirement age are now asking themselves if retirement is “when you stop working completely or retire from one job and begin another.” Independence, passion, flexibility, and additional income are significant motivators to “mid-life entrepreneurs.” In fact, boomers compose nearly half of the nation’s self-employed workers.Susan L. Reid, “Take Control of Your Retirement: Become a Midlife Entrepreneur,” American Express, August 25, 2009, www.openforum.com/idea-hub/topics/innovation/article/take-control-of-your-retirement-become-a-midlife-entrepreneur-susan-l-reid (accessed September 28, 2009).
Leaving corporate life behind for a chance to build a business and chase a dream is a trend that is occurring across the country in industries of all kinds from technology to personal services. Ali Galgano recently traded in her high-powered job as corporate recruiter doing work for firms such as Goldman Sachs for her own jewelry company, Charm & Chain. Brent Bouchez, David Page, and Nancy McNally gave up their glitzy, perk-filled jobs at an advertising agency on Madison Avenue in New York to start their own marketing firm named Five-0, which is focused on marketing to baby boomers. Kelly Elvin gave up a lucrative career as a lawyer to become a dog trainer. All have learned that life as an entrepreneur is very different from life in the corporate world. The thrill of running your own show and making your dream come alive is a learning process. There are no bosses, no corporate initiatives, no departmental deadlines, and no performance reviews. Everything is all you, all the time. Skills such as time management, understanding financial statements, building a good support system, and overcoming the isolation of working alone are all challenges in the entrepreneurial world. “Entrepreneurs have to be willing to listen and learn and make judgments and be adaptive,” according to Monica Doss, director of the Kauffman Foundation’s FastTrac entrepreneur training programs.Laura Petrecca, “From Corporation to Start-up: Who Is Going to Fix the Printer?” USA Today, September 21, 2009, www.usatoday.com/money/smallbusiness/startup/week2-corporation-to-startup.htm (accessed September 21, 2009). As for the issue of timing, some ideas might not work in the current economic environment. Entrepreneurs have to understand when it’s no longer feasible to keep a business running.Laura Petrecca, “Leaving Corporate Life Behind,” USA Today, September 21, 2009, B1, 2.
Who’s the Boss?
Ali Galgano, CEO of Charm & Chain, realized that working on her own without a boss had its strong points and challenges. Galgano formed an “advisory board of people who are smarter and more accomplished than I am.” She has monthly meetings with the advisory board to help her formulate strategies and make good business decisions. The group includes her father who is an investment banker. Read about her perceptions of life as an entrepreneur in this article that appeared in USA Today at www.usatoday.com/money/smAllBusiness/startup/week2-corporation-to-startup.htm?loc=interstitialskip.Laura Petrecca, “Leaving Corporate Life Behind,” USA Today, September 21, 2009, B1, 2. Check out the Web site for Ali’s company at http://www.charmandchain.com.
What Does It Take?
Entrepreneurialism is based on dreams and risk. Not every idea is commercially viable or economically feasible, and not every dream comes true. But there are some common ingredients that are part of being an entrepreneur.
Video Clip
Do You Have What It Takes?
Hear from some of the best entrepreneurs from around the world about what it takes to be an entrepreneur.
www.ey.com/GL/en/Issues/Driving-growth/WEOY-2009---What-does-it-take-to-be-an-entrepreneur
What’s the Big Idea?
Every business starts with an idea—that unique product or service that will serve customers better than any other. “Entrepreneurs are often so passionate about their ideas, they can lose objectivity,” according to Nancy A. Shenker, president of theONswitch LLC. “Rather than taking the time to thoroughly plan and research, they sometimes plow ahead with execution, only to spend valuable dollars on unfocused or untargeted activities,” she adds.Karen E. Spaeder, “How to Research Your Business Idea,” Entrepreneur, http://www.entrepreneur.com/article/printthis/70518.html (accessed September 19, 2009). That’s why it’s important to research the viability of a business idea starting with the size of the market and if the idea will be compelling enough to meet an unmet need in the marketplace. Checking out the competition can be extremely educational. You might be surprised about what you learn by visiting your competitors and asking questions to their customers.Karen E. Spaeder, “How to Research Your Business Idea” Entrepreneur, http://www.entrepreneur.com/article/printthis/70518.html (accessed September 19, 2009). The bottom line is that there has to be some recognition by the customer that there is a need for the product or service you want to bring to market. Without demand, it will be virtually impossible to have a successful business.
You might be wondering if there’s more to being an entrepreneur than simply selling a product or service for a profit. Social entrepreneurialism uses the concepts of entrepreneurship to bring about social change. While some social entrepreneurial efforts are nonprofit organizations, others are for-profit companies that focus on adding value to society. For example, City Year is a nonprofit organization that provides full-time year of service for young people from the United States and South Africa with the objective that they will go on to use their skills to better the world. Social entrepreneurialism is recognized and supported by several mainstream organizations.
Link
Best Social Entrepreneurs
View the Fast Company magazine annual list of “The Best Social Entrepreneurs.”Ilya Bodner, “Social Entrepreneurship,” Fast Company, June 2, 2009, www.fastcompany.com/blog/ilya-bodner/true-business-credit-card/social-entrepreneurship (accessed September 21, 2009).
www.fastcompany.com/social/2008/index.html
Link
Most Promising Social Entrepreneurs
BusinessWeek published its inaugural list of “The Most Promising Social Entrepreneurs” in May 2009.
http://www.businessweek.com/smallbiz/content/may2009/sb2009051_730988.htm
The Difference Is in the Questions
No one plans to bring a mediocre product or service to market, but the best way to avoid that fate (and ultimately failure) is to ask yourself the right questions before you start your business. For example, can you really answer the question “What sets your product or service apart from what the competitors offer?” A claim like “the best burger in Seattle” doesn’t offer any real point of difference to the customer. Asking the right questions helps identify important opportunities or explain the lack of them.Max Chafkin, “The Wexley Way: How to Think Creatively in 8 Easy Steps,” Inc., February 19, 2009, www.inc.com/articles/2009/02/wexley.html (accessed September 19, 2009). A good start is to ask yourself these three questions:
1. What are you selling?
2. To whom are you selling it?
3. Why would they buy it from you?
If answered honestly and specifically, these questions help identify the validity of a new business idea.Ridgely Evers, “The Three Toughest Questions,” Inc., April 1, 2008, www.inc.com/resources/startup/articles/20080401/revers.html (accessed September 19, 2009).
Hard Work, Long Hours
You might consider the concept of being your own boss to be a good deal. After all, you can do what you want, when you want, and work as hard as you want on what you want do to because you’re the boss. Well, that’s not really completely true. The challenge of bringing an idea to life is hard work and there’s no guarantee of success. According to the U.S. Census Bureau, only 48.8 percent of the new businesses that were started in 1977 were still around in 2000.Scott A. Shane, “Failure Is a Constant in Entrepreneurship,” New York Times, July 17, 2009, boss.blogs.nytimes.com/2009/07/15/failure-is-a-constant-in-entrepreneurship (accessed January 3, 2010). Being an entrepreneur is hard work. Think about Melissa Carter, the owner of San Diego’s first CiCi’s Pizza. She works about seventy hours a week and put in even more hours before the grand opening in August 2009.Laura Petrecca, “Leaving Corporate Life Behind,” USA Today, September 21, 2009, B1, 2. Dan Sanker, president and founder of CaseStack, a logistics outsourcing company, admits that he really doesn’t take any time off, despite his good intentions. Sanker strongly encourages people to follow their dreams and do something entrepreneurial as he has done. But he also reminds the aspiring entrepreneur to keep in mind that business ownership does not provide complete freedom and flexibility because you will ultimately “be beholden to investors, clients, and employees.”“Interview with an Entrepreneur—Dan Sanker of CaseStack,” E-Shadow.com, www.e-shadow.com/interview-with-an-entrepreneur-dan-sanker-of-casestack (accessed September 19, 2009). Vinny Lingham, founder of do-it-yourself Web site building company Yola, who recently secured \$20 million in investor funding and was featured on the cover of the July 2009 issue of Entrepreneur magazine, says, “Success may look like it happened overnight but that’s rarely the case in reality. You have to be prepared to put in long hours, take risks, and make personal sacrifices.” But he goes on to say, “And ideally the best time to make them is when you’re young, which is why I encourage young entrepreneurs to go for it.”Juliette Pitman, “Persistence Pays: Vinny Lingham,” Entrepreneur, July 2009, www.entrepreneurmag.co.za/article/h/?a=1516&z=161&title=Persistence+Pays:+Vinny+Lingham (accessed May 16, 2010). Read the entire article about Vinny Lingham’s entrepreneurial journey and success at www.vinnylingham.com/cover-story-entrepreneur-magazine-july-2009.html.
The Best and the Worst
(click to see video)
Listen to David Fox, founder and CEO of Brave Spirits, share his thoughts about the best and the worst of being an entrepreneur.
Get Rich Quick? Probably Not
Entrepreneurs are motivated by discovery, creativity, and innovation. While almost three-quarters of current business owners surveyed by the Kauffman Foundation said that “building wealth” is the reason they became an entrepreneur, it may take a long time to realize the financial benefit of entrepreneurship.Laura Petrecca, “Leaving Corporate Life Behind,” USA Today, September 21, 2009, B2. For example, a franchise, a form of business organization in which a person, or franchisee, pays a company to use its name and market its products, can cost hundreds of thousands of dollars in up-front fees.“Franchise,” InvestorWords.com, www.investorwords.com/2078/franchise.html (accessed September 28, 2009). A Subway franchise can cost as much as \$250,000, not including any royalty fees, rent, product, or labor costs.“Subway Franchise for Sale,” Docstoc, http://www.docstoc.com/docs/2418199/Subway-Franchise-for-Sale (accessed September 28, 2009).,“2009 Franchise 500 Rankings,” Entepreneur, http://www.entrepreneur.com/franchises/rankings/franchise500-115608/2009,.html (accessed September 28, 2009). Meanwhile, businesses that require inventory, such as retail stores or restaurants, require an investment in inventory, real estate, or even technology before the doors even open. But if the franchise or business idea is right, and the business is well run, the payback can be significant financially and personally.
Interestingly, it’s more than money that motivates many entrepreneurs. It’s more than “riches” according to Scott Laughlin, director of the University of Maryland’s Tech Entrepreneurship Program. Entrepreneurs are more interested in “wealth”; he points to two of the most famous entrepreneurs, Bill Gates from Microsoft and Warren Buffet from Berkshire Hathaway, who have pooled their resources into the \$60 billion philanthropy called the Bill & Melinda Gates Foundation. “Wealth is broader, encompassing less tangible rewards such as respect and independence,” says Laughlin.Jim Hopkins, “Starting a Business: What It Takes,” USA Today, October 25, 2006, www.usatoday.com/money/smAllBusiness/2006-07-30-starting-your-business_x.htm (accessed September 19, 2009).
Top Ten Franchise Opportunities
View Entrepreneur magazine’s list in slide show format including number of franchises in operation, start-up costs, and other statistics.
www.entrepreneur.com/slideshow/199084.html
Key Takeaways
• Entrepreneurship is the practice of selling ideas and having the passion and perseverance to make it become a reality. Entrepreneurs are willing to take risks to bring a product or service to market.
• Some of the world’s largest companies were started by entrepreneurs.
• Entrepreneurs have a significant impact on the economy of the United States and the world.
• Entrepreneurs protect their ideas by applying for a patent from the United States government.
• Entrepreneurs flourish in a free market system, one in which an individual’s success is dictated by demand on the part of the consumer, not the government.
• Being an entrepreneur requires a unique idea, passion, and hard work to bring it to fruition.
• Social entrepreneurship includes nonprofit organizations as well as for-profit companies that focus on impacting society in a positive way.
• An entrepreneur may start a business based on a new idea or expand an existing brand by buying a franchise.
Exercise \(1\)
1. Watch this video interview with Joe Kennedy, CEO of Pandora (the online personalized music Web site), to see how the company started and evolved: http://www.entrepreneur.com/video/index.html. Identify three key lessons that he talked about that brought the company to where it is today.
2. Review the list of Entrepreneur magazine’s fastest-growing franchises: http://www.entrepreneur.com/franchises/rankings/fastestgrowing-115162/2009,.html. Visit the Web sites of at least three of the franchises and answer these three key questions about their business: What do they sell? To whom do they sell it? Why do people want to buy it from them?
3. Visit the Web site of Entrepreneur magazine: http://www.entrepreneur.com. Read a current article about an entrepreneur and discuss his or her unique idea. | textbooks/biz/Marketing/The_Power_of_Selling/15%3A_Entrepreneurial_Selling-_The_Power_of_Running_Your_Own_Business/15.01%3A_Chapter_Introduction.txt |
Learning Objectives
• Understand how entrepreneurs sell themselves and their business ideas to secure funding to grow their businesses.
So you have a unique idea, the passion and perseverance to bring it to market, and the willingness to take the risk. Now what?
Being an entrepreneur in some ways is like being a student: you have to do your homework. In the business setting, that means creating a business plan: a road map of the who, what, when, where, why, and how of your business. A business plan is a document that details everything about the business from the product position in the marketplace to the financial information for the next three years. But a business plan is not simply like a term paper—a project that’s completed and then put on the shelf. A business plan is a dynamic document and should serve four purposes:
1. Sell you on the business. While this might sound like a no-brainer, the business plan development process includes rigorous research that can be a good eye-opener about the feasibility of your idea. Ideally, your business plan helps put your idea and its potential into perspective and gives you the details you need to move from concept to reality. However, even if you determine that your idea doesn’t have as much potential as you thought or might cost more than you anticipated, the process of creating a business plan helped you reach that conclusion.
2. Sell others on the business. In many cases, a business needs some kind of support—financial, consultative, or other resources. In this case, the business plan plays the role of the selling and marketing materials for your idea. How you make your idea come alive and support it with the necessary research and financial data can be the difference between someone becoming a stakeholder in your new venture or taking a pass.
3. Give you confidence. Having a great idea is one thing, doing the research to understand exactly what it will take to make the idea real is quite another. Having a better understanding of what it takes to launch and manage the business puts you in control to solicit investors and other supporters and start your journey.
4. Improve your chances of success. A business plan is a lot of planning and work, but it’s worth it. According to a study conducted by AT&T, 42 percent of entrepreneurs who had written a business plan rated themselves as more successful than the 58 percent who hadn’t written one.David E. Gumpert, “The Basics of Business Plans: Sell, Sell, Sell,” Inc., October 24, 2000, www.inc.com/articles/2000/10/14871.html (accessed September 19, 2009).
Writing Your Business Plan
Every business or organization is different, but a business plan is a common method of planning the launch and management of the business.SCORE, “Business Plan for a Startup Business,” www.score.org/resources/business-plan-startup-business (accessed September 29, 2009). While there is no single business plan format, the elements of a business plan are standard. The following business plan outline serves as a guide to developing a business plan. Keep in mind that the order in which you write your business plan should not necessarily follow the order in which you present your plan.
Business Plan Outline
1. Table of contents. Page numbers for each section.
2. Executive summary. Write this section after the plan is completed; this should be a compelling summary of the plan and how it will work.
3. General company description. A high-level description of the product, service, or organization and the unmet need it meets.
4. Products and services. A detailed description of the product, service, or organization; how it works; manufacturing costs; and so on.
5. Marketing plan. A detailed description of the current state of the market, including competition, your positioning, target audience, and how customers will learn about your product, service, or organization and the cost to get the word out.
6. Operational plan. A detailed description of how you will run the day-to-day operations, including product costs, real estate, inventory levels, labor, credit, and so on.
7. Management and organization. A detailed description of the principals of the company, including bios, board of directors, advisory board, banker, attorney, insurance agent, mentors, and so on.
8. Personal financial statement. A personal financial statement for each partner in the business. This is important as business owners often provide capital to start up or support the business; investors want to see the financial standing of the principal individuals.
9. Start-up expenses and capitalization. Accurate accounting of the expenses that are required to get the business started.
10. Financial plan. A twelve-month profit and loss statement, three-year financial projection, projected cash flow, and opening day balance sheet.
11. Appendices. Supporting information such as brochures and advertising, blueprints, leases, equipment, list of assets available, letters of recommendation, and any information that will help support your plan.
Link
Business Plan Template
A complete template with detailed descriptions for each section is available at the SCORE Web site.
http://www.score.org/resources/score-business-plan-template
Presenting Your Business Plan
Once you have identified your breakthrough idea (think iPod as the standard for a breakthrough idea), conducted your research, and written your business plan, it’s time to put everything to use. Whether you plan to fund the business yourself or find an investor to provide some capital (money), you will need your business plan to secure resources from your bank, insurance company, lawyer, attorney, and other support areas. Your business plan is the universal document for discussions with each of these people. In fact, you should first present your business plan to family, friends, and mentors to get some feedback before you take it out “on the road.”
Types of Investors
When it comes to investors, people who are willing to invest financial support based on the potential for the success of your business, there are several different types. Here is a summary of the types of investors:
• Banks. Banks are a common source of lending, and most offer several different types of business loans, including Small Business Administration (SBA) guaranteed loans. (The SBA does not actually provide loans; they simply guarantee them to banks that make the loans.) Banks are the most regulated form of lending; ratios must meet their requirements, and all paperwork must be in order.“Small Business Loan Sources, Take Aim,” Business Plan Master, www.businessplanmaster.com/small-business-loan-sources.html (accessed September 28, 2009).
• Private investors (or angel investors). These are people who are willing to invest money or resources to seed the business or get it up and running. Private investors can include anyone from your uncle who invests \$5,000 to a friend of the family who lets you use their second home for office space. Private investors may want to have a say in major business decisions.“How to Get Funding from Angel Investors,” Wall Street Journal, guides.wsj.com/small-business/funding/how-to-get-funding-from-angel-investors (accessed September 19, 2009). Each deal is negotiated separately; be sure to agree to terms with a contract. This is the least regulated area so it’s best to be informed about your angel’s background.“Small Business Loan Sources, Take Aim,” Business Plan Master, www.businessplanmaster.com/small-business-loan-sources.html (accessed September 28, 2009).
• Venture capital firms (also referred to as VCs). VCs usually specialize in investments of \$1 million and above, although there are no hard-and-fast rules. They are looking for a fast return on their investment, especially with the opportunity for an initial public offering (IPO) to take the company public. VCs look for a strong management team and an idea with market potential. Most want a return within three to five years and want to have a say in major decisions that impact the company.“Small Business Loan Sources, Take Aim,” Business Plan Master, www.businessplanmaster.com/small-business-loan-sources.html (accessed September 28, 2009).
• Equipment leasing companies. If you business requires equipment, leasing can be an option that frees up cash and provides an option to buy at the end of the lease.“Small Business Loan Sources, Take Aim,” Business Plan Master, www.businessplanmaster.com/small-business-loan-sources.html (accessed September 28, 2009).
• Government programs. There are many programs at the local, state, and national level designed to support the growth of small businesses. The SBA is only one of the many programs available. Many programs offer opportunities for minorities, business loans, tax incentives, and grants, just to name a few. Research is key to find the program that can potentially support your business; they are not all listed in one place.“Small Business Loan Sources, Take Aim,” Business Plan Master, www.businessplanmaster.com/small-business-loan-sources.html (accessed September 28, 2009).
A complete summary of types of investors is available at www.businessplanmaster.com/small-business-loan-sources.html.
Video Clip
The Best Partner Is No Partner
Investors aren’t for everyone. Bob Parsons, founder and CEO of GoDaddy.com, the world’s largest domain-name registrar, founded his company with his own money and runs it his way. “Nobody’s going to do things like I do,” says Parsons. Everything to support his business is done in-house from the computer software and award-winning customer service to the radio and video recording studios where Parsons records his frequent video blogs and hosts his weekly radio show.Wilson Harrell, “The Way I Work: Bob Parsons, Go Daddy,” Inc., January 1, 2009, www.inc.com/magazine/20090101/the-way-i-work-bob-parsons-go-daddy.html (accessed September 19, 2009).
Watch Bob’s video blog titled “5 Things I Wish I Learned in Business School.”
http://www.bobparsons.me/index.php?ci=13338&id=-1
Selling Your Business Plan—and Yourself
When you present your business plan to anyone—a banker, a lawyer, an accountant, you are asking her to make a commitment to support your business idea. While the contents and details of your business plan are critical to gaining support, you are selling more than your business idea—you are selling yourself. How you communicate your vision and supporting details in a clear, concise, and confident manner can make the difference between getting financial or other support or walking away empty handed.
All the concepts that you have learned in the Selling U section of each chapter apply when you are selling your business plan. Thinking about yourself and your business idea as a brand is where it all starts. Remember from Chapter 1 that a brand is unique, consistent, and relevant and has an emotional connection with its customers. Your personal brand and your business brand need to accomplish the same goal. Fast Company magazine identifies personal marketing as one of the first steps for Gen Y entrepreneurs to start their own business: “One of the most important requirements of entrepreneurship is the ability to sell yourself and your ideas.”Lindsey Pollak, “Gen Y Entrepreneurs: Here Are the First Steps to Starting Your Own Business,” Fast Company, March 15, 2009, www.fastcompany.com/blog/lindsey-pollak/next-generation-career-advice/are-you-gen-y-considering-entrepreneurship-first-s (accessed September 28, 2009).
Evolution of a Business Plan
(click to see video)
Listen to David Fox, founder and CEO of Brave Spirits, discuss the role and evolution of his company’s business plan.
Just as you use your résumé to tell “stories” about your three brand positioning points for your personal brand, your business plan pitch should be equally concise and powerful. Although you have worked for hours (probably months, if not years) on your business plan, your presentation or pitch should focus on the key points and demonstrate not only that it is a potentially profitable business idea (or nonprofit organization that can achieve its goals) but also that you are the right person to make the concept come alive.
Gen Y Entrepreneurs: Get Ready to Sell…Yourself
Fast Company magazine suggests Gen Yers start marketing themselves as a brand even before they start their business. Here are some things you can do now:
• Join professional organizations and become visible, especially to high-profile people in the industry.
• Volunteer at a nonprofit organization that is related to the business you want to start. Demonstrate the quality of your work by working on a committee or major project that is important to moving the organization forward.
• Get a mentor who will give you personal guidance and advice on your career, business idea, and resources.
• Write a blog, post entries to Twitter, share your observations and theories, and get feedback.
• Read everything you can about the industry you want to enter.Lindsey Pollak, “Gen Y Entrepreneurs: Here Are the First Steps to Starting Your Own Business,” Fast Company, March 15, 2009, www.fastcompany.com/blog/lindsey-pollak/next-generation-career-advice/are-you-gen-y-considering-entrepreneurship-first-s (accessed September 28, 2009).
Key Takeaways
• A business plan is a road map for your business and a tool to present your business idea to potential resources and investors.
• A business plan has certain key elements, including a statement of purpose and marketing, operational, and financial plans.
• An investor is a person or organization that provides financial or other support to your business.
• Types of investors include banks, private investors, venture capitalists, equipment-leasing companies, and government programs.
• When you present your business plan to prospective investors, you are selling more than your idea; you are selling yourself.
Exercise \(1\)
1. Assume you are starting an online business called FitMePerfect.com, a Web site where customers can order jeans made to their exact body measurements. What kind of information would you include in the marketing plan section of your business plan?
2. Contact a local bank and talk to the commercial lending officer to find out the process for applying for a business loan.
3. Discuss three things you can do now to prepare for a career as an entrepreneur. | textbooks/biz/Marketing/The_Power_of_Selling/15%3A_Entrepreneurial_Selling-_The_Power_of_Running_Your_Own_Business/15.03%3A_Selling_Yourself_and_Your_Idea.txt |
Learning Objectives
• Meet some experienced entrepreneurs and learn from their successes and failures.
• Understand the resources available to help you pursue your entrepreneurial dream.
Meet Some Entrepreneurs
This chapter is only the beginning of thinking about selling your idea and yourself. It’s designed to inspire the entrepreneurial spirit in you. Selling U is a resource section for each chapter that provides information about how to use the concepts in “real life.” This chapter is no different. The previous chapter sections of Selling U have included resources to build your personal brand; this chapter includes insights and information about how to apply those concepts to pursue a career in an entrepreneurial environment.
Hear and read about the success and lessons from some of the best in the business. If experience is the best teacher, these entrepreneurs are excellent professors.
Business: Gotham Dream Cars
Entrepreneur: Noah Lehmann-Haupt
Video:
How leasing luxury cars became a multimillion-dollar business
www.openforum.com/idea-hub/topics/innovation/video/small-business-rules-gotham-dream-cars-small-business-rules
Business: Jaguar Builders
Entrepreneur: Sherman “West” Westmoreland
Article:
How a quadriplegic became one of Florida’s most successful homebuilders
findarticles.com/p/articles/mi_hb4927/is_2_57/ai_n32018572/
Business: BTTR Ventures (stands for Back to the Roots)
Entrepreneurs: Nikhil Arora and Alex Velez
Article:
Interview with two college seniors started a company to recycle coffee waste
www.fastcompany.com/blog/tom-musbach /control-your-career-destiny/2-green -entrepreneurs-take-coffee-waste
Business: Live Green
Entrepreneur: Steve Ma
Article:
How the organization that connects customers with eco-friendly businesses in the Washington, DC, area got started
findarticles.com/p/articles/mi_7560/is_200907/ ai_n35629263/?tag=content;col1
Business: Deano’s Italian Grille and Brick Oven Pizza, Smokin’ Lobos Barbecue and Grill, Scoops, downtown lofts, and more
Entrepreneur: Josh Nichols
Article:
How Nichols brought entrepreneurship and urban development to Dublin, Florida.
www.allbusiness.com/real-estate/commercial-residential-property-property/12346813-1.html
Business: Chic Gems, Etc.
Entrepreneur: Sarah Radford
Article:
This former top medical sales rep shares how she turned life’s lemons into lemonade by starting her own jewelry design business and her tips for success
www.toiletpaperentrepreneur.com/blog/ guest-post-from-sales-rep-to-fashion-designer- a-story-about-making-lifes-lemons-into-lemonade- by-sarah-radford-of-chic-gems
Business: Sweat Fitness Gyms
Entrepreneur: Scott Caplan
Video:
How Sweat Fitness does what it takes to keep memberships during the recession
www.openforum.com/idea-hub/topics/money/video/msnbc-whats-the-deal-msnbcs-your-business-small-biz-inspiration
Business: CaseStack, a logistics outsourcing company
Entrepreneur: Dan Sanker
Article:
A virtual ride-along with Sanker as he talks about how the business got started and the thrills and challenges of being an entrepreneur
www.e-shadow.com/interview-with-an-entrepreneur- dan-sanker-of-casestack
Business: Stonyfield Farms, organic yogurt products
Entrepreneur: Gary Hirshberg
Video:
How the company became a 25-year overnight success
www.openforum.com/idea-hub/topics/money/video/ make-a-superior-product-ny-times-small-business-summit
Business: Wish of a Lifetime
Entrepreneur: Jeremy Bloom
Article:
How Bloom started his nonprofit organization for seniors
sportsillustrated.cnn.com/vault/article/magazine/MAG1164278/index.htm
Business: Brave Spirits
Entrepreneur: David Fox
Videos:
How Brave Spirits was started
http://www.youtube.com/watch?v=xm-9evHv2wA
Brave Spirits
http://www.bravespirits.com
Video Clip
Generation Next
Global entrepreneurs share their advice for the next generation of entrepreneurs.
www.ey.com/GL/en/Issues/Driving-growth/WEOY-2009---Conversations-with-entrepreneurs---advice-for-future-generations---part-1
www.ey.com/GL/en/Issues/Driving-growth/WEOY-2009---Conversations-with-entrepreneurs---advice-for-future-generations---part-2
Video Clip
Balancing Act
Restaurateur Tom Colicchio (Top Chef) and Sarah Molten (formerly of Gourmet magazine) discuss the balance between being a CEO and entrepreneur.
www.openforum.com/idea-hub/topics/managing/video/ceo-vs-entrepreneur-inside-the-entrepreneurial-mind-series
Link
Secrets of Success
Seven famous entrepreneurs share the secrets of their success…and their mistakes. Wally Amos (Famous Amos Cookie Co.), Daymond John (FUBU), David Liu (The Knot), Jim Koch (The Boston Beer Co.), Liz Lange (Liz Lange Maternity), Adam Lowry and Eric Ryan (Method), and Gary Hirshberg (Stonyfield Farm) share what they learned the hard way in this article that appeared in USA Today.
www.usatoday.com/money/smAllBusiness/startup/week3-famous-founders.htm
Link
Even the Best Fail
Even Steve Jobs, CEO of Apple, has had his share of failures. The article highlights the ten Apple products that didn’t make it.
http://www.newlaunches.com/archives/top_10_apple_products_which_flopped.php
Link
In the Beginning
Ten great entrepreneurs, including Gordon Segal (Crate and Barrel), Robert Redford (Sundance Institute), Mary Kay Ash (Mary Kay Cosmetics), and Jerry Yang (Yahoo!) talk about their start-up days in this slide show.
www.inc.com/ss/10-great-entrepreneurs-talk-about-their-start-days#0
Best in Class
Want to know the best of everything entrepreneurial? Check out these lists, articles, and slide shows. These are just a few of the many resources available. You may also find it helpful to do a Google search to read about successful entrepreneurs in your city.
• Top home-based businesses (slide show). www.inc.com/ss/11-businesses-you-can-start-your-pajamas-2009
• Best industries for starting a business (slide show). www.inc.com/ss/best-industries-for-starting-a-business#0
• Top green companies (slide show). www.inc.com/ss/2009-inc-500-top-green-companies
• Top ten Asian-run companies (slide show). www.inc.com/ss/2009-inc-500-top-10-asian-run-companies
• Top ten Hispanic entrepreneurs of 2009 (article). www.poder360.com/article_detail.php?id_article=2715
• Top ten black entrepreneurs (slide show). www.inc.com/ss/2009-inc-500-top-10-black-run-companies
• Top women entrepreneurs in tech (article). www.fastcompany.com/magazine/132/the-most-influential-women-in-technology-the-entrepreneurs.html
• Top five business books for college entrepreneurs (article). www.college-startup.com/entrepreneur/top-5-business-books-for-college-entrepreneurs
Entrepreneurial Resources
No matter what stage you are in your entrepreneurial journey, there are resources to help you get to the next step. Table \(1\) includes some of the best places to begin your research.
Table \(1\): Entrepreneurial Resources
Resource Name and Web Site Description
Government Resources
Small Business Development Centers
www.sba.gov/aboutsba/sbaprograms/sbdc/
1,100 branch offices of the Small Business Administration; provide free one-on-one counseling and other resources
SCORE (Service Corp of Retired Executives)
www.score.org/index.html
SBA partner; counselors to America’s small businesses; free one-on-one counseling and other resources
SCORE Resources for Young Entrepreneurs
www.score.org/topics/young-entrepreneurs
Free tools and advice to start your journey
Small Business Administration
http://www.sba.gov/
Free tools and resources for small businesses including online Start-up Assessment Tool
United States Chamber Small Business Center
www.uschambersmallbusinessnation.com/toolkits/start-up
Arm of the United States Chamber of Commerce; comprehensive free resources for small businesses
Government Resources for Small Businesses
http://www.businessweek.com/smallbiz/content/ mar2009/sb2009032_175015.htm ?chan=smallbiz_smallbiz +index+page_top+small+business+stories
Free resources from the government to support small businesses
Trade Journals
Inc. magazine
www.inc.com
Online articles and hard copy magazine with news, tips, and insights for and about entrepreneurs
Entrepreneur magazine
http://www.entrepreneur.com/
Online articles and hard copy magazine with news, tips, and insights for and about entrepreneurs
Organizations and Networks
Entrepreneur’s Organization (EO)
www.eonetwork.org/Pages/default.aspx
Online network and chapters throughout the United States
Collegiate Entrepreneurs’ Organization (CEO)
www.c-e-o.org
Online network and chapters throughout the United States
Entrepreneur America
http://www.entrepreneur-america.com
Mentoring services for small businesses and start-ups
Springboard Enterprises
https://www.springboardenterprises.org
National nonprofit dedicated to accelerating women entrepreneurs
Women Entrepreneurs in Science and Technology (WEST)
http://www.westorg.org
Nonprofit organization dedicated to supporting women entrepreneurs in science and technology
International Organizations
The Indus Entrepreneurs (TIE)
www.tie.org
International nonprofit for the advancement of entrepreneurialism
International Council for Small Businesses
http://www.icsb.org
Advancing entrepreneurship worldwide
Research and Resources
Kauffman Foundation
http://www.kauffman.org
Organization to connect entrepreneurs and provide resources
Global Entrepreneurship Monitor
http://www.gemconsortium.org
Organization to connect entrepreneurs and provide resources
A Few Blogs and Twitter Posts Worth Reading
Guy Kawasaki
http://holykaw.alltop.com/
twitter.com/GuyKawasaki
Blog and microblog on entrepreneurial issues, tips, new technology, and more
Seth Godin
http://sethgodin.typepad.com/
Entrepreneurial thoughts of the day
Michael Simmons
twitter.com/michaeldsimmons
http://www.facebook.com/michaeldsimmons#/michaeldsimmons?v=wall
Part motivation, part entrepreneurial insights
Start-up Spark
www.bizzia.com/start-upspark/
Daily insights on entrepreneurial businesses
Mark Cuban
http://blogmaverick.com/
Entrepreneurial insights from the owner of the Dallas Mavericks
Books
Guy Kawasaki’s books
www.guykawasaki.com/books/index.shtml
Seth Godin’s books
www.amazon.com/s/ref=nb_ss?url=search-alias%3Daps&field-keywords=seth+godin&x=0&y=0
Jeffrey Gitomer’s books
www.amazon.com/s/ref=nb_ss?url=search-alias%3Daps&field-keywords=jeffrey+gitomer&x=0&y=0
The best business books by entrepreneurs
www.inc.com/multimedia/slideshows/content/ceo-bookshelf_pagen_1.html
Free and Low-Cost Business Tools and Software
10 Free or Cheap Tools for Start-ups
www.inc.com/ss/10-free-or-cheap-tools-start-ups?nav=mostpopular#0
Great No-Cost Software
www.inc.com/magazine/20090901/great-no-cost-software.html?nav=mostpopular
Miscellaneous (worth checking out)
How-to Guides for Start-ups
www.inc.com/resources/start-up
E-commerce Starter Kit
www.inc.com/guides/sales/20696.html
American Express Open
http://www.openforum.com
Articles, videos, and forum for and about entrepreneurs
Don’t stop here! Look up local entrepreneurial organizations and get involved. Meet people, learn the ropes, and share your passion. If you have the entrepreneurial spirit, go for it and enjoy the journey! | textbooks/biz/Marketing/The_Power_of_Selling/15%3A_Entrepreneurial_Selling-_The_Power_of_Running_Your_Own_Business/15.04%3A_Selling_U_-_Inspiration_Resources_and_Assistance_for_Your_Entrepreneurial_Journey.txt |
Power Wrap-Up
Now that you have read this chapter, you should be able to understand the opportunities and resources available to pursue a career as an entrepreneur.
• You understand that entrepreneurs have a dream and are willing to take the risk to change the way things are done and the way people think.
• You can describe the impact that entrepreneurs have on the economy.
• You can discuss the free market system in which entrepreneurs can thrive.
• You can identify reasons and motivations for being an entrepreneur.
• You can recognize opportunities for social entrepreneurialism.
• You can describe the elements of a business plan.
• You can list the different types of investors.
• You can appreciate the lessons learned from experienced entrepreneurs.
• You can understand what resources are available to help you realize your dream and start your business.
TEST YOUR POWER KNOWLEDGE (AnswerS ARE BELOW)
1. What is entrepreneurship?
2. Name the government agency that oversees all policies and protects the interests of small businesses.
3. Why is a free market system a good environment for entrepreneurs?
4. Describe social entrepreneurialism.
5. Describe a business plan.
6. Name five different types of investors.
7. Identify at least one entrepreneur who inspires you.
8. List three resources that can help you to start your entrepreneurial journey.
POWER (ROLE) PLAY
Now it’s time to put what you’ve learned into practice. The following are two roles that are involved in the same selling situation; one role is that of the investor and the other is that of the entrepreneur. This will give you the opportunity to think about this selling situation from the point of view of both the investor and the entrepreneur.
Read each role carefully along with the discussion questions. Then, be prepared to play either of the roles in class using the concepts covered in this chapter. You may be asked to discuss the roles and do a role-play in groups or individually.
Pitch Your Plan
Role: Potential investor from Gateway Investment Partners
You have invested in several entrepreneurial start-ups over the years, and you are looking for the next great business idea. You’ve been talking to young entrepreneurs, but you haven’t seen a business that you think is compelling. You’re looking for a great idea and a smart, passionate entrepreneur who knows what it takes to build a business.
• What are the characteristics you are looking for in an aspiring entrepreneur?
• Is the idea a viable one? Does it have the potential to be profitable? Has the entrepreneur clearly communicated the opportunity in the business plan?
• Why would you be willing (or not willing) to invest in this business idea?
Role: Entrepreneur
You have an idea for a new business called FILL. It is a store that sells eco-friendly household cleaning products in bulk; customers buy a container or bring their own and fill each with products. Customers may also recycle any containers at the store. The products are sold by the ounce. Your philosophy is to make it easy for customers to save the planet. You have an opportunity to get some seed money to start your business if you successfully pitch your business idea to the potential investor from Gateway Investment Partners.
• How would you sell yourself and your idea to a potential investor?
• How do you make your passion for the idea come alive?
• What are the key points in your business plan?
• How do you use your selling skills to secure funding from this investor for your new business idea?
ACTIVITIES
1. Identify at least one local entrepreneur. Make an appointment to meet her and learn about how she started her business.
2. Watch the video book brief for Guy Kawasaki’s book Reality Check: An Irreverent Guide to Outsmarting, Outmanaging, and Outmarketing Your Competition at www.bnet.com/2422-13724_23-243321.html. Describe the concept of “frame or be framed.”
TEST YOUR POWER KNOWLEDGE AnswerS
1. Starting a new concept as a result of an idea that fills a need and taking the risk to bring the idea to market.
2. Small Business Administration (SBA).
3. Free market system is an economy in which an individual’s success is dictated by demand on the part of the consumer, not by the government, so it encourages entrepreneurialism.
4. Social entrepreneurialism: using the concepts of entrepreneurialism to bring about social change.
5. A business plan is a road map that includes the who, what, when, where, why, and how about the business or organization.
6. Bank, private investor (also called angel investor), venture capitalist, equipment leasing company, and government program.
7. Describe one of the entrepreneurs discussed in the chapter who inspires you.
8. Identify at least one resource from Table 15.1 .
15.06: Epilogue- Youve Got the Power
What a Journey!
Hear some parting thoughts from the author, Kim Richmond.
(click to see video)
You’ve met sales professionals and learned the importance of ethics, communication, and relationships. You’ve practiced the seven steps of the selling process and participated in role-plays to hone your skills. You’ve even learned how to market yourself as a brand and sell yourself to get the internship or job you want.
This is it. This is where it all comes together…the real world. Whether you are graduating or getting ready for your next semester, you’ll be able to use your selling skills to get what you want in life. And don’t forget the tips you learned in Selling U about selling the most important product of all: yourself.
In addition to all the things you learned in this book, here’s the one thing you should remember every day: believe.
Believe in the products and services you sell, believe in your company, believe in your customers. But most of all, believe in yourself. You can do anything you want to do in life with your newfound selling skills and a true belief that you can do it. The fact is you have the skills and the knowledge and now you have the power to achieve. Even on those days when the wind is at your face, remember that you are the most important product you will ever sell. Customers and prospective employers buy you before they buy your products, services, or even your skills. They want to believe in you, and that’s why it’s so important that you believe in yourself.
Put your selling skills to work every day and remember…you’ve got the power! | textbooks/biz/Marketing/The_Power_of_Selling/15%3A_Entrepreneurial_Selling-_The_Power_of_Running_Your_Own_Business/15.05%3A_Review_and_Practice.txt |
Mass Spectrometry is a powerful technique for identifying unknowns, studying molecular structure, and probing the fundamental principles of chemistry. Applications of mass spectrometry include identifying and quantitating pesticides in water samples, identifying steroids in athletes, determining metals at ppq (Parts Per Quadrillion) levels in water samples, carbon-14 dating the Shroud of Turin using only 40 mg of sample (1), looking for life on Mars, determining the mass of an 28Si atom with an accuracy of 70 ppt (2), and studying the effect of molecular collision angle on reaction mechanisms.
Mass spectrometry is essentially a technique for finding the mass by weighing molecules. Obviously, this is not done with a conventional balance or scale. Instead, mass spectrometry is based up on the motion of a charged particle, called an ion, in an electric or magnetic field. The mass-to-charge ratio m/z* of the ion effects this motion and it is actually the mass-to-charge ratio that is determined by the experiment. Since the charge of an electron is known, the mass to charge ratio a measurement of an ion’s mass. Typical mass spectrometry research focuses on the formation of gas phase ions, the chemistry of ions, and applications of mass spectrometry.
This paper covers the basics of mass spectrometry instrumentation and introduces the interpretation of mass spectra. It is only an introduction and interested readers are encouraged to consult more specialized books and journal articles for additional details. The articles and books referenced in this paper should be available at most college and university libraries.
Figure \(1\) is a block diagram that shows the basic parts of a mass spectrometer. The inlet transfers the sample into the vacuum of the mass spectrometer. In the source region, neutral sample molecules are ionized and then accelerated into the mass analyzer. The mass analyzer is the heart of the mass spectrometer. This section separates ions, either in space or in time, according to their mass to charge ratio. After the ions are separated, they are detected and the signal is transferred to a data system for analysis. All mass spectrometers also have a vacuum system to maintain the low pressure, which is also called high vacuum, required for operation. High vacuum minimizes ion-molecule reactions, scattering, and neutralization of the ions. In some experiments, the pressure in the source region or a part of the mass spectrometer is intentionally increased to study these ion-molecule reactions. Under normal operation, however, any collisions will interfere with the analysis.
* The mass to charge ratio, m/z, is used to describe ions observed in mass spectrometry. By convention, m is the numerical value for the mass of the ion and z is the numerical value for the charge of the ion. The unified atomic mass (u) and the elementary charge units e are used for these values. The unified atomic mass is defined as \(1 / 12\) the mass of an atom of 12C. Note: the amu is no longer an accepted term because there are conflicting definitions. Another unit, the dalton, is frequently used for polymers, peptides and other large molecules.
The elementary charge unit is defined as z is an integer equal to the number of electrons lost (or gained for negative ions). For many experiments one electron is lost during ionization so z is +1 and the m/z value is equivalent to the relative molecular mass of the ion. Because the unified atomic mass and the charge number are pure numbers the mass-to-charge ratio is a number and does not have any units. For calculations of the physical behavior of ions it is often necessary to use the actual mass (SI units of kilogram) and charge (SI units of coulomb). | textbooks/chem/Analytical_Chemistry/An_Introduction_to_Mass_Spectrometry_(Van_Bramer)/01%3A_INTRODUCTION.txt |
The selection of a sample inlet depends up on the sample and the sample matrix. Most ionization techniques are designed for gas phase molecules so the inlet must transfer the analyte into the source as a gas phase molecule. If the analyte is sufficiently volatile and thermally stable, a variety of inlets are available. Gases and samples with high vapor pressure are introduced directly into the source region. Liquids and solids are usually heated to increase the vapor pressure for analysis. If the analyte is thermally labile (it decomposes at high temperatures) or if it does not have a sufficient vapor pressure, the sample must be directly ionized from the condensed phase. These direct ionization techniques require special instrumentation and are more difficult to use. However, they greatly extend the range of compounds that may be analyzed by mass spectrometry. Commercial instruments are available that use direct ionization techniques to routinely analyze proteins and polymers with molecular weights greater than 100,000 dalton.
02: SAMPLE INTRODUCTION
Direct Vapor Inlet. The simplest sample introduction method is a direct vapor inlet. The gas phase analyte is introduced directly into the source region of the mass spectrometer through a needle valve. Pump out lines are usually included to remove air from the sample. This inlet works well for gases, liquids, or solids with a high vapor pressure. Samples with low vapor pressure are heated to increase the vapor pressure. Since this inlet is limited to stable compounds and modest temperatures, it only works for some samples.
2.02: Gas Chromatography
Gas chromatography is probably the most common technique for introducing samples into a mass spectrometer. Complex mixtures are routinely separated by gas chromatography and mass spectrometry is used to identify and quantitate the individual components. Several different interface designs are used to connect these two instruments. The most significant characteristics of the inlets are the amount of GC carrier gas that enters the mass spectrometer and the amount of analyte that enters the mass spectrometer. If a large flow of GC carrier gas enters the mass spectrometer it will increase the pressure in the source region.
Probably the most common GC/MS interface uses a capillary GC column. Since the carrier gas flow rate is very small for these columns, the end of the capillary is inserted directly into the source region of the mass spectrometer. The entire flow from the GC enters the mass spectrometer. Since capillary columns are now very common, this inlet is widely used. However this design is not well suited for experiments with wide bore capillaries and packed GC columns which have higher flow rates. The increase in the flow rate significantly increases the pressure in the mass spectrometer and maintaining the required source pressure will require larger and more expensive vacuum pumps. Several inlet designs are available to reduce the gas flow into the source. The simplest design splits the GC effluent so that only a small portion of the total flow enters the mass spectrometer. Although this inlet reduces the gas load on the vacuum system, it also reduces the amount of analyte and thus the sensitivity. Effusive separators and membrane inlets are more selective and transport a higher fraction of the analyte into the source region. Each of these methods has efficiency and resolution drawbacks but they are necessary for some experiments.
2.03: Liquid Chromatography
Liquid Chromatography. Liquid chromatography inlets are used to introduce thermally labile compounds not easily separated by gas chromatography. These inlets have undergone considerable development and LC/MS is now fairly routine. Because these inlets are used for temperature sensitive compounds, the sample is ionized directly from the condensed phase. These inlets are discussed in greater detail in the section on ionization techniques.
2.04: Direct Insertion Probe
Direct Insertion Probe. The Direct Insertion Probe (DIP) is widely used to introduce low vapor pressure liquids and solids into the mass spectrometer. The sample is loaded into a short capillary tube at the end of a heated sleeve. This sleeve is then inserted through a vacuum lock so the sample is inside the source region. After the probe is positioned, the temperature of the capillary tube is increased to vaporize the sample. This probe design allows higher temperatures than are possible with a direct vapor inlet. In addition, the sample is under vacuum and located close to the source so that lower temperatures are required for analysis. This is important for analyzing temperature sensitive compounds. Although the direct insertion probe is more cumbersome than the direct vapor inlet, it is useful for a wider range of samples.
2.05: Direct Ionization of Sample
Direct Ionization of Sample. Unfortunately, some compounds either decompose when heated or have no significant vapor pressure. These samples may be introduced to the mass spectrometer by direct ionization from the condensed phase. These direct ionization techniques include electrospray, matrix assisted laser desorption (MALDI), glow discharge mass spectrometry, fast atom bombardment and laser ablation. The development of new ionization techniques is an active research area and these techniques are rapidly evolving. Direct ionization is discussed in greater detail in the next section. | textbooks/chem/Analytical_Chemistry/An_Introduction_to_Mass_Spectrometry_(Van_Bramer)/02%3A_SAMPLE_INTRODUCTION/2.01%3A_Direct_Vapor_Inlet.txt |
A variety of ionization techniques are used for mass spectrometry. Most ionization techniques excite the neutral analyte molecule which then ejects an electron to form a radical cation M•+. This radical cation is the molecular ion and is produced by removing a single electron from from a neutral molecule. Other ionization techniques involve ion molecule reactions between an ion and a neutral molecule that produce an adduct ion like [M+H]+. Many of these reactions cause the addition of a proton H+ to the analyte molecule but other ions can also be formed. The most important considerations for selecting an ionization technique are the physical state of the analyte and the ionization energy needed. Electron ionization and chemical ionization are only suitable for gas phase ionization. Fast atom bombardment, secondary ion mass spectrometry, electrospray, and matrix assisted laser desorption are used to ionize condensed phase samples. The ionization energy is significant because it controls the amount of fragmentation observed in the mass spectrum. Although this fragmentation complicates the mass spectrum it provides structural information for the identification of unknown compounds. Some ionization techniques are very soft and only produce molecular ions, the intact ionized analyte molecule, while other techniques are very energetic and cause ions to undergo extensive fragmentation.
03: IONIZATION TECHNIQUES
Electron Ionization
Electron Ionization (EI) is the most common ionization technique used for mass spectrometry.* EI works well for many gas phase molecules, but it does have some limitations. Although the mass spectra are very reproducible and are widely used for spectral libraries, EI causes extensive fragmentation so that the molecular ion is not observed for many compounds. The fragmentation is useful because it provides structural information for interpreting unknown spectra. Fragmentation patterns are discussed in more detail in the chapter on Interpretation.
Figure \(1\): Electron Ionization Source
The electrons used for ionization are produced by passing a current through a wire filament (Figure \(1\)). The amount of current controls the number of electrons emitted by the filament. An electric field accelerates these electrons across the source region to produce a beam of high energy electrons. When an analyte molecule passes through this electron beam, a valence shell electron can be removed from the molecule to produce an ion.
Ionization does not occur by electron capture, which is highly dependent upon molecular structure. Instead, EI produces positive ions by knocking a valence electron off the analyte molecule (Figure \(2\)). As the electron passes close to the molecule the negative charge of the electron repels and distorts the electron cloud surrounding the molecule. This distortion transfers kinetic energy from the fast-moving electron to the electron cloud of the molecule. If enough energy is transferred by the process, the molecule will eject a valence electron and form a radical cation M•+.
Since the ionization is produced by a single electron that is accelerated to 70 V, this is commonly referred to as 70 eV EI.** This is enough energy to cause extensive fragmentation, and at this level small changes in the electron energy do not significantly effect the fragmentation patterns. The amount of energy transferred during this process depends up on how fast the electron is traveling and how close it passes to the molecule. In most 70 eV EI experiments, approximately 1400 kJ (15 eV) of energy is transferred during the ionization process. There is, however, a distribution of energy and as much as 2800 kJ (30 eV) is transferred to some molecules. Since approximately 960 kJ/mole (10 eV) of energy is required to ionize most organic compounds and a typical chemical bond energy is 290 kJ/mole (3 eV), extensive fragmentation is often observed in 70 eV EI mass spectra. The distribution of energy transferred during ionization and the large number of fragmentation pathways results in a variety of products for a given analyte. Other electron voltages may be used to vary the amount of fragmentation produced during ionization. For most organic compounds the threshold energy for EI is about 20 eV.
Because a mass spectrum is produced by ionizing many molecules, the spectrum is a distribution of the possible product ions. Intact molecular ions are observed from ions produced with little excess energy. Other molecular ions are formed with more energy and undergo fragmentation in the source region. The abundance of the resulting fragments, often called product ions, is determined by the kinetics of the fragmentation pathways and the ionization energy. Changing the ionization energy changes the observed distribution of fragment ions. This distribution provides the structural information for interpreting mass spectra and is discussed in detail in the section on interpretation.
* Some older literature will refer to EI as electron impact, but this term is not considered accurate. Electron Ionization is the currently accepted term.
** The SI unit for energy is the Joule. The energetics of chemical reactions are typically expressed in kilojoules per mole. In many gas phase experiments (like mass spectrometry), the mole is not a convenient unit. The electron volt is frequently used as an energy unit for single molecules or atoms. 1 eV = 1.60217733(49) x 10-19 J. So that: 1 eV (per molecule or atom) = 96.4152206 kJ/mole.
3.02: Chemical Ionization
Chemical Ionization (1, 2)
Chemical Ionization (CI) is a soft ionization technique that produces ions with little excess energy. As a result, less fragmentation is observed in the mass spectrum. Since this increases the abundance of the molecular ion, the technique is complimentary to 70 eV EI. CI is often used to verify the molecular mass of an unknown. Only slight modifications of an EI source region are required for CI experiments.
In Chemical Ionization the source is enclosed in a small cell with openings for the electron beam, the reagent gas and the sample. The reagent gas is added to this cell at a pressure of approximately 10 Pa (0.1 torr). This is higher than the pressure of 10-3 Pa (10-5 torr) typical for a mass spectrometer source. At 10-3 Pa the mean free path between collisions is approximately 2 meters and ion-molecule reactions are unlikely. In the CI source, however, the mean free path between collisions is only 10-4 meters and analyte molecules undergo many collisions with the reagent gas. The reagent gas in the CI source is ionized with an electron beam to produce a cloud of ions. The reagent gas ions in this cloud react and produce adduct ions like $\mathrm{CH}_{5}^{+}$ (Figure $1$), which are excellent proton donors.
When analyte molecules are introduced to a source region with this cloud of ions, the reagent gas ions donate a proton to the analyte molecule and produce adduct ions, [M+H]+. The energetics of the proton transfer is controlled by using different reagent gases. The most common reagent gases are methane, isobutane and ammonia. Methane is the strongest proton donor commonly used with a proton affinity (PA) of 5.7 eV. For softer ionization, isobutane (PA 8.5 eV) and ammonia (PA 9.0 eV) are frequently used. Acid base chemistry useful for describing these chemical ionization reactions. The reagent gas must be a strong enough Brønsted acid to transfer a proton to the analyte. Fragmentation is minimized in CI by reducing the amount of excess energy produced by the reaction. Because the adduct ion have little excess energy and are relatively stable, CI is very useful for molecular mass determination. Some typical reactions in a CI source are shown in Figure $2$. | textbooks/chem/Analytical_Chemistry/An_Introduction_to_Mass_Spectrometry_(Van_Bramer)/03%3A_IONIZATION_TECHNIQUES/3.01%3A_Electron_Ionization.txt |
Atmospheric Pressure Ionization and Electrospray Ionization. (1, 2, 3)
Atmospheric Pressure Ionization (API) sources ionize the sample at atmospheric pressure and then transfer the ions into the mass spectrometer. These techniques are used to ionize thermally labile samples such as peptides, proteins and polymers directly from the condensed phase. The sample is dissolved in an appropriate solvent and this solution is introduced into the mass spectrometer. With conventional inlets the solvent increases the pressure in the source region of the mass spectrometer. In addition, Joule-Thompson cooling of the liquid as it enters the vacuum causes the solvent droplets to freeze. The frozen clusters trap analyte molecules and reduce the sensitivity of the experiment. No matrix is used and the ionizing beam is focused directly on the sample. Although this makes sampling more difficult, it is useful for studying surface chemistry.
API sources introduce the sample through a series of differentially pumped stages. This maintains the large pressure difference between the ion source and the mass spectrometer (Figure \(1\)) without using extremely large vacuum pumps. In addition a drying gas is used to break up the clusters that form as the solvent evaporates. Because the analyte molecules have more momentum than the solvent and air molecules, they travel through the pumping stages to the mass analyzer.
ElectroSpray Ionization (ESI) is the most common API application. It has undergone remarkable growth in recent y ears and is frequently used for LC/MS of thermally labile and high molecular weight compounds. The electrospray is created by apply ing a large potential between the metal inlet needle and the first skimmer in an API source (Figure \(1\)). The mechanism for the ionization process is not well understood and there are several different theories that explain this ionization process. One theory is that as the liquid leaves the nozzle, the electric field induces a net charge on the small droplets. As the solvent evaporates, the droplet shrinks and the charge density at the surface of the droplet increases. The droplet finally reaches a point where the coulombic repulsion from this electric charge is greater than the surface tension holding it together. This causes the droplet to explode and produce multiply charged analyte ions. A typical ESI spectrum shows a distribution of molecular ions with different charge numbers.
Because electrospray produces multiply charged ions, high molecular weight compounds are observed at lower m/z value. This increases the mass range of the analyzer so that higher molecular weight compounds may be analyzed with a less expensive mass spectrometer. An ion with a mass of 5000 u and a charge of \(+10\) is observed at 500 m/z and is easily analyzed with an inexpensive quadrupole analyzer.
API Sources are also used for Inductively Coupled Plasma Mass Spectrometry (ICP/MS) and glow discharge experiments (4, 5, 6). In ICP/MS a nebulizer is used to introduce liquid samples into a high temperature plasma. The temperature of the plasma is high enough to efficiently ionize most elements. These ions are introduced to the mass spectrometer using an series of differentially pumped regions similar to the electrospray source discussed above. Glow discharge experiments are similar, but used for solid samples. The high sensitivity and selectivity of the mass spectrometer provides rapid multi-element detection at very low levels. Because the high temperature of the plasma destroys any chemical bonds, these techniques are used for elemental analysis.
3.04: Matrix Assisted Laser Desorption Ionization
Matrix Assisted Laser Desorption/Ionization. (1,2)
Matrix Assisted Laser Desorption/Ionization (MALDI) is used to analyze extremely large molecules. This technique directly ionizes and vaporizes the analyte from the condensed phase. MALDI is often used for the analysis of synthetic and natural polymers, proteins, and peptides. Analysis of compounds with molecular weights up to 200,000 dalton is possible and this high mass limit is continually increasing.
In MALDI, both desorption and ionization are induced by a single laser pulse (Figure \(1\) ). The sample is prepared by mixing the analyte and a matrix compound chosen to absorb the laser wavelength. This is placed on a probe tip and dried. A vacuum lock is used to insert the probe into the source region of the mass spectrometer. A laser beam is then focused on this dried mixture and the energy from a laser pulse is absorbed by the matrix. This energy ejects analyte ions from the surface so that a mass spectrum is acquired for each laser pulse. The mechanism for this process is not well understood and is the subject of much controversy in the literature. This technique is more universal (works with more compounds) than other laser ionization techniques because the matrix absorbs the laser pulse. With other laser ionization techniques, the analyte must absorb at the laser wavelength. Typical MALDI spectra include the molecular ion, some multiply charged ions, and very few fragments.
Figure \(1\) : MALDI Ionization Target with Sample and Matrix
Other Ionization Methods. There are several other ionization methods used for mass spectrometry and interested readers are referred to the chemical literature for additional information about other techniques. Field Desorption (3) was used for ionization and vaporization of moderate sized molecules before the development of FAB, electrospray, and MALDI. It is still an important technique for some analysis and is typically used for non-polar polymers and petroleum samples. Plasma Desorption (PD) (4, 5) is a technique used to analyze high molecular weight compounds before the development of MALDI and electrospray. However, it is very complex and has not found widespread application. Resonance Ionization Mass Spectrometry (RIMS) is used for selective atomic and molecular ionization. (6) Photoionization with lasers, lamps, and synchrotron sources is used to study the photochemistry and energetics of many compounds. (7) Lasers are used to ionize surface samples with Laser Microprobe Mass Analysis (LAMMA). (8, 9) | textbooks/chem/Analytical_Chemistry/An_Introduction_to_Mass_Spectrometry_(Van_Bramer)/03%3A_IONIZATION_TECHNIQUES/3.03%3A_Atmospheric_Pressure_Ionization_and_Electrospray_Ionization.txt |
Fast Atom Bombardment and Secondary Ion Mass Spectrometry. (1)
Fast Atom Bombardment (FAB) and Secondary Ion Mass Spectrometry (SIMS) both use high energy atoms to sputter and ionize the sample in a single step. In these techniques, a beam of rare gas neutrals (FAB) or ions (SIMS) is focused on the liquid or solid sample. The impact of this high energy beam causes the analyte molecules to sputter into the gas phase and ionize in a single step (Figure \(1\): Fast Atom Bombardment Source. ). The exact mechanism of this process is not well understood, but these techniques work well for compounds with molecular weights up to a few thousand dalton. Since no heating is required, sputtering techniques (especially FAB) are useful for study ing thermally labile compounds that decompose in conventional inlets (2, 3).
The most significant difference between FAB and SIMS is the sample preparation. In FAB the analyte is dissolved in a liquid matrix. A drop of the sample/matrix mixture is placed at the end of an insertion probe and introduced to the source region. The fast atom beam is focused on this droplet to produce analyte ions. Glycerol or similar low vapor pressure liquids are typically used for the matrix. Ideally, the analyte is soluble in the liquid matrix and a monolayer of analyte forms on the surface of the droplet. According to one theory, this monolayer concentrates the analyte while the dissolved sample provides a reservoir to replenish the monolayer as the analyte is depleted. Without this constant replenishment from the bulk solution, the ionizing beam will rapidly deplete the analyte and the signal is difficult to observe.
SIMS experiments(4) are used to study surface species and solid samples. Liquid SIMS (LSIMS) is very similar to FAB except cesium ions are used for higher energy collisions. No matrix is used and the ionizing beam is focused directly on the sample. Although this makes sampling more difficult, it is useful for studying surface chemistry. High resolution chemical maps are produced by scanning a tightly focused ionizing beam across the surface and depth profiles are produced by probing a single location(5,6). Although SIMS is a very sensitive and powerful technique for surface chemistry and materials analysis, the results are often difficult to quantitate.
3.06: Inductively Coupled Plasma
In addition to use for atomic emission spectroscopy, Inductively Coupled Plasma (ICP) is also used as an ionization method for elemental analysis. A liquid or slurry sample is introduced into an inductively coupled plasma torch and the ions produced are extracted and analyzed by mass spectrometry. These instruments are capable of extremely low detection limits and simultaneous detection of multiple elements.
3.07: Self-Test 1
Exercise \(1\)
What ionization technique would be appropriate for analyzing the following compounds:
1. gasoline fractions,
2. pesticide residue,
3. ibuprofen and acetaminophen,
4. insulin,
5. tripeptides,
6. heavy metals in water.
Answer
1) Gasoline fractions. Since these are very volatile, EI would be very easy to use and would provide abundant fragment information. CI may help to identify the molecular ions.
2) Pesticide residue. These are usually volatile enough to use with EI. Once again CI may provide some useful information that would compliment the fragmentation in the EI spectrum. If the pesticide is thermally labile it may be appropriate to use electrospray to avoid sample decomposition.
3) Ibuprofen and acetaminophen. These pharmaceutics are often analyzed by liquid chromatography, so electrospray would be an ideal interface for ionization.
4) Insulin. This is a large protein molecule. MALDI is probably required.
5) Tripeptides. These are generally small enough to be readily ionized by FAB.
6) Heavy metals in water. Atmospheric pressure ionization in a ICP torch will provide very low limits of detection. | textbooks/chem/Analytical_Chemistry/An_Introduction_to_Mass_Spectrometry_(Van_Bramer)/03%3A_IONIZATION_TECHNIQUES/3.05%3A_Fast_Atom_Bombardment_and_Secondary_Ion_Mass_Spectrometry.txt |
After ions are formed in the source region they are accelerated into the mass analyzer by an electric field. The mass analyzer separates these ions according to their m/z value. Each analyzer design has very different operating characteristics and the selection of an instrument involves important tradeoffs.The selection of a mass analyzer depends up on the resolution, (1) mass range, scan rate and detection limits required for an application.
• Resolution in mass spectrometry refers to the separation of two ions where $\mathrm{R}=\mathrm{m} / \mathrm{\Delta} \mathrm{m}$. These terms are defined several different ways. The most common are the $10 \%$ valley definition "Let two peaks of equal height in a mass spectrum at masses $\mathrm{m}$ and $\mathrm{\Delta} \mathrm{m}$ be separated by a valley that at its lowest point is just $10 \%$ of the height of either peak." and the peak width definition "For a single peak made up of singly charged ions at mass m in a mass spectrum, the resolution may be expressed as $\mathrm{m} / \mathrm{\Delta} \mathrm{m}$, where $\mathrm{\Delta} \mathrm{m}$ is the width of the peak at a height that is a specified fraction of the maximum peak height. It is recommended that one of three values $50 \%, 5 \%$ or $0.5 \%$ be used."(2)
• Mass range refers to the highest mass to charge ratio transmitted by the mass spectrometer.
• The scan rate of a mass spectrometer refers to how fast it scans a mass spectrum. This is important for chromatography applications where the entire mass spectrum must be scanned faster than the elution time of the chromatographic peak. Ideally, a minimum of ten complete mass spectra are acquired for a single chromatographic peak
Analyzers are typically described as either continuous or pulsed. Continuous analyzers include quadrupole filters and magnetic sectors. These analyzers are similar to a filter or monochromator used for optical spectroscopy. They transmit a single selected m/z to the detector and the mass spectrum is obtained by scanning the analyzer so that different mass to charge ratio ions are detected. While a certain m/z is selected, any ions at other m/z ratios are lost, reducing the S/N for continuous analyzers. Single Ion Monitoring(SIM) enhances the S/N by setting the mass spectrometer at the m/z for an ion of interest. Since the instrument is not scanned the S/N improves, but any information about other ions is lost. Pulsed mass analyzers are the other major class of mass analyzer. These are less common but they have some distinct advantages. These instruments collect an entire mass spectrum from a single pulse of ions. This results in a signal to noise advantage similar to Fourier transform or multichannel spectroscopic techniques. Pulsed analyzers include time-of-flight, ion cyclotron resonance, and quadrupole ion trap mass spectrometers.
04: MASS ANALYZERS
The quadrupole mass spectrometer (1) is the most common mass analyzer. Its compact size, fast scan rate, high transmission efficiency,* and modest vacuum requirements are ideal for small inexpensive instruments. Most quadrupole instruments are limited to unit m/z resolution** and have a mass range of 1000 m/z . Many bench-top instruments have a mass range of 500 m/z but research instruments are available with mass range up to 4000 m/z.
In the mass spectrometer, an electric field accelerates ions out of the source region and into the quadrupole analyzer. The analyzer consists of four rods or electrodes arranged across from each other ( Figure $1$). As the ions travel through the quadrupole they are filtered according to their m/z value so that only a single m/z value ion can strike the detector. The m/z value transmitted by the quadrupole is determined by the Radio Frequency (RF) and Direct Current (DC) voltages applied to the electrodes. These voltages produce an oscillating electric field that functions as a bandpass filter to transmit the selected mass to charge value.
The RF voltage rejects or transmits ions according to their m/z value by alternately focusing them in different planes (Figure 9). The four electrodes are connected in pairs and the RF potential is applied between these two pairs of electrodes. During the first part of the RF cycle the top and bottom rods are at a positive potential and the left and right rods are at a negative potential. This squeezes positive ions into the horizontal plane. During the second half of the RF cy cle the polarity of the rods is reversed. This changes the electric field and focuses the ions in the vertical plane. The quadrupole field continues to alternate as the ions travel through the mass analyzer. This causes the ions to undergo a complex set of motions that produces a three-dimensional wave.
The quadrupole field transmits selected ions because the amplitude of this three dimensional wave dep ends up on the m/z value of the ion, the potentials applied, and the RF frequency. By selecting an appropriate RF frequency and potential, the quadrupole acts like a high pass filter, transmitting high m/z ions and rejecting low m/z ions. The low m/z ions have a greater acceleration rate so the wave for these ions has a greater amplitude. If this amplitude is great enough the ions will collide with the electrodes and can not reach the detector. The low m/z value cutoff of the quadrupole is changed by adjusting the RF potential or the RF frequency. Any ions with a m/z greater than this cutoff are transmitted by the quadrupole.
A DC voltage is also applied across the rods of the analyzer. This potential combined with the RF potential acts like a low pass filter to reject high m/z ions. Because they respond quickly to the changing RF field the motion of the low m/z ions is dominated by the RF potential. This motion stabilizes their trajectory by refocusing each time the RF potential changes polarity. Because low m/z ions are quickly refocused, the DC potential does not affect these ions. High m/z ions, however, do not refocus as quickly during the RF cycle. The DC potential has a greater influence on their trajectory and they slowly drift away from the center of the quadrupole. At the end of the analyzer, they are too far off-axis to strike the detector.
The combination of high and low pass filters produced by the RF and DC potentials is adjusted to only transmit the selected m/z value. All ions above or below the set m/z value are rejected by the quadrupole filter. The RF and DC fields are scanned (either by potential or frequency) to collect a complete mass spectrum. Quadrupole mass analyzers are often called mass filters because of the similarity between m/z selection by a quadrupole and wavelength selection by an optical filter or frequency selection by an electronic filter.
*Transmission efficiency refers to how many of the ions produced in the source region actually reach the detector. This is an important measure of sensitivity for mass spectrometers.
**Unit resolution (or low resolution) mass spectra distinguish between ions separated by $1 \mathrm{~m} / \mathrm{z}$ unit. The spectra, like those presented here, are commonly displayed as histograms. This is a common method for presenting spectra because it results in much smaller data file size. Some mass analyzers can obtain spectra at much higher resolution. This is discussed in detail in the interpretation section.
References
1. Steel, C.; Henchman, M. J. Chem. Educ., 1998, 75, 1049-1054. | textbooks/chem/Analytical_Chemistry/An_Introduction_to_Mass_Spectrometry_(Van_Bramer)/04%3A_MASS_ANALYZERS/4.01%3A_Quadrupole.txt |
Magnetic Sector. The first mass spectrometer, built by J.J. Thompson in 1897 , used a magnet to measure the m/z value of an electron. Magnetic sector instruments have evolved from this same concept. Sector instruments have higher resolution and greater mass range than quadrupole instruments, but they require larger vacuum pumps and often scan more slowly. The typical mass range is to 5000 m/z, but this may be extended to 30,000 m/z. Magnetic sector instruments are often used in series with an electric sector, described below, for high resolution and tandem mass spectrometry experiments.
Magnetic sector instruments (Figure $1$) separate ions in a magnetic field according to the momentum and charge of the ion. Ions are accelerated from the source region into the magnetic sector by a 1 to 10 kV electric field. This acceleration is significantly greater than the 100 V acceleration typical for a quadrupole instrument. Since the ions are charged, as they move through the magnetic sector, the magnetic field bends the ion beam in an arc. This is the same principal that causes electric motors to turn. The radius of this arc (r) depends upon the momentum of the ion $\mathrm{\mu}$, the charge of the ion (C) and the magnetic field strength (B) according to Equation $1$.
$r=\frac{\mu}{C \times B} \nonumber$
Ions with greater momentum will follow an arc with a larger radius. This separates ions according to their momentum, so magnetic sectors are often called momentum analyzers. The momentum of the ion is the product of the mass $(m)$ and the velocity $(v)$. The charge of the ion is the product of the charge number of the ion (z) and the charge of an electron (e). Substituting these variables into Equation $1$ yields:
$r=\frac{m / z \times v}{\mathrm{~B} \times \mathrm{e}} \nonumber$
The velocity of an ion is determined by the acceleration voltage in the source region (V) and the mass to charge ratio (m/z) of the ion. Equation $2$ rearranges to give the m/z ion transmitted for a given radius, magnetic field, and acceleration voltage as: $m / z=\frac{r^{2} \mathrm{~B}^{2} \mathrm{e}}{2 \mathrm{~V}} \nonumber$
Only one m/z value will satisfy Equation $3$ for a given radius, magnetic field, and acceleration voltage. Other m/z ions will travel a different radius in the magnetic sector.
Older magnetic sector instruments use a photographic plate to simultaneously detect ions at different radii. Since each m/z has a different radius, they strike the photographic plate at a different location. Modern instruments have a set of slits at a fixed radius to transmit a single m/z to the detector. The mass spectrum is scanned by changing the magnetic field or the acceleration voltage to transmit different m/z ions. Some new instruments use multichannel diode array detectors to simultaneously detect ions over a range of m/z values.
4.03: Electric Sector Double Focusing Mass Spectrometers
Electric Sector/Double Focusing Mass Spectrometers (1). An electric sector consists of two concentric curved plates. A voltage is applied across these plates to bend the ion beam as it travels through the analyzer. The voltage is set so that the beam follows the curve of the analyzer. The radius of the ion trajectory (r) depends up on the kinetic energy of the ion (V) and the potential field (E) applied across the plates.
$r=\frac{2 V}{E} \nonumber$
Equation $1$ shows that an electric sector will not separate ions accelerated to a uniform kinetic energy. The radius of the ion beam is indep endent of the ion’s mass to charge ratio so the electric sector is not useful as a standalone mass analyzer.* An electric sector is, however, useful in series with a magnetic sector. The mass resolution of a magnetic sector is limited by the kinetic energy distribution ( $\mathrm{V}$ ) of the ion beam. This kinetic energy distribution results from variations in the acceleration of ions produced at different locations in the source and from the initial kinetic energy distribution of the molecules. An electric sector significantly imp roves the resolution of the magnetic sector by reducing the kinetic energy distribution of the ions**. These high resolutions experiments are discussed in the section on mass spectral interpretation. The effect of the electric sector is shown in Figure $1$ for a reverse geometry (BE) instrument with the magnetic sector $(B)$ located before the electric sector $(\mathrm{E})$.
*The electric sector is a kinetic energy analyzer. In the source region of the mass spectrometer all ions are accelerated to the same kinetic energy. Because they have the same kinetic energy, they are not separated by an electric sector. A magnetic sector will resolve different mass ions accelerated to a uniform kinetic energy because it separates ions based upon their momentum (See 4.2: Magnetic Sector).
**Ion optics are complex and interested readers are referred to the literature for more detail. The model presented here provides a framework for understanding many high resolution and tandem mass spectrometry experiments. The article by Nier (1) provides an excellent introduction, a historical perspective, and many references for the development and theory of these instruments. | textbooks/chem/Analytical_Chemistry/An_Introduction_to_Mass_Spectrometry_(Van_Bramer)/04%3A_MASS_ANALYZERS/4.02%3A_Magnetic_Sector.txt |
Time-of-Flight. The time-of-flight (TOF) mass analyzer separates ions in time as they travel down a flight tube (Figure $1$). This is a very simple mass spectrometer that uses fixed voltages and does not require a magnetic field. The greatest drawback is that TOF instruments have poor mass resolution, usually less than 500 . These instruments have high transmission efficiency, no upper m/z limit, very low detection limits, and fast scan rates. For some applications these advantages outweigh the low resolution. Recent developments in pulsed ionization techniques and new instrument designs with improved resolution have renewed interest in TOF-MS. (1)
In the source of a TOF analyzer, a packet of ions is formed by a very fast (ns) ionization pulse. These ions are accelerated into the flight tube by an electric field (typically 2-25 kV) applied between the backing plate and the acceleration grid. Since all the ions are accelerated across the same distance by the same force, they have the same kinetic energy. Because velocity $(v)$ is dependent upon the kinetic energy, Equation $1$ shows $\left(\mathrm{E}_{\text {kinetic }}\right)$ and mass $(m)$ lighter ions will travel faster.
$\mathrm{E}_{\text {kinetic }}=\frac{1}{2} m v^{2} \nonumber$
$\mathrm{E}_{\text {kinetic }}$ is determined by the acceleration voltage of the instrument $(\mathrm{V}$ ) and the charge of the ion (e $\times z)$. Equation $2$ rearranges to give the velocity of an ion $(v)$ as a function of acceleration voltage and m/z value.
$v=\sqrt{\frac{2 \mathrm{~V} \times \mathrm{e}}{m / z}} \nonumber$
After the ions accelerate, they enter a 1 to 2 meter flight tube. The ions drift through this field free region at the velocity reached during acceleration. At the end of the flight tube they strike a detector. The time delay ( $\mathrm{t}$ ) from the formation of the ions to the time they reach the detector dependents up on the length of the drift region (L), the mass to charge ratio of the ion, and the acceleration voltage in the source.
$\mathrm{t}=\frac{\mathrm{L}}{\sqrt{\sqrt{2 \times \mathrm{C} \times}}} \sqrt{m / z} \nonumber$
Equation $3$ shows that low m/z ions will reach the detector first. The mass spectrum is obtained by measuring the detector signal as a function of time for each pulse of ions produced in the source region. Because all the ions are detected, TOF instruments have very high transmission efficiency which increases the $\mathrm{S} / \mathrm{N}$ level.
References
1. Cotter, R.J. Anal. Chem. 1992, 64, 1027A-1039A.
4.05: Quadrupole Ion Trap
Quadrupole Ion Trap. (1, 2, 3)
The Quadrupole ion storage trap mass spectrometer (QUISTOR) is a recently developed mass analyzer with some special capabilities. Several commercial instruments are available and this analyzer is becoming more popular. QUISTORs are very sensitive, relatively inexpensive, and scan fast enough for GC/MS exp eriments. The sensitivity of the QUISTOR results from trapping and then analyzing all the ions produced in the source. Since all the ions are detected, the $\mathrm{S} / \mathrm{N}$ is high.
The QUISTOR consists of a doughnut shaped ring electrode and two endcap electrodes. A cutaway view of this arrangement is shown in Figure $1$. A combination of RF and DC voltages is applied to the electrodes to create a quadrup ole electric field similar to the electric field for the quadrupole mass analyzer. This electric field traps ions in a potential energy well at the center of the analyzer. The mass spectrum is acquired by scanning the RF and DC fields to destabilize low mass to charge ions. These destabilized ions are ejected through a hole in one endcap electrode and strike a detector. The mass spectrum is generated by scanning the fields so that ions of increasing $m / z$ value are ejected from the cell and detected. The trap is then refilled with a new batch of ions to acquire the next mass spectrum. The mass resolution of the ion trap is increased by adding a small amount $0.1 \mathrm{~Pa}\left(10^{-3}\right.$ torr $)$ of Helium as a bath gas. Collisions between the analyte ions and the inert bath gas dampen the motion of the ions and increases the trapping efficiency of the analyzer. | textbooks/chem/Analytical_Chemistry/An_Introduction_to_Mass_Spectrometry_(Van_Bramer)/04%3A_MASS_ANALYZERS/4.04%3A_Time-of-Flight.txt |
Ion Cyclotron Resonance. (1, 2, 3) The Ion Cy clotron Resonance (ICR) mass spectrometer uses a superconducting magnet to trap ions in a small sample cell. This type of mass analyzer has extremely high mass resolution (ca. $10^{9}$ ) and is also useful for tandem mass spectrometry experiments. These instruments are very expensive and are typically used for specialized research applications. The ICR traps ions in a magnetic field that causes ions travel in a circular path (Figure $1$). This is similar to the path of an ion in a magnetic sector, but the ions are not traveling as fast and the magnetic field is stronger. As a result the ions are contained in the small volume of the trap.
The ion’s cyclotron frequency $(\omega)$, is the angular frequency* of an ion’s orbit. Equation $1$ shows this frequency is determined by the magnetic field strength $(B)$ and the m/z value of the ion.
$\omega=\frac{\mathrm{B} \times \mathrm{e}}{m / z} \nonumber$
After ions are trapped in this cell they are detected by measuring the signal at this cyclotron frequency. This signal is measured by placing electrodes on each side of the ions circular orbit. An RF voltage is applied to the transmitter electrodes at the cyclotron frequency of the ion of interest. This RF energy moves ions at the applied frequency to a larger orbit. As these ions travel around the ICR cell they are close enough to the receiver electrodes to induce a capacitive current. This capacitive current oscillates at the cyclotron frequency and is detected as the signal.
The ICR is also used as a Fourier Transform Mass Spectrometer (FT-MS). Instead of using a single excitation frequency, a fast RF pulse is applied to the transmitter electrodes. This simultaneously excites all the ions and produces a signal at the cyclotron frequency of each m/z ion present. This signal is similar to the Free Induction Decay (FID) produced in an FT-NMR experiment. A complete mass spectrum is obtained by using the Fourier transform to convert this signal from the time domain to the frequency domain.
*The angular frequency $(\omega)$ is in radians per second. The unit Hertz (Hz) is in cycles per second where there are $2 \mathrm{\pi}$ radians per cycle.
4.07: Self-Test 2
Exercise \(1\)
Self-Test #2: Which mass analyzer is appropriate for the following analysis:
1. Routine analysis of drug testing samples
2. Analysis of small, 2000 dalton, peptides
3. Analysis of 50,000 dalton polymers
4. High sensitivity for detecting chemical warfare agents
5. High resolution analysis.
Answer
Self-Test #2: Which mass analyzer would be appropriate for the following analysis:
1) Routine analysis of drug testing samples. A quadrupole mass analyzer would provide the necessary mass range and resolution. It is also fast enough for use with high resolution chromatography.
2) Analysis of small, 2000 dalton, proteins. This will push the limits of a quadrupole (unless electrospray ionization is used to create multiply charged ions). A sector instrument with FAB ionization would work well.
3) Analysis of polymers up to 50,000 dalton. The value of singly charged ions is probably to high for a sector instrument (It might work with electrospray ionization to form multiply charged ions). A TOF analyzer does not have any mass limit so it would be ideal for this analysis.
4) High sensitivity testing for chemical warfare agents. For this experiment the high sensitivity of a QUISTOR would be beneficial.
5) High resolution analysis. This is usually done with a double focusing sector instrument, although even higher resolution is possible with an ICR. | textbooks/chem/Analytical_Chemistry/An_Introduction_to_Mass_Spectrometry_(Van_Bramer)/04%3A_MASS_ANALYZERS/4.06%3A_Ion_Cyclotron_Resonance.txt |
Deciding the appropriate sample introduction system, ionization technique, and mass analyzer are the major variables in deciding what type of mass spectrometer to use. Operating a mass spectrometer also requires understanding how these are integrated into an operating spectrometer system for acquiring and processing mass spectra. The details of how the data acquisition system work will vary from instrument to instrument and from one manufacturer to another. However, the basics will include control over the vacuum system, the source region, the mass analyzer and the detector.
05: MASS SPECTROMETER SYSTEMS
All mass spectrometers operate at very low pressure (high vacuum). This reduces the chance of ions colliding with other molecules in the mass analyzer. Any collision can cause the ions to react, neutralize, scatter, or fragment. All these processes will interfere with the mass spectrum. To minimize collisions, experiments are conducted under high vacuum conditions, typically 10−2 to 10−5 Pa (10−4 to 10−7 torr) depending up on the geometry of the instrument. This high vacuum requires two pumping stages. The first stage is a mechanical pump that provides rough vacuum down to 0.1 Pa (10−3 torr). The second stage uses diffusion pumps or turbomolecular pump s to provide high vacuum. ICR instruments have even higher vacuum requirements and often include a cryogenic pump for a third pumping stage.
The pumping system is an important part of any mass spectrometer and the control software will allow the user to turn the pumps off and on and monitor the pressure in different parts of the spectrometer. The pumpdown sequence for turning on a spectrometer starts by operating the roughing pumps to establish the initial vacuum and check for major leaks. After the roughing pumps get the system down to a pressure of approximately 0.1 Pa the high vacuum pumps are turned on to establish operating pressure. This sequence is more important with diffusion pumps for the high vacuum system because they do not tolerate atmospheric pressure.
The vacuum system will also include different types of gauges for measuring pressure in different parts of the system. Thermocouple or convectron gauges are used with the roughing pumps to measure pressure down to 0.01 Pa. Ion gauges are used to measure high vacuum down to 10-8 Pa but they cannot be used above 0.1 Pa. To protect the ion gauges and other high voltage electronics the instrument will typically include an interlock system that does not allow power to these components until the roughing pumps have reduced the pressure below a certain threshold. If there is a leak or loss of vacuum the interlock will also turn off power to these systems to protect the components. The thermocouple gauges are normally located at the entrance to any mechanical pumps and ion gauges are normally located in the source and analyzer regions. Depending on the ionization method additional pressure gauges may also be used to monitor the ionization system or any collision regions.
5.02: Source Region Control
The mass spectrometer system will also include controls for the source region. These controls will vary depending upon the ionization technique being used for analysis. For GC/MS and LC/MS systems this software will also control the chromatography system. In general, this requires setting parameters that control the temperature of the sample inlet, determine the ionization energy and efficiency along with parameters that control the efficiency of extracting ions from the source region and transferring them into the analyzer region. These parameters often interact with each other so acquiring spectra with good signal to noise levels requires careful optimization. Typically, this is done using a reference sample and an automated tuning program. The automated tuning program allow the user to set some parameters, like the ionization energy, and the software then varies the other parameters, including voltages on the ion extraction lens systems, to get the best signal possible. The source region for an electrospray mass spectrometer is shown in Figure \(1\)
The reference sample is often a fluorinated compound used for calibration and tuning since fluorine has a single isotope, which simplifies the spectra, and they have relatively high vapor pressure for their mass. Perfluorotributylamine and prefluorokerosene are two common reference standards for gas phase samples. Ultramark 1621, a mixture of fluorinated compounds, is often used for electrospray and FAB.
5.03: Mass Analyzer Control
Control of the mass analyzer requires adjustment of voltages and currents described for the operation of the analyzer and control over voltages on the ion lenses that focus and direct the ions to the detector. The user will need to set the mass range for the analysis. The mass range is limited by the design of the instrument and is an important specification. The range scanned and the speed of the scan are typically adjusted for an experiment. The mass range should cover the expected range for the analyte but scanning a larger range than needed should be avoided. Scanning outside the range needed will either slow down the analysis or just results in the collection of background noise. The scanning speed should be adjusted to balance the speed of the experiment with the signal to noise. Scanning too fast may distort the spectra and reduce S/N levels. Scanning too slow is a significant problem with chromatography or other experiments where the signal is transient. For GC/MS in particular the chromatography peak may only be several seconds wide. The mass spectrum needs to be acquired quickly enough that the analyte concentration is constant for the entire scan. If the concentration varies during the scan it will distort the relative intensity of ions collected at different times.
The mass analyzer needs to be tuned to optimize the efficiency of ion transmission and calibrated to determine the mass scale. This is typically done using a reference sample as part of the tuning process described for the source region. | textbooks/chem/Analytical_Chemistry/An_Introduction_to_Mass_Spectrometry_(Van_Bramer)/05%3A_MASS_SPECTROMETER_SYSTEMS/5.01%3A_Vacuum_System.txt |
Detection of ions is based up on their charge or momentum. For large signals a faraday cup is used to collect ions and measure the current. Older instruments used photographic plates to measure the ion abundance at each mass to charge ratio. Most detectors currently used amplify the ion signal using a collector similar to a photomultiplier tube. These amplifying detectors include: electron multipliers (shown in Figure \(1\)), channeltrons and multichannel plates. The gain is controlled by changing the high voltage applied to the detector. A detector is selected for it’s speed, dynamic range, gain, and geometry. Some detectors are sensitive enough to detect single ions.
The mass spectrometer system will include controls for the gain of the detector. Typically, the gain is adjusted by changing the potential applied to the detector. These voltages are controlled by the software and care should be taken to balance the sensitivity required for the analysis. If the gain is set too low, signal will not be detected, if the gain is set too high the signal will include a lot of noise, the response may not be linear, and the detector life will be shortened.
For GC/MS systems it is typical to use a solvent delay so that the detector is turned off at the start of a run. After the solvent has gone through the system the detector is turned on. This protects the detector from being overloaded by the signal from the solvent but set the gain high enough to see analytes at very low concentration.
5.05: Data System
The final component of a mass spectrometer is the data system. This part of the instrument has undergone revolutionary changes. It has evolved from photographic plates and strip chart recorders to data systems that control the instrument, acquire hundreds of spectra in a minute and search tens of thousands of reference spectra to identify an unknown. Important features of the data system include control over data acquisition and effective data processing.
Critical features for data processing include averaging, subtracting, and deconvolution of spectra. Figure \(1\) shows the background spectra for a mass spectrometer using 70 eV electron ionization. The peak at 44 m/z corresponds to carbon dioxide. Water (18 m/z), nitrogen (28 m/z), and oxygen (32 m/z) are normally observed if they are included in the mass range scanned by the spectrometer. The peak at 207 m/z is from a siloxane compound that is commonly observed in mass spectra and is likely caused by the GC septum used for injection. Other background peaks may be carryover from previous experiments or from the vacuum pump oil. It is a good idea to be familiar with the background peaks and levels for an instrument since changes in the background often indicate possible problems with the instrument.
The data processing software for the system can be used to reduce the background signal in mass spectrum. Figure \(2\) shows two mass spectra from the same time in the chromatogram. The top spectrum is the raw data from the spectrometer. In the bottom spectrum the background signal was subtracted. The background peaks at 77 m/z and 207 m/z are removed and a large number of smaller peaks are also eliminated.
Another important data processing feature is shown in Figure \(3\). This figure shows data for the analysis of caffeine by GC/MS. The top trace is the total ion chromatogram – the sum of the intensity for all masses as a function of time. The bottom trace is the extracted ion chromatograph that only shows the intensity of the 194 m/z signal as a function of time. Since caffeine is the only compound with an ion observed at 194 m/z this is the only peak in the chromatogram. This chromatogram shows a significant reduction in the background noise. This chromatogram was extracted from a full scan at each time in the chromatogram. It is also possible to set up the spectrometer to only monitor a single ion, this is called selective ion monitoring and the technique can significantly enhance the sensitivity of a mass spectrum analysis.
The final data processing technique for discussion here is database searching. Figure \(4\) shows the library search results using NIST MS Search 2.0 for the caffeine peak at 8.54 minutes in the chromatogram. After sending the mass spectrum to the search routine, the program displays likely matches and shows the reference spectra for easy comparison. Search routines like this make it possible to compare an unknown with a large database of target compounds for quick identification. | textbooks/chem/Analytical_Chemistry/An_Introduction_to_Mass_Spectrometry_(Van_Bramer)/05%3A_MASS_SPECTROMETER_SYSTEMS/5.04%3A_Detector_Control.txt |
Although mass spectrometry is a very sensitive instrumental technique, there are other techniques with picogram detection limits. In addition to sensitivity, however, mass spectrometry also is also useful for identifying the chemical structure of this picogram sample. Since the mass spectrum is a fingerprint of the molecular structure, comparison to a computer databases can be used to identify an unknown compound. This is often done using Probability Based Matching (PBM), a popular pattern recognition technique. Although these computer searches are convenient and powerful, it is important to understand how to interpret a mass spectrum. A computer only compares the unknown spectrum to the library spectra and offers a selection of compounds in the database that produce similar spectra. This computer search is very useful and it makes interpretation much easier, but there are limits to the computer search.
Molecular structure is important for understanding mass spectral interpretation. To get the most from this section, draw out the structures of the molecules discussed. During the discussion find which bonds break and calculate the mass of the fragments. Actively reading this section will result in a much greater understanding of and appreciation for mass spectrometry. You can interpret the spectrum but it will take some effort. One common mistake made in mass spectrometry is to blindly trust the results of a computer library match. You need to learn how to interpret and understand the mass spectrum to effectively use these computer searches. This section should help you get started.
06: INTERPRETATION
Molecular Ion
The molecular ion provides the molecular mass of the analyte and is the first clue used to interpret a mass spectrum. The mass to charge ratio of the molecular ion is based up on the mass of the most abundant isotope for each element in the molecule. This is not the relative atomic mass from the periodic table. Since many mass spectrometers have unit mass resolution, the isotope mass is normally rounded to the nearest whole number, this is called the nominal mass. For example the molecular ion for CHBr3 is observed at 250 m/z;(12 + 1 + 3 $\times$ 79)=250), not at the relative molecular weight of 253. The mass of the molecular ion is based upon the mass of the isotope with the highest natural abundance. The most common bromine isotope is 79Br. Do not use the weighted average atomic weight for Br (79.9) which is based upon the natural abundance of 79Br and 81Br. The mass spectrum of CHBr3 includes ions for all the naturally occurring isotopes and the intensity of each peak depends upon the probability for that combination of isotopes. These patterns are discussed in detail in the section on isotope abundance.
In many mass spectra, the molecular ion is easily identified as the ion with the highest mass to charge ratio. However, this assignment should be made with caution because the highest mass to charge ion may be an impurity, an isotope of the molecular ion, or a fragment. Many compounds fragment easily and no molecular ion is observed in the 70 eV EI spectrum. It is important to clarify that the molecular ion IS NOT necessarily the ion with the greatest abundance, the ion with the greatest abundance is called the base peak. The base peak is the peak with the greatest abundance. The mass spectrum is usually normalized so that this peak has an intensity of 100.
A list of molecular ion characteristics are in Table $1$ to help you identify them in a mass spectrum. Low energy EI, where the ionization energy is reduced, often increases in intensity of the molecular ion. Chemical Ionization, CI, is also useful for identifying the molecular ion since the the adduct ion is often more stable than the radical cation produced by electron ionization. The adduct ion is often formed by protonating the analyte to form $(\mathrm{M}+\mathrm{H})$ and is observed at a mass to charge ratio of M+1.
Table $1$: Characteristics of Molecular Ions
The mass to charge ratio must correspond to a reasonable molecular formula with the proper isotope abundance.
Most compounds have an even molecular mass. The one common exception to this is the "Nitrogen Rule" discussed below.
The Nitrogen Rule: Any compound with an odd number of nitrogen atoms will have an odd molecular mass. Any compound with an even number of nitrogen atoms (including zero) will have an even molecular mass. This is because nitrogen is the only common atom where the most common isotope has an odd valence and an even mass. For example: the molecular ion for CH4 is observed at 16 m/z, the molecular ion for NH3 is observed at 17 m/z, and the molecular ion for N2H4 is observed at 32 m/z.
If a peak is the molecular ion, the next highest mass fragment must correspond to the loss of a possible neutral fragment. For example, a peak that corresponds to loss of 5 u from the molecular ion is highly unlikely
Figure $1$ shows the mass spectrum of acetone (CH3COCH3). The molecular ion is clearly shown at 58 m/z (12 x 3 + 6 x 1 + 16 = 58). The base peak is at 43 m/z and corresponds to loss of 15 m/z from the intact molecule, this is caused by breaking a C-C bond for loss of a CH3 radical to give CH3CO+ at 43 m/z (12 x 2 + 3 x 1 + 16 = 43). The mass spectrum also includes several other minor peaks - the peak at 59 m/z is caused by the small abundance of C-13 that gives a small fraction of the acetone molecules a mass of 59; the peak at 15 m/z is caused by the CH3 fragment retaining the charge when the C-C bond breaks. These fragmentation and isotope patterns are discussed in more detail in the following sections. | textbooks/chem/Analytical_Chemistry/An_Introduction_to_Mass_Spectrometry_(Van_Bramer)/06%3A_INTERPRETATION/6.01%3A_Molecular_Ion.txt |
Fragmentation
Although the molecular ion is useful for identification, it does not provide any structural information about an unknown. The structural information is obtained from the fragmentation patterns of the mass spectrum. Identifying an unknown without analyzing the fragmentation patterns is like putting together a jigsaw puzzle without the picture. Fragmentation patterns are often complex, but they fit together like pieces of the puzzle to identify the structure of the molecule.
After a molecule is ionized, the molecular ion retains the excess ionization energy. If this excess energy is greater than the energy required to break a chemical bond, the molecule can fragment. The fragmentation processes are typically categorized as direct cleavage where a single bond is broken or rearrangement where bonds are broken and created simultaneously (Figure $1$ ).
The molecular ion formed by electron ionization is an odd electron ion, a radical species with an unpaired electron. These ions are formed by removing a lone pair electron or a bonding electron from a molecule during ionization. For example, water is ionized by removing a non-bonding electron from oxygen to produces $\mathrm{H}_{2} \mathrm{O}^{+1}$. This is an example of an odd electron ion. Odd electron ions have an even mass to charge value. The exception to this is if the ion has an odd number of nitrogen atoms. Calculate the mass to charge value for some molecular ions to verify this statement.
When a molecular ion fragments by direct cleavage a single bond is broken to produce two fragments. This usually separates the charge and the radical of the molecular ion. Direct cleavage produces an even electron ion, AB+, and a neutral odd electron radical, CD. The even electron ion is detected at an odd mass to charge value (assuming no nitrogen) and since the radical is a neutral fragment it is not observed in the mass spectrum. Even electron ions have all paired electrons. An example of this was shown in the mass spectrum of acetone where the molecular ion, CH3–CO–CH3+•, fragments to form CO–CH3+ - an even electron ion observed at an 43 m/z. The radical CH3 has an odd mass but since it is neutral this fragment is not observed in the mass spectrum. It is possible for the charge and radical species to switch. As a result, cleavage of CH3–CO–CH3+• can also form CH3+ which is an even electron ion observed at 15 m/z, an odd mass. The radical formed by this cleavage, CO–CH3, would not be observed. In the mass spectrum of acetone the 43 m/z peak is much larger than the 15 m/z peak, so the formation of CO–CH3+ at 43 m/z is clearly favored. When you are interpreting mass spectra look for possible cleavage fragments but keep in mind that either or both of the fragments may be observed in the mass spectrum.
Rearrangements are more complex reactions that involve both making and breaking bonds. These reactions are thermodynamically favorable because they require less energy. However they also require a concerted mechanism that is not as kinetically favorable when compared to a simple cleavage reaction. Rearrangement ions are easily identified because they are observed as odd electron ions with an even m/z value. These fragments often provide important clues about the location and identity of certain functional groups. Rearrangements are discussed in more detail in the next section.
The mass spectra of 4 different $\mathrm{C}_{4} \mathrm{H}_{10} \mathrm{O}$ isomers are shown in Figures 16 - 19. These spectra show how cleavage patterns help to identify the structure of a compound. It is important to remember that determining the molecular formula is just the first step in interpretation of mass spectra.
1-Butanol
At this point get a piece of scratch paper, draw a Lewis dot structure for 1-butanol, find the mass of the molecular ion, break some bonds and find the mass of some possible fragments. Then look to see which of these fragments are observed in the mass spectrum. This exercise will take some time, but the practice will help you learn how to interpret mass spectra. After you have some possibilities, take a look at Figure $2$ and see what you can find.
For 1-butanol the molecular ion should be observed at 74 m/z (4x12 + 10x1 + 1x16 = 74). There is a very small peak at this location, which is not unusual alcohols - and many other compound classes. If you look at the mass spectra for a large number of alcohols you will notice that they often show little or no molecular ion intensity. This makes interpreting their spectra challenging and IR spectra - which very clearly show OH functional groups - compliment mass spectra by helping to identify these functional groups. If you know from the IR that a compound is an alcohol you can be careful about identifying the molecular ion, knowing that it may not be observed.
Next look at possible cleavage fragments from the molecular ion. One possibility is loss of a hydrogen to give 73 m/z. There is a small peak at 73 m/z in the mass spectrum - that indicates that this fragmentation is possible but it is is not common. The same is also true for loss of OH - which is observed with a small peak at 57 m/z (74 - 17). The next loss is alpha-clevage, breaking the C-C bond next to the OH functional group, to form CH2OH+, observed at 31 m/z (CH2OH+) or the compliment, CH3CH2CH2+, observed at 43 m/z. Since the charge could be retained by either fragment both are observed in the spectrum. Alpha-cleavage is a common fragmentation pattern for alcohols, so observing a peak at 31 m/z is useful for identifying primary alcohols.
The 1-butanol spectrum also has a major peak at 56 m/z. This is an even mass ion so it is not formed by breaking a single bond. Looking at the loss from the molecular ion to this fragment (74 - 56 = 18) is a clue to the identity. Alcohols often undergo loss of water (H2O - 18 m/z, so 56 m/z is a likely peak for 1-butanol. This rearrangement is favorable because water is very stable and the resulting radical ion, CH2=CH-CH2-CH3+, has the same structure as an alkene. Rearrangements are much more likely when they create a stable species. The other significant peak in this mass spectrum is at 41 m/z. It is not possible to get this mass from breaking a single bond so it must also involve some sort of rearrangement. It is not unusual for fragmentation and loss of H2 to occur so this ion could be formed by alpha-cleavage followed by H2 loss. Since H2 is an intact molecule, this fragmentation is energetically favorable although it also requires some rearrangement.
2-Butanol
The next spectrum to examine is 2-butanol. Before looking at the mass spectrum draw the Lewis dot structure for 2-butanol and determine the mass of the possible alpha-cleavage fragments. Then compare your results with the spectrum in Figure $3$.
Since 2-butanol has the same molecular formula as 1-butanol, C4H10O, it also has the same molecular ion at 74 m/z. The molecular ion is not seen in Figure 17 but there are several very informative fragment ions observed in the mass spectrum. From the analysis of 1-butanol, it is reasonable to look for alpha cleavage fragments. Since this is a secondary alcohol, there are two possible alpha-cleavage locations for 2-butanol. Alpha-cleavage could result in loss of CH3 or C2H5 to produce ions observed at 59 m/z (74 - 15) and 45 m/z (74 - 29) respectively. Both of these peaks are observed in Figure 17 and their high intensity clearly distinguish this mass spectrum from 1-butanol. The compliment ions, CH3+ or C2H5+ are observed at 15 m/z and 29 m/z but are not particularly useful for identification since they are present in almost all organic mass spectra.
2-Methyl-1-Propanol
The next spectrum to examine is 2-methyl-1-propanol. Before looking at the mass spectrum in Figure $4$, draw the Lewis dot structure for 2-methyl-1-propanol and determine the mass of the possible alpha cleavage fragments. Then compare what you find with the spectrum below.
Since 2-methyl-1-propanol has the same molecular formula as 1-butanol and 2-butanol, C4H10O, it also has the same molecular ion at 74 m/z. Although the molecular ion at 74 m/z was not readily observed in the previous two spectra, it is clearly seen for 2-methyl-1-propanol in Figure 18. Based on the discussion of the previous two spectra we should also look for alpha cleavage fragments. Based on the structure for 2-methyl-1-propanol alpha-cleavage would result in loss of C3H7 to form CH2OH+ which is observed at 31 m/z and is characteristic of a primary alcohol. The complement ion C3H7+ at 43 m/z is also observed in Figure 18. Since the C3H7+ at 43 m/z is a secondary carbocation it is more stable than the C3H7+ ion formed from the fragmentation of 1-butanol. As a result the peak at 43 m/z is the base peak in the spectum of 2-methyl-1-propanol. The relative intensity of peaks like this is very important for distinguishing the mass spectra of similar compounds. You can also compare the relative intensity of the peaks at 31 m/z and 43 m/z in the spectra of 1-butanol and 2-methyl-1-propanol.
2-Methyl-2-Propanol
The next spectrum to examine is 2-methyl-2-propanol. Before looking at the mass spectrum draw the Lewis dot structure for 2-methyl-2-propanol and determine the mass of the possible alpha cleavage fragments. Then compare what you find with the spectrum in Figure $5$.
The molecular ion is not observed for 2-methyl-1-propanol in Figure 19. However, the alpha-cleavage peak showing loss of CH3 at 59 m/z is the base peak and is far more abundant than any other ion in the spectrum. There are two reasons for this, the first is that there are three different locations in the structure where alpha-cleavage results in loss of CH3. This alone would increase the probability of forming 59 m/z but the additional consideration is that the C3H7O+ carbocation produced by alpha-cleavage is a tertiary carbocation. As a result it is much more stable and therefore less likely to undergo further fragmentation.
C4H10O Summary
It is clear from the spectra shown in Figures 16-19 that the mass spectra for these four different C4H10O structures are readily distinguished based on the alpha-cleavage patterns. Learning common fragmentation patterns for different functional groups is very helpful for identifying unknowns and for distinguishing the spectra for similar compounds.
Toluene
The mass spectrum for toluene is shown in Figure $6$. Given the stability of aromatic compounds it should not be surprising that the molecular ion at 92 m/z has a high intensity. The base peak observed at 91 m/z is interesting because loss of H is not typically this intense. It turns out that the tropylium ion, C7H7+, is also aromatic and so this fragment is very stable and often has a high intensity. | textbooks/chem/Analytical_Chemistry/An_Introduction_to_Mass_Spectrometry_(Van_Bramer)/06%3A_INTERPRETATION/6.02%3A_Fragmentation.txt |
Heptane
The mass spectrum of heptane is shown in Figure \(1\). This mass spectrum is consistent with the fragmentation patterns discussed in the previous section. The molecular ion, C7H16•+ is observed at 100 m/z and a series of cleavage peaks are observed for loss of CH3 (M - 15), C2H5 (M - 29), and C3H7 (M - 43). These peaks are observed at 85 m/z , 71 m/z , and 57 m/z respectively. This fragmentation is characteristic for a linear hydrocarbon.
McLafferty Rearrangement
Some functional groups, however, can undergo very different fragmentation processes than the direct cleavage discussed so far. One common example is the McLafferty rearrangement (Figure \(2\)) which results in formation of an intact neutral molecule and a radical ion, both with an even mass to charge ratio. Since the most intense direct cleavage fragments have odd mass to charge ratios, this fragmentation pattern is very useful for identifying carbonyl compounds and for determining their structure. The McLafferty rearrangement is often observed for carbonyl compounds that contain a linear alkyl chain. If this alkyl chain is long enough, a six-membered ring forms from the carbonyl oxygen to the hydrogen on the fourth carbon. This spacing allows the hydrogen to transfer to the carbonyl oxygen via a six membered ring. This is followed by a rearrangement of the electrons to break the beta C-C bond, the second bond from the carbonyl carbon, to form an alkene and resonance stabilized radical with the carbonyl group. The McLafferty rearrangement is energetically favorable because it results in loss of a neutral alkene and formation of a resonance stabilized radical. Both these fragments may be observed in the mass spectrum, depending upon which fragment retains the charge. Figure \(2\) shows the charge on the resonance stabilized radical, this is the McLafferty ion. The alkene is referred to as the McLafferty compliment.
The products from the McLafferty rearrangement are observed in the mass spectra of C-7 carbonyl compounds shown in Figures 22-26. Draw structures for the following compounds and use the mechanism shown in Figure 21 to predict the mass of the two fragments formed by the McLafferty rearrangement. Then compare these predictions with the mass spectra shown in Figures \(3\) - \(7\).
• heptanal
• 2-heptanone
• 3-heptanone
• 4-heptanone
• heptanoic acid
Heptanal
The mass spectrum of heptanal shown in Figure \(3\) contains two even mass ions. C2H4O+ m/z 44 is produced by the McLafferty rearrangement of an aldehyde and is a characteristic peak that is very useful for interpretation of aldehydes. The McLafferty compliment, C5H10+, is observed at 70 m/z . The McLafferty compliment is produced when the charge is transferred to the alkene fragment during the rearrangement.
2-Heptanone
The mass spectrum of 2-hepanone shown in Figure \(4\) is easily distinguished from heptanal because the McLafferty rearrangement breaks the C-C bond between C-3 and C-4. This results in loss C4H8 to give the McLafferty ion for a 2-ketone, C3H6O+, at 58 m/z. The McLafferty compliment, C4H8+ (56 m/z) is not observed for 2-heptanone.
3-Heptanone
The mass spectrum of 3-hepanone in Figure \(5\) is easily distinguished from heptanal and 2-heptanone because the McLafferty rearrangement breaks the C-C bond between C-4 and C-5. This results in loss C3H6 to give the McLafferty ion for a 3-ketone, C4H8O+, at 72 m/z. The McLafferty compliment, C3H6+ (42 m/z) is not observed for 3-heptanone.
4-Heptanone
The mass spectrum of 4-hepanone shown in Figure \(6\) is easily distinguished from heptanal, 2-heptanone, and 3-heptanone. The McLafferty rearrangement would break the C-C bond between C-2 and C-3. This results in loss C2H4 to give the McLafferty ion for a 4-ketone, C5H10O+, at 86 m/z - which has a very low intensity in the mass spectrum of 4-heptanone shown in figure 25. The two major peaks in this spectrum 43 m/z and 71 m/z correspond to alpha cleavage to produce C3H7 and C4H7O which would be observed at 43 m/z and 71 m/z respectively. In this molecule the direct cleavage is highly favored over the McLafferty rearrangement. In this case the faster kinetics of the direct cleavage are favored over the concerted mechanism required for the rearrangement.
Heptanoic Acid
The mass spectrum of heptanoic acid shown in Figure \(7\) is easily distinguished from heptanal, 2-heptanone, 3-heptanone, and 4-heptanone because the McLafferty rearrangement produces C2H4O2+ observed at 60 m/z and characteristic of a carboxylic acid. In this case the McLafferty compliment, C5H8+, is not observed in the mass spectrum.
Based up on the discussion so far you should be able to identify many of the other fragments in these three mass spectra. Spend some time with a piece of scratch paper and see what you come up with. | textbooks/chem/Analytical_Chemistry/An_Introduction_to_Mass_Spectrometry_(Van_Bramer)/06%3A_INTERPRETATION/6.03%3A_Rearangement.txt |
Isotope Abundance.
The existence of isotopes was first observed by Aston using a mass spectrometer to study neon ions. When interpreting mass spectra it is important to remember that the relative atomic mass or atomic weight of an element is a weighted average of the naturally occurring isotopes. Mass spectrometers separate these isotopes and they are each observed at their respective mass to charge ratio. The relative abundance used to determine the relative atomic mass is determined using mass spectrometry. Although this complicates the mass spectrum, it also provides useful information for identifying the elements in an ion. Chlorine is an excellent example of how isotope distributions are useful for interpretation. The molecular weight of chlorine is $35.45 \mathrm{u}$. This is calculated from the natural abundance of ${ }^{35} \mathrm{Cl}(75 \%)$ and ${ }^{37} \mathrm{Cl}(25 \%)$. To avoid ambiguity the molecular ion is defined as the ion with the most commonly occurring isotopes. For $\mathrm{CH}_{3} \mathrm{Cl}$ the molecular ion is ${ }^{12} \mathrm{C}^{1} \mathrm{H}_{3}{ }^{35} \mathrm{Cl}$ at 50 m/z .
Chlorine Isotope Abundance
The natural abundance of these two isotopes is observed in the mass spectrum as two peaks separated by 2 m/z with a relative intensity of $3: 1$. The mass spectrum of chlorobenzene C6H5Cl in Figure $1$ clearly shows the chlorine isotope distribution at 112 m/z and 114 m/z . These peaks correspond to the molecular ion - the molecular ion has the most abundant isotope for each element - at 112 m/z (6x12 + 5x1 + 35) and the 37Cl isotope peak at 114 m/z (6x12 + 5x1 + 37) and the relative intensity is determined by the natural abundance of the 37Cl isotope. The other major peak in this spectrum at 77 m/z corresponds to the loss of chlorine from the molecular ion or the 37Cl isotope peak to give C6H5+ (112 - 35 = 77 OR 114 - 37 = 77).
If more than one chlorine atom is present, the isotope abundance is more complex. An ion with two chlorine atoms has three possible isotope combinations. This pattern is apparent in the mass spectrum of CH2Cl2 shown in Figure $2$. Ions are observed for $\mathrm{CH}_{2}{ }^{35} \mathrm{Cl}_{2}^{+}$ at 84 m/z, $\mathrm{CH}_{2}{ }^{35} \mathrm{Cl}^{37} \mathrm{Cl}^{+}$ at 86 m/z , and $\mathrm{CH}_{2}{ }^{37} \mathrm{Cl}_{2}^{+}$ at 88 m/z . Based up on the probability of each combination of isotopes, the relative intensity of these peaks is $10: 6: 1$. The $3: 1$ isotope ratio for an ion with a single chlorine atom is observed at 49 m/z and 51 m/z . This corresponds to $\mathrm{CH}_{2}{ }^{35} \mathrm{Cl}^{+}$and $\mathrm{CH}_{2}{ }^{37} \mathrm{Cl}^{+}$fragments formed by loss of $\mathrm{Cl}$ from the molecular ion. Careful examination of the spectrum also shows ions produced by loss of H and $\mathrm{H}_{2}$.
Bromine Isotope Abundance
Bromine also has two naturally occurring isotopes, 79Br is the most abundant and 81Br has a relative abundance of 98% which results in a relative intensity for these two peaks of 1:1. This is observed in the mass spectrum of bromobenzene shown in Figure $3$. The bromine isotope pattern is seen in the peaks at 156 m/z and 158 m/z which have the 1:1 relative abundance characteristic of bromine. These two peaks correspond to the molecular ion C6H579Br at 156 m/z and C6H581Br at 158 m/z . The base peak in this spectrum is from loss of Br to form C6H5 observed at 77 m/z .
Carbon 13 isotope peak
The $1.1 \%$ of natural abundance of ${ }^{13} \mathrm{C}$ is another useful tool for interpreting mass spectra. The abundance of a peak one m/z value higher, where a single ${ }^{12} C$ is replaced by a ${ }^{13} C$, is determined by the number of carbons in the ion. The rule of thumb for small compounds is that each carbon atom in the ion increases the abundance of the $M+1$ peak by $1 \%$. This effect is seen in all the spectra discussed in this paper. For example, in the $n$-decane mass spectrum (Figure $4$ ) compare the peak for ${ }^{12} \mathrm{C}_{9}{ }^{13} \mathrm{C}^{1} \mathrm{H}_{22}$ at 143 m/z (0.38 % relative abundance) to the peak for ${ }^{12} \mathrm{C}_{10}{ }^{1} \mathrm{H}_{22}$ at 142 m/z (3.96% relative abundance). The abundance of the $13 \mathrm{C}$ peak is $10 \%$ the abundance of the ${ }^{12} \mathrm{C}$ peak, consistent with a compound containing 10 carbon atoms. Now look at some previous spectra to find more examples of this pattern. Be aware that for compounds with low molecular ion abundances the uncertainty in measuring this ratio may be +/- several carbon atoms.
Isotope Abundances
Because all atoms have several naturally occurring isotopes, the patterns discussed here become more complex. Fortunately, most elements common in organic mass spectrometry have one predominant isotope. The high abundance of the two chlorine isotopes is unusual, so they are easy to identify. The relative abundances for isotopes of frequently encountered elements are given in Table $1$. For molecules with more complex isotope patterns there are a number of programs and websites available for modeling the distributions. The calculator provided by Scientific Instrument Services is available at: https://www.sisweb.com/mstools/isotope.htm.
Table $1$: Isotope Abundances. Adapted from McLafferty, F. Interpretation of Mass Spectra (University Science, Mill Valley CA: 1980.
Atom Isotope A Istope A+1 Isotope A+2
mass % mass % mass %
H 1 100 2 0.015
C 12 100 13 1.1
N 14 100 15 0.37
O 16 100 17 0.04 18 0.20
F 19 100
Si 28 100 29 5.1 30 3.4
P 31 100
S 32 100 33 0.80 34 4.4
Cl 35 100 37 32.5
Br 79 100 81 98.0
I 127 100 | textbooks/chem/Analytical_Chemistry/An_Introduction_to_Mass_Spectrometry_(Van_Bramer)/06%3A_INTERPRETATION/6.04%3A_Isotope_Abundance.txt |
Functional groups can have a significant effect the fragmentation patterns observed in mass spectrometry and textbooks on mass spectrometry cover a large range of common fragmentation patters for different functional groups. For a detailed discussion of this, interested readers are encouraged to look at any of the following books:
As one final example aliphatic amines often undergo cleavage at the $\alpha { } \mathrm{C}-\mathrm{C}$ bond to produce a relatively stable $\mathrm{CH}_{2} \mathrm{NH}_{2}{ }^{+}$ion (Figure $1$ ). The resulting fragments distinguish primary, secondary, and tertiary amines.
Figure $1$: $\alpha$-Cleavage fragmentation of an amine.
This fragmentation is useful for distinguishing mass spectra of the three different C4H11N isomers. Draw the structure of 1-butanamine, 2-butanamine, 2-methyl-1-propanamine, and 2-methyl-2-propanamine. Determine the mass to charge ratio for the molecular ion, identify the site for alpha-cleavage for each molecule, and determine the mass to charge ratio for the expected fragments. After you have done this, look up the mass spectra for these four compounds in the NIST Chemistry WebBook (https://webbook.nist.gov/) which contains mass spectra for a large number of compounds.
All four compounds have the same molecular formula, C4H11N with an odd number of nitrogen atoms so the molecular ion has an odd mass to charge ratio. The molecular ion is observed for all four compounds at 73 m/z .
1-butanamine. The $\alpha$-cleavage fragment for 1-butanamine produces CH2NH2+ at 30 m/z and C3H7. The C3H7 fragment has a very low intensity in the mass spectrum because since the charge is retained by the nitrogen containing fragment. See NIST Webbook for the mass spectrum of 1-butanamine.
2-butanamine. There are two $\alpha$-cleavage sites for 2-butanamine. Loss of $\mathrm{CH}_{3}^{\prime}$ produces $\mathrm{C}_{3} \mathrm{H}_{6} \mathrm{NH}_{2}{ }^{+}$ (58 m/z) and loss of $\mathrm{C}_{2} \mathrm{H}_{5}^{\prime}$ produces $\mathrm{C}_{2} \mathrm{H}_{4} \mathrm{NH}_{2}{ }^{+}$ (44 m/z). Both of these ions are observed but the greater abundance of the 44 m/z signal indicates that loss of $\mathrm{C}_{2} \mathrm{H}_{5}^{\prime \prime}$ is favored. See NIST Webbook for the mass spectrum of 2-butanamine.
2-methyl-1-propanaimne. The $\alpha$-cleavage fragment for 2-methyl-1-propanamine produces CH2NH2+ at 30 m/z and C3H7. The C3H7 fragment has a very low intensity in the mass spectrum because since the charge is retained by the nitrogen containing fragment. The resulting mass spectrum is very similar to 1-butanamine and distinguishing these two isomers by mass spectrometry will depend on careful comparison of the relative intensity of the molecular ion and other fragments observed in the mass spectrum. The See NIST Webbook for the mass spectrum of 2-methyl-1-propanamine.
2-methyl-2-propanaimne. The $\alpha$-cleavage fragment for 2-methyl-2-propanaimne produces C3H6NH2+ at 58 m/z and CH3. The CH3 fragment has a very low intensity in the mass spectrum because since the charge is retained by the nitrogen containing fragment. See NIST Webbook for the mass spectrum of 1-butanamine.
6.06: Exact Mass
Exact Mass. In most mass spectrometry experiments the nominal mass is used and the mass to charge ratio of an ion is rounded to the nearest whole number. High resolution instruments, including double focusing and FT-ICR mass spectrometers, are capable of determining the "exact mass" of an ion. This is useful for interpretation because each element has a slightly different mass defect. This "mass defect" is the difference between the mass of the isotope and the nominal mass (which is equivalent to the number of protons and neutrons).
Recall that the atomic mass scale is defined by carbon-12 with a mass of exactly $12.0000$ u. The exact mass of a specific isotope is determined relative to ${ }^{12} \mathrm{C}$ by high resolution mass spectrometry (see Table $1$). High resolution mass spectrometry can distinguish compounds with the same nominal mass but different exact mass caused by different elemental composition.
For example, $\mathrm{C}_{2} \mathrm{H}_{6}, \mathrm{CH}_{2} \mathrm{O}$, and $\mathrm{NO}$ all have a nominal mass of 30 u. Because they have the same nominal mass, a mass spectrometer with unit mass resolution can not distinguish these three ions. However, the exact masses for $\mathrm{C}_{2} \mathrm{H}_{6}(30.04695039), \mathrm{CH}_{2} \mathrm{O}(30.01056487)$ and $\mathrm{NO}^{2}$ (29.99798882) are different and a high resolution mass spectrometer can distinguish these three compounds.
Table $1$ lists the exact mass for the most abundant isotopes of several common elements. The Isotope Distribution Calculator on the SIS website will also calculate the exact mass for any chemical formula. This is available online at: https://www.sisweb.com/mstools/isotope.htm
Table $1$: Adopted from DiFlippo, F.; et. al. Phys Rev Lett. 1994, 73, 1482 .
Element Isotoope Mass
H 1H 1.007 825 031 6 (5)
2H 2.014 101 777 9 (5)
He 4He 4.002 603 36
3He 3.016 0
C 12C 12.000 000 000 0 (0)
13C 13.003 354 838 1 (10)
N 14N 14.003 074 004 0 (12)
15N 15.000 108 897 7 (11)
O 16O 15.994 914 619 5 (21)
17O 17.999 2
P 31P 30.973 763 3
S 32S 31.972 072 8
34S 33.967 9
Values in parentheses indicate error in last digit.
This section is only an introduction to the interpretation of mass spectra. A full analysis of fragmentation patterns is beyond the scope of this text but with practice interpretation becomes much easier. Several excellent references include McLafferty’s book (35) and the ACOL book on mass spectrometry (36). These contain additional information on mass spectral interpretation and many more practice problems. | textbooks/chem/Analytical_Chemistry/An_Introduction_to_Mass_Spectrometry_(Van_Bramer)/06%3A_INTERPRETATION/6.05%3A_Amine_Fragmentation.txt |
1. You work for a mass spectrometer vendor who sells all the instruments described here. Make a product table that shows all the different options for each major part of the mass spectrometer.
2. Using thermochemical data, find the energy required to remove an electron from the following species: $\mathrm{H}_{2}, \mathrm{Na}, \mathrm{C}, \mathrm{CH}_{4}$, and Fe. Express this energy in $\mathrm{kJ} / \mathrm{mole}$ and eV (per atom).
3. Using thermochemical data, determine the energy required to break the following bonds: $\mathrm{H}-\mathrm{H}, \mathrm{C}-\mathrm{C}, \mathrm{C}=\mathrm{C}, \mathrm{C}-\mathrm{H}$, and $\mathrm{C}-\mathrm{O}$. Express this energy in $\mathrm{kJ} / \mathrm{mole}$ and $\mathrm{eV}$ (per atom).
4. Determine the kinetic energy, velocity, and momentum of the following ions: $m / z 10,50$, 100 , and 1000. The ions are formed in the center of the source region, which is $10.0 \mathrm{~mm}$ long and has a $5000 \mathrm{~V}$ potential applied across the two ends.
5. How long would it take each of the ions from question 4 to travel through a 1 meter flight tube in a TOF mass spectrometer?
6. What magnetic field is required to select each of the ions from question 4 in a magnetic sector with a radius of 1 meter?
7. What is the cy clotron frequency of each ion from question 4 in an ICR with a 3 T magnetic field?
8. What electric field strength is required for each of the ions from question 4 to be selected by an electric sector with a radius of $0.5$ meter?
9. What is the $m / z$ value for the molecular ion produced by EI of the following molecules: A) benzene, B) octane, C) trinitrotoluene, D) acetone, E) t-butyl amine.
10. The highest mass ion observed in a mass spectrum is at $m / z 127$. If the compound contains a single $\mathrm{N}$ atom, could this be the molecular ion? If the compound contains four $\mathrm{N}$ atoms, could this be the molecular ion? Why?
11. The molecular ion in a high resolution mass spectrometry experiment is observed at $m / z$ 58.0055. What is the molecular formula for this compound? If this was a low resolution mass spectrometer, what other molecular formula’s are possible?
08: ACKNOWLEDGMENTS
ACKNOWLEDGMENTS:
I would like to thank all the people who helped to review this paper. Including: Nate Bower, Murray Johnston, Gordon Nicol, Gary Kinsel, Phil Ross, Pat McKeown, Curt Mowry, and an anonymous reviewer. Their comments and suggestions have been invaluable. Mass Spectra are all original spectra.
LITERATURE CITED:
1. Warner, M. Anal. Chem. 1989, 61, 101A-103A.
2. 2.DiFlippo, F.; et. al. Phys Rev Lett. 1994, 73,1482.
3. Munson, B. Anal. Chem. 1977, 49, 772A-778A.
4. Munson, B.; Field, F. J. Am. Chem. Soc., 1966, 88, 2621-2630.
5. Barber, M.; Bordoli, R.S.; Elliott, G.J.; Sedgwick, R.D.; Tyler, A.N. Anal. Chem. 1982, 54, 645A-657A.
6. Fenselau, C. Anal. Chem. 1982, 54, 105A-114A.
7. Biemann, K. Anal. Chem. 1986, 58, 1288A-1300A.
8. Day, R.J.; Unger, S.e.; Cooks, R.G. Anal. Chem. 1980, 82, 557A-572A.
9. Winograd, N. Anal. Chem. 1993, 65, 622A-629A.
10. Benninghoven, A.; Hagenhoff, B.; Niehuis, E. Anal. Chem. 1993, 65, 630A-640A.
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12. Smith, R.D.; Wahl. J.H.; Goodlett, D.R.; Hofstadler, S.A. Anal. Chem. 1993, 65, 574A-584A.
13. Hofstadler, S.; Bakhtiar, R.; Smith, R. J. Chem. Educ. 1996, 73, A82-A88.
14. Harrison, W.W.; Hess, K.R.; Marcus, R.K.; King, F.L. Anal. Chem. 1986, 58, 341A-356A.
15. Houk, R.S. Anal. Chem. 1986, 58, 97A-105A.
16. Vela, N.P.; Olson, L.K.; Caruso, J.A. Anal. Chem. 1993, 65, 585A-597A.
17. Karas, M.; Hillenkamp, F. Anal. Chem. 1988, 60, 2299-2301.
18. Fenselau, C. Anal. Chem. 1997, 69, 661A-665A.
19. Lattimer, R.P.; Schulten, H.R. Anal. Chem. 1989, 61, 1201A-1215A.
20. M acfarlane, R.D. Anal. Chem. 1983, 55, 1247A-1264A.
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23. Syage Anal. Chem. 1990, 62, 505A.
24. Van Grieken, R.; Adams, F.; Natusch, D. Anal. Chem. 1982, 54, 26A-41A.
25. Hercules, D.M.; Day, R.J.; Balasanmugam, K.; Dang, T.A.; Li, C.P. Anal. Chem. 1982, 54, 280A-305A.
26. Price, P. J. Am. Soc. Mass Spectrum., 1991, 2, 336-348.
27. Steel, C.; Henchman, M. J. Chem. Educ., 1998, 75, 1049-1054.
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29. Cotter, R.J. Anal. Chem. 1992, 64, 1027A-1039A.
30. Allison, J.; Stepnowski, R.M. Anal. Chem. 1987, 59, 1072A-1088A.
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36. Davis, R. Mass Spectrometry/Analytical Chemistry by Open Learning; Wiley: New York, 1987. | textbooks/chem/Analytical_Chemistry/An_Introduction_to_Mass_Spectrometry_(Van_Bramer)/07%3A_END_OF_CHAPTER_QUESTIONS-.txt |
Chemistry is the study of matter, including its composition, its structure, its physical properties, and its reactivity. Although there are many ways to study chemistry, traditionally we divide it into five areas: organic chemistry, inorganic chemistry, biochemistry, physical chemistry, and analytical chemistry. This division is historical and, perhaps, arbitrary, as suggested by current interest in interdisciplinary areas, such as bioanalytical chemistry and organometallic chemistry. Nevertheless, these five areas remain the simplest division that spans the discipline of chemistry.
Each of these traditional areas of chemistry brings a unique perspective to how a chemist makes sense of the diverse array of elements, ions, and molecules (both small and large) that make up our physical environment. An undergraduate chemistry course, therefore, is much more than a collection of facts; it is, instead, the means by which we learn to see the chemical world from a different perspective. In keeping with this spirit, this chapter introduces you to the field of analytical chemistry and highlights the unique perspectives that analytical chemists bring to the study of chemistry.
• 1.1: What is Analytical Chemistry
Let’s begin with a deceptively simple question: What is analytical chemistry? Like all areas of chemistry, analytical chemistry is so broad in scope and so much in flux that it is difficult to find a simple definition more revealing than that quoted above. In this chapter we will try to expand upon this simple definition by saying a little about what analytical chemistry is, as well as a little about what analytical chemistry is not.
• 1.2: The Analytical Perspective
Having noted that each area of chemistry brings a unique perspective to the study of chemistry, let’s ask a second deceptively simple question: What is the analytical perspective? Many analytical chemists describe this perspective as an analytical approach to solving problems.
• 1.3: Common Analytical Problems
Typical problems on which analytical chemists work include qualitative analyses (Is lead present in this sample ?), quantitative analyses (How much lead is present in this sample?), characterization analyses (What are the sample’s chemical and physical properties?), and fundamental analyses (How does this method work and how can it be improved?).
• 1.4: Problems
End-of-chapter problems to test your understanding of topics in this chapter.
• 1.5: Additional Resources
A compendium of resources to accompany topics in this chapter.
• 1.6: Chapter Summary and Key Terms
Summary of chapter's main topics and a list of key terms introduced in the chapter.
Thumbnail: Several graduated cylinders of various thickness and heights with white side markings in front of a large beaker. They are all filled about halfway with red or blue chemical compounds. The blue ink is showing signs of Brownian motion when dissolving into water. (CC BY-SA 3.0; Horia Varlan from Bucharest, Romania).
01: Introduction to Analytical Chemistry
“Analytical chemistry is what analytical chemists do.”
This quote is attributed to C. N. Reilly (1925-1981) on receipt of the 1965 Fisher Award in Analytical Chemistry. Reilly, who was a professor of chemistry at the University of North Carolina at Chapel Hill, was one of the most influential analytical chemists of the last half of the twentieth century.
For another view of what constitutes analytical chemistry, see the article “Quo Vadis, Analytical Chemistry?”, the full reference for which is Valcárcel, M. Anal. Bioanal. Chem. 2016, 408, 13-21.
Let’s begin with a deceptively simple question: What is analytical chemistry? Like all areas of chemistry, analytical chemistry is so broad in scope and so much in flux that it is difficult to find a simple definition more revealing than that quoted above. In this chapter we will try to expand upon this simple definition by saying a little about what analytical chemistry is, as well as a little about what analytical chemistry is not.
Analytical chemistry often is described as the area of chemistry responsible for characterizing the composition of matter, both qualitatively (Is there lead in this paint chip?) and quantitatively (How much lead is in this paint chip?). As we shall see, this description is misleading.
Most chemists routinely make qualitative and quantitative measurements. For this reason, some scientists suggest that analytical chemistry is not a separate branch of chemistry, but simply the application of chemical knowledge [Ravey, M. Spectroscopy, 1990, 5(7), 11]. In fact, you probably have performed many such quantitative and qualitative analyses in other chemistry courses.
You might, for example, have determined the concentration of acetic acid in vinegar using an acid–base titration, or used a qual scheme to identify which of several metal ions are in an aqueous sample.
Defining analytical chemistry as the application of chemical knowledge ignores the unique perspective that an analytical chemist bring to the study of chemistry. The craft of analytical chemistry is found not in performing a routine analysis on a routine sample—a task we appropriately call chemical analysis—but in improving established analytical methods, in extending these analytical methods to new types of samples, and in developing new analytical methods to measure chemical phenomena [de Haseth, J. Spectroscopy, 1990, 5(7), 11].
Here is one example of the distinction between analytical chemistry and chemical analysis. A mining engineers evaluates an ore by comparing the cost of removing the ore from the earth with the value of its contents, which they estimate by analyzing a sample of the ore. The challenge of developing and validating a quantitative analytical method is the analytical chemist’s responsibility; the routine, daily application of the analytical method is the job of the chemical analyst.
The Seven Stages of an Analytical Method
1. Conception of analytical method (birth).
2. Successful demonstration that the analytical method works.
3. Establishment of the analytical method’s capabilities.
4. Widespread acceptance of the analytical method.
5. Continued development of the analytical method leads to significant improvements.
6. New cycle through steps 3–5.
7. Analytical method can no longer compete with newer analytical methods (death).
Steps 1–3 and 5 are the province of analytical chemistry; step 4 is the realm of chemical analysis.
The seven stages of an analytical method listed here are modified from Fassel, V. A. Fresenius’ Z. Anal. Chem. 1986, 324, 511–518 and Hieftje, G. M. J. Chem. Educ. 2000, 77, 577–583.
Another difference between analytical chemistry and chemical analysis is that an analytical chemist works to improve and to extend established analytical methods. For example, several factors complicate the quantitative analysis of nickel in ores, including nickel’s unequal distribution within the ore, the ore’s complex matrix of silicates and oxides, and the presence of other metals that may interfere with the analysis. Figure 1.1.1 outlines one standard analytical method in use during the late nineteenth century [Fresenius. C. R. A System of Instruction in Quantitative Chemical Analysis; John Wiley and Sons: New York, 1881]. The need for many reactions, digestions, and filtrations makes this analytical method both time-consuming and difficult to perform accurately.
The discovery, in 1905, that dimethylglyoxime (dmg) selectively precipitates Ni2+ and Pd2+ led to an improved analytical method for the quantitative analysis of nickel [Kolthoff, I. M.; Sandell, E. B. Textbook of Quantitative Inorganic Analysis, 3rd Ed., The Macmillan Company: New York, 1952]. The resulting analysis, which is outlined in Figure 1.1.2 , requires fewer manipulations and less time. By the 1970s, flame atomic absorption spectrometry replaced gravimetry as the standard method for analyzing nickel in ores, resulting in an even more rapid analysis [Van Loon, J. C. Analytical Atomic Absorption Spectroscopy, Academic Press: New York, 1980]. Today, the standard analytical method utilizes an inductively coupled plasma optical emission spectrometer.
Perhaps a more appropriate description of analytical chemistry is “the science of inventing and applying the concepts, principles, and...strategies for measuring the characteristics of chemical systems” [Murray, R. W. Anal. Chem. 1991, 63, 271A]. Analytical chemists often work at the extreme edges of analysis, extending and improving the ability of all chemists to make meaningful measurements on smaller samples, on more complex samples, on shorter time scales, and on species present at lower concentrations. Throughout its history, analytical chemistry has provided many of the tools and methods necessary for research in other traditional areas of chemistry, as well as fostering multidisciplinary research in, to name a few, medicinal chemistry, clinical chemistry, toxicology, forensic chemistry, materials science, geochemistry, and environmental chemistry.
To an analytical chemist, the process of making a useful measurement is critical; if the measurement is not of central importance to the work, then it is not analytical chemistry.
You will come across numerous examples of analytical methods in this textbook, most of which are routine examples of chemical analysis. It is important to remember, however, that nonroutine problems prompted analytical chemists to develop these methods.
An editorial in Analytical Chemistry entitled “Some Words about Categories of Manuscripts” highlights nicely what makes a research endeavor relevant to modern analytical chemistry. The full citation is Murray, R. W. Anal. Chem. 2008, 80, 4775; for a more recent editorial, see “The Scope of Analytical Chemistry” by Sweedler, J. V. et. al. Anal. Chem. 2015, 87, 6425. | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/01%3A_Introduction_to_Analytical_Chemistry/1.01%3A_What_is_Analytical_Chemistry.txt |
Having noted that each area of chemistry brings a unique perspective to the study of chemistry, let’s ask a second deceptively simple question: What is the analytical perspective? Many analytical chemists describe this perspective as an analytical approach to solving problems.
For different viewpoints on the analytical approach see (a) Beilby, A. L. J. Chem. Educ. 1970, 47, 237-238; (b) Lucchesi, C. A. Am. Lab. 1980, October, 112-119; (c) Atkinson, G. F. J. Chem. Educ. 1982, 59, 201-202; (d) Pardue, H. L.; Woo, J. J. Chem. Educ. 1984, 61, 409-412; (e) Guarnieri, M. J. Chem. Educ. 1988, 65, 201-203, (f) Strobel, H. A. Am. Lab. 1990, October, 17-24.
Although there likely are as many descriptions of the analytical approach as there are analytical chemists, it is convenient to define it as the five-step process shown in Figure 1.2.1 .
Three general features of this approach deserve our attention. First, in steps 1 and 5 analytical chemists have the opportunity to collaborate with individuals outside the realm of analytical chemistry. In fact, many problems on which analytical chemists work originate in other fields. Second, the heart of the analytical approach is a feedback loop (steps 2, 3, and 4) in which the result of one step requires that we reevaluate the other steps. Finally, the solution to one problem often suggests a new problem.
Analytical chemistry begins with a problem, examples of which include evaluating the amount of dust and soil ingested by children as an indicator of environmental exposure to particulate based pollutants, resolving contradictory evidence regarding the toxicity of perfluoro polymers during combustion, and developing rapid and sensitive detectors for chemical and biological weapons. At this point the analytical approach involves a collaboration between the analytical chemist and the individual or agency working on the problem. Together they determine what information is needed and clarify how the problem relates to broader research goals or policy issues, both essential to the design of an appropriate experimental procedure.
These examples are taken from a series of articles, entitled the “Analytical Approach,” which for many years was a regular feature of the journal Analytical Chemistry.
To design the experimental procedure the analytical chemist considers criteria, such as the required accuracy, precision, sensitivity, and detection limit, the urgency with which results are needed, the cost of a single analysis, the number of samples to analyze, and the amount of sample available for analysis. Finding an appropriate balance between these criteria frequently is complicated by their interdependence. For example, improving precision may require a larger amount of sample than is available. Consideration also is given to how to collect, store, and prepare samples, and to whether chemical or physical interferences will affect the analysis. Finally a good experimental procedure may yield useless information if there is no method for validating the results.
The most visible part of the analytical approach occurs in the laboratory. As part of the validation process, appropriate chemical and physical standards are used to calibrate equipment and to standardize reagents.
The data collected during the experiment are then analyzed. Frequently the data first is reduced or transformed to a more readily analyzable form and then a statistical treatment of the data is used to evaluate accuracy and precision, and to validate the procedure. Results are compared to the original design criteria and the experimental design is reconsidered, additional trials are run, or a solution to the problem is proposed. When a solution is proposed, the results are subject to an external evaluation that may result in a new problem and the beginning of a new cycle.
Chapter 3 introduces you to the language of analytical chemistry. You will find terms such accuracy, precision, and sensitivity defined there. Chapter 4 introduces the statistical analysis of data. Calibration and standardization methods, including a discussion of linear regression, are covered in Chapter 5. See Chapter 7 for a discussion of how to collect, store, and prepare samples for analysis. See Chapter 14 for a discussion about how to validate an analytical method.
As noted earlier some scientists question whether the analytical approach is unique to analytical chemistry. Here, again, it helps to distinguish between a chemical analysis and analytical chemistry. For an analytically-oriented scientist, such as a physical organic chemist or a public health officer, the primary emphasis is how the analysis supports larger research goals that involve fundamental studies of chemical or physical processes, or that improve access to medical care. The essence of analytical chemistry, however, is in developing new tools for solving problems, and in defining the type and quality of information available to other scientists.
Exercise 1.2.1
As an exercise, let’s adapt our model of the analytical approach to the development of a simple, inexpensive, portable device for completing bioassays in the field. Before continuing, locate and read the article
“Simple Telemedicine for Developing Regions: Camera Phones and Paper-Based Microfluidic Devices for Real-Time, Off-Site Diagnosis”
by Andres W. Martinez, Scott T. Phillips, Emanuel Carriho, Samuel W. Thomas III, Hayat Sindi, and George M. Whitesides. You will find it on pages 3699-3707 in Volume 80 of the journal Analytical Chemistry, which was published in 2008. As you read the article, pay particular attention to how it emulates the analytical approach and consider the following questions:
1. What is the analytical problem and why is it important?
2. What criteria did the authors consider in designing their experiments? What is the basic experimental procedure?
3. What interferences were considered and how did they overcome them? How did the authors calibrate the assay?
4. How did the authors validate their experimental method?
5. Is there evidence that steps 2, 3, and 4 in Figure 1.2.1 are repeated?
6. Was there a successful conclusion to the analytical problem?
Don’t let the technical details in the paper overwhelm you; if you skim over these you will find the paper both well-written and accessible.
Answer
What is the analytical problem and why is it important?
A medical diagnoses often relies on the results of a clinical analysis. When you visit a doctor, they may draw a sample of your blood and send it to the lab for analysis. In some cases the result of the analysis is available in 10-15 minutes. What is possible in a developed country, such as the United States, may not be feasible in a country with less access to expensive lab equipment and with fewer trained personnel available to run the tests and to interpret the results. The problem addressed in this paper, therefore, is the development of a reliable device for rapidly performing a clinical assay under less than ideal circumstances.
What criteria did the authors consider in designing their experiments?
In considering a solution to this problem, the authors identify seven important criteria for the analytical method: (1) it must be inexpensive; (2) it must operate without the need for much electricity, so that it can be used in remote locations; (3) it must be adaptable to many types of assays; (4) its must not require a highly skilled technician; (5) it must be quantitative; (6) it must be accurate; and (7) it must produce results rapidly.
What is the basic experimental procedure?
The authors describe how they developed a paper-based microfluidic device that allows anyone to run an analysis simply by dipping the device into a sample (synthetic urine, in this case). The sample moves by capillary action into test zones containing reagents that react with specific species (glucose and protein, for this prototype device). The reagents react to produce a color whose intensity is proportional to the species’ concentration. A digital photograph of the microfluidic device is taken using a cell phone camera and sent to an off-site physician who uses image editing software to analyze the photograph and to interpret the assay’s result.
What interferences were considered and how did they overcome them?
In developing this analytical method the authors considered several chemical or physical interferences. One concern was the possibility of non-specific interactions between the paper and the glucose or protein, which might lead to non-uniform image in the test zones. A careful analysis of the distribution of glucose and protein in the text zones showed that this was not a problem. A second concern was the possibility that particulate materials in the sample might interfere with the analyses. Paper is a natural filter for particulate materials and the authors found that samples containing dust, sawdust, and pollen do not interfere with the analysis for glucose. Pollen, however, is an interferent for the protein analysis, presumably because it, too, contains protein.
How did the author’s calibrate the assay?
To calibrate the device the authors analyzed a series of standard solutions that contained known concentrations of glucose and protein. Because an image’s intensity depends upon the available light, a standard sample is run with the test samples, which allows a single calibration curve to be used for samples collected under different lighting conditions.
How did the author’s validate their experimental method?
The test device contains two test zones for each analyte, which allows for duplicate analyses and provides one level of experimental validation. To further validate the device, the authors completed 12 analyses at each of three known concentrations of glucose and protein, obtaining acceptable accuracy and precision in all cases.
Is there any evidence of repeating steps 2, 3, and 4 in Figure 1.2.1?
Developing this analytical method required several cycles through steps 2, 3, and 4 of the analytical approach. Examples of this feedback loop include optimizing the shape of the test zones and evaluating the importance of sample size.
Was there a successful conclusion to the analytical problem?
Yes. The authors were successful in meeting their goals by developing and testing an inexpensive, portable, and easy-to-use device for running clinical samples in developing countries.
This exercise provides you with an opportunity to think about the analytical approach in the context of a real analytical problem. Practice exercises such as this provide you with a variety of challenges ranging from simple review problems to more open-ended exercises. You will find answers to practice exercises at the end of each chapter.
Use this link to access the article’s abstract from the journal’s web site. If your institution has an on-line subscription you also will be able to download a PDF version of the article. | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/01%3A_Introduction_to_Analytical_Chemistry/1.02%3A_The_Analytical_Perspective.txt |
Many problems in analytical chemistry begin with the need to identify what is present in a sample. This is the scope of a qualitative analysis, examples of which include identifying the products of a chemical reaction, screening an athlete’s urine for a performance-enhancing drug, or determining the spatial distribution of Pb on the surface of an airborne particulate. An early challenge for analytical chemists was developing simple chemical tests to identify inorganic ions and organic functional groups. The classical laboratory courses in inorganic and organic qualitative analysis, still taught at some schools, are based on this work.
See, for example, the following laboratory textbooks: (a) Sorum, C. H.; Lagowski, J. J. Introduction to Semimicro Qualitative Analysis, 5th Ed.; Prentice-Hall: Englewood, NJ, 1977; (b) Shriner, R. L.; Fuson, R. C.; Curtin, D. Y. The Systematic Identification of Organic Compounds, 5th Ed.; John Wiley and Sons: New York, 1964.
Modern methods for qualitative analysis rely on instrumental techniques, such as infrared (IR) spectroscopy, nuclear magnetic resonance (NMR) spectroscopy, and mass spectrometry (MS). Because these qualitative applications are covered adequately elsewhere in the undergraduate curriculum, typically in organic chemistry, they receive no further consideration in this text.
Perhaps the most common analytical problem is a quantitative analysis, examples of which include the elemental analysis of a newly synthesized compound, measuring the concentration of glucose in blood, or determining the difference between the bulk and the surface concentrations of Cr in steel. Much of the analytical work in clinical, pharmaceutical, environmental, and industrial labs involves developing new quantitative methods to detect trace amounts of chemical species in complex samples. Most of the examples in this text are of quantitative analyses.
Another important area of analytical chemistry, which receives some attention in this text, are methods for characterizing physical and chemical properties. The determination of chemical structure, of equilibrium constants, of particle size, and of surface structure are examples of a characterization analysis.
The purpose of a qualitative, a quantitative, or a characterization analysis is to solve a problem associated with a particular sample. The purpose of a fundamental analysis, on the other hand, is to improve our understanding of the theory that supports an analytical method and to understand better an analytical method’s limitations.
A good resource for current examples of qualitative, quantitative, characterization, and fundamental analyses is Analytical Chemistry’s annual review issue that highlights fundamental and applied research in analytical chemistry. Examples of review articles in the 2015 issue include “Analytical Chemistry in Archaeological Research,” “Recent Developments in Paper-Based Microfluidic Devices,” and “Vibrational Spectroscopy: Recent Developments to Revolutionize Forensic Science.”
1.04: Problems
1. For each of the following problems indicate whether its solution requires a qualitative analysis, a quantitative analysis, a characterization analysis, and/or a fundamental analysis. More than one type of analysis may be appropriate for some problems.
1. The residents in a neighborhood near a hazardous-waste disposal site are concerned that it is leaking contaminants into their groundwater.
2. An art museum is concerned that a recently acquired oil painting is a forgery.
3. Airport security needs a more reliable method for detecting the presence of explosive materials in luggage.
4. The structure of a newly discovered virus needs to be determined.
5. A new visual indicator is needed for an acid–base titration.
6. A new law requires a method for evaluating whether automobiles are emitting too much carbon monoxide.
2. Read the article “When Machine Tastes Coffee: Instrumental Approach to Predict the Sensory Profile of Espresso Coffee,” which discusses work completed at the Nestlé Research Center in Lausanne, Switzerland. You will find the article on pages 1574-1581 in Volume 80 of Analytical Chemistry, published in 2008. Prepare an essay that summarizes the nature of the problem and how it was solved. Do not worry about the nitty-gritty details of the mathematical model developed by the authors, which relies on a combination of an analysis of variance (ANOVA), a topic we will consider in Chapter 14, and a principle component regression (PCR), at topic that we will not consider in this text. Instead, focus on the results of the model by examining the visualizations in Figure 3 and Figure 4 of the paper. As a guide, refer to Figure 1.2.1 in this chapter for a model of the analytical approach to solving problems. Use this link to access the article’s abstract from the journal’s web site. If your institution has an on-line subscription you also will be able to download a PDF version of the article. | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/01%3A_Introduction_to_Analytical_Chemistry/1.03%3A_Common_Analytical_Problems.txt |
The role of analytical chemistry within the broader discipline of chemistry has been discussed by many prominent analytical chemists; several notable examples are listed here.
• Baiulescu, G. E.; Patroescu, C; Chalmers, R. A. Education and Teaching in Analytical Chemistry, Ellis Horwood: Chichester, 1982.
• de Haseth, J. “What is Analytical Chemistry?,” Spectroscopy 1990, 5, 19–21.
• Heiftje, G. M. “The Two Sides of Analytical Chemistry,” Anal. Chem. 1985, 57, 256A–267A.
• Heiftje, G. M. “But is it analytical chemistry?,” Am. Lab. 1993, October, 53–61.
• Kissinger, P. T. “Analytical Chemistry—What is It? Why Teach It?,” Trends Anal. Chem. 1992, 11, 57–57.
• Laitinen, H. A.; Ewing, G. (eds.) A History of Analytical Chemistry, The Division of Analytical Chemistry of the American Chemical Society: Washington, D. C., 1972.
• Laitinen, H. A. “Analytical Chemistry in a Changing World,” Anal. Chem. 1980, 52, 605A–609A.
• Laitinen, H. A. “History of Analytical Chemistry in the U. S. A.,” Talanta, 1989, 36, 1–9.
• McLafferty, F. W. “Analytical Chemistry: Historic and Modern,” Acc. Chem. Res. 1990, 23, 63–64.
• Mottola, H. A. “The Interdisciplinary and Multidisciplinary Nature of Contemporary Analytical Chemistry and its Core Components,” Anal. Chim. Acta 1991, 242, 1–3.
• Noble, D. “From Wet Chemistry to Instrumental Analysis: A Perspective on Analytical Sciences,” Anal. Chem. 1994, 66, 251A–263A.
• Tyson, J. Analysis: What Analytical Chemists Do, Royal Society of Chemistry: Cambridge, England 1988.
For additional discussion of clinical assays based on paper-based microfluidic devices, see the following papers.
• Ellerbee, A. K.; Phillips, S. T.; Siegel, A. C.; Mirica, K. A.; Martinez, A. W.; Striehl, P.; Jain, N.; Prentiss, M.; Whitesides, G. M. “Quantifying Colorimetric Assays in Paper-Based Microfluidic Devices by Measuring the Transmission of Light Through Paper,” Anal. Chem. 2009, 81, 8447–8452.
• Martinez, A. W.; Phillips, S. T.; Whitesides, G. M. “Diagnostics for the Developing World: Microfluidic Paper-Based Analytical Devices,” Anal. Chem. 2010, 82, 3–10.
This textbook provides one introduction to the discipline of analytical chemistry. There are other textbooks for introductory courses in analytical chemistry and you may find it useful to consult them when you encounter a difficult concept; often a fresh perspective will help crystallize your understanding. The textbooks listed here are excellent resources.
• Enke, C. The Art and Science of Chemical Analysis, Wiley: New York.
• Christian, G. D.; Dasgupta, P, K.; Schug; K. A. Analytical Chemistry, Wiley: New York.
• Harris, D. Quantitative Chemical Analysis, W. H. Freeman and Company: New York.
• Kellner, R.; Mermet, J.-M.; Otto, M.; Valcárcel, M.; Widmer, H. M. Analytical Chemistry, Wiley- VCH: Weinheim, Germany.
• Rubinson, J. F.; Rubinson, K. A. Contemporary Chemical Analysis, Prentice Hall: Upper Saddle River, NJ.
• Skoog, D. A.; West, D. M.; Holler, F. J. Fundamentals of Analytical Chemistry, Saunders: Philadelphia.
To explore the practice of modern analytical chemistry there is no better resource than the primary literature. The following journals publish broadly in the area of analytical chemistry.
1.06: Chapter Summary and Key Terms
Chapter Summary
Analytical chemists work to improve the ability of chemists and other scientists to make meaningful measurements. The need to work with smaller samples, with more complex materials, with processes occurring on shorter time scales, and with species present at lower concentrations challenges analytical chemists to improve existing analytical methods and to develop new ones.
Typical problems on which analytical chemists work include qualitative analyses (What is present?), quantitative analyses (How much is present?), characterization analyses (What are the sample’s chemical and physical properties?), and fundamental analyses (How does this method work and how can it be improved?).
Key Terms
• characterization analysis
• fundamental analysis
• qualitative analysis
• quantitative analysis | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/01%3A_Introduction_to_Analytical_Chemistry/1.05%3A_Additional_Resources.txt |
In the chapters that follow we will explore many aspects of analytical chemistry. In the process we will consider important questions, such as “How do we extract useful results from experimental data?”, “How do we ensure our results are accurate?”, “How do we obtain a representative sample?”, and “How do we select an appropriate analytical technique?” Before we consider these and other questions, we first must review some basic tools of importance to analytical chemists.
• 2.1: Measurements in Analytical Chemistry
Analytical chemistry is a quantitative science. Whether determining the concentration of a species, evaluating an equilibrium constant, measuring a reaction rate, or drawing a correlation between a compound’s structure and its reactivity, analytical chemists engage in “measuring important chemical things.” In this section we review briefly the basic units of measurement and the proper use of significant figures.
• 2.2: Concentration
Concentration is a general measurement unit that reports the amount of solute present in a known amount of solution, which we can express in a variety of ways.
• 2.3: Stoichiometric Calculations
A balanced reaction, which defines the stoichiometric relationship between the moles of reactants and the moles of products, provides the basis for many analytical calculations.
• 2.4: Basic Equipment
The array of equipment available for making analytical measurements and working with analytical samples is impressive, ranging from the simple and inexpensive, to the complex and expensive. With three exceptions— the measurement of mass, the measurement of volume, and the drying of materials—we will postpone the discussion of equipment to later chapters where its application to specific analytical methods is relevant.
• 2.5: Preparing Solutions
Preparing a solution of known concentration is perhaps the most common activity in any analytical lab. The method for measuring out the solute and the solvent depend on the desired concentration and how exact the solu- tion’s concentration needs to be known.
• 2.6: Spreadsheets and Computational Software
Analytical chemistry is an inherently quantitative discipline. Whether you are completing a statistical analysis, trying to optimize experimental conditions, or exploring how a change in pH affects a compound’s solubility, the ability to work with complex mathematical equations is essential. Spreadsheets, such as Microsoft Excel, and computational software, such as R, are important tools for analyzing your data and for preparing graphs of your results.
• 2.7: The Laboratory Notebook
A laboratory notebook is your most important tool when working in the lab. If kept properly, you should be able to look back at your laboratory notebook several years from now and reconstruct the experiments on which you worked.
• 2.8: Problems
End-of-chapter problems to test your understanding of topics in this chapter.
• 2.9: Additional Resources
A compendium of resources to accompany topics in this chapter.
• 2.10: Chapter Summary and Key Terms
Summary of chapter's main topics and a list of key terms introduced in the chapter.
02: Basic Tools of Analytical Chemistry
Analytical chemistry is a quantitative science. Whether determining the concentration of a species, evaluating an equilibrium constant, measuring a reaction rate, or drawing a correlation between a compound’s structure and its reactivity, analytical chemists engage in “measuring important chemical things” [Murray, R. W. Anal. Chem. 2007, 79, 1765]. In this section we review briefly the basic units of measurement and the proper use of significant figures.
Units of Measurement
A measurement usually consists of a unit and a number that expresses the quantity of that unit. We can express the same physical measurement with different units, which creates confusion if we are not careful to specify the unit. For example, the mass of a sample that weighs 1.5 g is equivalent to 0.0033 lb or to 0.053 oz. To ensure consistency, and to avoid problems, scientists use the common set of fundamental base units listed in Table 2.1.1 . These units are called SI units after the Système International d’Unités.
It is important for scientists to agree upon a common set of units. In 1999, for example, NASA lost a Mar’s Orbiter spacecraft because one engineering team used English units in their calculations and another engineering team used metric units. As a result, the spacecraft came too close to the planet’s surface, causing its propulsion system to overheat and fail.
Some measurements, such as absorbance, do not have units. Because the meaning of a unitless number often is unclear, some authors include an artificial unit. It is not unusual to see the abbreviation AU—short for absorbance unit—following an absorbance value, which helps clarify that the measurement is an absorbance value.
Table 2.1.1 : Fundamental Base SI Units
Measurement
Unit
Symbol
Definition (1 unit is...)
mass
kilogram
kg
...the mass of the international prototype, a Pt-Ir object housed at the Bureau International de Poids and Measures at Sèvres, France. (Note: The mass of the international prototype changes at a rate of approximately 1 μg per year due to reversible surface contamination. The reference mass, therefore, is determined immediately after its cleaning using a specified procedure. Current plans call for retiring the international prototype and defining the kilogram in terms of Planck’s constant; see this link for more details.)
distance
meter
m
...the distance light travels in (299 792 458)–1 seconds.
temperature
Kelvin
K
...equal to (273.16)–1, where 273.16 K is the triple point of water (where its solid, liquid, and gaseous forms are in equilibrium).
time
second
s
...the time it takes for 9 192 631 770 periods of radiation corresponding to a specific transition of the 133Cs atom.
current
ampere
A
...the current producing a force of 2 $\times$ 10–7 N/m between two straight parallel conductors of infinite length separated by one meter (in a vacuum).
amount of substance
mole
mol
...the amount of a substance containing as many particles as there are atoms in exactly 0.012 kilogram of 12C.
light
candela
cd
...the luminous intensity of a source with a monochromatic frequency of 540 $\times$ 1012 hertz and a radiant power of (683)–1 watts per steradian.
There is some disagreement on the use of “amount of substance” to describe the measurement for which the mole is the base SI unit; see “What’s in a Name? Amount of Substance, Chemical Amount, and Stoichiometric Amount,” the full reference for which is Giunta, C. J. J. Chem. Educ. 2016, 93, 583–586.
We define other measurements using these fundamental SI units. For example, we measure the quantity of heat produced during a chemical reaction in joules, (J), where 1 J is equivalent to 1 m kg/s . Table 2.1.2 provides a list of some important derived SI units, as well as a few common non-SI units.
Table 2.1.2 : Derived SI Units and Non-SI Units of Importance to Analytical Chemistry
Measurement Unit Symbol Equivalent SI Units
length angstrom (non-SI) Å 1 Å = 1 $\times$ 10–10 m
volume liter (non-SI) L 1 L = 10–3 m3
force newton (SI) N 1 N = 1 m$\cdot$kg/s2
pressure
pascal (SI)
atmosphere (non-SI)
Pa
atm
1 Pa = 1 N/m3 = 1 kg/(m$\cdot$s2)
1 atm = 101 325 Pa
energy, work, heat
joule (SI)
calorie (non-SI)
electron volt (non-SI)
J
cal
eV
1 J = 1 N$\cdot$m = 1 m2$\cdot$kg/s2
1 cal = 4.184 J
1 eV = 1.602 177 33 $\times$ 10–19 J
power watt (SI) W 1 W = 1 J/s = 1 m2$\cdot$kg/s3
charge coulomb (SI) C 1 C = 1 A$\cdot$s
potential volt (SI) V 1 V = 1 W/A = 1 m2$\cdot$kg/(s3$\cdot$A)
frequency hertz (SI) Hz 1 Hz = s–1
temperature Celcius (non-SI) oC oC = K – 273.15
Chemists frequently work with measurements that are very large or very small. A mole contains 602 213 670 000 000 000 000 000 particles and some analytical techniques can detect as little as 0.000 000 000 000 001 g of a compound. For simplicity, we express these measurements using scientific notation; thus, a mole contains 6.022 136 7 $\times$ 1023 particles, and the detected mass is 1 $\times$ 10–15 g. Sometimes we wish to express a measurement without the exponential term, replacing it with a prefix (Table 2.1.3 ). A mass of $1 \times 10^{-15}$ g, for example, is the same as 1 fg, or femtogram.
Writing a lengthy number with spaces instead of commas may strike you as unusual. For a number with more than four digits on either side of the decimal point, however, the recommendation from the International Union of Pure and Applied Chemistry is to use a thin space instead of a comma.
Table 2.1.3 : Common Prefixes for Exponential Notation
Prefix Symbol Factor Prefix Symbol Factor Prefix Symbol Factor
yotta Y 1024 kilo k 103 micro µ 10–6
zetta Z 1021 hecto h 102 nano n 10–9
eta E 1018 deka da 101 pico p 10–12
peta P 1015 100 femto f 10–15
tera T 1012 deci d 10–1 atto a 10–18
giga G 109 centi c 10–2 zepto z 10–21
mega M 106 milli m 10–3 yocto y 10–24
Uncertainty in Measurements
A measurement provides information about both its magnitude and its uncertainty. Consider, for example, the three photos in Figure 2.1.1 , taken at intervals of approximately 1 sec after placing a sample on the balance. Assuming the balance is properly calibrated, we are certain that the sample’s mass is more than 0.5729 g and less than 0.5731 g. We are uncertain, however, about the sample’s mass in the last decimal place since the final two decimal places fluctuate between 29, 30, and 31. The best we can do is to report the sample’s mass as 0.5730 g ± 0.0001 g, indicating both its magnitude and its absolute uncertainty.
Figure 2.1.1 : When weighing an sample on a balance, the measurement fluctuates in the final decimal place. We record this sample’s mass as 0.5730 g ± 0.0001 g.
Significant Figures
A measurement’s significant figures convey information about a measurement’s magnitude and uncertainty. The number of significant figures in a measurement is the number of digits known exactly plus one digit whose value is uncertain. The mass shown in Figure 2.1.1 , for example, has four significant figures, three which we know exactly and one, the last, which is uncertain.
Suppose we weigh a second sample, using the same balance, and obtain a mass of 0.0990 g. Does this measurement have 3, 4, or 5 significant figures? The zero in the last decimal place is the one uncertain digit and is significant. The other two zeros, however, simply indicates the decimal point’s location. Writing the measurement in scientific notation, $9.90 \times 10^{-2}$, clarifies that there are three significant figures in 0.0990.
In the measurement 0.0990 g, the zero in green is a significant digit and the zeros in red are not significant digits.
Example 2.1.1
How many significant figures are in each of the following measurements? Convert each measurement to its equivalent scientific notation or decimal form.
1. 0.0120 mol HCl
2. 605.3 mg CaCO3
3. $1.043 \times 10^{-4}$ mol Ag+
4. $9.3 \times 10^4$ mg NaOH
Solution
(a) Three significant figures; $1.20 \times 10^{-2}$ mol HCl.
(b) Four significant figures; $6.053 \times 10^2$ mg CaCO3.
(c) Four significant figures; 0.000 104 3 mol Ag+.
(d) Two significant figures; 93 000 mg NaOH.
There are two special cases when determining the number of significant figures in a measurement. For a measurement given as a logarithm, such as pH, the number of significant figures is equal to the number of digits to the right of the decimal point. Digits to the left of the decimal point are not significant figures since they indicate only the power of 10. A pH of 2.45, therefore, contains two significant figures.
The log of $2.8 \times 10^2$ is 2.45. The log of 2.8 is 0.45 and the log of 102 is 2. The 2 in 2.45, therefore, only indicates the power of 10 and is not a significant digit.
An exact number, such as a stoichiometric coefficient, has an infinite number of significant figures. A mole of CaCl2, for example, contains exactly two moles of chloride ions and one mole of calcium ions. Another example of an exact number is the relationship between some units. There are, for example, exactly 1000 mL in 1 L. Both the 1 and the 1000 have an infinite number of significant figures.
Using the correct number of significant figures is important because it tells other scientists about the uncertainty of your measurements. Suppose you weigh a sample on a balance that measures mass to the nearest ±0.1 mg. Reporting the sample’s mass as 1.762 g instead of 1.7623 g is incorrect because it does not convey properly the measurement’s uncertainty. Reporting the sample’s mass as 1.76231 g also is incorrect because it falsely suggests an uncertainty of ±0.01 mg.
Significant Figures in Calculations
Significant figures are also important because they guide us when reporting the result of an analysis. When we calculate a result, the answer cannot be more certain than the least certain measurement in the analysis. Rounding an answer to the correct number of significant figures is important.
For addition and subtraction, we round the answer to the last decimal place in common for each measurement in the calculation. The exact sum of 135.621, 97.33, and 21.2163 is 254.1673. Since the last decimal place common to all three numbers is the hundredth’s place
\begin{align*} &135.6{\color{Red} 2}1\ &\phantom{1}97.3{\color{Red} 3}\ &\underline{\phantom{1}21.2{\color{Red} 1}63}\ &254.1673 \end{align*}
we round the result to 254.17.
The last common decimal place shared by 135.621, 97.33, and 21.2163 is shown in red.
When working with scientific notation, first convert each measurement to a common exponent before determining the number of significant figures. For example, the sum of $6.17 \times 10^7$, $4.3 \times 10^5$, and $3.23 \times 10^4$ is $6.22 \times 10^7$.
\begin{align*} &6.1{\color{Red} 7} \phantom{323} \times 10^7\ &0.0{\color{Red} 4}3 \phantom{23} \times 10^7\ &\underline{0.0{\color{Red} 0}323 \times 10^7}\ &6.21623 \times 10^7 \end{align*}
The last common decimal place shared by $6.17 \times 10^7$, $4.3 \times 10^5$ and $3.23 \times 10^4$ is shown in red.
For multiplication and division, we round the answer to the same number of significant figures as the measurement with the fewest number of significant figures. For example, when we divide the product of 22.91 and 0.152 by 16.302, we report the answer as 0.214 (three significant figures) because 0.152 has the fewest number of significant figures.
$\frac {22.91 \times 0.{\color{Red} 152}} {16.302} = 0.2136 = 0.214\nonumber$
There is no need to convert measurements in scientific notation to a common exponent when multiplying or dividing.
It is important to recognize that the rules presented here for working with significant figures are generalizations. What actually is conserved is uncertainty, not the number of significant figures. For example, the following calculation
101/99 = 1.02
is correct even though it violates the general rules outlined earlier. Since the relative uncertainty in each measurement is approximately 1% (101 ± 1 and 99 ± 1), the relative uncertainty in the final answer also is approximately 1%. Reporting the answer as 1.0 (two significant figures), as required by the general rules, implies a relative uncertainty of 10%, which is too large. The correct answer, with three significant figures, yields the expected relative uncertainty. Chapter 4 presents a more thorough treatment of uncertainty and its importance in reporting the result of an analysis.
Finally, to avoid “round-off” errors, it is a good idea to retain at least one extra significant figure throughout any calculation. Better yet, invest in a good scientific calculator that allows you to perform lengthy calculations without the need to record intermediate values. When your calculation is complete, round the answer to the correct number of significant figures using the following simple rules.
1. Retain the least significant figure if it and the digits that follow are less than halfway to the next higher digit. For example, rounding 12.442 to the nearest tenth gives 12.4 since 0.442 is less than half way between 0.400 and 0.500.
2. Increase the least significant figure by 1 if it and the digits that follow are more than halfway to the next higher digit. For example, rounding 12.476 to the nearest tenth gives 12.5 since 0.476 is more than halfway between 0.400 and 0.500.
3. If the least significant figure and the digits that follow are exactly halfway to the next higher digit, then round the least significant figure to the nearest even number. For example, rounding 12.450 to the nearest tenth gives 12.4, while rounding 12.550 to the nearest tenth gives 12.6. Rounding in this manner ensures that we round up as often as we round down.
Exercise 2.1.1
For a problem that involves both addition and/or subtraction, and multiplication and/or division, be sure to account for significant figures at each step of the calculation. With this in mind, report the result of this calculation to the correct number of significant figures.
$\frac {0.250 \times (9.93 \times 10^{-3}) - 0.100 \times (1.927 \times 10^{-2})} {9.93 \times 10^{-3} + 1.927 \times 10^{-2}} = \nonumber$
Answer
The correct answer to this exercise is $1.9 \times 10^{-2}$. To see why this is correct, let’s work through the problem in a series of steps. Here is the original problem
$\frac {0.250 \times (9.93 \times 10^{-3}) - 0.100 \times (1.927 \times 10^{-2})} {9.93 \times 10^{-3} + 1.927 \times 10^{-2}} = \nonumber$
Following the correct order of operations we first complete the two multiplications in the numerator. In each case the answer has three significant figures, although we retain an extra digit, highlight in red, to avoid round-off errors.
$\frac {2.48{\color{Red} 2} \times 10^{-3} - 1.92{\color{Red} 7} \times 10^{-3}} {9.93 \times 10^{-3} + 1.927 \times 10^{-2}} = \nonumber$
Completing the subtraction in the numerator leaves us with two significant figures since the last significant digit for each value is in the hundredths place.
$\frac {0.55{\color{Red} 5} \times 10^{-3}} {9.93 \times 10^{-3} + 1.927 \times 10^{-2}} = \nonumber$
The two values in the denominator have different exponents. Because we are adding together these values, we first rewrite them using a common exponent.
$\frac {0.55{\color{Red} 5} \times 10^{-3}} {0.993 \times 10^{-2} + 1.927 \times 10^{-2}} = \nonumber$
The sum in the denominator has four significant figures since each of the addends has three decimal places.
$\frac {0.55{\color{Red} 5} \times 10^{-3}} {2.92{\color{Red} 0} \times 10^{-2}} = \nonumber$
Finally, we complete the division, which leaves us with a result having two significant figures.
$\frac {0.55{\color{Red} 5} \times 10^{-3}} {2.92{\color{Red} 0} \times 10^{-2}} = 1.9 \times 10^{-2} \nonumber$ | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/02%3A_Basic_Tools_of_Analytical_Chemistry/2.01%3A_Measurements_in_Analytical_Chemistry.txt |
Concentration is a general measurement unit that reports the amount of solute present in a known amount of solution
$\text{concentration} = \dfrac {\text{amount of solute}} {\text{amount of solution}} \label{2.1}$
Although we associate the terms “solute” and “solution” with liquid samples, we can extend their use to gas-phase and solid-phase samples as well. Table 2.2.1 lists the most common units of concentration.
Table 2.2.1 : Common Units for Reporting Concentration
Name Units Symbol
molarity $\dfrac {\text{moles solute}} {\text{liters solution}}$ M
formality $\dfrac {\text{moles solute}} {\text{liters solution}}$ F
normality $\dfrac {\text{equivalents solute}} {\text{liters solution}}$ N
molality $\dfrac {\text{moles solute}} {\text{kilograms solvent}}$ m
weight percent $\dfrac {\text{grams solute}} {\text{100 grams solution}}$ % w/w
volume percent $\dfrac {\text{mL solute}} {\text{100 mL solution}}$ % v/v
weight-to-volume percent $\dfrac {\text{grams solute}} {\text{100 mL solution}}$ % w/v
parts per million $\dfrac {\text{grams solute}} {10^6 \text{ grams solution}}$ ppm
parts per billion $\dfrac {\text{grams solute}} {10^9 \text{ grams solution}}$ ppb
An alternative expression for weight percent is
$\dfrac {\text{grams solute}} {\text{grams solution}} \times 100\ \nonumber$
You can use similar alternative expressions for volume percent and for weight-to-volume percent.
Molarity and Formality
Both molarity and formality express concentration as moles of solute per liter of solution; however, there is a subtle difference between them. Molarity is the concentration of a particular chemical species. Formality, on the other hand, is a substance’s total concentration without regard to its specific chemical form. There is no difference between a compound’s molarity and formality if it dissolves without dissociating into ions. The formal concentration of a solution of glucose, for example, is the same as its molarity.
For a compound that ionizes in solution, such as CaCl2, molarity and formality are different. When we dissolve 0.1 moles of CaCl2 in 1 L of water, the solution contains 0.1 moles of Ca2+ and 0.2 moles of Cl. The molarity of CaCl2, therefore, is zero since there is no undissociated CaCl2 in solution; instead, the solution is 0.1 M in Ca2+ and 0.2 M in Cl. The formality of CaCl2, however, is 0.1 F since it represents the total amount of CaCl2 in solution. This more rigorous definition of molarity, for better or worse, largely is ignored in the current literature, as it is in this textbook. When we state that a solution is 0.1 M CaCl2 we understand it to consist of Ca2+ and Cl ions. We will reserve the unit of formality to situations where it provides a clearer description of solution chemistry.
Molarity is used so frequently that we use a symbolic notation to simplify its expression in equations and in writing. Square brackets around a species indicate that we are referring to that species’ molarity. Thus, [Ca2+] is read as “the molarity of calcium ions.”
For a solute that dissolves without undergoing ionization, molarity and formality have the same value. A solution that is 0.0259 M in glucose, for example, is 0.0259 F in glucose as well.
Normality
Normality is a concentration unit that no longer is in common use; however, because you may encounter normality in older handbooks of analytical methods, it is helpful to understand its meaning. Normality defines concentration in terms of an equivalent, which is the amount of one chemical species that reacts stoichiometrically with another chemical species. Note that this definition makes an equivalent, and thus normality, a function of the chemical reaction in which the species participates. Although a solution of H2SO4 has a fixed molarity, its normality depends on how it reacts. You will find a more detailed treatment of normality in Appendix 1.
One handbook that still uses normality is Standard Methods for the Examination of Water and Wastewater, a joint publication of the American Public Health Association, the American Water Works Association, and the Water Environment Federation. This handbook is one of the primary resources for the environmental analysis of water and wastewater.
Molality
Molality is used in thermodynamic calculations where a temperature independent unit of concentration is needed. Molarity is based on the volume of solution that contains the solute. Since density is a temperature dependent property, a solution’s volume, and thus its molar concentration, changes with temperature. By using the solvent’s mass in place of the solution’s volume, the resulting concentration becomes independent of temperature.
Weight, Volume, and Weight-to-Volume Percent
Weight percent (% w/w), volume percent (% v/v) and weight-to-volume percent (% w/v) express concentration as the units of solute present in 100 units of solution. A solution that is 1.5% w/v NH4NO3, for example, contains 1.5 gram of NH4NO3 in 100 mL of solution.
Parts Per Million and Parts Per Billion
Parts per million (ppm) and parts per billion (ppb) are ratios that give the grams of solute in, respectively, one million or one billion grams of sample. For example, a sample of steel that is 450 ppm in Mn contains 450 μg of Mn for every gram of steel. If we approximate the density of an aqueous solution as 1.00 g/mL, then we can express solution concentrations in ppm or ppb using the following relationships.
$\text{ppm} = \dfrac {\mu \text{g}} {\text{g}} = \dfrac {\text{mg}} {\text{L}} = \dfrac {\mu \text{g}} {\text{mL}} \quad \text{ppb} = \dfrac {\text{ng}} {\text{g}} = \dfrac {\mu \text{g}} {\text{L}} = \dfrac {\text{ng}} {\text{mL}} \nonumber$
For gases a part per million usually is expressed as a volume ratio; for example, a helium concentration of 6.3 ppm means that one liter of air contains 6.3 μL of He.
You should be careful when using parts per million and parts per billion to express the concentration of an aqueous solute. The difference between a solute’s concentration in mg/L and ng/g, for example, is significant if the solution’s density is not 1.00 g/mL. For this reason many organizations advise against using the abbreviation ppm and ppb (see section 7.10.3 at www.nist.gov). If in doubt, include the exact units, such as 0.53 μg Pb2+/L for the concentration of lead in a sample of seawater.
Converting Between Concentration Units
The most common ways to express concentration in analytical chemistry are molarity, weight percent, volume percent, weight-to-volume percent, parts per million and parts per billion. The general definition of concentration in Equation \ref{2.1} makes it is easy to convert between concentration units.
Example 2.2.1
A concentrated solution of ammonia is 28.0% w/w NH3 and has a density of 0.899 g/mL. What is the molar concentration of NH3 in this solution?
Solution
$\dfrac {28.0 \text{ g } \ce{NH3}} {100 \text{ g soln}} \times \dfrac {0.899 \text{ g soln}} {\text{ml soln}} \times \dfrac {1 \text{ mol } \ce{NH3}} {17.03 \text{ g } \ce{NH3}} \times \dfrac {1000 \text{mL}} {\text{L}} = 14.8 \text{ M} \nonumber$
Example 2.2.2
The maximum permissible concentration of chloride ion in a municipal drinking water supply is $2.50 \times 10^2$ ppm Cl. When the supply of water exceeds this limit it often has a distinctive salty taste. What is the equivalent molar concentration of Cl?
Solution
$\dfrac {2.50 \times 10^2 \text{ mg } \ce{Cl-}} {\text{L}} \times \dfrac {1 \text{ g}} {1000 \text{ mg}} \times \dfrac {1 \text{ mol } \ce{Cl-}} {35.453 \text{ g} \ce{Cl-}} = 7.05 \times 10^{-3} \text{ M} \nonumber$
Exercise 2.2.1
Which solution—0.50 M NaCl or 0.25 M SrCl2—has the larger concentration when expressed in mg/mL?
Answer
The concentrations of the two solutions are
$\dfrac {0.50 \text{ mol NaCl}} {\text{L}} \times \dfrac {58.44 \text{ g NaCl}} {\text{mol NaCl}} \times \dfrac {10^6 \: \mu \text{g}} {\text{g}} \times \dfrac {1 \text{L}} {1000 \text{ mL}} = 2.9 \times 10^{4} \: \mu \text{g/mL NaCl} \nonumber$
$\dfrac {0.25 \text{ mol } \ce{SrCl2}} {\text{L}} \times \dfrac {158.5 \text{ g } \ce{SrCl2}} {\text{mol } \ce{SrCl2}} \times \dfrac {10^6 \: \mu \text{g}} {\text{g}} \times \dfrac {1 \text{L}} {1000 \text{ mL}} = 4.0 \times 10^{4} \: \mu \text{g/ml } \ce{SrCl2} \nonumber$
The solution of SrCl2 has the larger concentration when it is expressed in μg/mL instead of in mol/L.
p-Functions
Sometimes it is inconvenient to use the concentration units in Table 2.2.1 . For example, during a chemical reaction a species’ concentration may change by many orders of magnitude. If we want to display the reaction’s progress graphically we might wish to plot the reactant’s concentration as a function of the volume of a reagent added to the reaction. Such is the case in Figure 2.2.1 for the titration of HCl with NaOH. The y-axis on the left-side of the figure displays the [H+] as a function of the volume of NaOH. The initial [H+] is 0.10 M and its concentration after adding 80 mL of NaOH is $4.3 \times 10^{-13}$ M. We easily can follow the change in [H+] for the addition of the first 50 mL of NaOH; however, for the remaining volumes of NaOH the change in [H+] is too small to see.
When working with concentrations that span many orders of magnitude, it often is more convenient to express concentration using a p-function. The p-function of X is written as pX and is defined as
$\text{p} X = - \log (X) \nonumber$
The pH of a solution that is 0.10 M H+ for example, is
$\text{pH} = - \log [\ce{H+}] = - \log (0.10) = 1.00 \nonumber$
and the pH of $4.3 \times 10^{-13}$ M H+ is
$\text{pH} = - \log [\ce{H+}] = - \log (4.3 \times 10^{-13}) = 12.37 \nonumber$
Figure 2.2.1 shows that plotting pH as a function of the volume of NaOH provides more useful information about how the concentration of H+ changes during the titration.
A more appropriate equation for pH is $\text{pH} = - \log (a_{\ce{H+}})$ where $a_{\ce{H+}}$ is the activity of the hydrogen ion. See Chapter 6.9 for more details. For now the approximate equation $\text{pH} = - \log [\ce{H+}]$ is sufficient.
Example 2.2.3
What is pNa for a solution of $1.76 \times 10^{-3}$ M Na3PO4?
Solution
Since each mole of Na3PO4 contains three moles of Na+, the concentration of Na+ is
$[\ce{Na+}] = (1.76 \times 10^{-3} \text{ M}) \times \dfrac {3 \text{ mol } \ce{Na+}} {\text{mol } \ce{Na3PO4}} = 5.28 \times 10^{-3} \text{ M} \nonumber$
and pNa is
$\text{pNa} = - \log [\ce{Na+}] = - \log (5.28 \times 10^{-3}) = 2.277 \nonumber$
Remember that a pNa of 2.777 has three, not four, significant figures; the 2 that appears in the one’s place indicates the power of 10 when we write [Na+] as $0.528 \times 10^{-2}$ M.
Example 2.2.4
What is the [H+] in a solution that has a pH of 5.16?
Solution
The concentration of H+ is
$\text{pH} = - \log [\ce{H+}] = 5.16 \nonumber$
$\log [\ce{H+}] = -5.16 \nonumber$
$[\ce{H+}] = 10^{-5.16} = 6.9 \times 10^{-6} \text{ M} \nonumber$
Recall that if log(X) = a, then X = 10a.
Exercise 2.2.2
What are the values for pNa and pSO4 if we dissolve 1.5 g Na2SO4 in a total solution volume of 500.0 mL?
Answer
The concentrations of Na+ and $\ce{SO4^{2-}}$ are
$\dfrac {1.5 \text{ g } \ce{Na2SO4}} {0.500 \text{L}} \times \dfrac {1 \text{ mol } \ce{Na2SO4}} {142.0 \text{ g } \ce{Na2SO4}} \times \dfrac {2 \text{ mol } \ce{Na+}} {\text{mol } \ce{mol } \ce{Na2SO4}} = 4.23 \times 10^{-2} \text{ M } \ce{Na+} \nonumber$
$\dfrac {1.5 \text{ g } \ce{Na2SO4}} {0.500 \text{L}} \times \dfrac {1 \text{ mol } \ce{Na2SO4}} {142.0 \text{ g } \ce{Na2SO4}} \times \dfrac {1 \text{ mol } \ce{SO4^{2-}}} {\text{mol } \ce{mol } \ce{Na2SO4}} = 2.11 \times 10^{-2} \text{ M } \ce{SO4^{2-}} \nonumber$
The pNa and pSO4 values are
$\text{pNa} = - \log (4.23 \times 10^{-2} \text{ M } \ce{Na+}) = 1.37 \nonumber$
$\text{pSO}_4 = - \log (2.11 \times 10^{-2} \text{ M } \ce{SO4^{2-}}) = 1.68 \nonumber$ | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/02%3A_Basic_Tools_of_Analytical_Chemistry/2.02%3A_Concentration.txt |
A balanced reaction, which defines the stoichiometric relationship between the moles of reactants and the moles of products, provides the basis for many analytical calculations. Consider, for example, an analysis for oxalic acid, H2C2O4, in which Fe3+ oxidizes oxalic acid to CO2
$2\ce{Fe^{3+}}(aq) + \ce{H2C2O4}(aq) + 2\ce{H2O}(l) \ce{->} 2\ce{Fe^{2+}}(aq) + 2\ce{CO2}(g) + 2\ce{H3O+}(aq) \nonumber$
The balanced reaction shows us that one mole of oxalic acid reacts with two moles of Fe3+. As shown in the following example, we can use this balanced reaction to determine the amount of H2C2O4 in a sample of rhubarb if we know the moles of Fe3+ needed to react completely with oxalic acid.
In sufficient amounts, oxalic acid, the structure for which is shown below, is toxic. At lower physiological concentrations it leads to the formation of kidney stones. The leaves of the rhubarb plant contain relatively high concentrations of oxalic acid. The stalk, which many individuals enjoy eating, contains much smaller concentrations of oxalic acid.
In the examples that follow, note that we retain an extra significant figure throughout the calculation, rounding to the correct number of significant figures at the end. We will follow this convention in any calculation that involves more than one step. If we forget that we are retaining an extra significant figure, we might report the final answer with one too many significant figures. Here we mark the extra digit in red for emphasis. Be sure you pick a system for keeping track of significant figures.
Example 2.3.1
The amount of oxalic acid in a sample of rhubarb was determined by reacting with Fe3+. After extracting a 10.62 g of rhubarb with a solvent, oxidation of the oxalic acid required 36.44 mL of 0.0130 M Fe3+. What is the weight percent of oxalic acid in the sample of rhubarb?
Solution
We begin by calculating the moles of Fe3+ used in the reaction
$\frac {0.0130 \text{ mol } \ce{Fe^{3+}}} {\text{L}} \times 0.03644 \text{ M} = 4.73{\color{Red} 7} \times 10^{-4} \text{ mol } \ce{Fe^{3+}} \nonumber$
The moles of oxalic acid reacting with the Fe3+, therefore, is
$4.73{\color{Red} 7} \times 10^{-4} \text{ mol } \ce{Fe^{3+}} \times \frac {1 \text{ mol } \ce{H2C2O4}} {2 \text{ mol } \ce{Fe^{3+}}} = 2.36{\color{Red} 8} \times 10^{-4} \text{ mol } \ce{H2C2O4} \nonumber$
Converting the moles of oxalic acid to grams of oxalic acid
$2.36{\color{Red} 8} \times 10^{-4} \text{ mol } \ce{H2C2O4} \times \frac {90.03 \text{ g } \ce{H2C2O4}} {\text{mol } \ce{H2C2O4}} = 2.13{\color{Red} 2} \times 10^{-2} \text{ g } \ce{H2C2O4} \nonumber$
and calculating the weight percent gives the concentration of oxalic acid in the sample of rhubarb as
$\frac {2.13{\color{Red} 2} \times 10^{-2} \text{ g } \ce{H2C2O4}} {10.62 \text{ g rhubarb}} \times 100 = 0.201 \text{% w/w } \ce{H2C2O4} \nonumber$
Exercise 2.3.1
You can dissolve a precipitate of AgBr by reacting it with Na2S2O3, as shown here.
$\ce{AgBr}(s) + 2\ce{Na2S2O3}(aq) \ce{->} \ce{Ag(S2O3)_2^{3-}}(aq) + \ce{Br-}(aq) + 4\ce{Na+}(aq) \nonumber$
How many mL of 0.0138 M Na2S2O3 do you need to dissolve 0.250 g of AgBr?
Answer
First, we find the moles of AgBr
$0.250 \text{ g AgBr} \times \frac {1 \text{ mol AgBr}} {187.8 \text{ g AgBr}} = 1.331 \times 10^{-3} \text{ mol AgBr} \nonumber$
and then the moles and volume of Na2S2O3
$1.331 \times 10^{-3} \text{ mol AgBr} \times \frac {2 \text{ mol } \ce{Na2S2O3}} {\text{mol AgBr}} = 2.662 \times 10^{-3} \text{ mol } \ce{Na2S2O3} \nonumber$
$2.662 \times 10^{-3} \text{ mol } \ce{Na2S2O3} \times \frac {1 \text{ L}} {0.0138 \text{ mol } \ce{Na2S2O3}} \times \frac {1000 \text{ mL}} {\text{L}} = 193 \text{ mL} \nonumber$
The analyte in Example 2.3.1 , oxalic acid, is in a chemically useful form because there is a reagent, Fe3+, that reacts with it quantitatively. In many analytical methods, we first must convert the analyte into a more accessible form before we can complete the analysis. For example, one method for the quantitative analysis of disulfiram, C10H20N2S4—the active ingredient in the drug Antabuse, and whose structure is shown below—requires that we first convert the sulfur to SO2 by combustion, and then oxidize the SO2 to H2SO4 by bubbling it through a solution of H2O2. When the conversion is complete, the amount of H2SO4 is determined by titrating with NaOH.
To convert the moles of NaOH used in the titration to the moles of disulfiram in the sample, we need to know the stoichiometry of each reaction. Writing a balanced reaction for H2SO4 and NaOH is straightforward
$\ce{H2SO4}(aq) + 2\ce{NaOH}(aq) \ce{->} 2\ce{H2O}(l) + \ce{Na2SO4}(aq) \nonumber$
but the balanced reactions for the oxidations of C10H20N2S4 to SO2, and of SO2 to H2SO4 are not as immediately obvious. Although we can balance these redox reactions, it is often easier to deduce the overall stoichiometry by use a little chemical logic.
Example 2.3.2
An analysis for disulfiram, C10H20N2S4, in Antabuse is carried out by oxidizing the sulfur to H2SO4 and titrating the H2SO4 with NaOH. If a 0.4613-g sample of Antabuse requires 34.85 mL of 0.02500 M NaOH to titrate the H2SO4, what is the %w/w disulfiram in the sample?
Solution
Calculating the moles of H2SO4 is easy—first, we calculate the moles of NaOH used in the titration
$(0.02500 \text{ M}) \times (0.03485 \text{ L}) = 8.712{\color{Red} 5} \times 10^{-4} \text{ mol NaOH} \nonumber$
and then we use the titration reaction’s stoichiometry to calculate the corresponding moles of H2SO4.
$8.712{\color{Red} 5} \times 10^{-4} \text{ mol NaOH} \times \frac {1 \text{ mol } \ce{H2SO4}} {2 \text{ mol NaOH}} = 4.356{\color{Red} 2} \times 10^{-4} \text{ mol } \ce{H2SO4} \nonumber$
Here is where we use a little chemical logic. Instead of balancing the reactions for the combustion of C10H20N2S4 to SO2 and for the subsequent oxidation of SO2 to H2SO4, we recognize that a conservation of mass requires that all the sulfur in C10H20N2S4 ends up in the H2SO4; thus
$4.356{\color{Red} 2} \times 10^{-4} \text{ mol } \ce{H2SO4} \times \frac {1 \text{ mol S}} {\text{mol } \ce{H2SO4}} \times \frac {1 \text{ mol } \ce{C10H20N2S4}} {4 \text{ mol S}} = 1.089{\color{Red} 0} \times 10^{-4} \text{ mol } \ce{C10H20N2S4} \nonumber$
$1.089{\color{Red} 0} \times 10^{-4} \text{ mol } \ce{C10H20N2S4} \times \frac {296.54 \text{ g } \ce{C10H20N2S4}} {\text{mol } \ce{C10H20N2S4}} = 0.03229{\color{Red} 3} \text{ g } \ce{C10H20N2S4} \nonumber$
$\frac {0.03229{\color{Red} 3} \text{ g } \ce{C10H20N2S4}} {0.4613 \text{ g sample}} \times 100 = 7.000 \text{% w/w } \ce{C10H20N2S4} \nonumber$
A conservation of mass is the essence of stoichiometry! | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/02%3A_Basic_Tools_of_Analytical_Chemistry/2.03%3A_Stoichiometric_Calculations.txt |
The array of equipment available for making analytical measurements and working with analytical samples is impressive, ranging from the simple and inexpensive, to the complex and expensive. With three exceptions—the measurement of mass, the measurement of volume, and the drying of materials—we will postpone the discussion of equipment to later chapters where its application to specific analytical methods is relevant.
Equipment for Measuring Mass
An object’s mass is measured using a digital electronic analytical balance (Figure 2.4.1 ). An electromagnet levitates the sample pan above a permanent cylindrical magnet. When we place an object on the sample pan, it displaces the sample pan downward by a force equal to the product of the sample’s mass and its acceleration due to gravity. The balance detects this downward movement and generates a counterbalancing force by increasing the current to the electromagnet. The current needed to return the balance to its original position is proportional to the object’s mass. A typical electronic balance has a capacity of 100–200 g, and can measure mass to the nearest ±0.01 mg to ±1 mg.
Although we tend to use interchangeably, the terms “weight” and “mass,” there is an important distinction between them. Mass is the absolute amount of matter in an object, measured in grams. Weight, W, is a measure of the gravitational force, g, acting on that mass, m:
$W = m \times g \nonumber$
An object has a fixed mass but its weight depends upon the acceleration due to gravity, which varies subtly from location-to-location.
A balance measures an object’s weight, not its mass. Because weight and mass are proportional to each other, we can calibrate a balance using a standard weight whose mass is traceable to the standard prototype for the kilogram. A properly calibrated balance gives an accurate value for an object’s mass; see Appendix 9 for more details on calibrating a balance.
If the sample is not moisture sensitive, a clean and dry container is placed on the balance. The container’s mass is called the tare and most balances allow you to set the container’s tare to a mass of zero. The sample is transferred to the container, the new mass is measured and the sample’s mass determined by subtracting the tare. A sample that absorbs moisture from the air is treated differently. The sample is placed in a covered weighing bottle and their combined mass is determined. A portion of the sample is removed and the weighing bottle and the remaining sample are reweighed. The difference between the two masses gives the sample’s mass.
Several important precautions help to minimize errors when we determine an object’s mass. To minimize the effect of vibrations, the balance is placed on a stable surface and in a level position. Because the sensitivity of an analytical balance is sufficient to measure the mass of a fingerprint, materials often are handled using tongs or laboratory tissues. Volatile liquid samples must be weighed in a covered container to avoid the loss of sample by evaporation. To minimize fluctuations in mass due to air currents, the balance pan often is housed within a wind shield, as seen in Figure 2.4.1 . A sample that is cooler or warmer than the surrounding air will create a convective air currents that affects the measurement of its mass. For this reason, bring your samples to room temperature before determining their mass. Finally, samples dried in an oven are stored in a desiccator to prevent them from reabsorbing moisture from the atmosphere.
Equipment for Measuring Volume
Analytical chemists use a variety of glassware to measure volume, including graduated cylinders, volumetric pipets, and volumetric flasks. The choice of what type of glassware to use depends on how accurately and how precisely we need to know the sample’s volume and whether we are interested in containing or delivering the sample.
A graduated cylinder is the simplest device for delivering a known volume of a liquid reagent (Figure 2.4.2 ). The graduated scale allows you to deliver any volume up to the cylinder’s maximum. Typical accuracy is ±1% of the maximum volume. A 100-mL graduated cylinder, for example, is accurate to ±1 mL.
A volumetric pipet provides a more accurate method for delivering a known volume of solution. Several different styles of pipets are available, two of which are shown in Figure 2.4.3 . Transfer pipets provide the most accurate means for delivering a known volume of solution. A transfer pipet delivering less than 100 mL generally is accurate to the hundredth of a mL. Larger transfer pipets are accurate to a tenth of a mL. For example, the 10-mL transfer pipet in Figure 2.4.3 will deliver 10.00 mL with an accuracy of ±0.02 mL.
Scientists at the Brookhaven National Laboratory used a germanium nanowire to make a pipet that delivers a 35 zeptoliter (10–21 L) drop of a liquid gold-germanium alloy. You can read about this work in the April 21, 2007 issue of Science News.
To fill a transfer pipet, use a rubber suction bulb to pull the solution up past the calibration mark (Never use your mouth to suck a solution into a pipet!). After replacing the bulb with your finger, adjust the solution’s level to the calibration mark and dry the outside of the pipet with a laboratory tissue. Allow the pipet’s contents to drain into the receiving container with the pipet’s tip touching the inner wall of the container. A small portion of the liquid remains in the pipet’s tip and is not be blown out. With some measuring pipets any solution remaining in the tip must be blown out.
Delivering microliter volumes of liquids is not possible using transfer or measuring pipets. Digital micropipets (Figure 2.4.4 ), which come in a variety of volume ranges, provide for the routine measurement of microliter volumes.
Graduated cylinders and pipets deliver a known volume of solution. A volumetric flask, on the other hand, contains a specific volume of solution (Figure 2.4.5 ). When filled to its calibration mark, a volumetric flask that contains less than 100 mL generally is accurate to the hundredth of a mL, whereas larger volumetric flasks are accurate to the tenth of a mL. For example, a 10-mL volumetric flask contains 10.00 mL ± 0.02 mL and a 250-mL volumetric flask contains 250.0 mL ± 0.12 mL.
Because a volumetric flask contains a solution, it is used to prepare a solution with an accurately known concentration. Transfer the reagent to the volumetric flask and add enough solvent to bring the reagent into solution. Continuing adding solvent in several portions, mixing thoroughly after each addition, and then adjust the volume to the flask’s calibration mark using a dropper. Finally, complete the mixing process by inverting and shaking the flask at least 10 times.
If you look closely at a volumetric pipet or a volumetric flask you will see markings similar to those shown in Figure 2.4.6 . The text of the markings, which reads
10 mL T. D. at 20 oC ± 0.02 mL
indicates that the pipet is calibrated to deliver (T. D.) 10 mL of solution with an uncertainty of ±0.02 mL at a temperature of 20 oC. The temperature is important because glass expands and contracts with changes in temperatures; thus, the pipet’s accuracy is less than ±0.02 mL at a higher or a lower temperature. For a more accurate result, you can calibrate your volumetric glassware at the temperature you are working by weighing the amount of water contained or delivered and calculating the volume using its temperature dependent density.
A volumetric flask has similar markings, but uses the abbreviation T. C. for “to contain” in place of T. D.
You should take three additional precautions when you work with pipets and volumetric flasks. First, the volume delivered by a pipet or contained by a volumetric flask assumes that the glassware is clean. Dirt and grease on the inner surface prevent liquids from draining evenly, leaving droplets of liquid on the container’s walls. For a pipet this means the delivered volume is less than the calibrated volume, while drops of liquid above the calibration mark mean that a volumetric flask contains more than its calibrated volume. Commercially available cleaning solutions are available for cleaning pipets and volumetric flasks.
Second, when filling a pipet or volumetric flask the liquid’s level must be set exactly at the calibration mark. The liquid’s top surface is curved into a meniscus, the bottom of which should align with the glassware’s calibration mark (Figure 2.4.7 ). When adjusting the meniscus, keep your eye in line with the calibration mark to avoid parallax errors. If your eye level is above the calibration mark you will overfill the pipet or the volumetric flask and you will underfill them if your eye level is below the calibration mark.
Finally, before using a pipet or volumetric flask rinse it with several small portions of the solution whose volume you are measuring. This ensures the removal of any residual liquid remaining in the pipet or volumetric flask.
Equipment for Drying Samples
Many materials need to be dried prior to their analysis to remove residual moisture. Depending on the material, heating to a temperature between 110 oC and 140 oC usually is sufficient. Other materials need much higher temperatures to initiate thermal decomposition.
Conventional drying ovens provide maximum temperatures of 160 oC to 325 oC, depending on the model. Some ovens include the ability to circulate heated air, which allows for a more efficient removal of moisture and shorter drying times. Other ovens provide a tight seal for the door, which allows the oven to be evacuated. In some situations a microwave oven can replace a conventional laboratory oven. Higher temperatures, up to as much as 1700 oC, require a muffle furnace (Figure 2.4.8 ).
After drying or decomposing a sample, it is cooled to room temperature in a desiccator to prevent the readsorption of moisture. A desiccator (Figure 2.4.9 ) is a closed container that isolates the sample from the atmosphere. A drying agent, called a desiccant, is placed in the bottom of the container. Typical desiccants include calcium chloride and silica gel. A perforated plate sits above the desiccant, providing a shelf for storing samples. Some desiccators include a stopcock that allows them to be evacuated. | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/02%3A_Basic_Tools_of_Analytical_Chemistry/2.04%3A_Basic_Equipment.txt |
Preparing a solution of known concentration is perhaps the most common activity in any analytical lab. The method for measuring out the solute and the solvent depend on the desired concentration and how exact the solution’s concentration needs to be known. Pipets and volumetric flasks are used when we need to know a solution’s exact concentration; graduated cylinders, beakers, and/or reagent bottles suffice when a concentrations need only be approximate. Two methods for preparing solutions are described in this section.
Preparing Stock Solutions
A stock solution is prepared by weighing out an appropriate portion of a pure solid or by measuring out an appropriate volume of a pure liquid, placing it in a suitable flask, and diluting to a known volume. Exactly how one measure’s the reagent depends on the desired concentration unit. For example, to prepare a solution with a known molarity you weigh out an appropriate mass of the reagent, dissolve it in a portion of solvent, and bring it to the desired volume. To prepare a solution where the solute’s concentration is a volume percent, you measure out an appropriate volume of solute and add sufficient solvent to obtain the desired total volume.
Example 2.5.1
Describe how to prepare the following three solutions: (a) 500 mL of approximately 0.20 M NaOH using solid NaOH; (b) 1 L of 150.0 ppm Cu2+ using Cu metal; and (c) 2 L of 4% v/v acetic acid using concentrated glacial acetic acid (99.8% w/w acetic acid).
Solution
(a) Because the desired concentration is known to two significant figures, we do not need to measure precisely the mass of NaOH or the volume of solution. The desired mass of NaOH is
$\frac {0.20 \text{ mol NaOH}} {\text{L}} \times \frac {40.0 \text{ g NaOH}} {\text{mol NaOH}} \times 0.50 \text{ L} = 4.0 \text{ g NaOH} \nonumber$
To prepare the solution, place 4.0 grams of NaOH, weighed to the nearest tenth of a gram, in a bottle or beaker and add approximately 500 mL of water.
(b) Since the desired concentration of Cu2+ is given to four significant figures, we must measure precisely the mass of Cu metal and the final solution volume. The desired mass of Cu metal is
$\frac {150.0 \text{ mg Cu}} {\text{L}} \times 1.000 \text{ M } \times \frac {1 \text{ g}} {1000 \text{ mg}} = 0.1500 \text{ g Cu} \nonumber$
To prepare the solution, measure out exactly 0.1500 g of Cu into a small beaker and dissolve it using a small portion of concentrated HNO3. To ensure a complete transfer of Cu2+ from the beaker to the volumetric flask—what we call a quantitative transfer—rinse the beaker several times with small portions of water, adding each rinse to the volumetric flask. Finally, add additional water to the volumetric flask’s calibration mark.
(c) The concentration of this solution is only approximate so it is not necessary to measure exactly the volumes, nor is it necessary to account for the fact that glacial acetic acid is slightly less than 100% w/w acetic acid (it is approximately 99.8% w/w). The necessary volume of glacial acetic acid is
$\frac {4 \text{ mL } \ce{CH3COOH}} {100 \text{ mL}} \times 2000 \text{ mL} = 80 \text{ mL } \ce{CH3COOH} \nonumber$
To prepare the solution, use a graduated cylinder to transfer 80 mL of glacial acetic acid to a container that holds approximately 2 L and add sufficient water to bring the solution to the desired volume.
Exercise 2.5.1
Provide instructions for preparing 500 mL of 0.1250 M KBrO3.
Answer
Preparing 500 mL of 0.1250 M KBrO3 requires
$0.5000 \text{ L} \times \frac {0.1250 \text{ mol } \ce{KBrO3}} {\text{L}} \times \frac {167.00 \text{ g } \ce{KBrO3}} {\text{mol } \ce{KBrO3}} = 10.44 \text{ g } \ce{KBrO3} \nonumber$
Because the concentration has four significant figures, we must prepare the solution using volumetric glassware. Place a 10.44 g sample of KBrO3 in a 500-mL volumetric flask and fill part way with water. Swirl to dissolve the KBrO3 and then dilute with water to the flask’s calibration mark.
Preparing Solutions by Dilution
Solutions are often prepared by diluting a more concentrated stock solution. A known volume of the stock solution is transferred to a new container and brought to a new volume. Since the total amount of solute is the same before and after dilution, we know that
$C_o \times V_o = C_d \times V_d \label{2.1}$
where $C_o$ is the stock solution’s concentration, $V_o$ is the volume of stock solution being diluted, $C_d$ is the dilute solution’s concentration, and $V_d$ is the volume of the dilute solution. Again, the type of glassware used to measure $V_o$ and $V_d$ depends on how precisely we need to know the solution’s concentration.
Note that Equation \ref{2.1} applies only to those concentration units that are expressed in terms of the solution’s volume, including molarity, formality, normality, volume percent, and weight-to-volume percent. It also applies to weight percent, parts per million, and parts per billion if the solution’s density is 1.00 g/mL. We cannot use Equation \ref{2.1} if we express concentration in terms of molality as this is based on the mass of solvent, not the volume of solution. See Rodríquez-López, M.; Carrasquillo, A. J. Chem. Educ. 2005, 82, 1327-1328 for further discussion.
Example 2.5.2
A laboratory procedure calls for 250 mL of an approximately 0.10 M solution of NH3. Describe how you would prepare this solution using a stock solution of concentrated NH3 (14.8 M).
Solution
Substituting known volumes into Equation \ref{2.1}
$14.8 \text{ M} \times V_o = 0.10 \text{ M} \times 250 \text{ mL} \nonumber$
and solving for $V_o$ gives 1.7 mL. Since we are making a solution that is approximately 0.10 M NH3, we can use a graduated cylinder to measure the 1.7 mL of concentrated NH3, transfer the NH3 to a beaker, and add sufficient water to give a total volume of approximately 250 mL.
Although usually we express molarity as mol/L, we can express the volumes in mL if we do so both for both $V_o$ and $V_d$.
Exercise 2.5.2
To prepare a standard solution of Zn2+ you dissolve a 1.004 g sample of Zn wire in a minimal amount of HCl and dilute to volume in a 500-mL volumetric flask. If you dilute 2.000 mL of this stock solution to 250.0 mL, what is the concentration of Zn2+, in μg/mL, in your standard solution?
Answer
The first solution is a stock solution, which we then dilute to prepare the standard solution. The concentration of Zn2+ in the stock solution is
$\frac {1.004 \text{ g } \ce{Zn^{2+}}} {500.0 \text{ mL}} \times \frac {10^6 \: \mu \text{g}} {\text{g}} = 2008 \: \mu \text{g } \ce{Zn^{2+}} \text{/mL} \nonumber$
To find the concentration of the standard solution we use Equation \ref{2.1}
$\frac {2008 \: \mu \text{g } \ce{Zn^{2+}}} {\text{mL}} \times 2.000 \text{ mL} = C_d \times 250.0 \text{ mL} \nonumber$
where Cd is the standard solution’s concentration. Solving gives a concentration of 16.06 μg Zn2+/mL.
As shown in the following example, we can use Equation \ref{2.1} to calculate a solution’s original concentration using its known concentration after dilution.
Example 2.5.3
A sample of an ore was analyzed for Cu2+ as follows. A 1.25 gram sample of the ore was dissolved in acid and diluted to volume in a 250-mL volumetric flask. A 20 mL portion of the resulting solution was transferred by pipet to a 50-mL volumetric flask and diluted to volume. An analysis of this solution gives the concentration of Cu2+ as 4.62 μg/mL. What is the weight percent of Cu in the original ore?
Solution
Substituting known volumes (with significant figures appropriate for pipets and volumetric flasks) into Equation \ref{2.1}
$(C_{\ce{Cu}})_o \times 20.00 \text{ mL} = 4.62 \: \mu \text{g/mL } \ce{Cu^{2+}} \times 50.00 \text{ mL} \nonumber$
and solving for $(C_{\ce{Cu}})_o$ gives the original concentration as 11.55 μg/mL Cu2+. To calculate the grams of Cu2+ we multiply this concentration by the total volume
$\frac {11.55 \mu \text{g } \ce{Cu^{2+}}} {\text{mL}} \times 250.0 \text{ mL} \times \frac {1 \text{ g}} {10^6 \: \mu \text{g}} = 2.888 \times 10^{-3} \text{ g } \ce{Cu^{2+}} \nonumber$
The weight percent Cu is
$\frac {2.888 \times 10^{-3} \text{ g } \ce{Cu^{2+}}} {1.25 \text{ g sample}} \times 100 = 0.231 \text{% w/w } \ce{Cu^{2+}} \nonumber$ | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/02%3A_Basic_Tools_of_Analytical_Chemistry/2.05%3A_Preparing_Solutions.txt |
Analytical chemistry is a quantitative discipline. Whether you are completing a statistical analysis, trying to optimize experimental conditions, or exploring how a change in pH affects a compound’s solubility, the ability to work with complex mathematical equations is essential. Spreadsheets, such as Microsoft Excel are an important tool for analyzing your data and for preparing graphs of your results. Scattered throughout this textbook you will find instructions for using spreadsheets.
If you do not have access to Microsoft Excel or another commercial spreadsheet package, you might considering using Calc, a freely available open-source spreadsheet that is part of the OpenOffice.org software package at www.openoffice.org, or Google Sheets.
Although spreadsheets are useful, they are not always well suited for working with scientific data. If you plan to pursue a career in chemistry, you may wish to familiarize yourself with a more sophisticated computational software package, such as the freely available open-source program that goes by the name R, or commercial programs such as Mathematica or Matlab. You will find instructions for using R scattered throughout this textbook.
You can download the current version of R from www.r-project.org. Click on the link for Download: CRAN and find a local mirror site. Click on the link for the mirror site and then use the link for Linux, MacOS X, or Windows under the heading “Download and Install R.”
Despite the power of spreadsheets and computational programs, don’t forget that the most important software is behind your eyes and between your ears. The ability to think intuitively about chemistry is a critically important skill. In many cases you will find that it is possible to determine if an analytical method is feasible or to approximate the optimum conditions for an analytical method without resorting to complex calculations. Why spend time developing a complex spreadsheet or writing software code when a “back-of-the-envelope” estimate will do the trick? Once you know the general solution to your problem, you can use a spreadsheet or a computational program to work out the specifics. Throughout this textbook we will introduce tools to help develop your ability to think intuitively.
For an interesting take on the importance of intuitive thinking, see Are You Smart Enough to Work at Google? by William Poundstone (Little, Brown and Company, New York, 2012).
2.07: The Laboratory Notebook
Finally, we can not end a chapter on the basic tools of analytical chemistry without mentioning the laboratory notebook. A laboratory notebook is your most important tool when working in the lab. If kept properly, you should be able to look back at your laboratory notebook several years from now and reconstruct the experiments on which you worked.
Your instructor will provide you with detailed instructions on how he or she wants you to maintain your notebook. Of course, you should expect to bring your notebook to the lab. Everything you do, measure, or observe while working in the lab should be recorded in your notebook as it takes place. Preparing data tables to organize your data will help ensure that you record the data you need, and that you can find the data when it is time to calculate and analyze your results. Writing a narrative to accompany your data will help you remember what you did, why you did it, and why you thought it was significant. Reserve space for your calculations, for analyzing your data, and for interpreting your results. Take your notebook with you when you do research in the library.
Maintaining a laboratory notebook may seem like a great deal of effort, but if you do it well you will have a permanent record of your work. Scientists working in academic, industrial and governmental research labs rely on their notebooks to provide a written record of their work. Questions about research carried out at some time in the past can be answered by finding the appropriate pages in the laboratory notebook. A laboratory notebook is also a legal document that helps establish patent rights and proof of discovery. | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/02%3A_Basic_Tools_of_Analytical_Chemistry/2.06%3A_Spreadsheets_and_Computational_Software.txt |
1. Indicate how many significant figures are in each of the following numbers.
a. 903
b. 0.903
c. 1.0903
d. 0.0903
e. 0.09030
f. 9.03 $\times$ 102
2. Round each of the following to three significant figures.
a. 0.89377
b. 0.89328
c. 0.89350
d. 0.8997
e. 0.08907
3. Round each to the stated number of significant figures.
a. the atomic weight of carbon to 4 significant figures
b. the atomic weight of oxygen to 3 significant figures
c. Avogadro’s number to 4 significant figures
d. Faraday’s constant to 3 significant figures
4. Report results for the following calculations to the correct number of significant figures.
a. 4.591 + 0.2309 + 67.1 =
b. 313 – 273.15 =
c. 712 $\times$ 8.6 =
d. 1.43/0.026 =
e. (8.314 $\times$ 298)/96 485 =
f. log(6.53$\times$10–5) =
g. $10^{–7.14} =$
h. (6.51 $\times$ 10–5) $\times$ (8.14 $\times$ 10–9)
5. A 12.1374 g sample of an ore containing Ni and Co is carried through Fresenius’ analytical scheme, as shown in Figure 1.1.1. At point A the combined mass of Ni and Co is 0.2306 g, while at point B the mass of Co is 0.0813 g. Report the weight percent Ni in the ore to the correct number of significant figures.
6. Figure 1.1.2 shows an analytical method for the analysis of Ni in ores based on the precipitation of Ni2+ using dimethylglyoxime. The formula for the precipitate is $\ce{Ni(C4H7N2O2)2}$. Calculate the precipitate’s formula weight to the correct number of significant figures.
7. An analyst wishes to add 256 mg of Cl to a reaction mixture. How many mL of 0.217 M BaCl2 is this?
8. The concentration of lead in an industrial waste stream is 0.28 ppm. What is its molar concentration?
9. Commercially available concentrated hydrochloric acid is 37.0% w/w HCl. Its density is 1.18 g/mL. Using this information calculate (a) the molarity of concentrated HCl, and (b) the mass and volume, in mL, of a solution that contains 0.315 moles of HCl.
10. The density of concentrated ammonia, which is 28.0% w/w NH3, is 0.899 g/mL. What volume of this reagent should you dilute to $1.0 \times 10^{3} \text{ mL}$ to make a solution that is 0.036 M in NH3?
11. A 250.0 mL aqueous solution contains 45.1 μg of a pesticide. Express the pesticide’s concentration in weight-to-volume percent, in parts per million, and in parts per billion.
12. A city’s water supply is fluoridated by adding NaF. The desired concentration of F is 1.6 ppm. How many mg of NaF should you add per gallon of treated water if the water supply already is 0.2 ppm in F?
13. What is the pH of a solution for which the concentration of H+ is $6.92 \times 10^{-6} \text{ M}$? What is the [H+] in a solution whose pH is 8.923?
14. When using a graduate cylinder, the absolute accuracy with which you can deliver a given volume is ±1% of the cylinder’s maximum volume. What are the absolute and the relative uncertainties if you deliver 15 mL of a reagent using a 25 mL graduated cylinder? Repeat for a 50 mL graduated cylinder.
15. Calculate the molarity of a potassium dichromate solution prepared by placing 9.67 grams of K2Cr2O7 in a 100-mL volumetric flask, dissolving, and diluting to the calibration mark.
16. For each of the following explain how you would prepare 1.0 L of a solution that is 0.10 M in K+. Repeat for concentrations of $1.0 \times 10^{2} \text{ ppm } \ce{K+}$ and 1.0% w/v K+.
a. KCl
b. K2SO4
c. K3Fe(CN)6
17. A series of dilute NaCl solutions are prepared starting with an initial stock solution of 0.100 M NaCl. Solution A is prepared by pipeting 10 mL of the stock solution into a 250-mL volumetric flask and diluting to volume. Solution B is prepared by pipeting 25 mL of solution A into a 100-mL volumetric flask and diluting to volume. Solution C is prepared by pipeting 20 mL of solution B into a 500-mL volumetric flask and diluting to volume. What is the molar concentration of NaCl in solutions A, B and C?
18. Calculate the molar concentration of NaCl, to the correct number of significant figures, if 1.917 g of NaCl is placed in a beaker and dissolved in 50 mL of water measured with a graduated cylinder. If this solution is quantitatively transferred to a 250-mL volumetric flask and diluted to volume, what is its concentration to the correct number of significant figures?
19. What is the molar concentration of $\ce{NO3-}$ in a solution prepared by mixing 50.0 mL of 0.050 M KNO3 with 40.0 mL of 0.075 M NaNO3? What is pNO3 for the mixture?
20. What is the molar concentration of Cl in a solution prepared by mixing 25.0 mL of 0.025 M NaCl with 35.0 mL of 0.050 M BaCl2? What is pCl for the mixture?
21. To determine the concentration of ethanol in cognac a 5.00 mL sample of the cognac is diluted to 0.500 L. Analysis of the diluted cognac gives an ethanol concentration of 0.0844 M. What is the molar concentration of ethanol in the undiluted cognac? | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/02%3A_Basic_Tools_of_Analytical_Chemistry/2.08%3A_Problems.txt |
The following two web sites contain useful information about the SI system of units.
For a chemist’s perspective on the SI units for mass and amount, consult the following papers.
• Davis, R. S. “What is a Kilogram in the Revised International System of Units (SI)?”, J. Chem. Educ. 2015, 92, 1604–1609.
• Freeman, R. D. “SI for Chemists: Persistent Problems, Solid Solutions,” J. Chem. Educ. 2003, 80, 16–20.
• Gorin, G. “Mole, Mole per Liter, and Molar: A Primer on SI and Related Units for Chemistry Students,” J. Chem. Educ. 2003, 80, 103–104.
Discussions regarding possible changes in the SI base units are reviewed in these articles.
• Chao, L. S.; Schlamminger, S.; Newell, D. B.; Pratt, J. R.; Seifert, F.; Zhang, X.; Sineriz, M. L.; Haddad, D. “A LEGO Watt Balance: An Apparatus to Determine a Mass Based on the New SI,” arXiv:1412.1699 [physics.ins-det].
• Fraundorf, P. “A Multiple of 12 for Avogadro,” arXiv:1201.5537 [physics.gen-ph].
• Kemsley, J. “Rethinking the Mole and Kilogram,” C&E News, August 25, 2014, p. 25.
The following are useful resources for maintaining a laboratory notebook and for preparing laboratory reports.
• Coghill, A. M.; Garson, L. M. (eds) The ACS Style Guide: Effective Communication of Scientific Information, 3rd Edition, American Chemical Society: Washington, D. C.; 2006.
• Kanare, H. M. Writing the Laboratory Notebook, American Chemical Society: Washington, D. C.; 1985.
The following texts provide instructions for using spreadsheets in analytical chemistry.
• de Levie, R. How to Use Excel® in Analytical Chemistry and in General Scientific Data Analysis, Cambridge University Press: Cambridge, UK, 2001.
• Diamond, D.; Hanratty, V. C. A., Spreadsheet Applications in Chemistry, Wiley-Interscience: New York, 1997.
• Feiser, H. Concepts and Calculations in Analytical Chemistry: A Spreadsheet Approach, CRC Press: Boca Raton, FL, 1992.
The following classic textbook emphasizes the application of intuitive thinking to the solving of problems.
• Harte, J. Consider a Spherical Cow: A Course in Environmental Problem Solving, University Science Books: Sausalito, CA, 1988.
2.10: Chapter Summary and Key Terms
Chapter Summary
There are a few basic numerical and experimental tools with which you must be familiar. Fundamental measurements in analytical chemistry, such as mass, use base SI units, such as the kilogram. Other units, such as energy, are defined in terms of these base units. When reporting a measurement, we must be careful to include only those digits that are significant, and to maintain the uncertainty implied by these significant figures when trans- forming measurements into results.
The relative amount of a constituent in a sample is expressed as a concentration. There are many ways to express concentration, the most common of which are molarity, weight percent, volume percent, weight-to-volume percent, parts per million and parts per billion. Concentrations also can be expressed using p-functions.
Stoichiometric relationships and calculations are important in many quantitative analyses. The stoichiometry between the reactants and the products of a chemical reaction are given by the coefficients of a balanced chemical reaction.
Balances, volumetric flasks, pipets, and ovens are standard pieces of equipment that you will use routinely in the analytical lab. You should be familiar with the proper way to use this equipment. You also should be familiar with how to prepare a stock solution of known concentration, and how to prepare a dilute solution from a stock solution.
Key Terms
analytical balance
desiccator
graduated cylinder
molarity
parts per billion
scientific notation
stock solution
volumetric flask
weight-to-volume percent
concentration
dilution
meniscus
normality
p-function
significant figures
tare
volumetric pipet
desiccant
formality
molality
parts per million
quantitative transfer
SI units
volume percent
weight percent | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/02%3A_Basic_Tools_of_Analytical_Chemistry/2.09%3A_Additional_Resources.txt |
If you browse through an issue of the journal Analytical Chemistry, you will discover that the authors and readers share a common vocabulary of analytical terms. You probably are familiar with some of these terms, such as accuracy and precision, but other terms, such as analyte and matrix, are perhaps less familiar to you. In order to participate in any community, one must first understand its vocabulary; the goal of this chapter, therefore, is to introduce some important analytical terms. Becoming comfortable with these terms will make the chapters that follow easier to read and to understand.
• 3.1: Analysis, Determination, and Measurement
The first important distinction we will make is among the terms analysis, determination, and measurement. An analysis provides chemical or physical information about a sample. The component in the sample of interest to us is called the analyte, and the remainder of the sample is the matrix. In an analysis we determine the identity, the concentration, or the properties of an analyte. To make this determination we measure one or more of the analyte’s chemical or physical properties.
• 3.2: Techniques, Methods, Procedures, and Protocols
Suppose you are asked to develop an analytical method to determine the concentration of lead in drinking water. How would you approach this problem? To provide a structure for answering this question, it is helpful to consider four levels of analytical methodology: techniques, methods, procedures, and protocols.
• 3.3: Classifying Analytical Techniques
The analysis of a sample generates a chemical or physical signal that is proportional to the amount of analyte in the sample. This signal may be anything we can measure, such as volume or absorbance. It is convenient to divide analytical techniques into two general classes based on whether the signal is proportional to the mass or moles of analyte, or is proportional to the analyte’s concentration
• 3.4: Selecting an Analytical Method
A method is the application of a technique to a specific analyte in a specific matrix. Ultimately, the requirements of the analysis determine the best method. In choosing among the available methods, we give consideration to some or all the following design criteria: accuracy, precision, sensitivity, selectivity, robustness, ruggedness, scale of operation, analysis time, availability of equipment, and cost.
• 3.5: Developing the Procedure
After selecting a method, the next step is to develop a procedure that accomplish our goals for the analysis. In developing a procedure we give attention to compensating for interferences, to selecting and calibrating equipment, to acquiring a representative sample, and to validating the method.
• 3.6: Protocols
Earlier we defined a protocol as a set of stringent written guidelines that specify an exact procedure that we must follow if an agency is to accept the results of our analysis. In addition to the considerations that went into the procedure’s design, a protocol also contains explicit instructions regarding internal and external quality assurance and quality control (QA/QC) procedures.
• 3.7: The Importance of Analytical Methodology
The importance of the issues raised in this chapter is evident if we examine environmental monitoring programs. The purpose of a monitoring program is to determine the present status of an environmental system, and to assess long term trends in the system’s health. Without careful planning, however, a poor experimental design may result in data that has little value.
• 3.8: Problems
End-of-chapter problems to test your understanding of topics in this chapter.
• 3.9: Additional Resources
A compendium of resources to accompany this chapter.
• 3.10: Chapter Summary and Key Terms
Summary of chapter's main topics and a list of key terms introduced in the chapter.
03: The Vocabulary of Analytical Chemistry
The first important distinction we will make is among the terms analysis, determination, and measurement. An analysis provides chemical or physical information about a sample. The component in the sample of interest to us is called the analyte, and the remainder of the sample is the matrix. In an analysis we determine the identity, the concentration, or the properties of an analyte. To make this determination we measure one or more of the analyte’s chemical or physical properties.
An example will help clarify the difference between an analysis, a determination and a measurement. In 1974 the federal government enacted the Safe Drinking Water Act to ensure the safety of the nation’s public drinking water supplies. To comply with this act, municipalities monitor their drinking water supply for potentially harmful substances, such as fecal coliform bacteria. Municipal water departments collect and analyze samples from their water supply. To determine the concentration of fecal coliform bacteria an analyst passes a portion of water through a membrane filter, places the filter in a dish that contains a nutrient broth, and incubates the sample for 22–24 hrs at 44.5 oC ± 0.2 oC. At the end of the incubation period the analyst counts the number of bacterial colonies in the dish and reports the result as the number of colonies per 100 mL (Figure 3.1.1 ). Thus, a municipal water department analyzes samples of water to determine the concentration of fecal coliform bacteria by measuring the number of bacterial colonies that form during a carefully defined incubation period
A fecal coliform count provides a general measure of the presence of pathogenic organisms in a water supply. For drinking water, the current maximum contaminant level (MCL) for total coliforms, including fecal coliforms is less than 1 colony/100 mL. Municipal water departments must regularly test the water supply and must take action if more than 5% of the samples in any month test positive for coliform bacteria. | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/03%3A__The_Vocabulary_of_Analytical_Chemistry/3.01%3A_Analysis_Determination_and_Measurement.txt |
Suppose you are asked to develop an analytical method to determine the concentration of lead in drinking water. How would you approach this problem? To provide a structure for answering this question, it is helpful to consider four levels of analytical methodology: techniques, methods, procedures, and protocols [Taylor, J. K. Anal. Chem. 1983, 55, 600A–608A].
A technique is any chemical or physical principle that we can use to study an analyte. There are many techniques for that we can use to determine the concentration of lead in drinking water [Fitch, A.; Wang, Y.; Mellican, S.; Macha, S. Anal. Chem. 1996, 68, 727A–731A]. In graphite furnace atomic absorption spectroscopy (GFAAS), for example, we first convert aqueous lead ions into free atoms—a process we call atomization. We then measure the amount of light absorbed by the free atoms. Thus, GFAAS uses both a chemical principle (atomization) and a physical principle (absorption of light).
See Chapter 10 for a discussion of graphite furnace atomic absorption spectroscopy.
A method is the application of a technique for a specific analyte in a specific matrix. As shown in Figure 3.2.1 , the GFAAS method for determining the concentration of lead in water is different from that for lead in soil or blood.
A procedure is a set of written directions that tell us how to apply a method to a particular sample, including information on how to collect the sample, how to handle interferents, and how to validate results. A method may have several procedures as each analyst or agency adapts it to a specific need. As shown in Figure 3.2.1 , the American Public Health Agency and the American Society for Testing Materials publish separate procedures for determining the concentration of lead in water.
Finally, a protocol is a set of stringent guidelines that specify a procedure that an analyst must follow if an agency is to accept the results. Protocols are common when the result of an analysis supports or defines public policy. When determining the concentration of lead in water under the Safe Drinking Water Act, for example, the analyst must use a protocol specified by the Environmental Protection Agency.
There is an obvious order to these four levels of analytical methodology. Ideally, a protocol uses a previously validated procedure. Before developing and validating a procedure, a method of analysis must be selected. This requires, in turn, an initial screening of available techniques to determine those that have the potential for monitoring the analyte.
3.03: Classifying Analytical Techniques
The analysis of a sample generates a chemical or physical signal that is proportional to the amount of analyte in the sample. This signal may be anything we can measure, such as volume or absorbance. It is convenient to divide analytical techniques into two general classes based on whether the signal is proportional to the mass or moles of analyte, or is proportional to the analyte’s concentration
Consider the two graduated cylinders in Figure 3.3.1 , each of which contains a solution of 0.010 M Cu(NO3)2. Cylinder 1 contains 10 mL, or $1.0 \times 10^{-4}$ moles of Cu2+, and cylinder 2 contains 20 mL, or $2.0 \times 10^{-4}$ moles of Cu2+. If a technique responds to the absolute amount of analyte in the sample, then the signal due to the analyte SA
$S_A = k_A n_A \label{3.1}$
where nA is the moles or grams of analyte in the sample, and kA is a proportionality constant. Because cylinder 2 contains twice as many moles of Cu2+ as cylinder 1, analyzing the contents of cylinder 2 gives a signal twice as large as that for cylinder 1.
A second class of analytical techniques are those that respond to the analyte’s concentration, CA
$S_A = k_A C_A \label{3.2}$
Since the solutions in both cylinders have the same concentration of Cu2+, their analysis yields identical signals.
A technique that responds to the absolute amount of analyte is a total analysis technique. Mass and volume are the most common signals for a total analysis technique, and the corresponding techniques are gravimetry (Chapter 8) and titrimetry (Chapter 9). With a few exceptions, the signal for a total analysis technique is the result of one or more chemical reactions, the stoichiometry of which determines the value of kA in Equation \ref{3.1}.
Historically, most early analytical methods used a total analysis technique. For this reason, total analysis techniques are often called “classical” techniques.
Spectroscopy (Chapter 10) and electrochemistry (Chapter 11), in which an optical or an electrical signal is proportional to the relative amount of analyte in a sample, are examples of concentration techniques. The relationship between the signal and the analyte’s concentration is a theoretical function that depends on experimental conditions and the instrumentation used to measure the signal. For this reason the value of kA in Equation \ref{3.2} is determined experimentally.
Since most concentration techniques rely on measuring an optical or electrical signal, they also are known as “instrumental” techniques. | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/03%3A__The_Vocabulary_of_Analytical_Chemistry/3.02%3A_Techniques_Methods_Procedures_and_Protocols.txt |
A method is the application of a technique to a specific analyte in a specific matrix. We can develop an analytical method to determine the concentration of lead in drinking water using any of the techniques mentioned in the previous section. A gravimetric method, for example, might precipiate the lead as PbSO4 or as PbCrO4, and use the precipitate’s mass as the analytical signal. Lead forms several soluble complexes, which we can use to design a complexation titrimetric method. As shown in Figure 3.2.1, we can use graphite furnace atomic absorption spectroscopy to determine the concentration of lead in drinking water. Finally, lead’s multiple oxidation states (Pb0, Pb2+, Pb4+) makes feasible a variety of electrochemical methods.
Ultimately, the requirements of the analysis determine the best method. In choosing among the available methods, we give consideration to some or all the following design criteria: accuracy, precision, sensitivity, selectivity, robustness, ruggedness, scale of operation, analysis time, availability of equipment, and cost.
Accuracy
Accuracy is how closely the result of an experiment agrees with the “true” or expected result. We can express accuracy as an absolute error, e
$e = \text{obtained result} - \text{expected result} \nonumber$
or as a percentage relative error, %er
$\% e_r = \frac {\text{obtained result} - \text{expected result}} {\text{expected result}} \times 100 \nonumber$
A method’s accuracy depends on many things, including the signal’s source, the value of kA in Equation 3.3.1 or Equation 3.3.2, and the ease of handling samples without loss or contamination. A total analysis technique, such as gravimetry and titrimetry, often produce more accurate results than does a concentration technique because we can measure mass and volume with high accuracy, and because the value of kA is known exactly through stoichiometry.
Because it is unlikely that we know the true result, we use an expected or accepted result to evaluate accuracy. For example, we might use a standard reference material, which has an accepted value, to establish an analytical method’s accuracy. You will find a more detailed treatment of accuracy in Chapter 4, including a discussion of sources of errors.
Precision
When a sample is analyzed several times, the individual results vary from trial-to-trial. Precision is a measure of this variability. The closer the agreement between individual analyses, the more precise the results. For example, the results shown in the upper half of Figure 3.4.1 for the concentration of K+ in a sample of serum are more precise than those in the lower half of Figure 3.4.1 . It is important to understand that precision does not imply accuracy. That the data in the upper half of Figure 3.4.1 are more precise does not mean that the first set of results is more accurate. In fact, neither set of results may be accurate.
A method’s precision depends on several factors, including the uncertainty in measuring the signal and the ease of handling samples reproducibly. In most cases we can measure the signal for a total analysis technique with a higher precision than is the case for a concentration method.
Confusing accuracy and precision is a common mistake. See Ryder, J.; Clark, A. U. Chem. Ed. 2002, 6, 1–3, and Tomlinson, J.; Dyson, P. J.; Garratt, J. U. Chem. Ed. 2001, 5, 16–23 for discussions of this and other common misconceptions about the meaning of error. You will find a more detailed treatment of precision in Chapter 4, including a discussion of sources of errors.
Sensitivity
The ability to demonstrate that two samples have different amounts of analyte is an essential part of many analyses. A method’s sensitivity is a measure of its ability to establish that such a difference is significant. Sensitivity is often confused with a method’s detection limit, which is the smallest amount of analyte we can determine with confidence.
Confidence, as we will see in Chapter 4, is a statistical concept that builds on the idea of a population of results. For this reason, we will postpone our discussion of detection limits to Chapter 4. For now, the definition of a detection limit given here is sufficient.
Sensitivity is equivalent to the proportionality constant, kA, in Equation 3.3.1 and Equation 3.3.2 [IUPAC Compendium of Chemical Terminology, Electronic version]. If $\Delta S_A$ is the smallest difference we can measure between two signals, then the smallest detectable difference in the absolute amount or the relative amount of analyte is
$\Delta n_A = \frac {\Delta S_A} {k_A} \quad \text{ or } \quad \Delta C_A = \frac {\Delta S_A} {k_A} \nonumber$
Suppose, for example, that our analytical signal is a measurement of mass using a balance whose smallest detectable increment is ±0.0001 g. If our method’s sensitivity is 0.200, then our method can conceivably detect a difference in mass of as little as
$\Delta n_A = \frac {\pm 0.0001 \text{ g}} {0.200} = \pm 0.0005 \text{ g} \nonumber$
For two methods with the same $\Delta S_A$, the method with the greater sensitivity—that is, the method with the larger kA—is better able to discriminate between smaller amounts of analyte.
Specificity and Selectivity
An analytical method is specific if its signal depends only on the analyte [Persson, B-A; Vessman, J. Trends Anal. Chem. 1998, 17, 117–119; Persson, B-A; Vessman, J. Trends Anal. Chem. 2001, 20, 526–532]. Although specificity is the ideal, few analytical methods are free from interferences. When an interferent contributes to the signal, we expand Equation 3.3.1 and Equation 3.3.2 to include its contribution to the sample’s signal, Ssamp
$S_{samp} = S_A + S_I = k_A n_A + k_I n_I \label{3.1}$
$S_{samp} = S_A + S_I = k_A C_A + k_I C_I \label{3.2}$
where SI is the interferent’s contribution to the signal, kI is the interferent’s sensitivity, and nI and CI are the moles (or grams) and the concentration of interferent in the sample, respectively.
Selectivity is a measure of a method’s freedom from interferences [Valcárcel, M.; Gomez-Hens, A.; Rubio, S. Trends Anal. Chem. 2001, 20, 386–393]. A method’s selectivity for an interferent relative to the analyte is defined by a selectivity coefficient, KA,I
$K_{A,I} = \frac {k_I} {k_A} \label{3.3}$
which may be positive or negative depending on the signs of kI and kA. The selectivity coefficient is greater than +1 or less than –1 when the method is more selective for the interferent than for the analyte.
Although kA and kI usually are positive, they can be negative. For example, some analytical methods work by measuring the concentration of a species that remains after is reacts with the analyte. As the analyte’s concentration increases, the concentration of the species that produces the signal decreases, and the signal becomes smaller. If the signal in the absence of analyte is assigned a value of zero, then the subsequent signals are negative.
Determining the selectivity coefficient’s value is easy if we already know the values for kA and kI. As shown by Example 3.4.1 , we also can determine KA,I by measuring Ssamp in the presence of and in the absence of the interferent.
Example 3.4.1
A method for the analysis of Ca2+ in water suffers from an interference in the presence of Zn2+. When the concentration of Ca2+ is 100 times greater than that of Zn2+, an analysis for Ca2+ has a relative error of +0.5%. What is the selectivity coefficient for this method?
Solution
Since only relative concentrations are reported, we can arbitrarily assign absolute concentrations. To make the calculations easy, we will let CCa = 100 (arbitrary units) and CZn = 1. A relative error of +0.5% means the signal in the presence of Zn2+ is 0.5% greater than the signal in the absence of Zn2+. Again, we can assign values to make the calculation easier. If the signal for Cu2+ in the absence of Zn2+ is 100 (arbitrary units), then the signal in the presence of Zn2+ is 100.5.
The value of kCa is determined using Equation 3.3.2
$k_\text{Ca} = \frac {S_\text{Ca}} {C_\text{Ca}} = \frac {100} {100} = 1 \nonumber$
In the presence of Zn2+ the signal is given by Equation 3.4.2; thus
$S_{samp} = 100.5 = k_\text{Ca} C_\text{Ca} + k_\text{Zn} C_\text{Zn} = (1 \times 100) + k_\text{Zn} \times 1 \nonumber$
Solving for kZn gives its value as 0.5. The selectivity coefficient is
$K_\text{Ca,Zn} = \frac {k_\text{Zn}} {k_\text{Ca}} = \frac {0.5} {1} = 0.5 \nonumber$
If you are unsure why, in the above example, the signal in the presence of zinc is 100.5, note that the percentage relative error for this problem is given by
$\frac {\text{obtained result} - 100} {100} \times 100 = +0.5 \% \nonumber$
Solving gives an obtained result of 100.5.
Exercise 3.4.1
Wang and colleagues describe a fluorescence method for the analysis of Ag+ in water. When analyzing a solution that contains $1.0 \times 10^{-9}$ M Ag+ and $1.1 \times 10^{-7}$ M Ni2+, the fluorescence intensity (the signal) was +4.9% greater than that obtained for a sample of $1.0 \times 10^{-9}$ M Ag+. What is KAg,Ni for this analytical method? The full citation for the data in this exercise is Wang, L.; Liang, A. N.; Chen, H.; Liu, Y.; Qian, B.; Fu, J. Anal. Chim. Acta 2008, 616, 170-176.
Answer
Because the signal for Ag+ in the presence of Ni2+ is reported as a relative error, we will assign a value of 100 as the signal for $1 \times 10^{-9}$ M Ag+. With a relative error of +4.9%, the signal for the solution of $1 \times 10^{-9}$ M Ag+ and $1.1 \times 10^{-7}$ M Ni2+ is 104.9. The sensitivity for Ag+ is determined using the solution that does not contain Ni2+; thus
$k_\text{Ag} = \frac {S_\text{Ag}} {C_\text{Ag}} = \frac {100} {1 \times 10^{-9} \text{ M}} = 1.0 \times 10^{11} \text{ M}^{-1} \nonumber$
Substituting into Equation \ref{3.2} values for kAg, Ssamp , and the concentrations of Ag+ and Ni2+
$104.9 = (1.0 \times 10^{11} \text{ M}^{-1}) \times (1 \times 10^{-9} \text{ M}) + k_\text{Ni} \times (1.1 \times 10^{-7} \text{ M}) \nonumber$
and solving gives kNi as $4.5 \times 10^7$ M–1. The selectivity coefficient is
$K_\text{Ag,Ni} = \frac {k_\text{Ni}} {k_\text{Ag}} = \frac {4.5 \times 10^7 \text{ M}^{-1}} {1.0 \times 10^{11} \text{ M}^{-1}} = 4.5 \times 10^{-4} \nonumber$
A selectivity coefficient provides us with a useful way to evaluate an interferent’s potential effect on an analysis. Solving Equation \ref{3.3} for kI
$k_I = K_{A,I} \times k_A \label{3.4}$
and substituting in Equation \ref{3.1} and Equation \ref{3.2}, and simplifying gives
$S_{samp} = k_A \{ n_A + K_{A,I} \times n_I \} \label{3.5}$
$S_{samp} = k_A \{ C_A + K_{A,I} \times C_I \} \label{3.6}$
An interferent will not pose a problem as long as the term $K_{A,I} \times n_I$ in Equation \ref{3.5} is significantly smaller than nA, or if $K_{A,I} \times C_I$ in Equation \ref{3.6} is significantly smaller than CA.
Example 3.4.2
Barnett and colleagues developed a method to determine the concentration of codeine (structure shown below) in poppy plants [Barnett, N. W.; Bowser, T. A.; Geraldi, R. D.; Smith, B. Anal. Chim. Acta 1996, 318, 309– 317]. As part of their study they evaluated the effect of several interferents. For example, the authors found that equimolar solutions of codeine and the interferent 6-methoxycodeine gave signals, respectively of 40 and 6 (arbitrary units).
(a) What is the selectivity coefficient for the interferent, 6-methoxycodeine, relative to that for the analyte, codeine.
(b) If we need to know the concentration of codeine with an accuracy of ±0.50%, what is the maximum relative concentration of 6-methoxy-codeine that we can tolerate?
Solution
(a) The signals due to the analyte, SA, and the interferent, SI, are
$S_A = k_A C_A \quad \quad S_I = k_I C_I \nonumber$
Solving these equations for kA and for kI, and substituting into Equation \ref{3.4} gives
$K_{A,I} = \frac {S_I / C_I} {S_A / C_I} \nonumber$
Because the concentrations of analyte and interferent are equimolar (CA = CI), the selectivity coefficient is
$K_{A,I} = \frac {S_I} {S_A} = \frac {6} {40} = 0.15 \nonumber$
(b) To achieve an accuracy of better than ±0.50% the term $K_{A,I} \times C_I$ in Equation \ref{3.6} must be less than 0.50% of CA; thus
$K_{A,I} \times C_I \le 0.0050 \times C_A \nonumber$
Solving this inequality for the ratio CI/CA and substituting in the value for KA,I from part (a) gives
$\frac {C_I} {C_A} \le \frac {0.0050} {K_{A,I}} = \frac {0.0050} {0.15} = 0.033 \nonumber$
Therefore, the concentration of 6-methoxycodeine must be less than 3.3% of codeine’s concentration.
When a method’s signal is the result of a chemical reaction—for example, when the signal is the mass of a precipitate—there is a good chance that the method is not very selective and that it is susceptible to an interference.
Exercise 3.4.2
Mercury (II) also is an interferent in the fluorescence method for Ag+ developed by Wang and colleagues (see Practice Exercise 3.4.1). The selectivity coefficient, KAg,Hg has a value of $-1.0 \times 10^{-3}$.
(a) What is the significance of the selectivity coefficient’s negative sign?
(b) Suppose you plan to use this method to analyze solutions with concentrations of Ag+ no smaller than 1.0 nM. What is the maximum concentration of Hg2+ you can tolerate if your percentage relative errors must be less than ±1.0%?
Answer
(a) A negative value for KAg,Hg means that the presence of Hg2+ decreases the signal from Ag+.
(b) In this case we need to consider an error of –1%, since the effect of Hg2+ is to decrease the signal from Ag+. To achieve this error, the term $K_{A,I} \times C_I$ in Equation \ref{3.6} must be less than –1% of CA; thus
$K_\text{Ag,Hg} \times C_\text{Hg} = -0.01 \times C_\text{Ag} \nonumber$
Substituting in known values for KAg,Hg and CAg, we find that the maximum concentration of Hg2+ is $1.0 \times 10^{-8}$ M.
Problems with selectivity also are more likely when the analyte is present at a very low concentration [Rodgers, L. B. J. Chem. Educ. 1986, 63, 3–6].
Look back at Figure 1.1.1, which shows Fresenius’ analytical method for the determination of nickel in ores. The reason there are so many steps in this procedure is that precipitation reactions generally are not very selective. The method in Figure 1.1.2 includes fewer steps because dimethylglyoxime is a more selective reagent. Even so, if an ore contains palladium, additional steps are needed to prevent the palladium from interfering.
Robustness and Ruggedness
For a method to be useful it must provide reliable results. Unfortunately, methods are subject to a variety of chemical and physical interferences that contribute uncertainty to the analysis. If a method is relatively free from chemical interferences, we can use it to analyze an analyte in a wide variety of sample matrices. Such methods are considered robust.
Random variations in experimental conditions introduces uncertainty. If a method’s sensitivity, k, is too dependent on experimental conditions, such as temperature, acidity, or reaction time, then a slight change in any of these conditions may give a significantly different result. A rugged method is relatively insensitive to changes in experimental conditions.
Scale of Operation
Another way to narrow the choice of methods is to consider three potential limitations: the amount of sample available for the analysis, the expected concentration of analyte in the samples, and the minimum amount of analyte that will produce a measurable signal. Collectively, these limitations define the analytical method’s scale of operations.
We can display the scale of operations visually (Figure 3.4.2 ) by plotting the sample’s size on the x-axis and the analyte’s concentration on the y-axis. For convenience, we divide samples into macro (>0.1 g), meso (10 mg–100 mg), micro (0.1 mg–10 mg), and ultramicro (<0.1 mg) sizes, and we divide analytes into major (>1% w/w), minor (0.01% w/w–1% w/w), trace (10–7% w/w–0.01% w/w), and ultratrace (<10–7% w/w) components. Together, the analyte’s concentration and the sample’s size provide a characteristic description for an analysis. For example, in a microtrace analysis the sample weighs between 0.1 mg and 10 mg and contains a concentration of analyte between 10–7% w/w and 10–2% w/w.
The diagonal lines connecting the axes show combinations of sample size and analyte concentration that contain the same absolute mass of analyte. As shown in Figure 3.4.2 , for example, a 1-g sample that is 1% w/w analyte has the same amount of analyte (10 mg) as a 100-mg sample that is 10% w/w analyte, or a 10-mg sample that is 100% w/w analyte.
We can use Figure 3.4.2 to establish limits for analytical methods. If a method’s minimum detectable signal is equivalent to 10 mg of analyte, then it is best suited to a major analyte in a macro or meso sample. Extending the method to an analyte with a concentration of 0.1% w/w requires a sample of 10 g, which rarely is practical due to the complications of carrying such a large amount of material through the analysis. On the other hand, a small sample that contains a trace amount of analyte places significant restrictions on an analysis. For example, a 1-mg sample that is 10–4% w/w in analyte contains just 1 ng of analyte. If we isolate the analyte in 1 mL of solution, then we need an analytical method that reliably can detect it at a concentration of 1 ng/mL.
It should not surprise you to learn that a total analysis technique typically requires a macro or a meso sample that contains a major analyte. A concentration technique is particularly useful for a minor, trace, or ultratrace analyte in a macro, meso, or micro sample.
Equipment, Time, and Cost
Finally, we can compare analytical methods with respect to their equipment needs, the time needed to complete an analysis, and the cost per sample. Methods that rely on instrumentation are equipment-intensive and may require significant operator training. For example, the graphite furnace atomic absorption spectroscopic method for determining lead in water requires a significant capital investment in the instrument and an experienced operator to obtain reliable results. Other methods, such as titrimetry, require less expensive equipment and less training.
The time to complete an analysis for one sample often is fairly similar from method-to-method. This is somewhat misleading, however, because much of this time is spent preparing samples, preparing reagents, and gathering together equipment. Once the samples, reagents, and equipment are in place, the sampling rate may differ substantially. For example, it takes just a few minutes to analyze a single sample for lead using graphite furnace atomic absorption spectroscopy, but several hours to analyze the same sample using gravimetry. This is a significant factor in selecting a method for a laboratory that handles a high volume of samples.
The cost of an analysis depends on many factors, including the cost of equipment and reagents, the cost of hiring analysts, and the number of samples that can be processed per hour. In general, methods that rely on instruments cost more per sample then other methods.
Making the Final Choice
Unfortunately, the design criteria discussed in this section are not mutually independent [Valcárcel, M.; Ríos, A. Anal. Chem. 1993, 65, 781A–787A]. Working with smaller samples or improving selectivity often comes at the expense of precision. Minimizing cost and analysis time may decrease accuracy. Selecting a method requires carefully balancing the various design criteria. Usually, the most important design criterion is accuracy, and the best method is the one that gives the most accurate result. When the need for a result is urgent, as is often the case in clinical labs, analysis time may become the critical factor.
In some cases it is the sample’s properties that determine the best method. A sample with a complex matrix, for example, may require a method with excellent selectivity to avoid interferences. Samples in which the analyte is present at a trace or ultratrace concentration usually require a concentration method. If the quantity of sample is limited, then the method must not require a large amount of sample.
Determining the concentration of lead in drinking water requires a method that can detect lead at the parts per billion concentration level. Selectivity is important because other metal ions are present at significantly higher concentrations. A method that uses graphite furnace atomic absorption spectroscopy is a common choice for determining lead in drinking water because it meets these specifications. The same method is also useful for determining lead in blood where its ability to detect low concentrations of lead using a few microliters of sample is an important consideration. | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/03%3A__The_Vocabulary_of_Analytical_Chemistry/3.04%3A_Selecting_an_Analytical_Method.txt |
After selecting a method, the next step is to develop a procedure that accomplish our goals for the analysis. In developing a procedure we give attention to compensating for interferences, to selecting and calibrating equipment, to acquiring a representative sample, and to validating the method.
Compensating for Interferences
A method’s accuracy depends on its selectivity for the analyte. Even the best method, however, may not be free from interferents that contribute to the measured signal. Potential interferents may be present in the sample itself or in the reagents used during the analysis.
When the sample is free of interferents, the total signal, Stotal, is a sum of the signal due to the analyte, SA, and the signal due to interferents in the reagents, Sreag,
$S_{total} = S_A + S_{reag} = k_A n_A + S_{reag} \label{3.1}$
$S_{total} = S_A + S_{reag} = k_A C_A + S_{reag} \label{3.2}$
Without an independent determination of Sreag we cannot solve Equation \ref{3.1} or \ref{3.2} for the moles or concentration of analyte.
To determine the contribution of Sreag in Equations \ref{3.1} and \ref{3.2} we measure the signal for a method blank, a solution that does not contain the sample. Consider, for example, a procedure in which we dissolve a 0.1-g sample in a portion of solvent, add several reagents, and dilute to 100 mL with additional solvent. To prepare the method blank we omit the sample and dilute the reagents to 100 mL using the solvent. Because the analyte is absent, Stotal for the method blank is equal to Sreag. Knowing the value for Sreag makes it is easy to correct Stotal for the reagent’s contribution to the total signal; thus
$(S_{total} - S_{reag}) = S_A = k_A n_A \nonumber$
$(S_{total} - S_{reag}) = S_A = k_A C_A \nonumber$
By itself, a method blank cannot compensate for an interferent that is part of the sample’s matrix. If we happen to know the interferent’s identity and concentration, then we can be add it to the method blank; however, this is not a common circumstance and we must, instead, find a method for separating the analyte and interferent before continuing the analysis.
A method blank also is known as a reagent blank. When the sample is a liquid, or is in solution, we use an equivalent volume of an inert solvent as a substitute for the sample.
Calibration
A simple definition of a quantitative analytical method is that it is a mechanism for converting a measurement, the signal, into the amount of analyte in a sample. Assuming we can correct for interferents, a quantitative analysis is nothing more than solving Equation 3.3.1 or Equation 3.3.2 for nA or for CA.
To solve these equations we need the value of kA. For a total analysis method usually we know the value of kA because it is defined by the stoichiometry of the chemical reactions responsible for the signal. For a concentration method, however, the value of kA usually is a complex function of experimental conditions. A calibration is the process of experimentally determining the value of kA by measuring the signal for one or more standard samples, each of which contains a known concentration of analyte.
With a single standard we can calculate the value of kA using Equation 3.3.1 or Equation 3.3.2. When using several standards with different concentrations of analyte, the result is best viewed visually by plotting SA versus the concentration of analyte in the standards. Such a plot is known as a calibration curve, an example of which is shown in Figure 3.5.1 .
Sampling
Selecting an appropriate method and executing it properly helps us ensure that our analysis is accurate. If we analyze the wrong sample, however, then the accuracy of our work is of little consequence.
A proper sampling strategy ensures that our samples are representative of the material from which they are taken. Biased or nonrepresentative sampling, and contaminating samples during or after their collection are two examples of sampling errors that can lead to a significant error in accuracy. It is important to realize that sampling errors are independent of errors in the analytical method. As a result, we cannot correct a sampling error in the laboratory by, for example, evaluating a reagent blank.
Chapter 7 provides a more detailed discussion of sampling, including strategies for obtaining representative samples.
Validation
If we are to have confidence in our procedure we must demonstrate that it can provide acceptable results, a process we call validation. Perhaps the most important part of validating a procedure is establishing that its precision and accuracy are appropriate for the problem we are trying to solve. We also ensure that the written procedure has sufficient detail so that different analysts or laboratories will obtain comparable results. Ideally, validation uses a standard sample whose composition closely matches the samples we will analyze. In the absence of appropriate standards, we can evaluate accuracy by comparing results to those obtained using a method of known accuracy.
You will find more details about validating analytical methods in Chapter 14. | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/03%3A__The_Vocabulary_of_Analytical_Chemistry/3.05%3A_Developing_the_Procedure.txt |
Earlier we defined a protocol as a set of stringent written guidelines that specify an exact procedure that we must follow if an agency is to accept the results of our analysis. In addition to the considerations that went into the procedure’s design, a protocol also contains explicit instructions regarding internal and external quality assurance and quality control (QA/QC) procedures [Amore, F. Anal. Chem. 1979, 51, 1105A–1110A; Taylor, J. K. Anal. Chem. 1981, 53, 1588A–1593A]. The goal of internal QA/QC is to ensure that a laboratory’s work is both accurate and precise. External QA/QC is a process in which an external agency certifies a laboratory.
As an example, let’s outline a portion of the Environmental Protection Agency’s protocol for determining trace metals in water by graphite furnace atomic absorption spectroscopy as part of its Contract Laboratory Program (CLP). The CLP protocol (see Figure 3.6.1 ) calls for an initial calibration using a method blank and three standards, one of which is at the detection limit. The resulting calibration curve is verified by analyzing initial calibration verification (ICV) and initial calibration blank (ICB) samples. The lab’s result for the ICV sample must fall within ±10% of its expected concentration. If the result is outside this limit the analysis is stopped and the problem identified and corrected before continuing.
After a successful analysis of the ICV and ICB samples, the lab reverifies the calibration by analyzing a continuing calibration verification (CCV) sample and a continuing calibration blank (CCB). Results for the CCV also must be within ±10% of its expected concentration. Again, if the lab’s result for the CCV is outside the established limits, the analysis is stopped, the problem identified and corrected, and the system recalibrated as described above. Additional CCV and the CCB samples are analyzed before the first sample and after the last sample, and between every set of ten samples. If the result for any CCV or CCB sample is unacceptable, the results for the last set of samples are discarded, the system is recalibrated, and the samples reanalyzed. By following this protocol, each result is bound by successful checks on the calibration. Although not shown in Figure 3.6.1 , the protocol also contains instructions for analyzing duplicate or split samples, and for using spike tests to verify accuracy.
3.07: The Importance of Analytical Methodology
The importance of the issues raised in this chapter is evident if we examine environmental monitoring programs. The purpose of a monitoring program is to determine the present status of an environmental system, and to assess long term trends in the system’s health. These are broad and poorly defined goals. In many cases, an environmental monitoring program begins before the essential questions are known. This is not surprising since it is difficult to formulate questions in the absence of results. Without careful planning, however, a poor experimental design may result in data that has little value.
These concerns are illustrated by the Chesapeake Bay Monitoring Program. This research program, designed to study nutrients and toxic pollutants in the Chesapeake Bay, was initiated in 1984 as a cooperative venture between the federal government, the state governments of Maryland, Virginia, and Pennsylvania, and the District of Columbia. A 1989 review of the program highlights the problems common to many monitoring programs [D’Elia, C. F.; Sanders, J. G.; Capone, D. G. Envrion. Sci. Technol. 1989, 23, 768–774].
At the beginning of the Chesapeake Bay monitoring program, little attention was given to selecting analytical methods, in large part because the eventual use of the data was not yet specified. The analytical methods initially chosen were standard methods already approved by the Environmental Protection Agency (EPA). In many cases these methods were not useful because they were designed to detect pollutants at their legally mandated maximum allowed concentrations. In unpolluted waters, however, the concentrations of these contaminants often are well below the detection limit of the EPA methods. For example, the detection limit for the EPA approved standard method for phosphate was 7.5 ppb. Since the actual phosphate concentrations in Chesapeake Bay were below the EPA method’s detection limit, it provided no useful information. On the other hand, the detection limit for a non-approved variant of the EPA method, a method routinely used by chemical oceanographers, was 0.06 ppb, a more realistic detection limit for their samples. In other cases, such as the elemental analysis for particulate forms of carbon, nitrogen and phosphorous, EPA approved procedures provided poorer reproducibility than nonapproved methods. | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/03%3A__The_Vocabulary_of_Analytical_Chemistry/3.06%3A_Protocols.txt |
1. When working with a solid sample, often it is necessary to bring the analyte into solution by digesting the sample with a suitable solvent. Any remaining solid impurities are removed by filtration before continuing with the analysis. In a typical total analysis method, the procedure might read
"After digesting the sample in a beaker using approximately 25 mL of solvent, remove any solid impurities that remain by passing the solution the analyte through filter paper, collecting the filtrate in a clean Erlenmeyer flask. Rinse the beaker with several small portions of solvent, passing these rinsings through the filter paper and collecting them in the same Erlenmeyer flask. Finally, rinse the filter paper with several portions of solvent, collecting the rinsings in the same Erlenmeyer flask."
For a typical concentration method, however, the procedure might state
"After digesting the sample in a beaker using 25.00 mL of solvent, remove any solid impurities by filtering a portion of the solution containing the analyte. Collect and discard the first several mL of filtrate before collecting a sample of 5.00 mL for further analysis."
Explain why these two procedures are different.
2. A certain concentration method works best when the analyte’s concentration is approximately 10 ppb.
(a) If the method requires a sample of 0.5 mL, about what mass of analyte is being measured?
(b) If the analyte is present at 10% w/v, how would you prepare the sample for analysis?
(c) Repeat for the case where the analyte is present at 10% w/w.
(d) Based on your answers to parts (a)–(c), comment on the method’s suitability for the determination of a major analyte.
3. An analyst needs to evaluate the potential effect of an interferent, I, on the quantitative analysis for an analyte, A. She begins by measuring the signal for a sample in which the interferent is absent and the analyte is present with a concentration of 15 ppm, obtaining an average signal of 23.3 (arbitrary units). When she analyzes a sample in which the analyte is absent and the interferent is present with a concentration of 25 ppm, she obtains an average signal of 13.7.
(a) What is the sensitivity for the analyte?
(b) What is the sensitivity for the interferent?
(c) What is the value of the selectivity coefficient?
(d) Is the method more selective for the analyte or the interferent?
(e) What is the maximum concentration of interferent relative to that of the analyte if the error in the analysis is to be less than 1%?
4. A sample is analyzed to determine the concentration of an analyte. Under the conditions of the analysis the sensitivity is $17.2 \text{ ppm}^{-1}$. What is the analyte’s concentration if Stotal is 35.2 and Sreag is 0.6?
5. A method for the analysis of Ca2+ in water suffers from an interference in the presence of Zn2+. When the concentration of Ca2+ is 50 times greater than that of Zn2+, an analysis for Ca2+ gives a relative error of –2.0%. What is the value of the selectivity coefficient for this method?
6. The quantitative analysis for reduced glutathione in blood is complicated by many potential interferents. In one study, when analyzing a solution of 10.0 ppb glutathione and 1.5 ppb ascorbic acid, the signal was 5.43 times greater than that obtained for the analysis of 10.0 ppb glutathione [Jiménez-Prieto, R.; Velasco, A.; Silva, M; Pérez-Bendito, D. Anal. Chem. Acta 1992, 269, 273– 279]. What is the selectivity coefficient for this analysis? The same study found that analyzing a solution of $3.5 \times 10^2$ ppb methionine and 10.0 ppb glutathione gives a signal that is 0.906 times less than that obtained for the analysis of 10.0 ppb glutathione. What is the selectivity coefficient for this analysis? In what ways do these interferents behave differently?
7. Oungpipat and Alexander described a method for determining the concentration of glycolic acid (GA) in a variety of samples, including physiological fluids such as urine [Oungpipat, W.; Alexander, P. W. Anal. Chim. Acta 1994, 295, 36–46]. In the presence of only GA, the signal is
$S_{samp,1} = k_\text{GA} C_\text{GA} \nonumber$
and in the presence of both glycolic acid and ascorbic acid (AA), the signal is
$S_{samp,2} = k_\text{GA} C_\text{GA} + k_\text{AA} C_\text{AA} \nonumber$
When the concentration of glycolic acid is $1.0 \times 10^{-4} \text{ M}$ and the concentration of ascorbic acid is $1.0 \times 10^{-5} \text{ M}$, the ratio of their signals is
$\frac {S_{samp,2}} {S_{samp,1}} = 1.44 \nonumber$
(a) Using the ratio of the two signals, determine the value of the selectivity ratio KGA,AA.
(b) Is the method more selective toward glycolic acid or ascorbic acid?
(c) If the concentration of ascorbic acid is $1.0 \times 10^{-5} \text{ M}$, what is the smallest concentration of glycolic acid that can be determined such that the error introduced by failing to account for the signal from ascorbic acid is less than 1%?
8. Ibrahim and co-workers developed a new method for the quantitative analysis of hypoxanthine, a natural compound of some nucleic acids [Ibrahim, M. S.; Ahmad, M. E.; Temerk, Y. M.; Kaucke, A. M. Anal. Chim. Acta 1996, 328, 47–52]. As part of their study they evaluated the method’s selectivity for hypoxanthine in the presence of several possible interferents, including ascorbic acid.
(a) When analyzing a solution of $1.12 \times 10^{-6} \text{ M}$ hypoxanthine the authors obtained a signal of $7.45 \times 10^{-5} \text{ amps}$. What is the sensitivity for hypoxanthine? You may assume the signal has been corrected for the method blank.
(b) When a solution containing $1.12 \times 10^{-6} \text{ M}$ hypoxanthine and $6.5 \times 10^{-5} \text{ M}$ ascorbic acid is analyzed a signal of $4.04 \times 10^{-5} \text{ amps}$ is obtained. What is the selectivity coefficient for this method?
(c) Is the method more selective for hypoxanthine or for ascorbic acid?
(d) What is the largest concentration of ascorbic acid that may be present if a concentration of $1.12 \times 10^{-6} \text{ M}$ hypoxanthine is to be determined within 1.0%?
9. Examine a procedure from Standard Methods for the Analysis of Waters and Wastewaters (or another manual of standard analytical methods) and identify the steps taken to compensate for interferences, to calibrate equipment and instruments, to standardize the method, and to acquire a representative sample. | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/03%3A__The_Vocabulary_of_Analytical_Chemistry/3.08%3A_Problems.txt |
The International Union of Pure and Applied Chemistry (IUPAC) maintains a web-based compendium of analytical terminology. You can find it at the following web site.
• old.iupac.org/publications/an...al_compendium/
The following papers provide alternative schemes for classifying analytical methods.
• Booksh, K. S.; Kowalski, B. R. “Theory of Analytical Chemistry,” Anal. Chem. 1994, 66, 782A– 791A.
• Phillips, J. B. “Classification of Analytical Methods,” Anal. Chem. 1981, 53, 1463A–1470A.
• Valcárcel, M.; Luque de Castro, M. D. “A Hierarchical Approach to Analytical Chemistry,” Trends Anal. Chem. 1995, 14, 242–250.
• Valcárcel, M.; Simonet, B. M. “Types of Analytical Information and Their Mutual Relationships,” Trends Anal. Chem. 1995, 14, 490–495.
Further details on criteria for evaluating analytical methods are found in the following series of papers.
• Wilson, A. L. “The Performance-Characteristics of Analytical Methods”, Part I-Talanta, 1970, 17, 21–29; Part II-Talanta, 1970, 17, 31–44; Part III-Talanta, 1973, 20, 725–732; Part IV-Talanta, 1974, 21, 1109–1121.
For a point/counterpoint debate on the meaning of sensitivity consult the following two papers and two letters of response.
• Ekins, R.; Edwards, P. “On the Meaning of ‘Sensitivity’,” Clin. Chem. 1997, 43, 1824–1831.
• Ekins, R.; Edwards, P. “On the Meaning of ‘Sensitivity:’ A Rejoinder,” Clin. Chem. 1998, 44, 1773–1776.
• Pardue, H. L. “The Inseparable Triangle: Analytical Sensitivity, Measurement Uncertainty, and Quantitative Resolution,” Clin. Chem. 1997, 43, 1831–1837.
• Pardue, H. L. “Reply to ‘On the Meaning of ‘Sensitivity:’ A Rejoinder’,” Clin. Chem. 1998, 44, 1776–1778.
Several texts provide analytical procedures for specific analytes in well-defined matrices.
• Basset, J.; Denney, R. C.; Jeffery, G. H.; Mendham, J. Vogel’s Textbook of Quantitative Inorganic Analysis, 4th Edition; Longman: London, 1981.
• Csuros, M. Environmental Sampling and Analysis for Technicians, Lewis: Boca Raton, 1994.
• Keith, L. H. (ed) Compilation of EPA’s Sampling and Analysis Methods, Lewis: Boca Raton, 1996
• Rump, H. H.; Krist, H. Laboratory Methods for the Examination of Water, Wastewater and Soil, VCH Publishers: NY, 1988.
• Standard Methods for the Analysis of Waters and Wastewaters, 21st Edition, American Public Health Association: Washington, D. C.; 2005.
For a review of the importance of analytical methodology in today’s regulatory environment, consult the following text.
• Miller, J. M.; Crowther, J. B. (eds) Analytical Chemistry in a GMP Environment, John Wiley & Sons: New York, 2000.
3.10: Chapter Summary and Key Terms
Chapter Summary
Every discipline has its own vocabulary and your success in studying ana- lytical chemistry will improve if you master this vocabulary. Be sure you understand the difference between an analyte and its matrix, between a technique and a method, between a procedure and a protocol, and between a total analysis technique and a concentration technique.
In selecting an analytical method we consider criteria such as accu- racy, precision, sensitivity, selectivity, robustness, ruggedness, the amount of available sample, the amount of analyte in the sample, time, cost, and the availability of equipment. These criteria are not mutually independent, and often it is necessary to find an acceptable balance between them.
In developing a procedure or protocol, we give consideration to compensating for interferences, calibrating the method, obtaining an appropriate sample, and validating the analysis. Poorly designed procedures and protocols produce results that are insufficient to meet the needs of the analysis.
Key Terms
accuracy
calibration
detection limit
matrix
method blank
protocol
rugged
sensitivity
technique
analysis
calibration curve
determination
measurement
precision
QA/QC
selectivity
signal
total analysis technique
analyte
concentration technique
interferent
method
procedure
robust
selectivity coefficient
specificity
validation | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/03%3A__The_Vocabulary_of_Analytical_Chemistry/3.09%3A_Additional_Resources.txt |
When we use an analytical method we make three separate evaluations of experimental error. First, before we begin the analysis we evaluate potential sources of errors to ensure they will not adversely effect our results. Second, during the analysis we monitor our measurements to ensure that errors remain acceptable. Finally, at the end of the analysis we evaluate the quality of the measurements and results, and compare them to our original design criteria. This chapter provides an introduction to sources of error, to evaluating errors in analytical measurements, and to the statistical analysis of data.
• 4.1: Characterizing Measurements and Results
One way to characterize data from multiple measurements/runs is to assume that the measurements are randomly scattered around a central value that provides the best estimate of expected, or “true” value. We describe the distribution of these results by reporting its central tendency and its spread.
• 4.2: Characterizing Experimental Errors
Two essential questions arise from any set of data. First, does our measure of central tendency agree with the expected result? Second, why is there so much variability in the individual results? The first of these questions addresses the accuracy of our measurements and the second addresses the precision of our measurements. In this section we consider the types of experimental errors that affect accuracy and precision.
• 4.3: Propagation of Uncertainty
A propagation of uncertainty allows us to estimate the uncertainty in a result from the uncertainties in the measurements used to calculate the result.
• 4.4: The Distribution of Measurements and Results
To compare two samples to each other, we need more than measures of their central tendencies and their spreads based on a small number of measurements. We need also to know how to predict the properties of the broader population from which the samples were drawn; in turn, this requires that we understand the distribution of samples within a population.
• 4.5: Statistical Analysis of Data
A confidence interval is a useful way to report the result of an analysis because it sets limits on the expected result. In the absence of determinate error, a confidence interval based on a sample’s mean indicates the range of values in which we expect to find the population’s mean. In this section we introduce a general approach to the statistical analysis of data. Specific statistical tests are presented in the next section.
• 4.6: Statistical Methods for Normal Distributions
The most common distribution for our results is a normal distribution. Because the area between any two limits of a normal distribution curve is well defined, constructing and evaluating significance tests is straightforward.
• 4.7: Detection Limits
The International Union of Pure and Applied Chemistry (IUPAC) defines a method’s detection limit as the smallest concentration or absolute amount of analyte that has a signal significantly larger than the signal from a suitable blank.
• 4.8: Using Excel and R to Analyze Data
Although the calculations in this chapter are relatively straightforward, it can be tedious to work problems using nothing more than a calculator. Both Excel and R include functions for many common statistical calculations. In addition, R provides useful functions for visualizing your data.
• 4.9: Problems
End-of-chapter problems to test your understanding of topics in this chapter.
• 4.10: Additional Resources
A compendium of resources to accompany topics in this chapter.
• 4.11: Chapter Summary and Key Terms
Summary of chapter's main topics and a list of key terms introduced in this chapter.
04: Evaluating Analytical Data
Let’s begin by choosing a simple quantitative problem that requires a single measurement: What is the mass of a penny? You probably recognize that our statement of the problem is too broad. For example, are we interested in the mass of a United States penny or of a Canadian penny, or is the difference relevant? Because a penny’s composition and size may differ from country to country, let’s narrow our problem to pennies from the United States.
There are other concerns we might consider. For example, the United States Mint produces pennies at two locations (Figure 4.1.1 ). Because it seems unlikely that a penny’s mass depends on where it is minted, we will ignore this concern. Another concern is whether the mass of a newly minted penny is different from the mass of a circulating penny. Because the answer this time is not obvious, let’s further narrow our question and ask “What is the mass of a circulating United States Penny?”
A good way to begin our analysis is to gather some preliminary data. Table 4.1.1 shows masses for seven pennies collected from my change jar. In examining this data we see that our question does not have a simple answer. That is, we can not use the mass of a single penny to draw a specific conclusion about the mass of any other penny (although we might reasonably conclude that all pennies weigh at least 3 g). We can, however, characterize this data by reporting the spread of the individual measurements around a central value.
Table 4.1.1 : Masses of Seven Circulating U. S. Pennies
Penny Mass (g)
1 3.080
2 3.094
3 3.107
4 3.056
5 3.112
6 3.174
7 3.198
Measures of Central Tendency
One way to characterize the data in Table 4.1.1 is to assume that the masses of individual pennies are scattered randomly around a central value that is the best estimate of a penny’s expected, or “true” mass. There are two common ways to estimate central tendency: the mean and the median.
Mean
The mean, $\overline{X}$, is the numerical average for a data set. We calculate the mean by dividing the sum of the individual values by the size of the data set
$\overline{X} = \frac {\sum_{i = 1}^n X_i} {n} \nonumber$
where $X_i$ is the ith measurement, and n is the size of the data set.
Example 4.1.1
What is the mean for the data in Table 4.1.1 ?
Solution
To calculate the mean we add together the results for all measurements
$3.080 + 3.094 + 3.107 + 3.056 + 3.112 + 3.174 + 3.198 = 21.821 \text{ g} \nonumber$
and divide by the number of measurements
$\overline{X} = \frac {21.821 \text{ g}} {7} = 3.117 \text{ g} \nonumber$
The mean is the most common estimate of central tendency. It is not a robust estimate, however, because a single extreme value—one much larger or much smaller than the remainder of the data—influences strongly the mean’s value [Rousseeuw, P. J. J. Chemom. 1991, 5, 1–20]. For example, if we accidently record the third penny’s mass as 31.07 g instead of 3.107 g, the mean changes from 3.117 g to 7.112 g!
An estimate for a statistical parameter is robust if its value is not affected too much by an unusually large or an unusually small measurement.
Median
The median, $\widetilde{X}$, is the middle value when we order our data from the smallest to the largest value. When the data has an odd number of values, the median is the middle value. For an even number of values, the median is the average of the n/2 and the (n/2) + 1 values, where n is the size of the data set.
When n = 5, the median is the third value in the ordered data set; for n = 6, the median is the average of the third and fourth members of the ordered data set.
Example 4.1.2
What is the median for the data in Table 4.1.1 ?
Solution
To determine the median we order the measurements from the smallest to the largest value
$3.056 \quad 3.080 \quad 3.094 \quad 3.107 \quad 3.112 \quad 3.174 \quad 3.198$
Because there are seven measurements, the median is the fourth value in the ordered data; thus, the median is 3.107 g.
As shown by Example 4.1.1 and Example 4.1.2 , the mean and the median provide similar estimates of central tendency when all measurements are comparable in magnitude. The median, however, is a more robust estimate of central tendency because it is less sensitive to measurements with extreme values. For example, if we accidently record the third penny’s mass as 31.07 g instead of 3.107 g, the median’s value changes from 3.107 g to 3.112 g.
Measures of Spread
If the mean or the median provides an estimate of a penny’s expected mass, then the spread of individual measurements about the mean or median provides an estimate of the difference in mass among pennies or of the uncertainty in measuring mass with a balance. Although we often define the spread relative to a specific measure of central tendency, its magnitude is independent of the central value. Although shifting all measurements in the same direction by adding or subtracting a constant value changes the mean or median, it does not change the spread. There are three common measures of spread: the range, the standard deviation, and the variance.
Problem 13 at the end of the chapter asks you to show that this is true.
Range
The range, w, is the difference between a data set’s largest and smallest values.
$w = X_\text{largest} - X_\text{smallest} \nonumber$
The range provides information about the total variability in the data set, but does not provide information about the distribution of individual values. The range for the data in Table 4.1.1 is
$w = 3.198 \text{ g} - 3.056 \text{ g} = 0.142 \text{ g} \nonumber$
Standard Deviation
The standard deviation, s, describes the spread of individual values about their mean, and is given as
$s = \sqrt{\frac {\sum_{i = 1}^{n} (X_i - \overline{X})^{2}} {n - 1}} \label{4.1}$
where $X_i$ is one of the n individual values in the data set, and $\overline{X}$ is the data set's mean value. Frequently, we report the relative standard deviation, sr, instead of the absolute standard deviation.
$s_r = \frac {s} {\overline{X}} \nonumber$
The percent relative standard deviation, %sr, is $s_r \times 100$.
The relative standard deviation is important because it allows for a more meaningful comparison between data sets when the individual measurements differ significantly in magnitude. Consider again the data in Table 4.1.1 . If we multiply each value by 10, the absolute standard deviation will increase by 10 as well; the relative standard deviation, however, is the same.
Example 4.1.3
Report the standard deviation, the relative standard deviation, and the percent relative standard deviation for the data in Table 4.1.1 ?
Solution
To calculate the standard deviation we first calculate the difference between each measurement and the data set’s mean value (3.117), square the resulting differences, and add them together to find the numerator of Equation \ref{4.1}
\begin{align*} (3.080-3.117)^2 = (-0.037)^2 = 0.001369\ (3.094-3.117)^2 = (-0.023)^2 = 0.000529\ (3.107-3.117)^2 = (-0.010)^2 = 0.000100\ (3.056-3.117)^2 = (-0.061)^2 = 0.003721\ (3.112-3.117)^2 = (-0.005)^2 = 0.000025\ (3.174-3.117)^2 = (+0.057)^2 = 0.003249\ (3.198-3.117)^2 = (+0.081)^2 = \underline{0.006561}\ 0.015554 \end{align*}
For obvious reasons, the numerator of Equation \ref{4.1} is called a sum of squares. Next, we divide this sum of squares by n – 1, where n is the number of measurements, and take the square root.
$s = \sqrt{\frac {0.015554} {7 - 1}} = 0.051 \text{ g} \nonumber$
Finally, the relative standard deviation and percent relative standard deviation are
$s_r = \frac {0.051 \text{ g}} {3.117 \text{ g}} = 0.016 \nonumber$
$\% s_r = (0.016) \times 100 = 1.6 \% \nonumber$
It is much easier to determine the standard deviation using a scientific calculator with built in statistical functions.
Many scientific calculators include two keys for calculating the standard deviation. One key calculates the standard deviation for a data set of n samples drawn from a larger collection of possible samples, which corresponds to Equation \ref{4.1}. The other key calculates the standard deviation for all possible samples. The latter is known as the population’s standard deviation, which we will cover later in this chapter. Your calculator’s manual will help you determine the appropriate key for each.
Variance
Another common measure of spread is the variance, which is the square of the standard deviation. We usually report a data set’s standard deviation, rather than its variance, because the mean value and the standard deviation share the same unit. As we will see shortly, the variance is a useful measure of spread because its values are additive.
Example 4.1.4
What is the variance for the data in Table 4.1.1 ?
Solution
The variance is the square of the absolute standard deviation. Using the standard deviation from Example 4.1.3 gives the variance as
$s^2 = (0.051)^2 = 0.0026 \nonumber$
Exercise 4.1.1
The following data were collected as part of a quality control study for the analysis of sodium in serum; results are concentrations of Na+ in mmol/L.
$140 \quad 143 \quad 141 \quad 137 \quad 132 \quad 157 \quad 143 \quad 149 \quad 118 \quad 145$
Report the mean, the median, the range, the standard deviation, and the variance for this data. This data is a portion of a larger data set from Andrew, D. F.; Herzberg, A. M. Data: A Collection of Problems for the Student and Research Worker, Springer-Verlag:New York, 1985, pp. 151–155.
Answer
Mean: To find the mean we add together the individual measurements and divide by the number of measurements. The sum of the 10 concentrations is 1405. Dividing the sum by 10, gives the mean as 140.5, or $1.40 \times 10^2$ mmol/L.
Median: To find the median we arrange the 10 measurements from the smallest concentration to the largest concentration; thus
$118 \quad 132 \quad 137 \quad 140 \quad 141 \quad 143 \quad 143 \quad 145 \quad 149 \quad 157$
The median for a data set with 10 members is the average of the fifth and sixth values; thus, the median is (141 + 143)/2, or 142 mmol/L.
Range: The range is the difference between the largest value and the smallest value; thus, the range is 157 – 118 = 39 mmol/L.
Standard Deviation: To calculate the standard deviation we first calculate the absolute difference between each measurement and the mean value (140.5), square the resulting differences, and add them together. The differences are
$–0.5 \quad 2.5 \quad 0.5 \quad –3.5 \quad –8.5 \quad 16.5 \quad 2.5 \quad 8.5 \quad –22.5 \quad 4.5$
and the squared differences are
$0.25 \quad 6.25 \quad 0.25 \quad 12.25 \quad 72.25 \quad 272.25 \quad 6.25 \quad 72.25 \quad 506.25 \quad 20.25$
The total sum of squares, which is the numerator of Equation \ref{4.1}, is 968.50. The standard deviation is
$s = \sqrt{\frac {968.50} {10 - 1}} = 10.37 \approx 10.4 \nonumber$
Variance: The variance is the square of the standard deviation, or 108. | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/04%3A_Evaluating_Analytical_Data/4.01%3A_Characterizing_Measurements_and_Results.txt |
Characterizing a penny’s mass using the data in Table 4.1.1 suggests two questions. First, does our measure of central tendency agree with the penny’s expected mass? Second, why is there so much variability in the individual results? The first of these questions addresses the accuracy of our measurements and the second addresses the precision of our measurements. In this section we consider the types of experimental errors that affect accuracy and precision.
Errors That Affect Accuracy
Accuracy is how close a measure of central tendency is to its expected value, $\mu$. We express accuracy either as an absolute error, e
$e = \overline{X} - \mu \label{4.1}$
or as a percent relative error, %e
$\% e = \frac {\overline{X} - \mu} {\mu} \times 100 \label{4.2}$
Although Equation \ref{4.1} and Equation \ref{4.2} use the mean as the measure of central tendency, we also can use the median.
The convention for representing a statistical parameter is to use a Roman letter for a value calculated from experimental data, and a Greek letter for its corresponding expected value. For example, the experimentally determined mean is $\overline{X}$ and its underlying expected value is $\mu$. Likewise, the experimental standard deviation is s and the underlying expected value is $\sigma$.
We identify as determinate an error that affects the accuracy of an analysis. Each source of a determinate error has a specific magnitude and sign. Some sources of determinate error are positive and others are negative, and some are larger in magnitude and others are smaller in magnitude. The cumulative effect of these determinate errors is a net positive or negative error in accuracy.
It is possible, although unlikely, that the positive and negative determinate errors will offset each other, producing a result with no net error in accuracy.
We assign determinate errors into four categories—sampling errors, method errors, measurement errors, and personal errors—each of which we consider in this section.
Sampling Errors
A determinate sampling error occurs when our sampling strategy does not provide a us with a representative sample. For example, if we monitor the environmental quality of a lake by sampling from a single site near a point source of pollution, such as an outlet for industrial effluent, then our results will be misleading. To determine the mass of a U. S. penny, our strategy for selecting pennies must ensure that we do not include pennies from other countries.
An awareness of potential sampling errors especially is important when we work with heterogeneous materials. Strategies for obtaining representative samples are covered in Chapter 5.
Method Errors
In any analysis the relationship between the signal, Stotal, and the absolute amount of analyte, nA, or the analyte’s concentration, CA, is
$S_{total} = k_A n_A + S_{mb} \label{4.3}$
$S_{total} = k_A C_A + S_{mb} \label{4.4}$
where kA is the method’s sensitivity for the analyte and Smb is the signal from the method blank. A method error exists when our value for kA or for Smb is in error. For example, a method in which Stotal is the mass of a precipitate assumes that k is defined by a pure precipitate of known stoichiometry. If this assumption is not true, then the resulting determination of nA or CA is inaccurate. We can minimize a determinate error in kA by calibrating the method. A method error due to an interferent in the reagents is minimized by using a proper method blank.
Measurement Errors
The manufacturers of analytical instruments and equipment, such as glassware and balances, usually provide a statement of the item’s maximum measurement error, or tolerance. For example, a 10-mL volumetric pipet (Figure 4.2.1 ) has a tolerance of ±0.02 mL, which means the pipet delivers an actual volume within the range 9.98–10.02 mL at a temperature of 20 oC. Although we express this tolerance as a range, the error is determinate; that is, the pipet’s expected volume, $\mu$, is a fixed value within this stated range.
Volumetric glassware is categorized into classes based on its relative accuracy. Class A glassware is manufactured to comply with tolerances specified by an agency, such as the National Institute of Standards and Technology or the American Society for Testing and Materials. The tolerance level for Class A glassware is small enough that normally we can use it without calibration. The tolerance levels for Class B glassware usually are twice that for Class A glassware. Other types of volumetric glassware, such as beakers and graduated cylinders, are not used to measure volume accurately. Table 4.2.1 provides a summary of typical measurement errors for Class A volumetric glassware. Tolerances for digital pipets and for balances are provided in Table 4.2.2 and Table 4.2.3 .
Table 4.2.1 : Measurement Errors for Type A Volumetric Glassware
Transfer Pipets Volumetric Flasks Burets
Capacity (mL) Tolerance (mL) Capacity (mL) Tolerance (mL) Capacity (mL) Tolerance (mL)
1 $\pm 0.006$ 5 $\pm 0.02$ 10 $\pm 0.02$
2 $\pm 0.006$ 10 $\pm 0.02$ 25 $\pm 0.03$
5 $\pm 0.01$ 25 $\pm 0.03$ 50 $\pm 0.05$
10 $\pm 0.02$ 50 $\pm 0.05$
20 $\pm 0.03$ 100 $\pm 0.08$
25 $\pm 0.03$ 250 $\pm 0.12$
50 $\pm 0.05$ 500 $\pm 0.20$
100 $\pm 0.08$ 1000 $\pm 0.30$
2000
Table 4.2.2 : Measurement Errors for Digital Pipets
Pipet Range Volume (mL or $\mu \text{L}$) Percent Measurement Error
10–100 $\mu \text{L}$ 10 $\pm 3.0\%$
50 $\pm 1.0\%$
100 $\pm 0.8\%$
100–1000 $\mu \text{L}$ 100 $\pm 3.0\%$
500 $\pm 1.0\%$
1000 $\pm 0.6\%$
1–10 mL 1 $\pm 3.0\%$
5 $\pm 0.8\%$
10 $\pm 0.6\%$
The tolerance values for the volumetric glassware in Table 4.2.1 are from the ASTM E288, E542, and E694 standards. The measurement errors for the digital pipets in Table 4.2.2 are from www.eppendorf.com.
Table 4.2.3 : Measurement Errors for Selected Balances
Balance Capacity (g) Measurement Error
Precisa 160M 160 $\pm 1 \text{ mg}$
A & D ER 120M 120 $\pm 0.1 \text{ mg}$
Metler H54 160 $\pm 0.01 \text{ mg}$
We can minimize a determinate measurement error by calibrating our equipment. Balances are calibrated using a reference weight whose mass we can trace back to the SI standard kilogram. Volumetric glassware and digital pipets are calibrated by determining the mass of water delivered or contained and using the density of water to calculate the actual volume. It is never safe to assume that a calibration does not change during an analysis or over time. One study, for example, found that repeatedly exposing volumetric glassware to higher temperatures during machine washing and oven drying, led to small, but significant changes in the glassware’s calibration [Castanheira, I.; Batista, E.; Valente, A.; Dias, G.; Mora, M.; Pinto, L.; Costa, H. S. Food Control 2006, 17, 719–726]. Many instruments drift out of calibration over time and may require frequent recalibration during an analysis.
Personal Errors
Finally, analytical work is always subject to personal error, examples of which include the ability to see a change in the color of an indicator that signals the endpoint of a titration, biases, such as consistently overestimating or underestimating the value on an instrument’s readout scale, failing to calibrate instrumentation, and misinterpreting procedural directions. You can minimize personal errors by taking proper care.
Identifying Determinate Errors
Determinate errors often are difficult to detect. Without knowing the expected value for an analysis, the usual situation in any analysis that matters, we often have nothing to which we can compare our experimental result. Nevertheless, there are strategies we can use to detect determinate errors.
The magnitude of a constant determinate error is the same for all samples and is more significant when we analyze smaller samples. Analyzing samples of different sizes, therefore, allows us to detect a constant determinate error. For example, consider a quantitative analysis in which we separate the analyte from its matrix and determine its mass. Let’s assume the sample is 50.0% w/w analyte. As we see in Table 4.2.4 , the expected amount of analyte in a 0.100 g sample is 0.050 g. If the analysis has a positive constant determinate error of 0.010 g, then analyzing the sample gives 0.060 g of analyte, or an apparent concentration of 60.0% w/w. As we increase the size of the sample the experimental results become closer to the expected result. An upward or downward trend in a graph of the analyte’s experimental concentration versus the sample’s mass (Figure 4.2.2 ) is evidence of a constant determinate error.
Table 4.2.4 : Effect of a Constant Determinate Error on the Analysis of a Sample That is 50.0% w/w Analyte
Mass of Sample (g) Expected Mass
of Analyte (g)
Constant Error (g) Experimental
Mass of Analyte (g)
Experimental
Concentration of Analyte (% w/w)
0.100 0.050 0.010 0.060 60.0
0.200 0.100 0.010 0.110 55.0
0.400 0.200 0.010 0.210 52.5
0.800 0.400 0.010 0.410 51.2
1.600 0.800 0.010 0.810 50.6
A proportional determinate error, in which the error’s magnitude depends on the amount of sample, is more difficult to detect because the result of the analysis is independent of the amount of sample. Table 4.2.5 outlines an example that shows the effect of a positive proportional error of 1.0% on the analysis of a sample that is 50.0% w/w in analyte. Regardless of the sample’s size, each analysis gives the same result of 50.5% w/w analyte.
Table 4.2.5 : Effect of a Proportional Determinate Error on the Analysis of a Sample That is 50.0% w/w Analyte
Mass of Sample (g) Expected Mass
of Analyte (g)
Proportional
Error (%)
Experimental
Mass of Analyte (g)
Experimental
Concentration of Analyte (% w/w)
0.100 0.050 1.00 0.0505 50.5
0.200 0.100 1.00 0.101 50.5
0.400 0.200 1.00 0.202 50.5
0.800 0.400 1.00 0.404 50.5
1.600 0.800 1.00 0.808 50.5
One approach for detecting a proportional determinate error is to analyze a standard that contains a known amount of analyte in a matrix similar to our samples. Standards are available from a variety of sources, such as the National Institute of Standards and Technology (where they are called Standard Reference Materials) or the American Society for Testing and Materials. Table 4.2.6 , for example, lists certified values for several analytes in a standard sample of Gingko biloba leaves. Another approach is to compare our analysis to an analysis carried out using an independent analytical method that is known to give accurate results. If the two methods give significantly different results, then a determinate error is the likely cause.
Table 4.2.6 : Certified Concentrations for SRM 3246: Gingko bilbo (Leaves)
Class of Analyte Analyte Mass Fraction (mg/g or ng/g)
Flavonoids/Ginkgolide B (mass fraction in mg/g) Qurecetin $2.69 \pm 0.31$
Kaempferol $3.02 \pm 0.41$
Isorhamnetin $0.517 \pm 0.0.99$
Total Aglycones $6.22 \pm 0.77$
Selected Terpenes (mass fraction in mg/g) Ginkgolide A $0.57 \pm 0.28$
Ginkgolide B $0.470 \pm 0.090$
Ginkgolide C $0.59 \pm 0.22$
Ginkgolide J $0.18 \pm 0.10$
Bilobalide $1.52 \pm 0.40$
Total Terpene Lactones $3.3 \pm 1.1$
Selected Toxic Elements (mass fraction in ng/g) Cadmium $20.8 \pm 1.0$
Lead $995 \pm 30$
Mercury $23.08 \pm 0.17$
The primary purpose of this Standard Reference Material is to validate analytical methods for determining flavonoids, terpene lactones, and toxic elements in Ginkgo biloba or other materials with a similar matrix. Values are from the official Certificate of Analysis available at www.nist.gov.
Constant and proportional determinate errors have distinctly different sources, which we can define in terms of the relationship between the signal and the moles or concentration of analyte (Equation \ref{4.3} and Equation \ref{4.4}). An invalid method blank, Smb, is a constant determinate error as it adds or subtracts the same value to the signal. A poorly calibrated method, which yields an invalid sensitivity for the analyte, kA, results in a proportional determinate error.
Errors that Affect Precision
As we saw in Section 4.1, precision is a measure of the spread of individual measurements or results about a central value, which we express as a range, a standard deviation, or a variance. Here we draw a distinction between two types of precision: repeatability and reproducibility. Repeatability is the precision when a single analyst completes an analysis in a single session using the same solutions, equipment, and instrumentation. Reproducibility, on the other hand, is the precision under any other set of conditions, including between analysts or between laboratory sessions for a single analyst. Since reproducibility includes additional sources of variability, the reproducibility of an analysis cannot be better than its repeatability.
The ratio of the standard deviation associated with reproducibility to the standard deviation associated with repeatability is called the Horowitz ratio. For a wide variety of analytes in foods, for example, the median Horowtiz ratio is 2.0 with larger values for fatty acids and for trace elements; see Thompson, M.; Wood, R. “The ‘Horowitz Ratio’–A Study of the Ratio Between Reproducibility and Repeatability in the Analysis of Foodstuffs,” Anal. Methods, 2015, 7, 375–379.
Errors that affect precision are indeterminate and are characterized by random variations in their magnitude and their direction. Because they are random, positive and negative indeterminate errors tend to cancel, provided that we make a sufficient number of measurements. In such situations the mean and the median largely are unaffected by the precision of the analysis.
Sources of Indeterminate Error
We can assign indeterminate errors to several sources, including collecting samples, manipulating samples during the analysis, and making measurements. When we collect a sample, for instance, only a small portion of the available material is taken, which increases the chance that small-scale inhomogeneities in the sample will affect repeatability. Individual pennies, for example, may show variations in mass from several sources, including the manufacturing process and the loss of small amounts of metal or the addition of dirt during circulation. These variations are sources of indeterminate sampling errors.
During an analysis there are many opportunities to introduce indeterminate method errors. If our method for determining the mass of a penny includes directions for cleaning them of dirt, then we must be careful to treat each penny in the same way. Cleaning some pennies more vigorously than others might introduce an indeterminate method error.
Finally, all measuring devices are subject to indeterminate measurement errors due to limitations in our ability to read its scale. For example, a buret with scale divisions every 0.1 mL has an inherent indeterminate error of ±0.01–0.03 mL when we estimate the volume to the hundredth of a milliliter (Figure 4.2.3 ).
Evaluating Indeterminate Error
Indeterminate errors associated with our analytical equipment or instrumentation generally are easy to estimate if we measure the standard deviation for several replicate measurements, or if we monitor the signal’s fluctuations over time in the absence of analyte (Figure 4.2.4 ) and calculate the standard deviation. Other sources of indeterminate error, such as treating samples inconsistently, are more difficult to estimate.
To evaluate the effect of an indeterminate measurement error on our analysis of the mass of a circulating United States penny, we might make several determinations of the mass for a single penny (Table 4.2.7 ). The standard deviation for our original experiment (see Table 4.1.1) is 0.051 g, and it is 0.0024 g for the data in Table 4.2.7 . The significantly better precision when we determine the mass of a single penny suggests that the precision of our analysis is not limited by the balance. A more likely source of indeterminate error is a variability in the masses of individual pennies.
Table 4.2.7 : Replicate Determinations of the Mass of a Single Circulating U. S. Penny
Replicate Mass (g) Replicate Mass (g)
1 3.025 6 3.023
2 3.024 7 3.022
3 3.028 8 3.021
4 3.027 9 3.026
5 3.028 10 3.024
In Section 4.5 we will discuss a statistical method—the F-test—that you can use to show that this difference is significant.
Error and Uncertainty
Analytical chemists make a distinction between error and uncertainty [Ellison, S.; Wegscheider, W.; Williams, A. Anal. Chem. 1997, 69, 607A–613A]. Error is the difference between a single measurement or result and its expected value. In other words, error is a measure of bias. As discussed earlier, we divide errors into determinate and indeterminate sources. Although we can find and correct a source of determinate error, the indeterminate portion of the error remains.
Uncertainty expresses the range of possible values for a measurement or result. Note that this definition of uncertainty is not the same as our definition of precision. We calculate precision from our experimental data and use it to estimate the magnitude of indeterminate errors. Uncertainty accounts for all errors—both determinate and indeterminate—that reasonably might affect a measurement or a result. Although we always try to correct determinate errors before we begin an analysis, the correction itself is subject to uncertainty.
Here is an example to help illustrate the difference between precision and uncertainty. Suppose you purchase a 10-mL Class A pipet from a laboratory supply company and use it without any additional calibration. The pipet’s tolerance of ±0.02 mL is its uncertainty because your best estimate of its expected volume is 10.00 mL ± 0.02 mL. This uncertainty primarily is determinate. If you use the pipet to dispense several replicate samples of a solution and determine the volume of each sample, the resulting standard deviation is the pipet’s precision. Table 4.2.8 shows results for ten such trials, with a mean of 9.992 mL and a standard deviation of ±0.006 mL. This standard deviation is the precision with which we expect to deliver a solution using a Class A 10-mL pipet. In this case the pipet’s published uncertainty of ±0.02 mL is worse than its experimentally determined precision of ±0.006 ml. Interestingly, the data in Table 4.2.8 allows us to calibrate this specific pipet’s delivery volume as 9.992 mL. If we use this volume as a better estimate of the pipet’s expected volume, then its uncertainty is ±0.006 mL. As expected, calibrating the pipet allows us to decrease its uncertainty [Kadis, R. Talanta 2004, 64, 167–173].
Table 4.2.8 : Experimental Results for Volume Dispensed by a 10–mL Class A Transfer Pipet
Replicate Volume (ml) Replicate Volume (mL)
1 10.002 6 9.983
2 9.993 7 9.991
3 9.984 8 9.990
4 9.996 9 9.988
5 9.989 10 9.999 | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/04%3A_Evaluating_Analytical_Data/4.02%3A_Characterizing_Experimental_Errors.txt |
Suppose we dispense 20 mL of a reagent using the Class A 10-mL pipet whose calibration information is given in Table 4.2.8. If the volume and uncertainty for one use of the pipet is 9.992 ± 0.006 mL, what is the volume and uncertainty if we use the pipet twice?
As a first guess, we might simply add together the volume and the maximum uncertainty for each delivery; thus
(9.992 mL + 9.992 mL) ± (0.006 mL + 0.006 mL) = 19.984 ± 0.012 mL
It is easy to appreciate that combining uncertainties in this way overestimates the total uncertainty. Adding the uncertainty for the first delivery to that of the second delivery assumes that with each use the indeterminate error is in the same direction and is as large as possible. At the other extreme, we might assume that the uncertainty for one delivery is positive and the other is negative. If we subtract the maximum uncertainties for each delivery,
(9.992 mL + 9.992 mL) ± (0.006 mL – 0.006 mL) = 19.984 ± 0.000 mL
we clearly underestimate the total uncertainty.
So what is the total uncertainty? From the discussion above, we reasonably expect that the total uncertainty is greater than ±0.000 mL and that it is less than ±0.012 mL. To estimate the uncertainty we use a mathematical technique known as the propagation of uncertainty. Our treatment of the propagation of uncertainty is based on a few simple rules.
A Few Symbols
A propagation of uncertainty allows us to estimate the uncertainty in a result from the uncertainties in the measurements used to calculate that result. For the equations in this section we represent the result with the symbol R, and we represent the measurements with the symbols A, B, and C. The corresponding uncertainties are uR, uA, uB, and uC. We can define the uncertainties for A, B, and C using standard deviations, ranges, or tolerances (or any other measure of uncertainty), as long as we use the same form for all measurements.
The requirement that we express each uncertainty in the same way is a critically important point. Suppose you have a range for one measurement, such as a pipet’s tolerance, and standard deviations for the other measurements. All is not lost. There are ways to convert a range to an estimate of the standard deviation. See Appendix 2 for more details.
Uncertainty When Adding or Subtracting
When we add or subtract measurements we propagate their absolute uncertainties. For example, if the result is given by the equation
$R = A + B - C \nonumber$
the the absolute uncertainty in R is
$u_R = \sqrt{u_A^2 + u_B^2 + u_C^2} \label{4.1}$
Example 4.3.1
If we dispense 20 mL using a 10-mL Class A pipet, what is the total volume dispensed and what is the uncertainty in this volume? First, complete the calculation using the manufacturer’s tolerance of 10.00 mL±0.02 mL, and then using the calibration data from Table 4.2.8.
Solution
To calculate the total volume we add the volumes for each use of the pipet. When using the manufacturer’s values, the total volume is
$V = 10.00 \text{ mL} + 10.00 \text{ mL} = 20.00 \text{ mL} \nonumber$
and when using the calibration data, the total volume is
$V = 9.992 \text{ mL} + 9.992 \text{ mL} = 19.984 \text{ mL} \nonumber$
Using the pipet’s tolerance as an estimate of its uncertainty gives the uncertainty in the total volume as
$u_R = (0.02)^2 + (0.02)^2 = 0.028 \text{ mL} = 0.028 \text{ mL} \nonumber$
and using the standard deviation for the data in Table 4.2.8 gives an uncertainty of
$u_R = (0.006)^2 + (0.006)^2 = 0.0085 \text{ mL} \nonumber$
Rounding the volumes to four significant figures gives 20.00 mL ± 0.03 mL when we use the tolerance values, and 19.98 ± 0.01 mL when we use the calibration data.
Uncertainty When Multiplying or Dividing
When we multiple or divide measurements we propagate their relative uncertainties. For example, if the result is given by the equation
$R = \frac {A \times B} {C} \nonumber$
then the relative uncertainty in R is
$\frac {u_R} {R}= \sqrt{\left( \frac {u_A} {A} \right)^2 + \left( \frac {u_B} {B} \right)^2 + \left( \frac {u_C} {C} \right)^2} \label{4.2}$
Example 4.3.2
The quantity of charge, Q, in coulombs that passes through an electrical circuit is
$Q = i \times t \nonumber$
where i is the current in amperes and t is the time in seconds. When a current of 0.15 A ± 0.01 A passes through the circuit for 120 s ± 1 s, what is the total charge and its uncertainty?
Solution
The total charge is
$Q = (0.15 \text{ A}) \times (120 \text{ s}) = 18 \text{ C} \nonumber$
Since charge is the product of current and time, the relative uncertainty in the charge is
$\frac {u_R} {R} = \sqrt{\left( \frac {0.01} {0.15} \right)^2 + \left( \frac {1} {120} \right)^2} = 0.0672 \nonumber$
and the charge’s absolute uncertainty is
$u_R = R \times 0.0672 = (18 \text{ C}) \times (0.0672) = 1.2 \text{ C} \nonumber$
Thus, we report the total charge as 18 C ± 1 C.
Uncertainty for Mixed Operations
Many chemical calculations involve a combination of adding and subtracting, and of multiply and dividing. As shown in the following example, we can calculate the uncertainty by separately treating each operation using Equation \ref{4.1} and Equation \ref{4.2} as needed.
Example 4.3.3
For a concentration technique, the relationship between the signal and the an analyte’s concentration is
$S_{total} = k_A C_A + S_{mb} \nonumber$
What is the analyte’s concentration, CA, and its uncertainty if Stotal is 24.37 ± 0.02, Smb is 0.96 ± 0.02, and kA is $0.186 \pm 0.003 \text{ ppm}^{-1}$?
Solution
Rearranging the equation and solving for CA
$C_A = \frac {S_{total} - S_{mb}} {k_A} = \frac {24.37 - 0.96} {0.186 \text{ ppm}^{-1}} = \frac {23.41} {0.186 \text{ ppm}^{-1}} = 125.9 \text{ ppm} \nonumber$
gives the analyte’s concentration as 126 ppm. To estimate the uncertainty in CA, we first use Equation \ref{4.1} to determine the uncertainty for the numerator.
$u_R = \sqrt{(0.02)^2 + (0.02)^2} = 0.028 \nonumber$
The numerator, therefore, is 23.41 ± 0.028. To complete the calculation we use Equation \ref{4.2} to estimate the relative uncertainty in CA.
$\frac {u_R} {R} = \sqrt{\left( \frac {0.028} {23.41} \right)^2 + \left( \frac {0.003} {0.186} \right)^2} = 0.0162 \nonumber$
The absolute uncertainty in the analyte’s concentration is
$u_R = (125.9 \text{ ppm}) \times (0.0162) = 2.0 \text{ ppm} \nonumber$
Thus, we report the analyte’s concentration as 126 ppm ± 2 ppm.
Exercise 4.3.1
To prepare a standard solution of Cu2+ you obtain a piece of copper from a spool of wire. The spool’s initial weight is 74.2991 g and its final weight is 73.3216 g. You place the sample of wire in a 500-mL volumetric flask, dissolve it in 10 mL of HNO3, and dilute to volume. Next, you pipet a 1 mL portion to a 250-mL volumetric flask and dilute to volume. What is the final concentration of Cu2+ in mg/L, and its uncertainty? Assume that the uncertainty in the balance is ±0.1 mg and that you are using Class A glassware.
Answer
The first step is to determine the concentration of Cu2+ in the final solution. The mass of copper is
$74.2991 \text{ g} - 73.3216 \text{ g} = 0.9775 \text{ g Cu} \nonumber$
The 10 mL of HNO3 used to dissolve the copper does not factor into our calculation. The concentration of Cu2+ is
$\frac {0.9775 \text{ g Cu}} {0.5000 \text{ L}} \times \frac {1.000 \text{ mL}} {250.0 \text{ mL}} \times \frac {1000 \text{ mg}} {\text{g}} = 7.820 \text{ mg } \ce{Cu^{2+}} \text{/L} \nonumber$
Having found the concentration of Cu2+, we continue with the propagation of uncertainty. The absolute uncertainty in the mass of Cu wire is
$u_\text{g Cu} = \sqrt{(0.0001)^2 + (0.0001)^2} = 0.00014 \text{ g} \nonumber$
The relative uncertainty in the concentration of Cu2+ is
$\frac {u_\text{mg/L}} {7.820 \text{ mg/L}} = \sqrt{\left( \frac {0.00014} {0.9775} \right)^2 + \left( \frac {0.20} {500.0} \right)^2 + \left( \frac {0.006} {1.000} \right)^2 + \left( \frac {0.12} {250.0} \right)^2} = 0.00603 \nonumber$
Solving for umg/L gives the uncertainty as 0.0472. The concentration and uncertainty for Cu2+ is 7.820 mg/L ± 0.047 mg/L.
Uncertainty for Other Mathematical Functions
Many other mathematical operations are common in analytical chemistry, including the use of powers, roots, and logarithms. Table 4.3.1 provides equations for propagating uncertainty for some of these function where A and B are independent measurements and where k is a constant whose value has no uncertainty.
Table 4.3.1 : Propagation of Uncertainty for Selected Mathematical Functions
Function $u_R$ Function $u_R$
$R = kA$ $u_R = ku_A$ $R = \ln (A)$ $u_R = \frac {u_A} {A}$
$R = A + B$ $u_R = \sqrt{u_A^2 + u_B^2}$ $R = \log (A)$ $u_R = 0.4343 \times \frac {u_A} {A}$
$R = A - B$ $u_R = \sqrt{u_A^2 + u_B^2}$ $R = e^A$ $\frac {u_R} {R} = u_A$
$R = A \times B$ $\frac {u_R} {R} = \sqrt{\left( \frac {u_A} {A} \right)^2 +\left( \frac {u_B} {B} \right)^2}$ $R = 10^A$ $\frac {u_R} {R} = 2.303 \times u_A$
$R = \frac {A} {B}$ $\frac {u_R} {R} = \sqrt{\left( \frac {u_A} {A} \right)^2 +\left( \frac {u_B} {B} \right)^2}$ $R = A^k$ $\frac {u_R} {R} = k \times \frac {u_A} {A}$
Example 4.3.4
If the pH of a solution is 3.72 with an absolute uncertainty of ±0.03, what is the [H+] and its uncertainty?
Solution
The concentration of H+ is
$[\ce{H+}] = 10^{-\text{pH}} = 10^{-3.72} = 1.91 \times 10^{-4} \text{ M} \nonumber$
or $1.9 \times 10^{-4}$ M to two significant figures. From Table 4.3.1 the relative uncertainty in [H+] is
$\frac {u_R} {R} = 2.303 \times u_A = 2.303 \times 0.03 = 0.069 \nonumber$
The uncertainty in the concentration, therefore, is
$(1.91 \times 10^{-4} \text{ M}) \times (0.069) = 1.3 \times 10^{-5} \text{ M} \nonumber$
We report the [H+] as $1.9 (\pm 0.1) \times 10^{-4}$ M, which is equivalent to $1.9 \times 10^{-4} \text{ M } \pm 0.1 \times 10^{-4} \text{ M}$.
Exercise 4.3.2
A solution of copper ions is blue because it absorbs yellow and orange light. Absorbance, A, is defined as
$A = - \log T = - \log \left( \frac {P} {P_\text{o}} \right) \nonumber$
where, T is the transmittance, Po is the power of radiation as emitted from the light source and P is its power after it passes through the solution. What is the absorbance if Po is $3.80 \times 10^2$ and P is $1.50 \times 10^2$? If the uncertainty in measuring Po and P is 15, what is the uncertainty in the absorbance?
Answer
The first step is to calculate the absorbance, which is
$A = - \log T = -\log \frac {P} {P_\text{o}} = - \log \frac {1.50 \times 10^2} {3.80 \times 10^2} = 0.4037 \approx 0.404 \nonumber$
Having found the absorbance, we continue with the propagation of uncertainty. First, we find the uncertainty for the ratio P/Po, which is the transmittance, T.
$\frac {u_{T}} {T} = \sqrt{\left( \frac {15} {3.80 \times 10^2} \right)^2 + \left( \frac {15} {1.50 \times 10^2} \right)^2 } = 0.1075 \nonumber$
Finally, from Table 4.3.1 the uncertainty in the absorbance is
$u_A = 0.4343 \times \frac {u_{T}} {T} = (0.4343) \times (0.1075) = 4.669 \times 10^{-2} \nonumber$
The absorbance and uncertainty is 0.40 ± 0.05 absorbance units.
Is Calculating Uncertainty Actually Useful?
Given the effort it takes to calculate uncertainty, it is worth asking whether such calculations are useful. The short answer is, yes. Let’s consider three examples of how we can use a propagation of uncertainty to help guide the development of an analytical method.
One reason to complete a propagation of uncertainty is that we can compare our estimate of the uncertainty to that obtained experimentally. For example, to determine the mass of a penny we measure its mass twice—once to tare the balance at 0.000 g and once to measure the penny’s mass. If the uncertainty in each measurement of mass is ±0.001 g, then we estimate the total uncertainty in the penny’s mass as
$u_R = \sqrt{(0.001)^2 + (0.001)^2} = 0.0014 \text{ g} \nonumber$
If we measure a single penny’s mass several times and obtain a standard deviation of ±0.050 g, then we have evidence that the measurement process is out of control. Knowing this, we can identify and correct the problem.
We also can use a propagation of uncertainty to help us decide how to improve an analytical method’s uncertainty. In Example 4.3.3 , for instance, we calculated an analyte’s concentration as 126 ppm ± 2 ppm, which is a percent uncertainty of 1.6%. Suppose we want to decrease the percent uncertainty to no more than 0.8%. How might we accomplish this? Looking back at the calculation, we see that the concentration’s relative uncertainty is determined by the relative uncertainty in the measured signal (corrected for the reagent blank)
$\frac {0.028} {23.41} = 0.0012 \text{ or } 0.12\% \nonumber$
and the relative uncertainty in the method’s sensitivity, kA,
$\frac {0.003 \text{ ppm}^{-1}} {0.186 \text{ ppm}^{-1}} = 0.016 \text{ or } 1.6\% \nonumber$
Of these two terms, the uncertainty in the method’s sensitivity dominates the overall uncertainty. Improving the signal’s uncertainty will not improve the overall uncertainty of the analysis. To achieve an overall uncertainty of 0.8% we must improve the uncertainty in kA to ±0.0015 ppm–1.
Exercise 4.3.3
Verify that an uncertainty of ±0.0015 ppm–1 for kA is the correct result.
Answer
An uncertainty of 0.8% is a relative uncertainty in the concentration of 0.008; thus, letting u be the uncertainty in kA
$0.008 = \sqrt{\left( \frac {0.028} {23.41} \right)^2 + \left( \frac {u} {0.186} \right)^2} \nonumber$
Squaring both sides of the equation gives
$6.4 \times 10^{-5} = \left( \frac {0.028} {23.41} \right)^2 + \left( \frac {u} {0.186} \right)^2 \nonumber$
Solving for the uncertainty in kA gives its value as $1.47 \times 10^{-3}$ or ±0.0015 ppm–1.
Finally, we can use a propagation of uncertainty to determine which of several procedures provides the smallest uncertainty. When we dilute a stock solution usually there are several combinations of volumetric glassware that will give the same final concentration. For instance, we can dilute a stock solution by a factor of 10 using a 10-mL pipet and a 100-mL volumetric flask, or using a 25-mL pipet and a 250-mL volumetric flask. We also can accomplish the same dilution in two steps using a 50-mL pipet and 100-mL volumetric flask for the first dilution, and a 10-mL pipet and a 50-mL volumetric flask for the second dilution. The overall uncertainty in the final concentration—and, therefore, the best option for the dilution—depends on the uncertainty of the volumetric pipets and volumetric flasks. As shown in the following example, we can use the tolerance values for volumetric glassware to determine the optimum dilution strategy [Lam, R. B.; Isenhour, T. L. Anal. Chem. 1980, 52, 1158–1161].
Example 4.3.5 :
Which of the following methods for preparing a 0.0010 M solution from a 1.0 M stock solution provides the smallest overall uncertainty?
(a) A one-step dilution that uses a 1-mL pipet and a 1000-mL volumetric flask.
(b) A two-step dilution that uses a 20-mL pipet and a 1000-mL volumetric flask for the first dilution, and a 25-mL pipet and a 500-mL volumetric flask for the second dilution.
Solution
The dilution calculations for case (a) and case (b) are
$\text{case (a): 1.0 M } \times \frac {1.000 \text { mL}} {1000.0 \text { mL}} = 0.0010 \text{ M} \nonumber$
$\text{case (b): 1.0 M } \times \frac {20.00 \text { mL}} {1000.0 \text { mL}} \times \frac {25.00 \text{ mL}} {500.0 \text{mL}} = 0.0010 \text{ M} \nonumber$
Using tolerance values from Table 4.2.1, the relative uncertainty for case (a) is
$u_R = \sqrt{\left( \frac {0.006} {1.000} \right)^2 + \left( \frac {0.3} {1000.0} \right)^2} = 0.006 \nonumber$
and for case (b) the relative uncertainty is
$u_R = \sqrt{\left( \frac {0.03} {20.00} \right)^2 + \left( \frac {0.3} {1000} \right)^2 + \left( \frac {0.03} {25.00} \right)^2 + \left( \frac {0.2} {500.0} \right)^2} = 0.002 \nonumber$
Since the relative uncertainty for case (b) is less than that for case (a), the two-step dilution provides the smallest overall uncertainty. Of course we must balance the smaller uncertainty for case (b) against the increased opportunity for introducing a determinate error when making two dilutions instead of just one dilution, as in case (a). | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/04%3A_Evaluating_Analytical_Data/4.03%3A_Propagation_of_Uncertainty.txt |
Earlier we reported results for a determination of the mass of a circulating United States penny, obtaining a mean of 3.117 g and a standard deviation of 0.051 g. Table 4.4.1 shows results for a second, independent determination of a penny’s mass, as well as the data from the first experiment. Although the means and standard deviations for the two experiments are similar, they are not identical. The difference between the two experiments raises some interesting questions. Are the results for one experiment better than the results for the other experiment? Do the two experiments provide equivalent estimates for the mean and the standard deviation? What is our best estimate of a penny’s expected mass? To answer these questions we need to understand how we might predict the properties of all pennies using the results from an analysis of a small sample of pennies. We begin by making a distinction between populations and samples.
Table 4.4.1 : Results for Two Determinations of the Mass of a Circulating U. S. Penny
First Experiment Second Experiment
Penny Mass (g) Penny Mass (g)
1 3.080 1 3.052
2 3.094 2 3.141
3 3.107 3 3.083
4 3.056 4 3.083
5 3.112 5 3.048
6 3.174
7 3.198
$\overline{X}$ 3.117 3.081
$s$ 0.051 0.037
Populations and Samples
A population is the set of all objects in the system we are investigating. For the data in Table 4.4.1 , the population is all United States pennies in circulation. This population is so large that we cannot analyze every member of the population. Instead, we select and analyze a limited subset, or sample of the population. The data in Table 4.4.1 , for example, shows the results for two such samples drawn from the larger population of all circulating United States pennies.
Probability Distributions for Populations
Table 4.4.1 provides the means and the standard deviations for two samples of circulating United States pennies. What do these samples tell us about the population of pennies? What is the largest possible mass for a penny? What is the smallest possible mass? Are all masses equally probable, or are some masses more common?
To answer these questions we need to know how the masses of individual pennies are distributed about the population’s average mass. We represent the distribution of a population by plotting the probability or frequency of obtaining a specific result as a function of the possible results. Such plots are called probability distributions.
There are many possible probability distributions; in fact, the probability distribution can take any shape depending on the nature of the population. Fortunately many chemical systems display one of several common probability distributions. Two of these distributions, the binomial distribution and the normal distribution, are discussed in this section.
The Binomial Distribution
The binomial distribution describes a population in which the result is the number of times a particular event occurs during a fixed number of trials. Mathematically, the binomial distribution is defined as
$P(X, N) = \frac {N!} {X!(N - X)!} \times p^X \times (1 - p)^{N - X} \nonumber$
where P(X , N) is the probability that an event occurs X times during N trials, and p is the event’s probability for a single trial. If you flip a coin five times, P(2,5) is the probability the coin will turn up “heads” exactly twice.
The term N! reads as N-factorial and is the product $N \times (N – 1) \times (N – 2) \times \cdots \times 1$. For example, 4! is $4 \times 3 \times 2 \times 1 = 24$. Your calculator probably has a key for calculating factorials.
A binomial distribution has well-defined measures of central tendency and spread. The expected mean value is
$\mu = Np \nonumber$
and the expected spread is given by the variance
$\sigma^2 = Np(1 - p) \nonumber$
or the standard deviation.
$\sigma = \sqrt{Np(1 - p)} \nonumber$
The binomial distribution describes a population whose members have only specific, discrete values. When you roll a die, for example, the possible values are 1, 2, 3, 4, 5, or 6. A roll of 3.45 is not possible. As shown in Worked Example 4.4.1 , one example of a chemical system that obeys the binomial distribution is the probability of finding a particular isotope in a molecule.
Example 4.4.1
Carbon has two stable, non-radioactive isotopes, 12C and 13C, with relative isotopic abundances of, respectively, 98.89% and 1.11%.
(a) What are the mean and the standard deviation for the number of 13C atoms in a molecule of cholesterol (C27H44O)?
(b) What is the probability that a molecule of cholesterol has no atoms of 13C?
Solution
The probability of finding an atom of 13C in a molecule of cholesterol follows a binomial distribution, where X is the number of 13C atoms, N is the number of carbon atoms in a molecule of cholesterol, and p is the probability that an atom of carbon in 13C.
For (a), the mean number of 13C atoms in a molecule of cholesterol is
$\mu = Np = 27 \times 0.0111 = 0.300 \nonumber$
with a standard deviation of
$\sigma = \sqrt{Np(1 - p)} = \sqrt{27 \times 0.0111 \times (1 - 0.0111)} = 0.544 \nonumber$
For (b), the probability of finding a molecule of cholesterol without an atom of 13C is
$P(0, 27) = \frac {27!} {0! \: (27 - 0)!} \times (0.0111)^0 \times (1 - 0.0111)^{27 - 0} = 0.740 \nonumber$
There is a 74.0% probability that a molecule of cholesterol will not have an atom of 13C, a result consistent with the observation that the mean number of 13C atoms per molecule of cholesterol, 0.300, is less than one.
A portion of the binomial distribution for atoms of 13C in cholesterol is shown in Figure 4.4.1 . Note in particular that there is little probability of finding more than two atoms of 13C in any molecule of cholesterol.
The Normal Distribution
A binomial distribution describes a population whose members have only certain discrete values. This is the case with the number of 13C atoms in cholesterol. A molecule of cholesterol, for example, can have two 13C atoms, but it can not have 2.5 atoms of 13C. A population is continuous if its members may take on any value. The efficiency of extracting cholesterol from a sample, for example, can take on any value between 0% (no cholesterol is extracted) and 100% (all cholesterol is extracted).
The most common continuous distribution is the Gaussian, or normal distribution, the equation for which is
$f(X) = \frac {1} {\sqrt{2 \pi \sigma^2}} e^{- \frac {(X - \mu)^2} {2 \sigma^2}} \nonumber$
where $\mu$ is the expected mean for a population with n members
$\mu = \frac {\sum_{i = 1}^n X_i} {n} \nonumber$
and $\sigma^2$ is the population’s variance.
$\sigma^2 = \frac {\sum_{i = 1}^n (X_i - \mu)^2} {n} \label{4.1}$
Examples of three normal distributions, each with an expected mean of 0 and with variances of 25, 100, or 400, respectively, are shown in Figure 4.4.2 . Two features of these normal distribution curves deserve attention. First, note that each normal distribution has a single maximum that corresponds to $\mu$, and that the distribution is symmetrical about this value. Second, increasing the population’s variance increases the distribution’s spread and decreases its height; the area under the curve, however, is the same for all three distributions.
The area under a normal distribution curve is an important and useful property as it is equal to the probability of finding a member of the population within a particular range of values. In Figure 4.4.2 , for example, 99.99% of the population shown in curve (a) have values of X between –20 and +20. For curve (c), 68.26% of the population’s members have values of X between –20 and +20.
Because a normal distribution depends solely on $\mu$ and $\sigma^2$, the probability of finding a member of the population between any two limits is the same for all normally distributed populations. Figure 4.4.3 , for example, shows that 68.26% of the members of a normal distribution have a value within the range $\mu \pm 1 \sigma$, and that 95.44% of population’s members have values within the range $\mu \pm 2 \sigma$. Only 0.27% members of a population have values that exceed the expected mean by more than ± 3$\sigma$. Additional ranges and probabilities are gathered together in the probability table included in Appendix 3. As shown in Example 4.4.2 , if we know the mean and the standard deviation for a normally distributed population, then we can determine the percentage of the population between any defined limits.
Example 4.4.2
The amount of aspirin in the analgesic tablets from a particular manufacturer is known to follow a normal distribution with $\mu$ = 250 mg and $\sigma$ = 5. In a random sample of tablets from the production line, what percentage are expected to contain between 243 and 262 mg of aspirin?
Solution
We do not determine directly the percentage of tablets between 243 mg and 262 mg of aspirin. Instead, we first find the percentage of tablets with less than 243 mg of aspirin and the percentage of tablets having more than 262 mg of aspirin. Subtracting these results from 100%, gives the percentage of tablets that contain between 243 mg and 262 mg of aspirin.
To find the percentage of tablets with less than 243 mg of aspirin or more than 262 mg of aspirin we calculate the deviation, z, of each limit from $\mu$ in terms of the population’s standard deviation, $\sigma$
$z = \frac {X - \mu} {\sigma} \nonumber$
where X is the limit in question. The deviation for the lower limit is
$z_{lower} = \frac {243 - 250} {5} = -1.4 \nonumber$
and the deviation for the upper limit is
$z_{upper} = \frac {262 - 250} {5} = +2.4 \nonumber$
Using the table in Appendix 3, we find that the percentage of tablets with less than 243 mg of aspirin is 8.08%, and that the percentage of tablets with more than 262 mg of aspirin is 0.82%. Therefore, the percentage of tablets containing between 243 and 262 mg of aspirin is
$100.00 \% - 8.08 \% - 0.82 \% = 91.10 \% \nonumber$
Figure 4.4.4 shows the distribution of aspiring in the tablets, with the area in blue showing the percentage of tablets containing between 243 mg and 262 mg of aspirin.
Exercise 4.4.1
What percentage of aspirin tablets will contain between 240 mg and 245 mg of aspirin if the population’s mean is 250 mg and the population’s standard deviation is 5 mg.
Answer
To find the percentage of tablets that contain less than 245 mg of aspirin we first calculate the deviation, z,
$z = \frac {245 - 250} {5} = -1.00 \nonumber$
and then look up the corresponding probability in Appendix 3, obtaining a value of 15.87%. To find the percentage of tablets that contain less than 240 mg of aspirin we find that
$z = \frac {240 - 250} {5} = -2.00 \nonumber$
which corresponds to 2.28%. The percentage of tablets containing between 240 and 245 mg of aspiring is 15.87% – 2.28% = 13.59%.
Confidence Intervals for Populations
If we select at random a single member from a population, what is its most likely value? This is an important question, and, in one form or another, it is at the heart of any analysis in which we wish to extrapolate from a sample to the sample’s parent population. One of the most important features of a population’s probability distribution is that it provides a way to answer this question.
Figure 4.4.3 shows that for a normal distribution, 68.26% of the population’s members have values within the range $\mu \pm 1\sigma$. Stating this another way, there is a 68.26% probability that the result for a single sample drawn from a normally distributed population is in the interval $\mu \pm 1\sigma$. In general, if we select a single sample we expect its value, Xi is in the range
$X_i = \mu \pm z \sigma \label{4.2}$
where the value of z is how confident we are in assigning this range. Values reported in this fashion are called confidence intervals. Equation \ref{4.2}, for example, is the confidence interval for a single member of a population. Table 4.4.2 gives the confidence intervals for several values of z. For reasons discussed later in the chapter, a 95% confidence level is a common choice in analytical chemistry.
When z = 1, we call this the 68.26% confidence interval.
Table 4.4.2 : Confidence Intervals for a Normal Distribution
z Confidence Interval
0.50 38.30
1.00 68.26
1.50 86.64
1.96 95.00
2.00 95.44
2.50 98.76
3.00 99.73
3.50 99.95
Example 4.4.3
What is the 95% confidence interval for the amount of aspirin in a single analgesic tablet drawn from a population for which $\mu$ is 250 mg and for which $\sigma$ is 5?
Solution
Using Table 4.4.2 , we find that z is 1.96 for a 95% confidence interval. Substituting this into Equation \ref{4.2} gives the confidence interval for a single tablet as
$X_i = \mu \pm 1.96\sigma = 250 \text{ mg} \pm (1.96 \times 5) = 250 \text{ mg} \pm 10 \text{mg} \nonumber$
A confidence interval of 250 mg ± 10 mg means that 95% of the tablets in the population contain between 240 and 260 mg of aspirin.
Alternatively, we can rewrite Equation \ref{4.2} so that it gives the confidence interval is for $\mu$ based on the population’s standard deviation and the value of a single member drawn from the population.
$\mu = X_i \pm z \sigma \label{4.3}$
Example 4.4.4
The population standard deviation for the amount of aspirin in a batch of analgesic tablets is known to be 7 mg of aspirin. If you randomly select and analyze a single tablet and find that it contains 245 mg of aspirin, what is the 95% confidence interval for the population’s mean?
Solution
The 95% confidence interval for the population mean is given as
$\mu = X_i \pm z \sigma = 245 \text{ mg} \pm (1.96 \times 7) \text{ mg} = 245 \text{ mg} \pm 14 \text{ mg} \nonumber$
Therefore, based on this one sample, we estimate that there is 95% probability that the population’s mean, $\mu$, lies within the range of 231 mg to 259 mg of aspirin.
Note the qualification that the prediction for $\mu$ is based on one sample; a different sample likely will give a different 95% confidence interval. Our result here, therefore, is an estimate for $\mu$ based on this one sample.
It is unusual to predict the population’s expected mean from the analysis of a single sample; instead, we collect n samples drawn from a population of known $\sigma$, and report the mean, X . The standard deviation of the mean, $\sigma_{\overline{X}}$, which also is known as the standard error of the mean, is
$\sigma_{\overline{X}} = \frac {\sigma} {\sqrt{n}} \nonumber$
The confidence interval for the population’s mean, therefore, is
$\mu = \overline{X} \pm \frac {z \sigma} {\sqrt{n}} \nonumber$
Example 4.4.5
What is the 95% confidence interval for the analgesic tablets in Example 4.4.4 , if an analysis of five tablets yields a mean of 245 mg of aspirin?
Solution
In this case the confidence interval is
$\mu = 245 \text{ mg} \pm \frac {1.96 \times 7} {\sqrt{5}} \text{ mg} = 245 \text{ mg} \pm 6 \text{ mg} \nonumber$
We estimate a 95% probability that the population’s mean is between 239 mg and 251 mg of aspirin. As expected, the confidence interval when using the mean of five samples is smaller than that for a single sample.
Exercise 4.4.2
An analysis of seven aspirin tablets from a population known to have a standard deviation of 5, gives the following results in mg aspirin per tablet:
$246 \quad 249 \quad 255 \quad 251 \quad 251 \quad 247 \quad 250$
What is the 95% confidence interval for the population’s expected mean?
Answer
The mean is 249.9 mg aspirin/tablet for this sample of seven tablets. For a 95% confidence interval the value of z is 1.96, which makes the confidence interval
$249.9 \pm \frac {1.96 \times 5} {\sqrt{7}} = 249.9 \pm 3.7 \approx 250 \text{ mg} \pm 4 \text { mg} \nonumber$
Probability Distributions for Samples
In Examples 4.4.2 –4.4.5 we assumed that the amount of aspirin in analgesic tablets is normally distributed. Without analyzing every member of the population, how can we justify this assumption? In a situation where we cannot study the whole population, or when we cannot predict the mathematical form of a population’s probability distribution, we must deduce the distribution from a limited sampling of its members.
Sample Distributions and the Central Limit Theorem
Let’s return to the problem of determining a penny’s mass to explore further the relationship between a population’s distribution and the distribution of a sample drawn from that population. The two sets of data in Table 4.4.1 are too small to provide a useful picture of a sample’s distribution, so we will use the larger sample of 100 pennies shown in Table 4.4.3 . The mean and the standard deviation for this sample are 3.095 g and 0.0346 g, respectively.
Table 4.4.3 : Masses for a Sample of 100 Circulating U. S. Pennies
Penny Weight (g) Penny Weight (g) Penny Weight (g) Penny Weight (g)
1 3.126 26 3.073 51 3.101 76 3.086
2 3.140 27 3.084 52 3.049 77 3.123
3 3.092 28 3.148 53 3.082 78 3.115
4 3.095 29 3.047 54 3.142 79 3.055
5 3.080 30 3.121 55 3.082 80 3.057
6 3.065 31 3.116 56 3.066 81 3.097
7 3.117 32 3.005 57 3.128 82 3.066
8 3.034 33 3.115 58 3.112 83 3.113
9 3.126 34 3.103 59 3.085 84 3.102
10 3.057 35 3.086 60 3.086 85 3.033
11 3.053 36 3.103 61 3.084 86 3.112
12 3.099 37 3.049 62 3.104 87 3.103
13 3.065 38 2.998 63 3.107 88 3.198
14 3.059 39 3.063 64 3.093 89 3.103
15 3.068 40 3.055 65 3.126 90 3.126
16 3.060 41 3.181 66 3.138 91 3.111
17 3.078 42 3.108 67 3.131 92 3.126
18 3.125 43 3.114 68 3.120 93 3.052
19 3.090 44 3.121 69 3.100 94 3.113
20 3.100 45 3.105 70 3.099 95 3.085
21 3.055 46 3.078 71 3.097 96 3.117
22 3.105 47 3.147 72 3.091 97 3.142
23 3.063 48 3.104 73 3.077 98 3.031
24 3.083 49 3.146 74 3.178 99 3.083
25 3.065 50 3.095 75 3.054 100 3.104
A histogram (Figure 4.4.5 ) is a useful way to examine the data in Table 4.4.3 . To create the histogram, we divide the sample into intervals, by mass, and determine the percentage of pennies within each interval (Table 4.4.4 ). Note that the sample’s mean is the midpoint of the histogram.
Table 4.4.4 : Frequency Distribution for the Data in Table 4.4.3
Mass Interval Frequency (as % of Sample) Mass Interval Frequency (as % of Sample)
2.991 – 3.009 2 3.105 – 3.123 19
3.010 – 3.028 0 3.124 – 3.142 12
3.029 – 3.047 4 3.143 – 3.161 3
3.048 – 3.066 19 3.162 – 3.180 1
3.067 – 3.085 14 3.181 – 3.199 2
3.086 – 3.104 24 3.200 – 3.218 0
Figure 4.4.5 also includes a normal distribution curve for the population of pennies, based on the assumption that the mean and the variance for the sample are appropriate estimates for the population’s mean and variance. Although the histogram is not perfectly symmetric in shape, it provides a good approximation of the normal distribution curve, suggesting that the sample of 100 pennies is normally distributed. It is easy to imagine that the histogram will approximate more closely a normal distribution if we include additional pennies in our sample.
We will not offer a formal proof that the sample of pennies in Table 4.4.3 and the population of all circulating U. S. pennies are normally distributed; however, the evidence in Figure 4.4.5 strongly suggests this is true. Although we cannot claim that the results of all experiments are normally distributed, in most cases our data are normally distributed. According to the central limit theorem, when a measurement is subject to a variety of indeterminate errors, the results for that measurement will approximate a normal distribution [Mark, H.; Workman, J. Spectroscopy 1988, 3, 44–48]. The central limit theorem holds true even if the individual sources of indeterminate error are not normally distributed. The chief limitation to the central limit theorem is that the sources of indeterminate error must be independent and of similar magnitude so that no one source of error dominates the final distribution.
An additional feature of the central limit theorem is that a distribution of means for samples drawn from a population with any distribution will approximate closely a normal distribution if the size of each sample is sufficiently large. For example, Figure 4.4.6 shows the distribution for two samples of 10 000 drawn from a uniform distribution in which every value between 0 and 1 occurs with an equal frequency. For samples of size n = 1, the resulting distribution closely approximates the population’s uniform distribution. The distribution of the means for samples of size n = 10, however, closely approximates a normal distribution.
You might reasonably ask whether this aspect of the central limit theorem is important as it is unlikely that we will complete 10 000 analyses, each of which is the average of 10 individual trials. This is deceiving. When we acquire a sample of soil, for example, it consists of many individual particles each of which is an individual sample of the soil. Our analysis of this sample, therefore, gives the mean for this large number of individual soil particles. Because of this, the central limit theorem is relevant. For a discussion of circumstances where the central limit theorem may not apply, see “Do You Reckon It’s Normally Distributed?”, the full reference for which is Majewsky, M.; Wagner, M.; Farlin, J. Sci. Total Environ. 2016, 548–549, 408–409.
Degrees of Freedom
Did you notice the differences between the equation for the variance of a population and the variance of a sample? If not, here are the two equations:
$\sigma^2 = \frac {\sum_{i = 1}^n (X_i - \mu)^2} {n} \nonumber$
$s^2 = \frac {\sum_{i = 1}^n (X_i - \overline{X})^2} {n - 1} \nonumber$
Both equations measure the variance around the mean, using $\mu$ for a population and $\overline{X}$ for a sample. Although the equations use different measures for the mean, the intention is the same for both the sample and the population. A more interesting difference is between the denominators of the two equations. When we calculate the population’s variance we divide the numerator by the population’s size, n; for the sample’s variance, however, we divide by n – 1, where n is the sample’s size. Why do we divide by n – 1 when we calculate the sample’s variance?
A variance is the average squared deviation of individual results relative to the mean. When we calculate an average we divide the sum by the number of independent measurements, or degrees of freedom, in the calculation. For the population’s variance, the degrees of freedom is equal to the population’s size, n. When we measure every member of a population we have complete information about the population.
When we calculate the sample’s variance, however, we replace $\mu$ with $\overline{X}$, which we also calculate using the same data. If there are n members in the sample, we can deduce the value of the nth member from the remaining n – 1 members and the mean. For example, if $n = 5$ and we know that the first four samples are 1, 2, 3 and 4, and that the mean is 3, then the fifth member of the sample must be
$X_5 = (\overline{X} \times n) - X_1 - X_2 - X_3 - X_4 = (3 \times 5) - 1 - 2 - 3 - 4 = 5 \nonumber$
Because we have just four independent measurements, we have lost one degree of freedom. Using n – 1 in place of n when we calculate the sample’s variance ensures that $s^2$ is an unbiased estimator of $\sigma^2$.
Here is another way to think about degrees of freedom. We analyze samples to make predictions about the underlying population. When our sample consists of n measurements we cannot make more than n independent predictions about the population. Each time we estimate a parameter, such as the population’s mean, we lose a degree of freedom. If there are n degrees of freedom for calculating the sample’s mean, then n – 1 degrees of freedom remain when we calculate the sample’s variance.
Confidence Intervals for Samples
Earlier we introduced the confidence interval as a way to report the most probable value for a population’s mean, $\mu$
$\mu = \overline{X} \pm \frac {z \sigma} {\sqrt{n}} \label{4.4}$
where $\overline{X}$ is the mean for a sample of size n, and $\sigma$ is the population’s standard deviation. For most analyses we do not know the population’s standard deviation. We can still calculate a confidence interval, however, if we make two modifications to Equation \ref{4.4}.
The first modification is straightforward—we replace the population’s standard deviation, $\sigma$, with the sample’s standard deviation, s. The second modification is not as obvious. The values of z in Table 4.4.2 are for a normal distribution, which is a function of $sigma^2$, not s2. Although the sample’s variance, s2, is an unbiased estimate of the population’s variance, $\sigma^2$, the value of s2 will only rarely equal $\sigma^2$. To account for this uncertainty in estimating $\sigma^2$, we replace the variable z in Equation \ref{4.4} with the variable t, where t is defined such that $t \ge z$ at all confidence levels.
$\mu = \overline{X} \pm \frac {t s} {\sqrt{n}} \label{4.5}$
Values for t at the 95% confidence level are shown in Table 4.4.5 . Note that t becomes smaller as the number of degrees of freedom increases, and that it approaches z as n approaches infinity. The larger the sample, the more closely its confidence interval for a sample (Equation \ref{4.5}) approaches the confidence interval for the population (Equation \ref{4.3}). Appendix 4 provides additional values of t for other confidence levels.
Table 4.4.5 : Values of t for a 95% Confidence Interval
Degrees of Freedom t Degrees of Freedom t Degrees of Freedom t Degrees of Freedom t
1 12.706 6 2.447 12 2.179 30 2.042
2 4.303 7 2.365 14 2.145 40 2.021
3 3.181 8 2.306 16 2.120 60 2.000
4 2.776 9 2.262 18 2.101 100 1.984
5 2.571 10 2.228 20 2.086 $\infty 1.960 Example 4.4.6 What are the 95% confidence intervals for the two samples of pennies in Table 4.4.1 ? Solution The mean and the standard deviation for first experiment are, respectively, 3.117 g and 0.051 g. Because the sample consists of seven measurements, there are six degrees of freedom. The value of t from Table 4.4.5 , is 2.447. Substituting into Equation \ref{4.5} gives $\mu = 3.117 \text{ g} \pm \frac {2.447 \times 0.051 \text{ g}} {\sqrt{7}} = 3.117 \text{ g} \pm 0.047 \text{ g} \nonumber$ For the second experiment the mean and the standard deviation are 3.081 g and 0.073 g, respectively, with four degrees of freedom. The 95% confidence interval is $\mu = 3.081 \text{ g} \pm \frac {2.776 \times 0.037 \text{ g}} {\sqrt{5}} = 3.081 \text{ g} \pm 0.046 \text{ g} \nonumber$ Based on the first experiment, the 95% confidence interval for the population’s mean is 3.070–3.164 g. For the second experiment, the 95% confidence interval is 3.035–3.127 g. Although the two confidence intervals are not identical—remember, each confidence interval provides a different estimate for \(\mu$—the mean for each experiment is contained within the other experiment’s confidence interval. There also is an appreciable overlap of the two confidence intervals. Both of these observations are consistent with samples drawn from the same population.
Note that our comparison of these two confidence intervals at this point is somewhat vague and unsatisfying. We will return to this point in the next section, when we consider a statistical approach to comparing the results of experiments.
Exercise 4.4.3
What is the 95% confidence interval for the sample of 100 pennies in Table 4.4.3 ? The mean and the standard deviation for this sample are 3.095 g and 0.0346 g, respectively. Compare your result to the confidence intervals for the samples of pennies in Table 4.4.1 .
Answer
With 100 pennies, we have 99 degrees of freedom for the mean. Although Table 4.4.3 does not include a value for t(0.05, 99), we can approximate its value by using the values for t(0.05, 60) and t(0.05, 100) and by assuming a linear change in its value.
$t(0.05, 99) = t(0.05, 60) - \frac {39} {40} \left\{ t(0.05, 60) - t(0.05, 100\} \right) \nonumber$
$t(0.05, 99) = 2.000 - \frac {39} {40} \left\{ 2.000 - 1.984 \right\} = 1.9844 \nonumber$
The 95% confidence interval for the pennies is
$3.095 \pm \frac {1.9844 \times 0.0346} {\sqrt{100}} = 3.095 \text{ g} \pm 0.007 \text{ g} \nonumber$
From Example 4.4.6 , the 95% confidence intervals for the two samples in Table 4.4.1 are 3.117 g ± 0.047 g and 3.081 g ± 0.046 g. As expected, the confidence interval for the sample of 100 pennies is much smaller than that for the two smaller samples of pennies. Note, as well, that the confidence interval for the larger sample fits within the confidence intervals for the two smaller samples.
A Cautionary Statement
There is a temptation when we analyze data simply to plug numbers into an equation, carry out the calculation, and report the result. This is never a good idea, and you should develop the habit of reviewing and evaluating your data. For example, if you analyze five samples and report an analyte’s mean concentration as 0.67 ppm with a standard deviation of 0.64 ppm, then the 95% confidence interval is
$\mu = 0.67 \text{ ppm} \pm \frac {2.776 \times 0.64 \text{ ppm}} {\sqrt{5}} = 0.67 \text{ ppm} \pm 0.79 \text{ ppm} \nonumber$
This confidence interval estimates that the analyte’s true concentration is between –0.12 ppm and 1.46 ppm. Including a negative concentration within the confidence interval should lead you to reevaluate your data or your conclusions. A closer examination of your data may convince you that the standard deviation is larger than expected, making the confidence interval too broad, or you may conclude that the analyte’s concentration is too small to report with confidence.
We will return to the topic of detection limits near the end of this chapter.
Here is a second example of why you should closely examine your data: results obtained on samples drawn at random from a normally distributed population must be random. If the results for a sequence of samples show a regular pattern or trend, then the underlying population either is not normally distributed or there is a time-dependent determinate error. For example, if we randomly select 20 pennies and find that the mass of each penny is greater than that for the preceding penny, then we might suspect that our balance is drifting out of calibration. | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/04%3A_Evaluating_Analytical_Data/4.04%3A_The_Distribution_of_Measurements_and_Results.txt |
A confidence interval is a useful way to report the result of an analysis because it sets limits on the expected result. In the absence of determinate error, a confidence interval based on a sample’s mean indicates the range of values in which we expect to find the population’s mean. When we report a 95% confidence interval for the mass of a penny as 3.117 g ± 0.047 g, for example, we are stating that there is only a 5% probability that the penny’s expected mass is less than 3.070 g or more than 3.164 g.
Because a confidence interval is a statement of probability, it allows us to consider comparative questions, such as these: “Are the results for a newly developed method to determine cholesterol in blood significantly different from those obtained using a standard method?” or “Is there a significant variation in the composition of rainwater collected at different sites downwind from a coal-burning utility plant?” In this section we introduce a general approach to the statistical analysis of data. Specific statistical tests are presented in Section 4.6.
The reliability of significance testing recently has received much attention—see Nuzzo, R. “Scientific Method: Statistical Errors,” Nature, 2014, 506, 150–152 for a general discussion of the issues—so it is appropriate to begin this section by noting the need to ensure that our data and our research question are compatible so that we do not read more into a statistical analysis than our data allows; see Leek, J. T.; Peng, R. D. “What is the Question? Science, 2015, 347, 1314-1315 for a use-ul discussion of six common research questions.
In the context of analytical chemistry, significance testing often accompanies an exploratory data analysis (Is there a reason to suspect that there is a difference between these two analytical methods when applied to a common sample?) or an inferential data analysis (Is there a reason to suspect that there is a relationship between these two independent measurements?). A statistically significant result for these types of analytical research questions generally leads to the design of additional experiments better suited to making predictions or to explaining an underlying causal relationship. A significance test is the first step toward building a greater understanding of an analytical problem, not the final answer to that problem.
Significance Testing
Let’s consider the following problem. To determine if a medication is effective in lowering blood glucose concentrations, we collect two sets of blood samples from a patient. We collect one set of samples immediately before we administer the medication, and collect the second set of samples several hours later. After analyzing the samples, we report their respective means and variances. How do we decide if the medication was successful in lowering the patient’s concentration of blood glucose?
One way to answer this question is to construct a normal distribution curve for each sample, and to compare the two curves to each other. Three possible outcomes are shown in Figure 4.5.1 . In Figure 4.5.1 a, there is a complete separation of the two normal distribution curves, which suggests the two samples are significantly different from each other. In Figure 4.5.1 b, the normal distribution curves for the two samples almost completely overlap, which suggests that the difference between the samples is insignificant. Figure 4.5.1 c, however, presents us with a dilemma. Although the means for the two samples seem different, the overlap of their normal distribution curves suggests that a significant number of possible outcomes could belong to either distribution. In this case the best we can do is to make a statement about the probability that the samples are significantly different from each other.
The process by which we determine the probability that there is a significant difference between two samples is called significance testing or hypothesis testing. Before we discuss specific examples we will first establish a general approach to conducting and interpreting a significance test.
Constructing a Significance Test
The purpose of a significance test is to determine whether the difference between two or more results is sufficiently large that it cannot be explained by indeterminate errors. The first step in constructing a significance test is to state the problem as a yes or no question, such as “Is this medication effective at lowering a patient’s blood glucose levels?” A null hypothesis and an alternative hypothesis define the two possible answers to our yes or no question. The null hypothesis, H0, is that indeterminate errors are sufficient to explain any differences between our results. The alternative hypothesis, HA, is that the differences in our results are too great to be explained by random error and that they must be determinate in nature. We test the null hypothesis, which we either retain or reject. If we reject the null hypothesis, then we must accept the alternative hypothesis and conclude that the difference is significant.
Failing to reject a null hypothesis is not the same as accepting it. We retain a null hypothesis because we have insufficient evidence to prove it incorrect. It is impossible to prove that a null hypothesis is true. This is an important point and one that is easy to forget. To appreciate this point let’s return to our sample of 100 pennies in Table 4.4.3. After looking at the data we might propose the following null and alternative hypotheses.
H0: The mass of a circulating U.S. penny is between 2.900 g and 3.200 g
HA: The mass of a circulating U.S. penny may be less than 2.900 g or more than 3.200 g
To test the null hypothesis we find a penny and determine its mass. If the penny’s mass is 2.512 g then we can reject the null hypothesis and accept the alternative hypothesis. Suppose that the penny’s mass is 3.162 g. Although this result increases our confidence in the null hypothesis, it does not prove that the null hypothesis is correct because the next penny we sample might weigh less than 2.900 g or more than 3.200 g.
After we state the null and the alternative hypotheses, the second step is to choose a confidence level for the analysis. The confidence level defines the probability that we will reject the null hypothesis when it is, in fact, true. We can express this as our confidence that we are correct in rejecting the null hypothesis (e.g. 95%), or as the probability that we are incorrect in rejecting the null hypothesis. For the latter, the confidence level is given as $\alpha$, where
$\alpha = 1 - \frac {\text{confidence interval (%)}} {100} \label{4.1}$
For a 95% confidence level, $\alpha$ is 0.05.
In this textbook we use $\alpha$ to represent the probability that we incorrectly reject the null hypothesis. In other textbooks this probability is given as p (often read as “p- value”). Although the symbols differ, the meaning is the same.
The third step is to calculate an appropriate test statistic and to compare it to a critical value. The test statistic’s critical value defines a breakpoint between values that lead us to reject or to retain the null hypothesis, which is the fourth, and final, step of a significance test. How we calculate the test statistic depends on what we are comparing, a topic we cover in Section 4.6. The last step is to either retain the null hypothesis, or to reject it and accept the alternative hypothesis.
The four steps for a statistical analysis of data using a significance test:
1. Pose a question, and state the null hypothesis, H0, and the alternative hypothesis, HA.
2. Choose a confidence level for the statistical analysis.
3. Calculate an appropriate test statistic and compare it to a critical value.
4. Either retain the null hypothesis, or reject it and accept the alternative hypothesis.
One-Tailed and Two-tailed Significance Tests
Suppose we want to evaluate the accuracy of a new analytical method. We might use the method to analyze a Standard Reference Material that contains a known concentration of analyte, $\mu$. We analyze the standard several times, obtaining a mean value, $\overline{X}$, for the analyte’s concentration. Our null hypothesis is that there is no difference between $\overline{X}$ and $\mu$
$H_0 \text{: } \overline{X} = \mu \nonumber$
If we conduct the significance test at $\alpha = 0.05$, then we retain the null hypothesis if a 95% confidence interval around $\overline{X}$ contains $\mu$. If the alternative hypothesis is
$H_\text{A} \text{: } \overline{X} \neq \mu \nonumber$
then we reject the null hypothesis and accept the alternative hypothesis if $\mu$ lies in the shaded areas at either end of the sample’s probability distribution curve (Figure 4.5.2 a). Each of the shaded areas accounts for 2.5% of the area under the probability distribution curve, for a total of 5%. This is a two-tailed significance test because we reject the null hypothesis for values of $\mu$ at either extreme of the sample’s probability distribution curve.
We also can write the alternative hypothesis in two additional ways
$H_\text{A} \text{: } \overline{X} > \mu \nonumber$
$H_\text{A} \text{: } \overline{X} < \mu \nonumber$
rejecting the null hypothesis if n falls within the shaded areas shown in Figure 4.5.2 b or Figure 4.5.2 c, respectively. In each case the shaded area represents 5% of the area under the probability distribution curve. These are examples of a one-tailed significance test.
For a fixed confidence level, a two-tailed significance test is the more conservative test because rejecting the null hypothesis requires a larger difference between the parameters we are comparing. In most situations we have no particular reason to expect that one parameter must be larger (or must be smaller) than the other parameter. This is the case, for example, when we evaluate the accuracy of a new analytical method. A two-tailed significance test, therefore, usually is the appropriate choice.
We reserve a one-tailed significance test for a situation where we specifically are interested in whether one parameter is larger (or smaller) than the other parameter. For example, a one-tailed significance test is appropriate if we are evaluating a medication’s ability to lower blood glucose levels. In this case we are interested only in whether the glucose levels after we administer the medication are less than the glucose levels before we initiated treatment. If a patient’s blood glucose level is greater after we administer the medication, then we know the answer—the medication did not work—and do not need to conduct a statistical analysis.
Error in Significance Testing
Because a significance test relies on probability, its interpretation is subject to error. In a significance test, a defines the probability of rejecting a null hypothesis that is true. When we conduct a significance test at $\alpha = 0.05$, there is a 5% probability that we will incorrectly reject the null hypothesis. This is known as a type 1 error, and its risk is always equivalent to $\alpha$. A type 1 error in a two-tailed or a one-tailed significance tests corresponds to the shaded areas under the probability distribution curves in Figure 4.5.2 .
A second type of error occurs when we retain a null hypothesis even though it is false. This is as a type 2 error, and the probability of its occurrence is $\beta$. Unfortunately, in most cases we cannot calculate or estimate the value for $\beta$. The probability of a type 2 error, however, is inversely proportional to the probability of a type 1 error.
Minimizing a type 1 error by decreasing $\alpha$ increases the likelihood of a type 2 error. When we choose a value for $\alpha$ we must compromise between these two types of error. Most of the examples in this text use a 95% confidence level ($\alpha = 0.05$) because this usually is a reasonable compromise between type 1 and type 2 errors for analytical work. It is not unusual, however, to use a more stringent (e.g. $\alpha = 0.01$) or a more lenient (e.g. $\alpha = 0.10$) confidence level when the situation calls for it. | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/04%3A_Evaluating_Analytical_Data/4.05%3A_Statistical_Analysis_of_Data.txt |
The most common distribution for our results is a normal distribution. Because the area between any two limits of a normal distribution curve is well defined, constructing and evaluating significance tests is straightforward.
Comparing $\overline{X}$ to $\mu$
One way to validate a new analytical method is to analyze a sample that contains a known amount of analyte, $\mu$. To judge the method’s accuracy we analyze several portions of the sample, determine the average amount of analyte in the sample, $\overline{X}$, and use a significance test to compare $\overline{X}$ to $\mu$. Our null hypothesis is that the difference between $\overline{X}$ and $\mu$ is explained by indeterminate errors that affect the determination of $\overline{X}$. The alternative hypothesis is that the difference between $\overline{X}$ and $\mu$ is too large to be explained by indeterminate error.
$H_0 \text{: } \overline{X} = \mu \nonumber$
$H_A \text{: } \overline{X} \neq \mu \nonumber$
The test statistic is texp, which we substitute into the confidence interval for $\mu$ given by Equation 4.4.5
$\mu = \overline{X} \pm \frac {t_\text{exp} s} {\sqrt{n}} \label{4.1}$
Rearranging this equation and solving for $t_\text{exp}$
$t_\text{exp} = \frac {|\mu - \overline{X}| \sqrt{n}} {s} \label{4.2}$
gives the value for $t_\text{exp}$ when $\mu$ is at either the right edge or the left edge of the sample's confidence interval (Figure 4.6.1 a)
To determine if we should retain or reject the null hypothesis, we compare the value of texp to a critical value, $t(\alpha, \nu)$, where $\alpha$ is the confidence level and $\nu$ is the degrees of freedom for the sample. The critical value $t(\alpha, \nu)$ defines the largest confidence interval explained by indeterminate error. If $t_\text{exp} > t(\alpha, \nu)$, then our sample’s confidence interval is greater than that explained by indeterminate errors (Figure 4.6.1 b). In this case, we reject the null hypothesis and accept the alternative hypothesis. If $t_\text{exp} \leq t(\alpha, \nu)$, then our sample’s confidence interval is smaller than that explained by indeterminate error, and we retain the null hypothesis (Figure 4.6.1 c). Example 4.6.1 provides a typical application of this significance test, which is known as a t-test of $\overline{X}$ to $\mu$.
You will find values for $t(\alpha, \nu)$ in Appendix 4.
Another name for the t-test is Student’s t-test. Student was the pen name for William Gossett (1876-1927) who developed the t-test while working as a statistician for the Guiness Brewery in Dublin, Ireland. He published under the name Student because the brewery did not want its competitors to know they were using statistics to help improve the quality of their products.
Example 4.6.1
Before determining the amount of Na2CO3 in a sample, you decide to check your procedure by analyzing a standard sample that is 98.76% w/w Na2CO3. Five replicate determinations of the %w/w Na2CO3 in the standard gave the following results
$98.71 \% \quad 98.59 \% \quad 98.62 \% \quad 98.44 \% \quad 98.58 \%$
Using $\alpha = 0.05$, is there any evidence that the analysis is giving inaccurate results?
Solution
The mean and standard deviation for the five trials are
$\overline{X} = 98.59 \quad \quad \quad s = 0.0973 \nonumber$
Because there is no reason to believe that the results for the standard must be larger or smaller than $\mu$, a two-tailed t-test is appropriate. The null hypothesis and alternative hypothesis are
$H_0 \text{: } \overline{X} = \mu \quad \quad \quad H_\text{A} \text{: } \overline{X} \neq \mu \nonumber$
The test statistic, texp, is
$t_\text{exp} = \frac {|\mu - \overline{X}|\sqrt{n}} {s} = \frac {|98.76 - 98.59| \sqrt{5}} {0.0973} = 3.91 \nonumber$
The critical value for t(0.05, 4) from Appendix 4 is 2.78. Since texp is greater than t(0.05, 4), we reject the null hypothesis and accept the alternative hypothesis. At the 95% confidence level the difference between $\overline{X}$ and $\mu$ is too large to be explained by indeterminate sources of error, which suggests there is a determinate source of error that affects the analysis.
There is another way to interpret the result of this t-test. Knowing that texp is 3.91 and that there are 4 degrees of freedom, we use Appendix 4 to estimate the $\alpha$ value corresponding to a t($\alpha$, 4) of 3.91. From Appendix 4, t(0.02, 4) is 3.75 and t(0.01, 4) is 4.60. Although we can reject the null hypothesis at the 98% confidence level, we cannot reject it at the 99% confidence level. For a discussion of the advantages of this approach, see J. A. C. Sterne and G. D. Smith “Sifting the evidence—what’s wrong with significance tests?” BMJ 2001, 322, 226–231.
Exercise 4.6.1
To evaluate the accuracy of a new analytical method, an analyst determines the purity of a standard for which $\mu$ is 100.0%, obtaining the following results.
$99.28 \% \quad 103.93 \% \quad 99.43 \% \quad 99.84 \% \quad 97.60 \% \quad 96.70 \% \quad 98.02 \%$
Is there any evidence at $\alpha = 0.05$ that there is a determinate error affecting the results?
Answer
The null hypothesis is $H_0 \text{: } \overline{X} = \mu$ and the alternative hypothesis is $H_\text{A} \text{: } \overline{X} \neq \mu$. The mean and the standard deviation for the data are 99.26% and 2.35%, respectively. The value for texp is
$t_\text{exp} = \frac {|100.0 - 99.26| \sqrt{7}} {2.35} = 0.833 \nonumber$
and the critical value for t(0.05, 6) is 2.477. Because texp is less than t(0.05, 6) we retain the null hypothesis and have no evidence for a significant difference between $\overline{X}$ and $\mu$.
Earlier we made the point that we must exercise caution when we interpret the result of a statistical analysis. We will keep returning to this point because it is an important one. Having determined that a result is inaccurate, as we did in Example 4.6.1 , the next step is to identify and to correct the error. Before we expend time and money on this, however, we first should examine critically our data. For example, the smaller the value of s, the larger the value of texp. If the standard deviation for our analysis is unrealistically small, then the probability of a type 2 error increases. Including a few additional replicate analyses of the standard and reevaluating the t-test may strengthen our evidence for a determinate error, or it may show us that there is no evidence for a determinate error.
Comparing $s^2$ to $\sigma^2$
If we analyze regularly a particular sample, we may be able to establish an expected variance, $\sigma^2$, for the analysis. This often is the case, for example, in a clinical lab that analyze hundreds of blood samples each day. A few replicate analyses of a single sample gives a sample variance, s2, whose value may or may not differ significantly from $\sigma^2$.
We can use an F-test to evaluate whether a difference between s2 and $\sigma^2$ is significant. The null hypothesis is $H_0 \text{: } s^2 = \sigma^2$ and the alternative hypothesis is $H_\text{A} \text{: } s^2 \neq \sigma^2$. The test statistic for evaluating the null hypothesis is Fexp, which is given as either
$F_\text{exp} = \frac {s^2} {\sigma^2} \text{ if } s^2 > \sigma^2 \text{ or } F_\text{exp} = \frac {\sigma^2} {s^2} \text{ if } \sigma^2 > s^2 \label{4.3}$
depending on whether s2 is larger or smaller than $\sigma^2$. This way of defining Fexp ensures that its value is always greater than or equal to one.
If the null hypothesis is true, then Fexp should equal one; however, because of indeterminate errors Fexp usually is greater than one. A critical value, $F(\alpha, \nu_\text{num}, \nu_\text{den})$, is the largest value of Fexp that we can attribute to indeterminate error given the specified significance level, $\alpha$, and the degrees of freedom for the variance in the numerator, $\nu_\text{num}$, and the variance in the denominator, $\nu_\text{den}$. The degrees of freedom for s2 is n – 1, where n is the number of replicates used to determine the sample’s variance, and the degrees of freedom for $\sigma^2$ is defined as infinity, $\infty$. Critical values of F for $\alpha = 0.05$ are listed in Appendix 5 for both one-tailed and two-tailed F-tests.
Example 4.6.2
A manufacturer’s process for analyzing aspirin tablets has a known variance of 25. A sample of 10 aspirin tablets is selected and analyzed for the amount of aspirin, yielding the following results in mg aspirin/tablet.
$254 \quad 249 \quad 252 \quad 252 \quad 249 \quad 249 \quad 250 \quad 247 \quad 251 \quad 252$
Determine whether there is evidence of a significant difference between the sample’s variance and the expected variance at $\alpha = 0.05$.
Solution
The variance for the sample of 10 tablets is 4.3. The null hypothesis and alternative hypotheses are
$H_0 \text{: } s^2 = \sigma^2 \quad \quad \quad H_\text{A} \text{: } s^2 \neq \sigma^2 \nonumber$
and the value for Fexp is
$F_\text{exp} = \frac {\sigma^2} {s^2} = \frac {25} {4.3} = 5.8 \nonumber$
The critical value for F(0.05, $\infty$, 9) from Appendix 5 is 3.333. Since Fexp is greater than F(0.05, $\infty$, 9), we reject the null hypothesis and accept the alternative hypothesis that there is a significant difference between the sample’s variance and the expected variance. One explanation for the difference might be that the aspirin tablets were not selected randomly.
Comparing Variances for Two Samples
We can extend the F-test to compare the variances for two samples, A and B, by rewriting Equation \ref{4.3} as
$F_\text{exp} = \frac {s_A^2} {s_B^2} \nonumber$
defining A and B so that the value of Fexp is greater than or equal to 1.
Example 4.6.3
Table 4.4.1 shows results for two experiments to determine the mass of a circulating U.S. penny. Determine whether there is a difference in the variances of these analyses at $\alpha = 0.05$.
Solution
The standard deviations for the two experiments are 0.051 for the first experiment (A) and 0.037 for the second experiment (B). The null and alternative hypotheses are
$H_0 \text{: } s_A^2 = s_B^2 \quad \quad \quad H_\text{A} \text{: } s_A^2 \neq s_B^2 \nonumber$
and the value of Fexp is
$F_\text{exp} = \frac {s_A^2} {s_B^2} = \frac {(0.051)^2} {(0.037)^2} = \frac {0.00260} {0.00137} = 1.90 \nonumber$
From Appendix 5, the critical value for F(0.05, 6, 4) is 9.197. Because Fexp < F(0.05, 6, 4), we retain the null hypothesis. There is no evidence at $\alpha = 0.05$ to suggest that the difference in variances is significant.
Exercise 4.6.2
To compare two production lots of aspirin tablets, we collect an analyze samples from each, obtaining the following results (in mg aspirin/tablet).
Lot 1: $256 \quad 248 \quad 245 \quad 245 \quad 244 \quad 248 \quad 261$
Lot 2: $241 \quad 258 \quad 241 \quad 244 \quad 256 \quad 254$
Is there any evidence at $\alpha = 0.05$ that there is a significant difference in the variances for these two samples?
Answer
The standard deviations are 6.451 mg for Lot 1 and 7.849 mg for Lot 2. The null and alternative hypotheses are
$H_0 \text{: } s_\text{Lot 1}^2 = s_\text{Lot 2}^2 \quad \quad \quad H_\text{A} \text{: } s_\text{Lot 1}^2 \neq s_\text{Lot 2}^2 \nonumber$
and the value of Fexp is
$F_\text{exp} = \frac {(7.849)^2} {(6.451)^2} = 1.480 \nonumber$
The critical value for F(0.05, 5, 6) is 5.988. Because Fexp < F(0.05, 5, 6), we retain the null hypothesis. There is no evidence at $\alpha = 0.05$ to suggest that the difference in the variances is significant.
Comparing Means for Two Samples
Three factors influence the result of an analysis: the method, the sample, and the analyst. We can study the influence of these factors by conducting experiments in which we change one factor while holding constant the other factors. For example, to compare two analytical methods we can have the same analyst apply each method to the same sample and then examine the resulting means. In a similar fashion, we can design experiments to compare two analysts or to compare two samples.
It also is possible to design experiments in which we vary more than one of these factors. We will return to this point in Chapter 14.
Before we consider the significance tests for comparing the means of two samples, we need to make a distinction between unpaired data and paired data. This is a critical distinction and learning to distinguish between these two types of data is important. Here are two simple examples that highlight the difference between unpaired data and paired data. In each example the goal is to compare two balances by weighing pennies.
• Example 1: We collect 10 pennies and weigh each penny on each balance. This is an example of paired data because we use the same 10 pennies to evaluate each balance.
• Example 2: We collect 10 pennies and divide them into two groups of five pennies each. We weigh the pennies in the first group on one balance and we weigh the second group of pennies on the other balance. Note that no penny is weighed on both balances. This is an example of unpaired data because we evaluate each balance using a different sample of pennies.
In both examples the samples of 10 pennies were drawn from the same population; the difference is how we sampled that population. We will learn why this distinction is important when we review the significance test for paired data; first, however, we present the significance test for unpaired data.
One simple test for determining whether data are paired or unpaired is to look at the size of each sample. If the samples are of different size, then the data must be unpaired. The converse is not true. If two samples are of equal size, they may be paired or unpaired.
Unpaired Data
Consider two analyses, A and B with means of $\overline{X}_A$ and $\overline{X}_B$, and standard deviations of sA and sB. The confidence intervals for $\mu_A$ and for $\mu_B$ are
$\mu_A = \overline{X}_A \pm \frac {t s_A} {\sqrt{n_A}} \label{4.4}$
$\mu_B = \overline{X}_B \pm \frac {t s_B} {\sqrt{n_B}} \label{4.5}$
where nA and nB are the sample sizes for A and for B. Our null hypothesis, $H_0 \text{: } \mu_A = \mu_B$, is that and any difference between $\mu_A$ and $\mu_B$ is the result of indeterminate errors that affect the analyses. The alternative hypothesis, $H_A \text{: } \mu_A \neq \mu_B$, is that the difference between $\mu_A$and $\mu_B$ is too large to be explained by indeterminate error.
To derive an equation for texp, we assume that $\mu_A$ equals $\mu_B$, and combine Equation \ref{4.4} and Equation \ref{4.5}
$\overline{X}_A \pm \frac {t_\text{exp} s_A} {\sqrt{n_A}} = \overline{X}_B \pm \frac {t_\text{exp} s_B} {\sqrt{n_B}} \nonumber$
Solving for $|\overline{X}_A - \overline{X}_B|$ and using a propagation of uncertainty, gives
$|\overline{X}_A - \overline{X}_B| = t_\text{exp} \times \sqrt{\frac {s_A^2} {n_A} + \frac {s_B^2} {n_B}} \label{4.6}$
Finally, we solve for texp
$t_\text{exp} = \frac {|\overline{X}_A - \overline{X}_B|} {\sqrt{\frac {s_A^2} {n_A} + \frac {s_B^2} {n_B}}} \label{4.7}$
and compare it to a critical value, $t(\alpha, \nu)$, where $\alpha$ is the probability of a type 1 error, and $\nu$ is the degrees of freedom.
Problem 9 asks you to use a propagation of uncertainty to show that Equation \ref{4.6} is correct.
Thus far our development of this t-test is similar to that for comparing $\overline{X}$ to $\mu$, and yet we do not have enough information to evaluate the t-test. Do you see the problem? With two independent sets of data it is unclear how many degrees of freedom we have.
Suppose that the variances $s_A^2$ and $s_B^2$ provide estimates of the same $\sigma^2$. In this case we can replace $s_A^2$ and $s_B^2$ with a pooled variance, $s_\text{pool}^2$, that is a better estimate for the variance. Thus, Equation \ref{4.7} becomes
$t_\text{exp} = \frac {|\overline{X}_A - \overline{X}_B|} {s_\text{pool} \times \sqrt{\frac {1} {n_A} + \frac {1} {n_B}}} = \frac {|\overline{X}_A - \overline{X}_B|} {s_\text{pool}} \times \sqrt{\frac {n_A n_B} {n_A + n_B}} \label{4.8}$
where spool, the pooled standard deviation, is
$s_\text{pool} = \sqrt{\frac {(n_A - 1) s_A^2 + (n_B - 1)s_B^2} {n_A + n_B - 2}} \label{4.9}$
The denominator of Equation \ref{4.9} shows us that the degrees of freedom for a pooled standard deviation is $n_A + n_B - 2$, which also is the degrees of freedom for the t-test. Note that we lose two degrees of freedom because the calculations for $s_A^2$ and $s_B^2$ require the prior calculation of $\overline{X}_A$ amd $\overline{X}_B$.
So how do you determine if it is okay to pool the variances? Use an F-test.
If $s_A^2$ and $s_B^2$ are significantly different, then we calculate texp using Equation \ref{4.7}. In this case, we find the degrees of freedom using the following imposing equation.
$\nu = \frac {\left( \frac {s_A^2} {n_A} + \frac {s_B^2} {n_B} \right)^2} {\frac {\left( \frac {s_A^2} {n_A} \right)^2} {n_A + 1} + \frac {\left( \frac {s_B^2} {n_B} \right)^2} {n_B + 1}} - 2 \label{4.10}$
Because the degrees of freedom must be an integer, we round to the nearest integer the value of $\nu$ obtained using Equation \ref{4.10}.
Equation \ref{4.10}, which is from Miller, J.C.; Miller, J.N. Statistics for Analytical Chemistry, 2nd Ed., Ellis-Horward: Chichester, UK, 1988. In the 6th Edition, the authors note that several different equations have been suggested for the number of degrees of freedom for t when sA and sB differ, reflecting the fact that the determination of degrees of freedom an approximation. An alternative equation—which is used by statistical software packages, such as R, Minitab, Excel—is
$\nu = \frac {\left( \frac {s_A^2} {n_A} + \frac {s_B^2} {n_B} \right)^2} {\frac {\left( \frac {s_A^2} {n_A} \right)^2} {n_A - 1} + \frac {\left( \frac {s_B^2} {n_B} \right)^2} {n_B - 1}} = \frac {\left( \frac {s_A^2} {n_A} + \frac {s_B^2} {n_B} \right)^2} {\frac {s_A^4} {n_A^2(n_A - 1)} + \frac {s_B^4} {n_B^2(n_B - 1)}} \nonumber$
For typical problems in analytical chemistry, the calculated degrees of freedom is reasonably insensitive to the choice of equation.
Regardless of whether we calculate texp using Equation \ref{4.7} or Equation \ref{4.8}, we reject the null hypothesis if texp is greater than $t(\alpha, \nu)$ and retain the null hypothesis if texp is less than or equal to $t(\alpha, \nu)$.
Example 4.6.4
Table 4.4.1 provides results for two experiments to determine the mass of a circulating U.S. penny. Determine whether there is a difference in the means of these analyses at $\alpha = 0.05$.
Solution
First we use an F-test to determine whether we can pool the variances. We completed this analysis in Example 4.6.3 , finding no evidence of a significant difference, which means we can pool the standard deviations, obtaining
$s_\text{pool} = \sqrt{\frac {(7 - 1)(0.051)^2 + (5 - 1)(0.037)^2} {7 + 5 - 2}} = 0.0459 \nonumber$
with 10 degrees of freedom. To compare the means we use the following null hypothesis and alternative hypotheses
$H_0 \text{: } \mu_A = \mu_B \quad \quad \quad H_A \text{: } \mu_A \neq \mu_B \nonumber$
Because we are using the pooled standard deviation, we calculate texp using Equation \ref{4.8}.
$t_\text{exp} = \frac {|3.117 - 3.081|} {0.0459} \times \sqrt{\frac {7 \times 5} {7 + 5}} = 1.34 \nonumber$
The critical value for t(0.05, 10), from Appendix 4, is 2.23. Because texp is less than t(0.05, 10) we retain the null hypothesis. For $\alpha = 0.05$ we do not have evidence that the two sets of pennies are significantly different.
Example 4.6.5
One method for determining the %w/w Na2CO3 in soda ash is to use an acid–base titration. When two analysts analyze the same sample of soda ash they obtain the results shown here.
Analyst A: $86.82 \% \quad 87.04 \% \quad 86.93 \% \quad 87.01 \% \quad 86.20 \% \quad 87.00 \%$
Analyst B: $81.01 \% \quad 86.15 \% \quad 81.73 \% \quad 83.19 \% \quad 80.27 \% \quad 83.93 \% \quad$
Determine whether the difference in the mean values is significant at $\alpha = 0.05$.
Solution
We begin by reporting the mean and standard deviation for each analyst.
$\overline{X}_A = 86.83\% \quad \quad s_A = 0.32\% \nonumber$
$\overline{X}_B = 82.71\% \quad \quad s_B = 2.16\% \nonumber$
To determine whether we can use a pooled standard deviation, we first complete an F-test using the following null and alternative hypotheses.
$H_0 \text{: } s_A^2 = s_B^2 \quad \quad \quad H_A \text{: } s_A^2 \neq s_B^2 \nonumber$
Calculating Fexp, we obtain a value of
$F_\text{exp} = \frac {(2.16)^2} {(0.32)^2} = 45.6 \nonumber$
Because Fexp is larger than the critical value of 7.15 for F(0.05, 5, 5) from Appendix 5, we reject the null hypothesis and accept the alternative hypothesis that there is a significant difference between the variances; thus, we cannot calculate a pooled standard deviation.
To compare the means for the two analysts we use the following null and alternative hypotheses.
$H_0 \text{: } \overline{X}_A = \overline{X}_B \quad \quad \quad H_A \text{: } \overline{X}_A \neq \overline{X}_B \nonumber$
Because we cannot pool the standard deviations, we calculate texp using Equation \ref{4.7} instead of Equation \ref{4.8}
$t_\text{exp} = \frac {|86.83 - 82.71|} {\sqrt{\frac {(0.32)^2} {6} + \frac {(2.16)^2} {6}}} = 4.62 \nonumber$
and calculate the degrees of freedom using Equation \ref{4.10}.
$\nu = \frac {\left( \frac {(0.32)^2} {6} + \frac {(2.16)^2} {6} \right)^2} {\frac {\left( \frac {(0.32)^2} {6} \right)^2} {6 + 1} + \frac {\left( \frac {(2.16)^2} {6} \right)^2} {6 + 1}} - 2 = 5.3 \approx 5 \nonumber$
From Appendix 4, the critical value for t(0.05, 5) is 2.57. Because texp is greater than t(0.05, 5) we reject the null hypothesis and accept the alternative hypothesis that the means for the two analysts are significantly different at $\alpha = 0.05$.
Exercise 4.6.3
To compare two production lots of aspirin tablets, you collect samples from each and analyze them, obtaining the following results (in mg aspirin/tablet).
Lot 1: $256 \quad 248 \quad 245 \quad 245 \quad 244 \quad 248 \quad 261$
Lot 2: $241 \quad 258 \quad 241 \quad 244 \quad 256 \quad 254$
Is there any evidence at $\alpha = 0.05$ that there is a significant difference in the variance between the results for these two samples? This is the same data from Exercise 4.6.2 .
Answer
To compare the means for the two lots, we use an unpaired t-test of the null hypothesis $H_0 \text{: } \overline{X}_\text{Lot 1} = \overline{X}_\text{Lot 2}$ and the alternative hypothesis $H_A \text{: } \overline{X}_\text{Lot 1} \neq \overline{X}_\text{Lot 2}$. Because there is no evidence to suggest a difference in the variances (see Exercise 4.6.2 ) we pool the standard deviations, obtaining an spool of
$s_\text{pool} = \sqrt{\frac {(7 - 1) (6.451)^2 + (6 - 1) (7.849)^2} {7 + 6 - 2}} = 7.121 \nonumber$
The means for the two samples are 249.57 mg for Lot 1 and 249.00 mg for Lot 2. The value for texp is
$t_\text{exp} = \frac {|249.57 - 249.00|} {7.121} \times \sqrt{\frac {7 \times 6} {7 + 6}} = 0.1439 \nonumber$
The critical value for t(0.05, 11) is 2.204. Because texp is less than t(0.05, 11), we retain the null hypothesis and find no evidence at $\alpha = 0.05$ that there is a significant difference between the means for the two lots of aspirin tablets.
Paired Data
Suppose we are evaluating a new method for monitoring blood glucose concentrations in patients. An important part of evaluating a new method is to compare it to an established method. What is the best way to gather data for this study? Because the variation in the blood glucose levels amongst patients is large we may be unable to detect a small, but significant difference between the methods if we use different patients to gather data for each method. Using paired data, in which the we analyze each patient’s blood using both methods, prevents a large variance within a population from adversely affecting a t-test of means.
Typical blood glucose levels for most non-diabetic individuals ranges between 80–120 mg/dL (4.4–6.7 mM), rising to as high as 140 mg/dL (7.8 mM) shortly after eating. Higher levels are common for individuals who are pre-diabetic or diabetic.
When we use paired data we first calculate the difference, di, between the paired values for each sample. Using these difference values, we then calculate the average difference, $\overline{d}$, and the standard deviation of the differences, sd. The null hypothesis, $H_0 \text{: } d = 0$, is that there is no difference between the two samples, and the alternative hypothesis, $H_A \text{: } d \neq 0$, is that the difference between the two samples is significant.
The test statistic, texp, is derived from a confidence interval around $\overline{d}$
$t_\text{exp} = \frac {|\overline{d}| \sqrt{n}} {s_d} \nonumber$
where n is the number of paired samples. As is true for other forms of the t-test, we compare texp to $t(\alpha, \nu)$, where the degrees of freedom, $\nu$, is n – 1. If texp is greater than $t(\alpha, \nu)$, then we reject the null hypothesis and accept the alternative hypothesis. We retain the null hypothesis if texp is less than or equal to t(a, o). This is known as a paired t-test.
Example 4.6.6
Marecek et. al. developed a new electrochemical method for the rapid determination of the concentration of the antibiotic monensin in fermentation vats [Marecek, V.; Janchenova, H.; Brezina, M.; Betti, M. Anal. Chim. Acta 1991, 244, 15–19]. The standard method for the analysis is a test for microbiological activity, which is both difficult to complete and time-consuming. Samples were collected from the fermentation vats at various times during production and analyzed for the concentration of monensin using both methods. The results, in parts per thousand (ppt), are reported in the following table.
Sample Microbiological Electrochemical
1 129.5 132.3
2 89.6 91.0
3 76.6 73.6
4 52.2 58.2
5 110.8 104.2
6 50.4 49.9
7 72.4 82.1
8 141.4 154.1
9 75.0 73.4
10 34.1 38.1
11 60.3 60.1
Is there a significant difference between the methods at $\alpha = 0.05$?
Solution
Acquiring samples over an extended period of time introduces a substantial time-dependent change in the concentration of monensin. Because the variation in concentration between samples is so large, we use a paired t-test with the following null and alternative hypotheses.
$H_0 \text{: } \overline{d} = 0 \quad \quad \quad H_A \text{: } \overline{d} \neq 0 \nonumber$
Defining the difference between the methods as
$d_i = (X_\text{elect})_i - (X_\text{micro})_i \nonumber$
we calculate the difference for each sample.
sample 1 2 3 4 5 6 7 8 9 10 11
$d_i$ 2.8 1.4 –3.0 6.0 –6.6 –0.5 9.7 12.7 –1.6 4.0 –0.2
The mean and the standard deviation for the differences are, respectively, 2.25 ppt and 5.63 ppt. The value of texp is
$t_\text{exp} = \frac {|2.25| \sqrt{11}} {5.63} = 1.33 \nonumber$
which is smaller than the critical value of 2.23 for t(0.05, 10) from Appendix 4. We retain the null hypothesis and find no evidence for a significant difference in the methods at $\alpha = 0.05$.
Exercise 4.6.4
Suppose you are studying the distribution of zinc in a lake and want to know if there is a significant difference between the concentration of Zn2+ at the sediment-water interface and its concentration at the air-water interface. You collect samples from six locations—near the lake’s center, near its drainage outlet, etc.—obtaining the results (in mg/L) shown in the table. Using this data, determine if there is a significant difference between the concentration of Zn2+ at the two interfaces at $\alpha = 0.05$. Complete this analysis treating the data as (a) unpaired and as (b) paired. Briefly comment on your results.
Location Air-Water Interface Sediment-Water Interface
1 0.430 0.415
2 0.266 0.238
3 0.457 0.390
4 0.531 0.410
5 0.707 0.605
6 0.716 0.609
Complete this analysis treating the data as (a) unpaired and as (b) paired. Briefly comment on your results.
Answer
Treating as Unpaired Data: The mean and the standard deviation for the concentration of Zn2+ at the air-water interface are 0.5178 mg/L and 0.1732 mg/L, respectively, and the values for the sediment-water interface are 0.4445 mg/L and 0.1418 mg/L, respectively. An F-test of the variances gives an Fexp of 1.493 and an F(0.05, 5, 5) of 7.146. Because Fexp is smaller than F(0.05, 5, 5), we have no evidence at $\alpha = 0.05$ to suggest that the difference in variances is significant. Pooling the standard deviations gives an spool of 0.1582 mg/L. An unpaired t-test gives texp as 0.8025. Because texp is smaller than t(0.05, 11), which is 2.204, we have no evidence that there is a difference in the concentration of Zn2+ between the two interfaces.
Treating as Paired Data: To treat as paired data we need to calculate the difference, di, between the concentration of Zn2+ at the air-water interface and at the sediment-water interface for each location, where
$d_i = \left( \text{[Zn}^{2+} \text{]}_\text{air-water} \right)_i - \left( \text{[Zn}^{2+} \text{]}_\text{sed-water} \right)_i \nonumber$
The mean difference is 0.07333 mg/L with a standard deviation of 0.0441 mg/L. The null hypothesis and the alternative hypothesis are
$H_0 \text{: } \overline{d} = 0 \quad \quad \quad H_A \text{: } \overline{d} \neq 0 \nonumber$
and the value of texp is
$t_\text{exp} = \frac {|0.07333| \sqrt{6}} {0.0441} = 4.073 \nonumber$
Because texp is greater than t(0.05, 5), which is 2.571, we reject the null hypothesis and accept the alternative hypothesis that there is a significant difference in the concentration of Zn2+ between the air-water interface and the sediment-water interface.
The difference in the concentration of Zn2+ between locations is much larger than the difference in the concentration of Zn2+ between the interfaces. Because out interest is in studying the difference between the interfaces, the larger standard deviation when treating the data as unpaired increases the probability of incorrectly retaining the null hypothesis, a type 2 error.
One important requirement for a paired t-test is that the determinate and the indeterminate errors that affect the analysis must be independent of the analyte’s concentration. If this is not the case, then a sample with an unusually high concentration of analyte will have an unusually large di. Including this sample in the calculation of $\overline{d}$ and sd gives a biased estimate for the expected mean and standard deviation. This rarely is a problem for samples that span a limited range of analyte concentrations, such as those in Example 4.6.6 or Exercise 4.6.4 . When paired data span a wide range of concentrations, however, the magnitude of the determinate and indeterminate sources of error may not be independent of the analyte’s concentration; when true, a paired t-test may give misleading results because the paired data with the largest absolute determinate and indeterminate errors will dominate $\overline{d}$. In this situation a regression analysis, which is the subject of the next chapter, is more appropriate method for comparing the data.
Outliers
Earlier in the chapter we examined several data sets consisting of the mass of a circulating United States penny. Table 4.6.1 provides one more data set. Do you notice anything unusual in this data? Of the 112 pennies included in Table 4.4.1 and Table 4.4.3, no penny weighed less than 3 g. In Table 4.6.1, however, the mass of one penny is less than 3 g. We might ask whether this penny’s mass is so different from the other pennies that it is in error.
Table 4.6.1 : Mass (g) for Additional Sample of Circulating U. S. Pennies
3.067 2.514 3.094
3.049 3.048 3.109
3.039 3.079 3.102
A measurement that is not consistent with other measurements is called outlier. An outlier might exist for many reasons: the outlier might belong to a different population (Is this a Canadian penny?); the outlier might be a contaminated or otherwise altered sample (Is the penny damaged or unusually dirty?); or the outlier may result from an error in the analysis (Did we forget to tare the balance?). Regardless of its source, the presence of an outlier compromises any meaningful analysis of our data. There are many significance tests that we can use to identify a potential outlier, three of which we present here.
Dixon's Q-Test
One of the most common significance tests for identifying an outlier is Dixon’s Q-test. The null hypothesis is that there are no outliers, and the alternative hypothesis is that there is an outlier. The Q-test compares the gap between the suspected outlier and its nearest numerical neighbor to the range of the entire data set (Figure 4.6.2 ).
The test statistic, Qexp, is
$Q_\text{exp} = \frac {\text{gap}} {\text{range}} = \frac {|\text{outlier's value} - \text{nearest value}|} {\text{largest value} - \text{smallest value}} \nonumber$
This equation is appropriate for evaluating a single outlier. Other forms of Dixon’s Q-test allow its extension to detecting multiple outliers [Rorabacher, D. B. Anal. Chem. 1991, 63, 139–146].
The value of Qexp is compared to a critical value, $Q(\alpha, n)$, where $\alpha$ is the probability that we will reject a valid data point (a type 1 error) and n is the total number of data points. To protect against rejecting a valid data point, usually we apply the more conservative two-tailed Q-test, even though the possible outlier is the smallest or the largest value in the data set. If Qexp is greater than $Q(\alpha, n)$, then we reject the null hypothesis and may exclude the outlier. We retain the possible outlier when Qexp is less than or equal to $Q(\alpha, n)$. Table 4.6.2 provides values for $Q(\alpha, n)$ for a data set that has 3–10 values. A more extensive table is in Appendix 6. Values for $Q(\alpha, n)$ assume an underlying normal distribution.
Table 4.6.2 : Dixon's Q-Test
n Q(0.05, n)
3 0.970
4 0.829
5 0.710
6 0.625
7 0.568
8 0.526
9 0.493
10 0.466
Grubb's Test
Although Dixon’s Q-test is a common method for evaluating outliers, it is no longer favored by the International Standards Organization (ISO), which recommends the Grubb’s test. There are several versions of Grubb’s test depending on the number of potential outliers. Here we will consider the case where there is a single suspected outlier.
For details on this recommendation, see International Standards ISO Guide 5752-2 “Accuracy (trueness and precision) of measurement methods and results–Part 2: basic methods for the determination of repeatability and reproducibility of a standard measurement method,” 1994.
The test statistic for Grubb’s test, Gexp, is the distance between the sample’s mean, $\overline{X}$, and the potential outlier, $X_\text{out}$, in terms of the sample’s standard deviation, s.
$G_\text{exp} = \frac {|X_\text{out} - \overline{X}|} {s} \nonumber$
We compare the value of Gexp to a critical value $G(\alpha, n)$, where $\alpha$ is the probability that we will reject a valid data point and n is the number of data points in the sample. If Gexp is greater than $G(\alpha, n)$, then we may reject the data point as an outlier, otherwise we retain the data point as part of the sample. Table 4.6.3 provides values for G(0.05, n) for a sample containing 3–10 values. A more extensive table is in Appendix 7. Values for $G(\alpha, n)$ assume an underlying normal distribution.
Table 4.6.3 : Grubb's Test
n G(0.05, n)
3 1.115
4 1.481
5 1.715
6 1.887
7 2.020
8 2.126
9 2.215
10 2.290
Chauvenet's Criterion
Our final method for identifying an outlier is Chauvenet’s criterion. Unlike Dixon’s Q-Test and Grubb’s test, you can apply this method to any distribution as long as you know how to calculate the probability for a particular outcome. Chauvenet’s criterion states that we can reject a data point if the probability of obtaining the data point’s value is less than (2n)–1, where n is the size of the sample. For example, if n = 10, a result with a probability of less than $(2 \times 10)^{-1}$, or 0.05, is considered an outlier.
To calculate a potential outlier’s probability we first calculate its standardized deviation, z
$z = \frac {|X_\text{out} - \overline{X}|} {s} \nonumber$
where $X_\text{out}$ is the potential outlier, $\overline{X}$ is the sample’s mean and s is the sample’s standard deviation. Note that this equation is identical to the equation for Gexp in the Grubb’s test. For a normal distribution, we can find the probability of obtaining a value of z using the probability table in Appendix 3.
Example 4.6.7
Table 4.6.1 contains the masses for nine circulating United States pennies. One entry, 2.514 g, appears to be an outlier. Determine if this penny is an outlier using a Q-test, Grubb’s test, and Chauvenet’s criterion. For the Q-test and Grubb’s test, let $\alpha = 0.05$.
Solution
For the Q-test the value for Qexp is
$Q_\text{exp} = \frac {|2.514 - 3.039|} {3.109 - 2.514} = 0.882 \nonumber$
From Table 4.6.2 , the critical value for Q(0.05, 9) is 0.493. Because Qexp is greater than Q(0.05, 9), we can assume the penny with a mass of 2.514 g likely is an outlier.
For Grubb’s test we first need the mean and the standard deviation, which are 3.011 g and 0.188 g, respectively. The value for Gexp is
$G_\text{exp} = \frac {|2.514 - 3.011} {0.188} = 2.64 \nonumber$
Using Table 4.6.3 , we find that the critical value for G(0.05, 9) is 2.215. Because Gexp is greater than G(0.05, 9), we can assume that the penny with a mass of 2.514 g likely is an outlier.
For Chauvenet’s criterion, the critical probability is $(2 \times 9)^{-1}$, or 0.0556. The value of z is the same as Gexp, or 2.64. Using Appendix 3, the probability for z = 2.64 is 0.00415. Because the probability of obtaining a mass of 0.2514 g is less than the critical probability, we can assume the penny with a mass of 2.514 g likely is an outlier.
You should exercise caution when using a significance test for outliers because there is a chance you will reject a valid result. In addition, you should avoid rejecting an outlier if it leads to a precision that is much better than expected based on a propagation of uncertainty. Given these concerns it is not surprising that some statisticians caution against the removal of outliers [Deming, W. E. Statistical Analysis of Data; Wiley: New York, 1943 (republished by Dover: New York, 1961); p. 171].
You also can adopt a more stringent requirement for rejecting data. When using the Grubb’s test, for example, the ISO 5752 guidelines suggests retaining a value if the probability for rejecting it is greater than $\alpha = 0.05$, and flagging a value as a “straggler” if the probability for rejecting it is between $\alpha = 0.05$ and $\alpha = 0.01$. A “straggler” is retained unless there is compelling reason for its rejection. The guidelines recommend using $\alpha = 0.01$ as the minimum criterion for rejecting a possible outlier.
On the other hand, testing for outliers can provide useful information if we try to understand the source of the suspected outlier. For example, the outlier in Table 4.6.1 represents a significant change in the mass of a penny (an approximately 17% decrease in mass), which is the result of a change in the composition of the U.S. penny. In 1982 the composition of a U.S. penny changed from a brass alloy that was 95% w/w Cu and 5% w/w Zn (with a nominal mass of 3.1 g), to a pure zinc core covered with copper (with a nominal mass of 2.5 g) [Richardson, T. H. J. Chem. Educ. 1991, 68, 310–311]. The pennies in Table 4.6.1 , therefore, were drawn from different populations. | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/04%3A_Evaluating_Analytical_Data/4.06%3A_Statistical_Methods_for_Normal_Distributions.txt |
The International Union of Pure and Applied Chemistry (IUPAC) defines a method’s detection limit as the smallest concentration or absolute amount of analyte that has a signal significantly larger than the signal from a suitable blank [IUPAC Compendium of Chemical Technology, Electronic Version]. Although our interest is in the amount of analyte, in this section we will define the detection limit in terms of the analyte’s signal. Knowing the signal you can calculate the analyte’s concentration, CA, or the moles of analyte, nA, using the equations
$S_A = k_A C_A \text{ or } S_A = k_A n_A \nonumber$
where k is the method’s sensitivity.
See Chapter 3 for a review of these equations.
Let’s translate the IUPAC definition of the detection limit into a mathematical form by letting Smb represent the average signal for a method blank, and letting $\sigma_{mb}$ represent the method blank’s standard deviation. The null hypothesis is that the analyte is not present in the sample, and the alternative hypothesis is that the analyte is present in the sample. To detect the analyte, its signal must exceed Smb by a suitable amount; thus,
$(S_A)_{DL} = S_{mb} \pm z \sigma_{mb} \label{4.1}$
where $(S_A)_{DL}$ is the analyte’s detection limit.
If $\sigma_{mb}$ is not known, we can replace it with smb; Equation \ref{4.1} then becomes
$(S_A)_{DL} = S_{mb} \pm t s_{mb} \nonumber$
You can make similar adjustments to other equations in this section. See, for example, Kirchner, C. J. “Estimation of Detection Limits for Environme tal Analytical Procedures,” in Currie, L. A. (ed) Detection in Analytical Chemistry: Importance, Theory, and Practice; American Chemical Society: Washington, D. C., 1988.
The value we choose for z depends on our tolerance for reporting the analyte’s concentration even if it is absent from the sample (a type 1 error). Typically, z is set to three, which, from Appendix 3, corresponds to a probability, $\alpha$, of 0.00135. As shown in Figure 4.7.1 a, there is only a 0.135% probability of detecting the analyte in a sample that actually is analyte-free.
A detection limit also is subject to a type 2 error in which we fail to find evidence for the analyte even though it is present in the sample. Consider, for example, the situation shown in Figure 4.7.1 b where the signal for a sample that contains the analyte is exactly equal to (SA)DL. In this case the probability of a type 2 error is 50% because half of the sample’s possible signals are below the detection limit. We correctly detect the analyte at the IUPAC detection limit only half the time. The IUPAC definition for the detection limit is the smallest signal for which we can say, at a significance level of $\alpha$, that an analyte is present in the sample; however, failing to detect the analyte does not mean it is not present in the sample.
The detection limit often is represented, particularly when discussing public policy issues, as a distinct line that separates detectable concentrations of analytes from concentrations we cannot detect. This use of a detection limit is incorrect [Rogers, L. B. J. Chem. Educ. 1986, 63, 3–6]. As suggested by Figure 4.7.1 , for an analyte whose concentration is near the detection limit there is a high probability that we will fail to detect the analyte.
An alternative expression for the detection limit, the limit of identification, minimizes both type 1 and type 2 errors [Long, G. L.; Winefordner, J. D. Anal. Chem. 1983, 55, 712A–724A]. The analyte’s signal at the limit of identification, (SA)LOI, includes an additional term, $z \sigma_A$, to account for the distribution of the analyte’s signal.
$(S_A)_\text{LOI} = (S_A)_\text{DL} + z \sigma_A = S_{mb} + z \sigma_{mb} + z \sigma_A \nonumber$
As shown in Figure 4.7.2 , the limit of identification provides an equal probability of a type 1 and a type 2 error at the detection limit. When the analyte’s concentration is at its limit of identification, there is only a 0.135% probability that its signal is indistinguishable from that of the method blank.
The ability to detect the analyte with confidence is not the same as the ability to report with confidence its concentration, or to distinguish between its concentration in two samples. For this reason the American Chemical Society’s Committee on Environmental Analytical Chemistry recommends the limit of quantitation, (SA)LOQ [“Guidelines for Data Acquisition and Data Quality Evaluation in Environmental Chemistry,” Anal. Chem. 1980, 52, 2242–2249 ].
$(S_A)_\text{LOQ} = S_{mb} + 10 \sigma_{mb} \nonumber$ | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/04%3A_Evaluating_Analytical_Data/4.07%3A_Detection_Limits.txt |
Although the calculations in this chapter are relatively straightforward, it can be tedious to work problems using nothing more than a calculator. Both Excel and R include functions for many common statistical calculations. In addition, R provides useful functions for visualizing your data.
Excel
Excel has built-in functions that we can use to complete many of the statistical calculations covered in this chapter, including reporting descriptive statistics, such as means and variances, predicting the probability of obtaining a given outcome from a binomial distribution or a normal distribution, and carrying out significance tests. Table 4.8.1 provides the syntax for many of these functions; you can information on functions not included here by using Excel’s Help menu.
Table 4.8.1 : Excel Functions for Statistics Calculations
Parameter Excel Function
Descriptive Statistics
mean = average(data)
median = median(data)
standard deviation for sample = stdev.s(data)
standard deviation for populations = stdev.p(data)
variance for sample = var.s(data)
variance for population = var.p(data)
maximum value = max(data)
minimum value = min(data)
Probability Distributions
binomial distribution = binom.dist(X, N, p, TRUE or FALSE)
normal distribution = norm.dist(x, $\mu$ $\sigma$, TRUE or FALSE)
Significance Tests
F-test = f.test(data set 1, data set 2)
t-test = t.test(data set 1, data set 2, tails = 1 or 2, type of t-test: 1 = paired; 2 = unpaired with equal variances; or 3 = unpaired with unequal variances)
Descriptive Statistics
Let’s use Excel to provide a statistical summary of the data in Table 4.1.1. Enter the data into a spreadsheet, as shown in Figure 4.8.1 . To calculate the sample’s mean, for example, click on any empty cell, enter the formula
= average(b2:b8)
and press Return or Enter to replace the cell’s content with Excel’s calculation of the mean (3.117285714), which we round to 3.117. Excel does not have a function for the range, but we can use the functions that report the maximum value and the minimum value to calculate the range; thus
= max(b2:b8) – min(b2:b8)
returns 0.142 as an answer.
Probability Distributions
In Example 4.4.2 we showed that 91.10% of a manufacturer’s analgesic tablets contained between 243 and 262 mg of aspirin. We arrived at this result by calculating the deviation, z, of each limit from the population’s expected mean, $\mu$, of 250 mg in terms of the population’s expected standard deviation, $\sigma$, of 5 mg. After we calculated values for z, we used the table in Appendix 3 to find the area under the normal distribution curve between these two limits.
We can complete this calculation in Excel using the norm.dist function As shown in Figure 4.8.2 , the function calculates the probability of obtaining a result less than x from a normal distribution with a mean of $\mu$ and a standard deviation of $\sigma$. To solve Example 4.4.2 using Excel enter the following formulas into separate cells
= norm.dist(243, 250, 5, TRUE)
= norm.dist(262, 250, 5, TRUE)
obtaining results of 0.080756659 and 0.991802464. Subtracting the smaller value from the larger value and adjusting to the correct number of significant figures gives the probability as 0.9910, or 99.10%.
Excel also includes a function for working with binomial distributions. The function’s syntax is
= binom.dist(X, N, p, TRUE or FALSE)
where X is the number of times a particular outcome occurs in N trials, and p is the probability that X occurs in a single trial. Setting the function’s last term to TRUE gives the total probability for any result up to X and setting it to FALSE gives the probability for X. Using Example 4.4.1 to test this function, we use the formula
= binom.dist(0, 27, 0.0111, FALSE)
to find the probability of finding no atoms of 13C atoms in a molecule of cholesterol, C27H44O, which returns a value of 0.740 after adjusting for significant figures. Using the formula
= binom.dist(2, 27, 0.0111, TRUE)
we find that 99.7% of cholesterol molecules contain two or fewer atoms of 13C.
Significance Tests
As shown in Table 4.8.1 , Excel includes functions for the following significance tests covered in this chapter:
• an F-test of variances
• an unpaired t-test of sample means assuming equal variances
• an unpaired t-test of sample means assuming unequal variances
• a paired t-test for of sample means
Let’s use these functions to complete a t-test on the data in Table 4.4.1, which contains results for two experiments to determine the mass of a circulating U. S. penny. Enter the data from Table 4.4.1 into a spreadsheet as shown in Figure 4.8.3 .
Because the data in this case are unpaired, we will use Excel to complete an unpaired t-test. Before we can complete the t-test, we use an F-test to determine whether the variances for the two data sets are equal or unequal.
To complete the F-test, we click on any empty cell, enter the formula
= f.test(b2:b8, c2:c6)
and press Return or Enter, which replaces the cell’s content with the value of $\alpha$ for which we can reject the null hypothesis of equal variances. In this case, Excel returns an $\alpha$ of 0.566 105 03; because this value is not less than 0.05, we retain the null hypothesis that the variances are equal. Excel’s F-test is two-tailed; for a one-tailed F-test, we use the same function, but divide the result by two; thus
= f.test(b2:b8, c2:c6)/2
Having found no evidence to suggest unequal variances, we next complete an unpaired t-test assuming equal variances, entering into any empty cell the formula
= t.test(b2:b8, c2:c6, 2, 2)
where the first 2 indicates that this is a two-tailed t-test, and the second 2 indicates that this is an unpaired t-test with equal variances. Pressing Return or Enter replaces the cell’s content with the value of $\alpha$ for which we can reject the null hypothesis of equal means. In this case, Excel returns an $\alpha$ of 0.211 627 646; because this value is not less than 0.05, we retain the null hypothesis that the means are equal.
See Example 4.6.3 and Example 4.6.4 for our earlier solutions to this problem.
The other significance tests in Excel work in the same format. The following practice exercise provides you with an opportunity to test yourself.
Exercise 4.8.1
Rework Example 4.6.5 and Example 4.6.6 using Excel.
Answer
You will find small differences between the values you obtain using Excel’s built in functions and the worked solutions in the chapter. These differences arise because Excel does not round off the results of intermediate calculations.
R
R is a programming environment that provides powerful capabilities for analyzing data. There are many functions built into R’s standard installation and additional packages of functions are available from the R web site (www.r-project.org). Commands in R are not available from pull down menus. Instead, you interact with R by typing in commands.
You can download the current version of R from www.r-project.org. Click on the link for Download: CRAN and find a local mirror site. Click on the link for the mirror site and then use the link for Linux, MacOS X, or Windows under the heading “Download and Install R.”
Descriptive Statistics
Let’s use R to provide a statistical summary of the data in Table 4.1.1. To do this we first need to create an object that contains the data, which we do by typing in the following command.
> penny1 = c(3.080, 3.094, 3.107, 3.056, 3.112, 3.174, 3.198)
In R, the symbol ‘>’ is a prompt, which indicates that the program is waiting for you to enter a command. When you press ‘Return’ or ‘Enter,’ R executes the command, displays the result (if there is a result to return), and returns the > prompt.
Table 4.8.2 lists some of the commands in R for calculating basic descriptive statistics. As is the case for Excel, R does not include stand alone commands for all descriptive statistics of interest to us, but we can calculate them using other commands. Using a command is easy—simply enter the appropriate code at the prompt; for example, to find the sample’s variance we enter
> var(penny1)
[1] 0.002221918
Table 4.8.2 : R Functions for Descriptive Statistics
Parameter Excel Function
mean mean(object)
median median(object)
standard deviation for sample sd(object)
standard deviation for populations sd(object) * ((length(object) – 1)/length(object))^0.5
variance for sample var(object)
variance for population var(object) * ((length(object) – 1)/length(object))
range max(object) – min(object)
Probability Distributions
In Example 4.4.2 we showed that 91.10% of a manufacturer’s analgesic tablets contained between 243 and 262 mg of aspirin. We arrived at this result by calculating the deviation, z, of each limit from the population’s expected mean, $\mu$, of 250 mg in terms of the population’s expected standard deviation, $\sigma$, of 5 mg. After we calculated values for z, we used the table in Appendix 3 to find the area under the normal distribution curve between these two limits.
We can complete this calculation in R using the function pnorm. The function’s general format is
pnorm($x, \mu, \sigma$)
where x is the limit of interest, $\mu$ is the distribution’s expected mean, and $\sigma$ is the distribution’s expected standard deviation. The function returns the probability of obtaining a result of less than x (Figure 4.8.4 ).
Figure 4.8.4 : Shown in blue is the area returned by the function pnorm($x, \mu, \sigma$).
Here is the output of an R session for solving Example 4.4.2.
> pnorm(243, 250, 5)
[1] 0.08075666
> pnorm(262, 250, 5)
[1] 0.9918025
Subtracting the smaller value from the larger value and adjusting to the correct number of significant figures gives the probability as 0.9910, or 99.10%.
R also includes functions for binomial distributions. To find the probability of obtaining a particular outcome, X, in N trials we use the dbinom function.
dbinom(X, N, p)
where X is the number of times a particular outcome occurs in N trials, and p is the probability that X occurs in a single trial. Using Example 4.4.1 to test this function, we find that the probability of finding no atoms of 13C atoms in a molecule of cholesterol, C27H44O is
> dbinom(0, 27, 0.0111)
[1] 0.7397997
0.740 after adjusting the significant figures. To find the probability of obtaining any outcome up to a maximum value of X, we use the pbinom function.
pbinom(X, N, p)
To find the percentage of cholesterol molecules that contain 0, 1, or 2 atoms of 13C, we enter
> pbinom(2, 27, 0.0111)
[1] 0.9967226
and find that the answer is 99.7% of cholesterol molecules.
Significance Tests
R includes commands for the following significance tests covered in this chapter:
• F-test of variances
• unpaired t-test of sample means assuming equal variances
• unpaired t-test of sample means assuming unequal variances
• paired t-test for of sample means
• Dixon’s Q-test for outliers
• Grubb’s test for outliers
Let’s use these functions to complete a t-test on the data in Table 4.4.1, which contains results for two experiments to determine the mass of a circulating U. S. penny. First, enter the data from Table 4.4.1 into two objects.
> penny1 = c(3.080, 3.094, 3.107, 3.056, 3.112, 3.174, 3.198)
> penny2 = c(3.052, 3.141, 3.083, 3.083, 3.048)
Because the data in this case are unpaired, we will use R to complete an unpaired t-test. Before we can complete a t-test we use an F-test to determine whether the variances for the two data sets are equal or unequal.
To complete a two-tailed F-test in R we use the command
var.test(X, Y)
where X and Y are the objects that contain the two data sets. Figure 4.8.5 shows the output from an R session to solve this problem.
Note that R does not provide the critical value for F(0.05, 6, 4); instead it reports the 95% confidence interval for Fexp. Because this confidence interval of 0.204 to 11.661 includes the expected value for F of 1.00, we retain the null hypothesis and have no evidence for a difference between the variances. R also provides the probability of incorrectly rejecting the null hypothesis, which in this case is 0.5561.
For a one-tailed F-test the command is one of the following
var.test(X, Y, alternative = “greater”)
var.test(X, Y, alternative = “less”)
where “greater” is used when the alternative hypothesis is $s_X^2 > s_Y^2$, and “less” is used when the alternative hypothesis is $s_X^2 < s_Y^2$.
Having found no evidence suggesting unequal variances, we now complete an unpaired t-test assuming equal variances. The basic syntax for a two-tailed t-test is
t.test(X, Y, mu = 0, paired = FALSE, var.equal = FALSE)
where X and Y are the objects that contain the data sets. You can change the underlined terms to alter the nature of the t-test. Replacing “var.equal = FALSE” with “var.equal = TRUE” makes this a two-tailed t-test with equal variances, and replacing “paired = FALSE” with “paired = TRUE” makes this a paired t-test. The term “mu = 0” is the expected difference between the means, which for this problem is 0. You can, of course, change this to suit your needs. The underlined terms are default values; if you omit them, then R assumes you intend an unpaired two-tailed t-test of the null hypothesis that X = Y with unequal variances. Figure 4.8.6 shows the output of an R session for this problem.
We can interpret the results of this t-test in two ways. First, the p-value of 0.2116 means there is a 21.16% probability of incorrectly rejecting the null hypothesis. Second, the 95% confidence interval of –0.024 to 0.0958 for the difference between the sample means includes the expected value of zero. Both ways of looking at the results provide no evidence for rejecting the null hypothesis; thus, we retain the null hypothesis and find no evidence for a difference between the two samples.
The other significance tests in R work in the same format. The following practice exercise provides you with an opportunity to test yourself.
Exercise 4.8.2
Rework Example 4.6.5 and Example 4.6.6 using R.
Answer
Shown here are copies of R sessions for each problem. You will find small differences between the values given here for texp and for Fexp and those values shown with the worked solutions in the chapter. These differences arise because R does not round off the results of intermediate calculations.
Example 4.6.5
> AnalystA = c(86.82, 87.04, 86.93, 87.01, 86.20, 87.00)
> AnalystB = c(81.01, 86.15, 81.73, 83.19, 80.27, 83.94)
> var.test(AnalystB, AnalystA)
F test to compare two variances
data: AnalystB and AnalystA
F = 45.6358, num df = 5, denom df = 5, p-value = 0.0007148
alternative hypothesis: true ratio of variances is not equal to 1
95 percent confidence interval:
6.385863 326.130970
sample estimates:
ratio of variances
45.63582
> t.test(AnalystA, AnalystB, var.equal=FALSE)
Welch Two Sample t-test
data: AnalystA and AnalystB
t = 4.6147, df = 5.219, p-value = 0.005177
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval: 1.852919 6.383748
sample estimates: mean of x mean of y
86.83333 82.71500
Example 4.21
> micro = c(129.5, 89.6, 76.6, 52.2, 110.8, 50.4, 72.4, 141.4, 75.0, 34.1, 60.3)
> elect = c(132.3, 91.0, 73.6, 58.2, 104.2, 49.9, 82.1, 154.1, 73.4, 38.1, 60.1)
> t.test(micro,elect,paired=TRUE)
Paired t-test
data: micro and elect
t = -1.3225, df = 10, p-value = 0.2155
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
-6.028684 1.537775
sample estimates:
mean of the differences
–2.245455
Unlike Excel, R also includes functions for evaluating outliers. These functions are not part of R’s standard installation. To install them enter the following command within R (note: you will need an internet connection to download the package of functions).
> install.packages(“outliers”)
After you install the package, you must load the functions into R by using the following command (note: you need to do this step each time you begin a new R session as the package does not automatically load when you start R).
> library(“outliers”)
You need to install a package once, but you need to load the package each time you plan to use it. There are ways to configure R so that it automatically loads certain packages; see An Introduction to R for more information (click here to view a PDF version of this document).
Let’s use this package to find the outlier in Table 4.6.1 using both Dixon’s Q-test and Grubb’s test. The commands for these tests are
dixon.test(X, type = 10, two.sided = TRUE)
grubbs.test(X, type = 10, two.sided = TRUE)
where X is the object that contains the data, “type = 10” specifies that we are looking for one outlier, and “two.sided = TRUE” indicates that we are using the more conservative two-tailed test. Both tests have other variants that allow for the testing of outliers on both ends of the data set (“type = 11”) or for more than one outlier (“type = 20”), but we will not consider these here. Figure 4.8.7 shows the output of a session for this problem. For both tests the very small p-value indicates that we can treat as an outlier the penny with a mass of 2.514 g.
Visualizing Data
One of R’s more useful features is the ability to visualize data. Visualizing data is important because it provides us with an intuitive feel for our data that can help us in applying and evaluating statistical tests. It is tempting to believe that a statistical analysis is foolproof, particularly if the probability for incorrectly rejecting the null hypothesis is small. Looking at a visual display of our data, however, can help us determine whether our data is normally distributed—a requirement for most of the significance tests in this chapter—and can help us identify potential outliers. There are many useful ways to look at data, four of which we consider here.
Visualizing data is important, a point we will return to in Chapter 5 when we consider the mathematical modeling of data.
To plot data in R, we will use the package “lattice,” which you will need to load using the following command.
> library(“lattice”)
To demonstrate the types of plots we can generate, we will use the object “penny,” which contains the masses of the 100 pennies in Table 4.4.3.
You do not need to use the command install.package this time because lattice was automatically installed on your computer when you downloaded R.
Our first visualization is a histogram. To construct the histogram we use mass to divide the pennies into bins and plot the number of pennies or the percent of pennies in each bin on the y-axis as a function of mass on the x-axis. Figure 4.8.8 shows the result of entering the command
> histogram(penny, type = “percent”, xlab = “Mass (g)”, ylab = “Percent of Pennies”, main = “Histogram of Data in Table 4.4.3”)
A histogram allows us to visualize the data’s distribution. In this example the data appear to follow a normal distribution, although the largest bin does not include the mean of 3.095 g and the distribution is not perfectly symmetric. One limitation of a histogram is that its appearance depends on how we choose to bin the data. Increasing the number of bins and centering the bins around the data’s mean gives a histogram that more closely approximates a normal distribution (Figure 4.4.5).
An alternative to the histogram is a kernel density plot, which basically is a smoothed histogram. In this plot each value in the data set is replaced with a normal distribution curve whose width is a function of the data set’s standard deviation and size. The resulting curve is a summation of the individual distributions. Figure 4.8.9 shows the result of entering the command
> densityplot(penny, xlab = “Mass of Pennies (g)”, main = “Kernel Density Plot of Data in Table 4.4.3”)
The circles at the bottom of the plot show the mass of each penny in the data set. This display provides a more convincing picture that the data in Table 4.4.3 are normally distributed, although we see evidence of a small clustering of pennies with a mass of approximately 3.06 g.
We analyze samples to characterize the parent population. To reach a meaningful conclusion about a population, the samples must be representative of the population. One important requirement is that the samples are random. A dot chart provides a simple visual display that allows us to examine the data for non-random trends. Figure 4.8.10 shows the result of entering
> dotchart(penny, xlab = “Mass of Pennies (g)”, ylab = “Penny Number”, main = “Dotchart of Data in Table 4.4.3”)
In this plot the masses of the 100 pennies are arranged along the y-axis in the order in which they were sampled. If we see a pattern in the data along the y-axis, such as a trend toward smaller masses as we move from the first penny to the last penny, then we have clear evidence of non-random sampling. Because our data do not show a pattern, we have more confidence in the quality of our data.
The last plot we will consider is a box plot, which is a useful way to identify potential outliers without making any assumptions about the data’s distribution. A box plot contains four pieces of information about a data set: the median, the middle 50% of the data, the smallest value and the largest value within a set distance of the middle 50% of the data, and possible outliers. Figure 4.8.11 shows the result of entering
> bwplot(penny, xlab = “Mass of Pennies (g)”, main = “Boxplot of Data in Table 4.4.3)”
The black dot (•) is the data set’s median. The rectangular box shows the range of masses spanning the middle 50% of the pennies. This also is known as the interquartile range, or IQR. The dashed lines, which are called “whiskers,” extend to the smallest value and the largest value that are within $\pm 1.5 \times \text{IQR}$ of the rectangular box. Potential outliers are shown as open circles (o). For normally distributed data the median is near the center of the box and the whiskers will be equidistant from the box. As is often the case in statistics, the converse is not true—finding that a boxplot is perfectly symmetric does not prove that the data are normally distributed.
To find the interquartile range you first find the median, which divides the data in half. The median of each half provides the limits for the box. The IQR is the median of the upper half of the data minus the median for the lower half of the data. For the data in Table 4.4.3 the median is 3.098. The median for the lower half of the data is 3.068 and the median for the upper half of the data is 3.115. The IQR is 3.115 – 3.068 = 0.047. You can use the command “summary(penny)” in R to obtain these values.
The lower “whisker” extend to the first data point with a mass larger than
3.068 – 1.5 $\times$ IQR = 3.068 – 1.5 $\times$ 0.047 = 2.9975
which for this data is 2.998 g. The upper “whisker” extends to the last data point with a mass smaller than
3.115 + 1.5 $\times$ IQR = 3.115 + 1.5 $\times$ 0.047 = 3.1855
which for this data is 3.181 g.
The box plot in Figure 4.8.11 is consistent with the histogram (Figure 4.8.8 ) and the kernel density plot (Figure 4.8.9 ). Together, the three plots provide evidence that the data in Table 4.4.3 are normally distributed. The potential outlier, whose mass of 3.198 g, is not sufficiently far away from the upper whisker to be of concern, particularly as the size of the data set (n = 100) is so large. A Grubb’s test on the potential outlier does not provide evidence for treating it as an outlier.
Exercise 4.8.3
Use R to create a data set consisting of 100 values from a uniform distribution by entering the command
> data = runif(100, min = 0, max = 100)
A uniform distribution is one in which every value between the minimum and the maximum is equally probable. Examine the data set by creating a histogram, a kernel density plot, a dot chart, and a box plot. Briefly comment on what the plots tell you about the your sample and its parent population.
Answer
Because we are selecting a random sample of 100 members from a uniform distribution, you will see subtle differences between your plots and the plots shown as part of this answer. Here is a record of my R session and the resulting plots.
> data = runif(100, min = 0, max = 0)
> data
[1] 18.928795 80.423589 39.399693 23.757624 30.088554
[6] 76.622174 36.487084 62.186771 81.115515 15.726404
[11] 85.765317 53.994179 7.919424 10.125832 93.153308
[16] 38.079322 70.268597 49.879331 73.115203 99.329723
[21] 48.203305 33.093579 73.410984 75.128703 98.682127
[26] 11.433861 53.337359 81.705906 95.444703 96.843476
[31] 68.251721 40.567993 32.761695 74.635385 70.914957
[36] 96.054750 28.448719 88.580214 95.059215 20.316015
[41] 9.828515 44.172774 99.648405 85.593858 82.745774
[46] 54.963426 65.563743 87.820985 17.791443 26.417481
[51] 72.832037 5.518637 58.231329 10.213343 40.581266
[56] 6.584000 81.261052 48.534478 51.830513 17.214508
[61] 31.232099 60.545307 19.197450 60.485374 50.414960
[66] 88.908862 68.939084 92.515781 72.414388 83.195206
[71] 74.783176 10.643619 41.775788 20.464247 14.547841
[76] 89.887518 56.217573 77.606742 26.956787 29.641171
[81] 97.624246 46.406271 15.906540 23.007485 17.715668
[86] 84.652814 29.379712 4.093279 46.213753 57.963604
[91] 91.160366 34.278918 88.352789 93.004412 31.055807
[96] 47.822329 24.052306 95.498610 21.089686 2.629948
> histogram(data, type = “percent”)
> densityplot(data)
> dotchart(data)
> bwplot(data)
The histogram (far left) divides the data into eight bins, each of which contains between 10 and 15 members. As we expect for a uniform distribution, the histogram’s overall pattern suggests that each outcome is equally probable. In interpreting the kernel density plot (second from left), it is important to remember that it treats each data point as if it is from a normally distributed population (even though, in this case, the underlying population is uniform). Although the plot appears to suggest that there are two normally distributed populations, the individual results shown at the bottom of the plot provide further evidence for a uniform distribution. The dot chart (second from right) shows no trend along the y-axis, which indicates that the individual members of this sample were drawn at random from the population. The distribution along the x-axis also shows no pattern, as expected for a uniform distribution, Finally, the box plot (far right) shows no evidence of outliers. | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/04%3A_Evaluating_Analytical_Data/4.08%3A_Using_Excel_and_R_to_Analyze_Data.txt |
1. The following masses were recorded for 12 different U.S. quarters (all given in grams):
5.683 5.549 5.548 5.552
5.620 5.536 5.539 5.684
5.551 5.552 5.554 5.632
Report the mean, median, range, standard deviation and variance for this data.
2. A determination of acetaminophen in 10 separate tablets of Excedrin Extra Strength Pain Reliever gives the following results (in mg)
224.3 240.4 246.3 239.4 253.1
261.7 229.4 255.5 235.5 249.7
(a) Report the mean, median, range, standard deviation and variance for this data.
(b) Assuming that $\overline{X}$ and s2 are good approximations for $\mu$ and for $\sigma^2$, and that the population is normally distributed, what percentage of tablets contain more than the standard amount of 250 mg acetaminophen per tablet?
The data in this problem are from Simonian, M. H.; Dinh, S.; Fray, L. A. Spectroscopy 1993, 8(6), 37–47.
3. Salem and Galan developed a new method to determine the amount of morphine hydrochloride in tablets. An analysis of tablets with different nominal dosages gave the following results (in mg/tablet).
100-mg tablets 60-mg tablets 30-mg tablets 10-mg tablets
99.17 54.21 28.51 9.06
94.31 55.62 26.25 8.83
95.92 57.40 25.92 9.08
94.55 57.51 28.62
93.83 52.59 24.93
(a) For each dosage, calculate the mean and the standard deviation for the mg of morphine hydrochloride per tablet.
(b) For each dosage level, and assuming that $\overline{X}$ and s2 are good approximations for $\mu$ and for $\sigma^2$, and that the population is normally distributed, what percentage of tablets contain more than the nominal amount of morphine hydro- chloride per tablet?
The data in this problem are from Salem, I. I.; Galan, A. C. Anal. Chim. Acta 1993, 283, 334–337.
4. Daskalakis and co-workers evaluated several procedures for digesting oyster and mussel tissue prior to analyzing them for silver. To evaluate the procedures they spiked samples with known amounts of silver and analyzed the samples to determine the amount of silver, reporting results as the percentage of added silver found in the analysis. A procedure was judged acceptable if its spike recoveries fell within the range 100±15%. The spike recoveries for one method are shown here.
105% 108% 92% 99%
101% 93% 93% 104%
Assuming a normal distribution for the spike recoveries, what is the probability that any single spike recovery is within the accepted range?
The data in this problem are from Daskalakis, K. D.; O’Connor, T. P.; Crecelius, E. A. Environ. Sci. Technol. 1997, 31, 2303– 2306. See Chapter 15 to learn more about using a spike recovery to evaluate an analytical method.
5. The formula weight (FW) of a gas can be determined using the following form of the ideal gas law
$FW = \frac {g \text{R} T} {P V} \nonumber$
where g is the mass in grams, R is the gas constant, T is the temperature in Kelvin, P is the pressure in atmospheres, and V is the volume in liters. In a typical analysis the following data are obtained (with estimated uncertainties in parentheses)
g = 0.118 g (± 0.002 g)
R = 0.082056 L atm mol–1 K–1 (± 0.000001 L atm mol–1 K–1)
T = 298.2 K (± 0.1 K)
P = 0.724 atm (± 0.005 atm)
V = 0.250 L (± 0.005 L)
(a) What is the compound’s formula weight and its estimated uncertainty?
(b) To which variable(s) should you direct your attention if you wish to improve the uncertainty in the compound’s molecular weight?
6. To prepare a standard solution of Mn2+, a 0.250 g sample of Mn is dissolved in 10 mL of concentrated HNO3 (measured with a graduated cylinder). The resulting solution is quantitatively transferred to a 100-mL volumetric flask and diluted to volume with distilled water. A 10-mL aliquot of the solution is pipeted into a 500-mL volumetric flask and diluted to volume.
(a) Express the concentration of Mn in mg/L, and estimate its uncertainty using a propagation of uncertainty.
(b) Can you improve the concentration’s uncertainty by using a pipet to measure the HNO3, instead of a graduated cylinder?
7. The mass of a hygroscopic compound is measured using the technique of weighing by difference. In this technique the compound is placed in a sealed container and weighed. A portion of the compound is removed and the container and the remaining material are reweighed. The difference between the two masses gives the sample’s mass. A solution of a hygroscopic compound with a gram formula weight of 121.34 g/mol (±0.01 g/mol) is prepared in the following manner. A sample of the compound and its container has a mass of 23.5811 g. A portion of the compound is transferred to a 100-mL volumetric flask and diluted to volume. The mass of the compound and container after the transfer is 22.1559 g. Calculate the compound’s molarity and estimate its uncertainty by a propagation of uncertainty.
8. Use a propagation of uncertainty to show that the standard error of the mean for n determinations is $\sigma / \sqrt{n}$.
9. Beginning with Equation 4.6.4 and Equation 4.6.5, use a propagation of uncertainty to derive Equation 4.6.6.
10. What is the smallest mass you can measure on an analytical balance that has a tolerance of ±0.1 mg, if the relative error must be less than 0.1%?
11. Which of the following is the best way to dispense 100.0 mL if we wish to minimize the uncertainty: (a) use a 50-mL pipet twice; (b) use a 25-mL pipet four times; or (c) use a 10-mL pipet ten times?
12. You can dilute a solution by a factor of 200 using readily available pipets (1-mL to 100-mL) and volumetric flasks (10-mL to 1000-mL) in either one step, two steps, or three steps. Limiting yourself to the glassware in Table 4.2.1, determine the proper combination of glassware to accomplish each dilution, and rank them in order of their most probable uncertainties.
13. Explain why changing all values in a data set by a constant amount will change $\overline{X}$ but has no effect on the standard deviation, s.
14. Obtain a sample of a metal, or other material, from your instructor and determine its density by one or both of the following methods:
Method A: Determine the sample’s mass with a balance. Calculate the sample’s volume using appropriate linear dimensions.
Method B: Determine the sample’s mass with a balance. Calculate the sample’s volume by measuring the amount of water it displaces by adding water to a graduated cylinder, reading the volume, adding the sample, and reading the new volume. The difference in volumes is equal to the sample’s volume.
Determine the density at least five times.
(a) Report the mean, the standard deviation, and the 95% confidence interval for your results.
(b) Find the accepted value for the metal’s density and determine the absolute and relative error for your determination of the metal’s density.
(c) Use a propagation of uncertainty to determine the uncertainty for your method of analysis. Is the result of this calculation consistent with your experimental results? If not, suggest some possible reasons for this disagreement.
15. How many carbon atoms must a molecule have if the mean number of 13C atoms per molecule is at least one? What percentage of such molecules will have no atoms of 13C?
16. In Example 4.4.1 we determined the probability that a molecule of cholesterol, C27H44O, had no atoms of 13C.
(a) Calculate the probability that a molecule of cholesterol, has 1 atom of 13C.
(b) What is the probability that a molecule of cholesterol has two or more atoms of 13C?
17. Berglund and Wichardt investigated the quantitative determination of Cr in high-alloy steels using a potentiometric titration of Cr(VI). Before the titration, samples of the steel were dissolved in acid and the chromium oxidized to Cr(VI) using peroxydisulfate. Shown here are the results ( as %w/w Cr) for the analysis of a reference steel.
16.968 16.922 16.840 16.883
16.887 16.977 16.857 16.728
Calculate the mean, the standard deviation, and the 95% confidence interval about the mean. What does this confidence interval mean?
The data in this problem are from Berglund, B.; Wichardt, C. Anal. Chim. Acta 1990, 236, 399–410.
18. Ketkar and co-workers developed an analytical method to determine trace levels of atmospheric gases. An analysis of a sample that is 40.0 parts per thousand (ppt) 2-chloroethylsulfide gave the following results
43.3 34.8 31.9
37.8 34.4 31.9
42.1 33.6 35.3
(a) Determine whether there is a significant difference between the experimental mean and the expected value at $\alpha = 0.05$.
(b) As part of this study, a reagent blank was analyzed 12 times giving a mean of 0.16 ppt and a standard deviation of 1.20 ppt. What are the IUPAC detection limit, the limit of identification, and limit of quantitation for this method assuming $\alpha = 0.05$?
The data in this problem are from Ketkar, S. N.; Dulak, J. G.; Dheandhanou, S.; Fite, W. L. Anal. Chim. Acta 1991, 245, 267–270.
19. To test a spectrophotometer’s accuracy a solution of 60.06 ppm K2Cr2O7 in 5.0 mM H2SO4 is prepared and analyzed. This solution has an expected absorbance of 0.640 at 350.0 nm in a 1.0-cm cell when using 5.0 mM H2SO4 as a reagent blank. Several aliquots of the solution produce the following absorbance values.
0.639 0.638 0.640 0.639 0.640 0.639 0.638
Determine whether there is a significant difference between the experimental mean and the expected value at $\alpha = 0.01$.
20. Monna and co-workers used radioactive isotopes to date sediments from lakes and estuaries. To verify this method they analyzed a 208Po standard known to have an activity of 77.5 decays/min, obtaining the following results.
77.09 75.37 72.42 76.84 77.84 76.69
78.03 74.96 77.54 76.09 81.12 75.75
Determine whether there is a significant difference between the mean and the expected value at $\alpha = 0.05$.
The data in this problem are from Monna, F.; Mathieu, D.; Marques, A. N.; Lancelot, J.; Bernat, M. Anal. Chim. Acta 1996, 330, 107–116.
21. A 2.6540-g sample of an iron ore, which is 53.51% w/w Fe, is dissolved in a small portion of concentrated HCl and diluted to volume in a 250-mL volumetric flask. A spectrophotometric determination of the concentration of Fe in this solution yields results of 5840, 5770, 5650, and 5660 ppm. Determine whether there is a significant difference between the experimental mean and the expected value at $\alpha = 0.05$.
22. Horvat and co-workers used atomic absorption spectroscopy to determine the concentration of Hg in coal fly ash. Of particular interest to the authors was developing an appropriate procedure for digesting samples and releasing the Hg for analysis. As part of their study they tested several reagents for digesting samples. Their results using HNO3 and using a 1 + 3 mixture of HNO3 and HCl are shown here. All concentrations are given as ppb Hg sample.
HNO3: 161 165 160 167 166
1 + 3 HNO3 – HCl: 159 145 140 147 143 156
Determine whether there is a significant difference between these methods at $\alpha = 0.05$.
The data in this problem are from Horvat, M.; Lupsina, V.; Pihlar, B. Anal. Chim. Acta 1991, 243, 71–79.
23, Lord Rayleigh, John William Strutt (1842-1919), was one of the most well known scientists of the late nineteenth and early twentieth centuries, publishing over 440 papers and receiving the Nobel Prize in 1904 for the discovery of argon. An important turning point in Rayleigh’s discovery of Ar was his experimental measurements of the density of N2. Rayleigh approached this experiment in two ways: first by taking atmospheric air and removing O2 and H2; and second, by chemically producing N2 by decomposing nitrogen containing compounds (NO, N2O, and NH4NO3) and again removing O2 and H2. The following table shows his results for the density of N2, as published in Proc. Roy. Soc. 1894, LV, 340 (publication 210); all values are the grams of gas at an equivalent volume, pressure, and temperature.
atmospheric origin chemical origin
2.31017 2.30143
2.30986 2.29890
2.31010 2.29816
2.31001 2.30182
2.31024 2.29869
2.31010 2.29940
2.31028 2.29849
2.29889
Explain why this data led Rayleigh to look for and to discover Ar. You can read more about this discovery here: Larsen, R. D. J. Chem. Educ. 1990, 67, 925–928.
24. Gács and Ferraroli reported a method for monitoring the concentration of SO2 in air. They compared their method to the standard method by analyzing urban air samples collected from a single location. Samples were collected by drawing air through a collection solution for 6 min. Shown here is a summary of their results with SO2 concentrations reported in $\mu \text{L/m}^3$.
standard method new method
21.62 21.54
22.20 20.51
24.27 22.31
23.54 21.30
24.25 24.62
23.09 25.72
21.02 21.54
Using an appropriate statistical test, determine whether there is any significant difference between the standard method and the new method at $\alpha = 0.05$.
The data in this problem are from Gács, I.; Ferraroli, R. Anal. Chim. Acta 1992, 269, 177–185.
25. One way to check the accuracy of a spectrophotometer is to measure absorbances for a series of standard dichromate solutions obtained from the National Institute of Standards and Technology. Absorbances are measured at 257 nm and compared to the accepted values. The results obtained when testing a newly purchased spectrophotometer are shown here. Determine if the tested spectrophotometer is accurate at $\alpha = 0.05$.
standard measured absorbance expected absorbance
1 0.2872 0.2871
2 0.5773 0.5760
3 0.8674 0.8677
4 1.1623 1.1608
5 1.4559 1.4565
26. Maskarinec and co-workers investigated the stability of volatile organics in environmental water samples. Of particular interest was establishing the proper conditions to maintain the sample’s integrity between its collection and its analysis. Two preservatives were investigated—ascorbic acid and sodium bisulfate—and maximum holding times were determined for a number of volatile organics and water matrices. The following table shows results for the holding time (in days) of nine organic compounds in surface water.
compound Ascorbic Acid Sodium Bisulfate
methylene chloride 77 62
carbon disulfide 23 54
trichloroethane 52 51
benzene 62 42
1,1,2-trichlorethane 57 53
1,1,2,2-tetrachloroethane 33 85
tetrachloroethene 32 94
chlorbenzene 36 86
Determine whether there is a significant difference in the effectiveness of the two preservatives at $\alpha = 0.10$.
The data in this problem are from Maxkarinec, M. P.; Johnson, L. H.; Holladay, S. K.; Moody, R. L.; Bayne, C. K.; Jenkins, R. A. Environ. Sci. Technol. 1990, 24, 1665–1670.
27. Using X-ray diffraction, Karstang and Kvalhein reported a new method to determine the weight percent of kaolinite in complex clay minerals using X-ray diffraction. To test the method, nine samples containing known amounts of kaolinite were prepared and analyzed. The results (as % w/w kaolinite) are shown here.
actual 5.0 10.0 20.0 40.0 50.0 60.0 80.0 90.0 95.0
found 6.8 11.7 19.8 40.5 53.6 61.7 78.9 91.7 94.7
Evaluate the accuracy of the method at $\alpha = 0.05$.
The data in this problem are from Karstang, T. V.; Kvalhein, O. M. Anal. Chem. 1991, 63, 767–772.
28. Mizutani, Yabuki and Asai developed an electrochemical method for analyzing l-malate. As part of their study they analyzed a series of beverages using both their method and a standard spectrophotometric procedure based on a clinical kit purchased from Boerhinger Scientific. The following table summarizes their results. All values are in ppm.
Sample Electrode Spectrophotometric
Apple Juice 1 34.0 33.4
Apple Juice 2 22.6 28.4
Apple Juice 3 29.7 29.5
Apple Juice 4 24.9 24.8
Grape Juice 1 17.8 18.3
Grape Juice 2 14.8 15.4
Mixed Fruit Juice 1 8.6 8.5
Mixed Fruit Juice 2 31.4 31.9
White Wine 1 10.8 11.5
White Wine 2 17.3 17.6
White Wine 3 15.7 15.4
White Wine 4 18.4 18.3
The data in this problem are from Mizutani, F.; Yabuki, S.; Asai, M. Anal. Chim. Acta 1991, 245,145–150.
29. Alexiev and colleagues describe an improved photometric method for determining Fe3+ based on its ability to catalyze the oxidation of sulphanilic acid by KIO4. As part of their study, the concentration of Fe3+ in human serum samples was determined by the improved method and the standard method. The results, with concentrations in $\mu \text{mol/L}$, are shown in the following table.
Sample Improved Method Standard Method
1 8.25 8.06
2 9.75 8.84
3 9.75 8.36
4 9.75 8.73
5 10.75 13.13
6 11.25 13.65
7 13.88 13.85
8 14.25 13.43
Determine whether there is a significant difference between the two methods at $\alpha = 0.05$.
The data in this problem are from Alexiev, A.; Rubino, S.; Deyanova, M.; Stoyanova, A.; Sicilia, D.; Perez Bendito, D. Anal. Chim. Acta, 1994, 295, 211–219.
30. Ten laboratories were asked to determine an analyte’s concentration of in three standard test samples. Following are the results, in $\mu \text{g/ml}$.
Laboratory Sample 1 Sample 2 Sample 3
1 22.6 13.6 16.0
2 23.0 14.2 15.9
3 21.5 13.9 16.9
4 21.9 13.9 16.9
5 21.3 13.5 16.7
6 22.1 13.5 17.4
7 23.1 13.5 17.5
8 21.7 13.5 16.8
9 22.2 12.9 17.2
10 21.7 13.8 16.7
Determine if there are any potential outliers in Sample 1, Sample 2 or Sample 3. Use all three methods—Dixon’s Q-test, Grubb’s test, and Chauvenet’s criterion—and compare the results to each other. For Dixon’s Q-test and for the Grubb’s test, use a significance level of $\alpha = 0.05$.
The data in this problem are adapted from Steiner, E. H. “Planning and Analysis of Results of Collaborative Tests,” in Statistical Manual of the Association of Official Analytical Chemists, Association of Official Analytical Chemists: Washington, D. C., 1975.
31.When copper metal and powdered sulfur are placed in a crucible and ignited, the product is a sulfide with an empirical formula of CuxS. The value of x is determined by weighing the Cu and the S before ignition and finding the mass of CuxS when the reaction is complete (any excess sulfur leaves as SO2). The following table shows the Cu/S ratios from 62 such experiments (note that the values are organized from smallest-to-largest by rows).
1.764 1.838 1.865 1.866 1.872 1.877
1.890 1.891 1.891 1.897 1.899 1.900
1.906 1.908 1.910 1.911 1.916 1.919
1.920 1.922 1.927 1.931 1.935 1.936
1.936 1.937 1.939 1.939 1.940 1.941
1.941 1.942 1.943 1.948 1.953 1.955
1.957 1.957 1.957 1.959 1.962 1.963
1.963 1.963 1.966 1.968 1.969 1.973
1.975 1.976 1.977 1.981 1.981 1.988
1.993 1.993 1.995 1.995 1.995 2.017
2.029 2.042
(a) Calculate the mean, the median, and the standard deviation for this data.
(b) Construct a histogram for this data. From a visual inspection of your histogram, do the data appear normally distributed?
(c) In a normally distributed population 68.26% of all members lie within the range $\mu \pm 1 \sigma$. What percentage of the data lies within the range $\overline{X} \pm 1 \sigma$? Does this support your answer to the previous question?
(d) Assuming that $\overline{X}$ and $s^2$ are good approximations for $\mu$ and for $\sigma^2$, what percentage of all experimentally determined Cu/S ratios should be greater than 2? How does this compare with the experimental data? Does this support your conclusion about whether the data is normally distributed?
(e) It has been reported that this method of preparing copper sulfide results in a non-stoichiometric compound with a Cu/S ratio of less than 2. Determine if the mean value for this data is significantly less than 2 at a significance level of $\alpha = 0.01$.
See Blanchnik, R.; Müller, A. “The Formation of Cu2S From the Elements I. Copper Used in Form of Powders,” Thermochim. Acta, 2000, 361, 31-52 for a discussion of some of the factors affecting the formation of non-stoichiometric copper sulfide. The data in this problem were collected by students at DePauw University.
32. Real-time quantitative PCR is an analytical method for determining trace amounts of DNA. During the analysis, each cycle doubles the amount of DNA. A probe species that fluoresces in the presence of DNA is added to the reaction mixture and the increase in fluorescence is monitored during the cycling. The cycle threshold, $C_t$, is the cycle when the fluorescence exceeds a threshold value. The data in the following table shows $C_t$ values for three samples using real-time quantitative PCR. Each sample was analyzed 18 times.
Sample X Sample Y Sample Z
24.24 25.14 24.41 28.06 22.97 23.43
23.97 24.57 27.21 27.77 22.93 23.66
24.44 24.49 27.02 28.74 22.95 28.79
24.79 24.68 26.81 28.35 23.12 23.77
23.92 24.45 26.64 28.80 23.59 23.98
24.53 24,48 27.63 27.99 23.37 23.56
24.95 24.30 28.42 28.21 24.17 22.80
24.76 24.60 25.16 28.00 23.48 23.29
25.18 24.57 28.53 28.21 23.80 23.86
Examine this data and write a brief report on your conclusions. Issues you may wish to address include the presence of outliers in the samples, a summary of the descriptive statistics for each sample, and any evidence for a difference between the samples.
The data in this problem is from Burns, M. J.; Nixon, G. J.; Foy, C. A.; Harris, N. BMC Biotechnol. 2005, 5:31 (open access publication). | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/04%3A_Evaluating_Analytical_Data/4.09%3A_Problems.txt |
The following experiments provide useful introductions to the statistical analysis of data in the analytical chemistry laboratory.
• Bularzik, J. “The Penny Experiment Revisited: An Illustration of Significant Figures, Accuracy, Precision, and Data Analysis,” J. Chem. Educ. 2007, 84, 1456–1458.
• Columbia, M. R. “The Statistics of Coffee: 1. Evaluation of Trace Metals for Establishing a Coffee’s Country of Origin Based on a Means Comparison,” Chem. Educator 2007, 12, 260–262.
• Cunningham, C. C.; Brown, G. R.; St Pierre, L. E. “Evaluation of Experimental Data,” J. Chem. Educ. 1981, 58, 509–511.
• Edminston, P. L.; Williams, T. R. “An Analytical Laboratory Experiment in Error Analysis: Repeated Determination of Glucose Using Commercial Glucometers,” J. Chem. Educ. 2000, 77, 377–379.
• Gordus, A. A. “Statistical Evaluation of Class Data for Two Buret Readings,” J. Chem. Educ. 1987, 64, 376–377.
• Harvey, D. T. “Statistical Evaluation of Acid/Base Indicators,” J. Chem. Educ. 1991, 68, 329–331.
• Hibbert, D. B. “Teaching modern data analysis with The Royal Austrian Chemical Institute’s titration competition,” Aust. J. Ed. Chem. 2006, 66, 5–11.
• Johll, M. E.; Poister, D.; Ferguson, J. “Statistical Comparison of Multiple Methods for the Determination of Dissolved Oxygen Levels in Natural Water,” Chem. Educator 2002, 7, 146–148.
• Jordon, A. D. “Which Method is Most Precise; Which is Most Accurate?,” J. Chem. Educ. 2007, 84, 1459–1460.
• Olsen, R. J. “Using Pooled Data and Data Visualization To Introduce Statistical Concepts in the General Chemistry Laboratory,” J. Chem. Educ. 2008, 85, 544–545.
• O’Reilley, J. E. “The Length of a Pestle,” J. Chem. Educ. 1986, 63, 894–896.
• Overway, K. “Population versus Sampling Statistics: A Spreadsheet Exercise,” J. Chem. Educ. 2008 85, 749.
• Paselk, R. A. “An Experiment for Introducing Statistics to Students of Analytical and Clinical Chem- istry,” J. Chem. Educ. 1985, 62, 536.
• Puignou, L.; Llauradó, M. “An Experimental Introduction to Interlaboratory Exercises in Analytical Chemistry,” J. Chem. Educ. 2005, 82, 1079–1081.
• Quintar, S. E.; Santagata, J. P.; Villegas, O. I.; Cortinez, V. A. “Detection of Method Effects on Quality of Analytical Data,” J. Chem. Educ. 2003, 80, 326–329.
• Richardson, T. H. “Reproducible Bad Data for Instruction in Statistical Methods,” J. Chem. Educ. 1991, 68, 310–311.
• Salzsieder, J. C. “Statistical Analysis Experiment for Freshman Chemistry Lab,” J. Chem. Educ. 1995, 72, 623.
• Samide, M. J. “Statistical Comparison of Data in the Analytical Laboratory,” J. Chem. Educ. 2004, 81, 1641–1643.
• Sheeran, D. “Copper Content in Synthetic Copper Carbonate: A Statistical Comparison of Experimental and Expected Results,” J. Chem. Educ. 1998, 75, 453–456.
• Spencer, R. D. “The Dependence of Strength in Plastics upon Polymer Chain Length and Chain Orientation,” J. Chem. Educ. 1984, 61, 555–563.
• Stolzberg, R. J. “Do New Pennies Lose Their Shells? Hypothesis Testing in the Sophomore Analytical Chemistry Laboratory,” J. Chem. Educ. 1998, 75, 1453–1455.
• Stone, C. A.; Mumaw, L. D. “Practical Experiments in Statistics,” J. Chem. Educ. 1995, 72, 518– 524.
• Thomasson, K.; Lofthus-Merschman, S.; Humbert, M.; Kulevsky, N. “Applying Statistics in the Undergraduate Chemistry Laboratory: Experiments with Food Dyes,” J. Chem. Educ. 1998, 75, 231–233.
• Vitha, M. F.; Carr, P. W. “A Laboratory Exercise in Statistical Analysis of Data,” J. Chem. Educ. 1997, 74, 998–1000.
A more comprehensive discussion of the analysis of data, which includes all topics considered in this chapter as well as additional material, are found in many textbook on statistics or data analysis; several such texts are listed here.
• Anderson, R. L. Practical Statistics for Analytical Chemists, Van Nostrand Reinhold: New York; 1987.
• Graham, R. C. Data Analysis for the Chemical Sciences, VCH Publishers: New York; 1993.
• Mark, H.; Workman, J. Statistics in Spectroscopy, Academic Press: Boston; 1991.
• Mason, R. L.; Gunst, R. F.; Hess, J. L. Statistical Design and Analysis of Experiments; Wiley: New York, 1989.
• Massart, D. L.; Vandeginste, B. G. M.; Buydens, L. M. C.; De Jong, S.; Lewi, P. J.; Smeyers-Verbeke, J. Handbook of Chemometrics and Qualimetrics, Elsevier: Amsterdam, 1997.
• Miller, J. C.; Miller, J. N. Statistics for Analytical Chemistry, Ellis Horwood PTR Prentice-Hall: New York; 3rd Edition, 1993.
• NIST/SEMATECH e-Handbook of Statistical Methods, http://www.itl.nist.gov/div898/handbook/, 2006.
• Sharaf, M. H.; Illman, D. L.; Kowalski, B. R. Chemometrics, Wiley-Interscience: New York; 1986.
The importance of defining statistical terms is covered in the following papers.
• Analytical Methods Committee “Terminology—the key to understanding analytical science. Part 1: Accuracy, precision and uncertainty,” AMC Technical Brief No. 13, Sept. 2003.
• Goedart, M. J.; Verdonk, A. H. “The Development of Statistical Concepts in a Design-Oriented Laboratory Course in Scientific Measuring,” J. Chem. Educ. 1991, 68, 1005–1009.
• Sánchez, J. M. “Teaching Basic Applied Statistics in University Chemistry Courses: Students’ Misconceptions,” Chem. Educator 2006, 11, 1–4.
• Thompson, M. “Towards a unified model of errors in analytical measurements,” Analyst 2000, 125, 2020–2025.
• Treptow, R. S. “Precision and Accuracy in Measurements,” J. Chem. Educ. 1998, 75, 992–995.
The detection of outliers, particularly when working with a small number of samples, is discussed in the following papers.
• Analytical Methods Committee “Robust Statistics—How Not To Reject Outliers Part 1. Basic Concepts,” Analyst 1989, 114, 1693–1697.
• Analytical Methods Committee “Robust Statistics—How Not to Reject Outliers Part 2. Inter-laboratory Trials,” Analyst 1989, 114, 1699–1702.
• Analytical Methods Committee “Rogues and Suspects: How to Tackle Outliers,” AMCTB 39, 2009.
• Analytical Methods Committee “Robust statistics: a method of coping with outliers,” AMCTB 6, 2001.
• Analytical Methods Committee “Using the Grubbs and Cochran tests to identify outliers,” Anal. Meth- ods, 2015, 7, 7948–7950.
• Efstathiou, C. “Stochastic Calculation of Critical Q-Test Values for the Detection of Outliers in Measurements,” J. Chem. Educ. 1992, 69, 773–736.
• Efstathiou, C. “Estimation of type 1 error probability from experimental Dixon’s Q parameter on testing for outliers within small data sets,” Talanta 2006, 69, 1068–1071.
• Kelly, P. C. “Outlier Detection in Collaborative Studies,” Anal. Chem. 1990, 73, 58–64.
• Mitschele, J. “Small Sample Statistics,” J. Chem. Educ. 1991, 68, 470–473.
The following papers provide additional information on error and uncertainty, including the propagation of uncertainty.
• Analytical Methods Committee “Optimizing your uncertainty—a case study,” AMCTB 32, 2008.
• Analytical Methods Committee “Dark Uncertainty,” AMCTB 53, 2012.
• Analytical Methods Committee “What causes most errors in chemical analysis?” AMCTB 56, 2013.
• Andraos, J. “On the Propagation of Statistical Errors for a Function of Several Variables,” J. Chem. Educ. 1996, 73, 150–154.
• Donato, H.; Metz, C. “A Direct Method for the Propagation of Error Using a Personal Computer Spreadsheet Program,” J. Chem. Educ. 1988, 65, 867–868.
• Gordon, R.; Pickering, M.; Bisson, D. “Uncertainty Analysis by the ‘Worst Case’ Method,” J. Chem. Educ. 1984, 61, 780–781.
• Guare, C. J. “Error, Precision and Uncertainty,” J. Chem. Educ. 1991, 68, 649–652.
• Guedens, W. J.; Yperman, J.; Mullens, J.; Van Poucke, L. C.; Pauwels, E. J. “Statistical Analysis of Errors: A Practical Approach for an Undergraduate Chemistry Lab Part 1. The Concept,” J. Chem. Educ. 1993, 70, 776–779
• Guedens, W. J.; Yperman, J.; Mullens, J.; Van Poucke, L. C.; Pauwels, E. J. “Statistical Analysis of Errors: A Practical Approach for an Undergraduate Chemistry Lab Part 2. Some Worked Examples,” J. Chem. Educ. 1993, 70, 838–841.
• Heydorn, K. “Detecting Errors in Micro and Trace Analysis by Using Statistics,” Anal. Chim. Acta 1993, 283, 494–499.
• Hund, E.; Massart, D. L.; Smeyers-Verbeke, J. “Operational definitions of uncertainty,” Trends Anal. Chem. 2001, 20, 394–406.
• Kragten, J. “Calculating Standard Deviations and Confidence Intervals with a Universally Applicable Spreadsheet Technique,” Analyst 1994, 119, 2161–2165.
• Taylor, B. N.; Kuyatt, C. E. “Guidelines for Evaluating and Expressing the Uncertainty of NIST Mea- surement Results,” NIST Technical Note 1297, 1994.
• Van Bramer, S. E. “A Brief Introduction to the Gaussian Distribution, Sample Statistics, and the Student’s t Statistic,” J. Chem. Educ. 2007, 84, 1231.
• Yates, P. C. “A Simple Method for Illustrating Uncertainty Analysis,” J. Chem. Educ. 2001, 78, 770–771.
Consult the following resources for a further discussion of detection limits.
• Boumans, P. W. J. M. “Detection Limits and Spectral Interferences in Atomic Emission Spectrometry,” Anal. Chem. 1984, 66, 459A–467A.
• Currie, L. A. “Limits for Qualitative Detection and Quantitative Determination: Application to Radiochemistry,” Anal. Chem. 1968, 40, 586–593.
• Currie, L. A. (ed.) Detection in Analytical Chemistry: Importance, Theory and Practice, American Chemical Society: Washington, D. C., 1988.
• Ferrus, R.; Egea, M. R. “Limit of discrimination, limit of detection and sensitivity in analytical systems,” Anal. Chim. Acta 1994, 287, 119–145.
• Fonollosa, J.; Vergara, A; Huerta, R.; Marco, S. “Estimation of the limit of detection using information theory measures,” Anal. Chim. Acta 2014, 810, 1–9.
• Glaser, J. A.; Foerst, D. L.; McKee, G. D.; Quave, S. A.; Budde, W. L. “Trace analyses for wastewaters,” Environ. Sci. Technol. 1981, 15, 1426–1435.
• Kimbrough, D. E.; Wakakuwa, J. “Quality Control Level: An Introduction to Detection Levels,” Environ. Sci. Technol. 1994, 28, 338–345.
The following articles provide thoughts on the limitations of statistical analysis based on significance testing.
• Analytical Methods Committee “Significance, importance, and power,” AMCTB 38, 2009.
• Analytical Methods Committee “An introduction to non-parametric statistics,” AMCTB 57, 2013.
• Berger, J. O.; Berry, D. A. “Statistical Analysis and the Illusion of Objectivity,” Am. Sci. 1988, 76, 159–165.
• Kryzwinski, M. “Importance of being uncertain,” Nat. Methods 2013, 10, 809–810.
• Kryzwinski, M. “Significance, P values, and t-tests,” Nat. Methods 2013, 10, 1041–1042.
• Kryzwinski, M. “Power and sample size,” Nat. Methods 2013, 10, 1139–1140.
• Leek, J. T.; Peng, R. D. “What is the question?,” Science 2015, 347, 1314–1315.
The following resources provide additional information on using Excel, including reports of errors in its handling of some statistical procedures.
• McCollough, B. D.; Wilson, B. “On the accuracy of statistical procedures in Microsoft Excel 2000 and Excel XP,” Comput. Statist. Data Anal. 2002, 40, 713–721.
• Morgon, S. L.; Deming, S. N. “Guide to Microsoft Excel for calculations, statistics, and plotting data,”
• (http://www.chem.sc.edu/faculty/morga...ide_Morgan.pdf ).
• Kelling, K. B.; Pavur, R. J. “A Comparative Study of the Reliability of Nine Statistical Software Pack-ages,” Comput. Statist. Data Anal. 2007, 51, 3811–3831.
To learn more about using R, consult the following resources.
• Chambers, J. M. Software for Data Analysis: Programming with R, Springer: New York, 2008.
• Maindonald, J.; Braun, J. Data Analysis and Graphics Using R, Cambridge University Press: Cambridge, UK, 2003.
• Sarkar, D. Lattice: Multivariate Data Visualization With R, Springer: New York, 2008.
The following papers provide insight into visualizing data.
• Analytical Methods Committee “Representing data distributions with kernel density estimates,” AMC Technical Brief, March 2006.
• Frigge, M.; Hoaglin, D. C.; Iglewicz, B. “Some Implementations of the Boxplot,” The American Statistician 1989, 43, 50–54.
4.11: Chapter Summary and Key Terms
Summary
The data we collect are characterized by their central tendency (where the values cluster), and their spread (the variation of individual values around the central value). We report our data’s central tendency by stating the mean or median, and our data’s spread using the range, standard deviation or variance. Our collection of data is subject to errors, including determinate errors that affect the data’s accuracy and indeterminate errors that affect its precision. A propagation of uncertainty allows us to estimate how these determinate and indeterminate errors affect our results.
When we analyze a sample several times the distribution of the results is described by a probability distribution, two examples of which are the binomial distribution and the normal distribution. Knowing the type of distribution allows us to determine the probability of obtaining a particular range of results. For a normal distribution we express this range as a confidence interval.
A statistical analysis allows us to determine whether our results are significantly different from known values, or from values obtained by other analysts, by other methods of analysis, or for other samples. We can use a t-test to compare mean values and an F-test to compare variances. To compare two sets of data you first must determine whether the data is paired or unpaired. For unpaired data you also must decide if you can pool the standard deviations. A decision about whether to retain an outlying value can be made using Dixon’s Q-test, Grubb’s test, or Chauvenet’s criterion.
You should be sure to exercise caution if you decide to reject an outlier. Finally, the detection limit is a statistical statement about the smallest amount of analyte we can detect with confidence. A detection limit is not exact since its value depends on how willing we are to falsely report the analyte’s presence or absence in a sample. When reporting a detection limit you should clearly indicate how you arrived at its value.
Key Terms
alternative hypothesis
box plot
confidence interval
detection limit
dot chart
Grubb’s test
kernel density plot
mean
method error
one-tailed significance test
paired t-test
probability distribution
range
sample
standard deviation
tolerance
type 1 error
unpaired data
bias
central limit theorem
constant determinate error
determinate error
error
histogram
limit of identification
median
normal distribution
outlier
personal error
propagation of uncertainty
repeatability
sampling error
standard error of the mean
t-test
type 2 error
variance
binomial distribution
Chauvenet’s criterion
degrees of freedom
Dixon’s Q-test
F-test
indeterminate error
limit of quantitation
measurement error
null hypothesis
paired data
population
proportional determinate error
reproducibility
significance test
Standard Reference Material
two-tailed significance test
uncertainty | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/04%3A_Evaluating_Analytical_Data/4.10%3A_Additional_Resources.txt |
The American Chemical Society’s Committee on Environmental Improvement defines standardization as the process of determining the relationship between the signal and the amount of analyte in a sample. In Chapter 3 we defined this relationship as
$S_{total} = k_A n_A + S_{reag} \text{ or } S_{total} = k_A C_A + S_{reag} \nonumber$
where Stotal is the signal, nA is the moles of analyte, CA is the analyte’s concentration, kA is the method’s sensitivity for the analyte, and Sreag is the contribution to Stotal from sources other than the sample. To standardize a method we must determine values for kA and Sreag. Strategies for accomplishing this are the subject of this chapter.
• 5.1: Analytical Signals
To standardize an analytical method we use standards that contain known amounts of analyte. The accuracy of a standardization, therefore, depends on the quality of the reagents and the glassware we use to prepare these standards.
• 5.2: Calibrating the Signal
The accuracy with which we can determine $k_A$ and $S_{reag}$ depends on how accurately we can measure the signal, $S_{total}$. We measure signals using equipment, such as glassware and balances, and instrumentation, such as spectrophotometers and pH meters. To minimize determinate errors that might affect the signal, we first calibrate our equipment and instrumentation.
• 5.3: Determining the Sensitivity
To standardize an analytical method we also must determine the value of $k_A$. In principle, it should be possible to derive the value of $k_A$ for any analytical method by considering the chemical and physical processes generating the signal. Unfortunately, such calculations are not feasible when we lack a sufficiently developed theoretical model of the physical processes, or are not useful because of non-ideal chemical behavior.
• 5.4: Linear Regression and Calibration Curves
How do we find the best estimate for the relationship between the signal and the concentration of analyte in a multiple-point standardization? The process of determining the best equation for the calibration curve is called linear regression, which is the focus of this section.
• 5.5: Compensating for the Reagent Blank
Thus far in our discussion of strategies for standardizing analytical methods, we have assumed that a suitable reagent blank is available to correct for signals arising from sources other than the analyte. We did not, however ask an important question: “What constitutes an appropriate reagent blank?” Surprisingly, the answer is not immediately obvious.
• 5.6: Using Excel and R for a Linear Regression
Although the calculations in this chapter are relatively straightforward— consisting, as they do, mostly of summations—it is tedious to work through problems using nothing more than a calculator. Both Excel and R include functions for completing a linear regression analysis and for visually evaluating the resulting model.
• 5.7: Problems
End-of-chapter problems to test your understanding of topics in this chapter.
• 5.8: Additional Resources
A compendium of resources to accompany topics in this chapter.
• 5.9: Chapter Summary and Key Terms
Summary of chapter's main topics and a list of key terms introduced in the chapter.
05: Standardizing Analytical Methods
To standardize an analytical method we use standards that contain known amounts of analyte. The accuracy of a standardization, therefore, depends on the quality of the reagents and the glassware we use to prepare these standards. For example, in an acid–base titration the stoichiometry of the acid–base reaction defines the relationship between the moles of analyte and the moles of titrant. In turn, the moles of titrant is the product of the titrant’s concentration and the volume of titrant used to reach the equivalence point. The accuracy of a titrimetric analysis, therefore, is never better than the accuracy with which we know the titrant’s concentration.
See Chapter 9 for a thorough discussion of titrimetric methods of analysis.
Primary and Secondary Standards
There are two categories of analytical standards: primary standards and secondary standards. A primary standard is a reagent that we can use to dispense an accurately known amount of analyte. For example, a 0.1250-g sample of K2Cr2O7 contains $4.249 \times 10^{-4}$ moles of K2Cr2O7. If we place this sample in a 250-mL volumetric flask and dilute to volume, the concentration of K2Cr2O7 in the resulting solution is $1.700 \times 10^{-3} \text{ M}$. A primary standard must have a known stoichiometry, a known purity (or assay), and it must be stable during long-term storage. Because it is difficult to establishing accurately the degree of hydration, even after drying, a hydrated reagent usually is not a primary standard.
Reagents that do not meet these criteria are secondary standards. The concentration of a secondary standard is determined relative to a primary standard. Lists of acceptable primary standards are available (see, for instance, Smith, B. W.; Parsons, M. L. J. Chem. Educ. 1973, 50, 679–681; or Moody, J. R.; Green- burg, P. R.; Pratt, K. W.; Rains, T. C. Anal. Chem. 1988, 60, 1203A–1218A). Appendix 8 provides examples of some common primary standards.
NaOH is one example of a secondary standard. Commercially available NaOH contains impurities of NaCl, Na2CO3, and Na2SO4, and readily absorbs H2O from the atmosphere. To determine the concentration of NaOH in a solution, we titrate it against a primary standard weak acid, such as potassium hydrogen phthalate, KHC8H4O4.
Other Reagents
Preparing a standard often requires additional reagents that are not primary standards or secondary standards, such as a suitable solvent or reagents needed to adjust the standard’s matrix. These solvents and reagents are potential sources of additional analyte, which, if not accounted for, produce a determinate error in the standardization. If available, reagent grade chemicals that conform to standards set by the American Chemical Society are used [Committee on Analytical Reagents, Reagent Chemicals, 8th ed., American Chemical Society: Washington, D. C., 1993]. The label on the bottle of a reagent grade chemical (Figure 5.1.1 ) lists either the limits for specific impurities or provides an assay for the impurities. We can improve the quality of a reagent grade chemical by purifying it, or by conducting a more accurate assay. As discussed later in the chapter, we can correct for contributions to Stotal from reagents used in an analysis by including an appropriate blank determination in the analytical procedure.
Preparing a Standard Solution
It often is necessary to prepare a series of standards, each with a different concentration of analyte. We can prepare these standards in two ways. If the range of concentrations is limited to one or two orders of magnitude, then each solution is best prepared by transferring a known mass or volume of the pure standard to a volumetric flask and diluting to volume.
When working with a larger range of concentrations, particularly a range that extends over more than three orders of magnitude, standards are best prepared by a serial dilution from a single stock solution. In a serial dilution we prepare the most concentrated standard and then dilute a portion of that solution to prepare the next most concentrated standard. Next, we dilute a portion of the second standard to prepare a third standard, continuing this process until we have prepared all of our standards. Serial dilutions must be prepared with extra care because an error in preparing one standard is passed on to all succeeding standards.
5.02: Calibrating the Signal
The accuracy with which we determine kA and Sreag depends on how accurately we can measure the signal, Stotal. We measure signals using equipment, such as glassware and balances, and instrumentation, such as spectrophotometers and pH meters. To minimize determinate errors that might affect the signal, we first calibrate our equipment and instrumentation by measuring Stotal for a standard with a known response of Sstd, adjusting Stotal until
Stotal = Sstd
Here are two examples of how we calibrate signals; other examples are provided in later chapters that focus on specific analytical methods.
When the signal is a measurement of mass, we determine Stotal using an analytical balance. To calibrate the balance’s signal we use a reference weight that meets standards established by a governing agency, such as the National Institute for Standards and Technology or the American Society for Testing and Materials. An electronic balance often includes an internal calibration weight for routine calibrations, as well as programs for calibrating with external weights. In either case, the balance automatically adjusts Stotal to match Sstd.
See Chapter 2.4 to review how an electronic balance works. Calibrating a balance is important, but it does not eliminate all sources of determinate error when measuring mass. See Appendix 9 for a discussion of correcting for the buoyancy of air.
We also must calibrate our instruments. For example, we can evaluate a spectrophotometer’s accuracy by measuring the absorbance of a carefully prepared solution of 60.06 mg/L K2Cr2O7 in 0.0050 M H2SO4, using 0.0050 M H2SO4 as a reagent blank [Ebel, S. Fresenius J. Anal. Chem. 1992, 342, 769]. An absorbance of $0.640 \pm 0.010$ absorbance units at a wavelength of 350.0 nm indicates that the spectrometer’s signal is calibrated properly.
Be sure to read and follow carefully the calibration instructions provided with any instrument you use. | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/05%3A_Standardizing_Analytical_Methods/5.01%3A_Analytical_Signals.txt |
To standardize an analytical method we also must determine the analyte’s sensitivity, kA, in Equation 5.3.1 or Equation 5.3.2 .
$S_{total} = k_A n_A + S_{reag} \label{5.1}$
$S_{total} = k_A C_A + S_{reag} \label{5.2}$
In principle, it is possible to derive the value of kA for any analytical method if we understand fully all the chemical reactions and physical processes responsible for the signal. Unfortunately, such calculations are not feasible if we lack a sufficiently developed theoretical model of the physical processes or if the chemical reaction’s evince non-ideal behavior. In such situations we must determine the value of kA by analyzing one or more standard solutions, each of which contains a known amount of analyte. In this section we consider several approaches for determining the value of kA. For simplicity we assume that Sreag is accounted for by a proper reagent blank, allowing us to replace Stotal in Equation \ref{5.1} and Equation \ref{5.2} with the analyte’s signal, SA.
$S_A = k_A n_A \label{5.3}$
$S_A = k_A C_A \label{5.4}$
Equation \ref{5.3} and Equation \ref{5.4} essentially are identical, differing only in whether we choose to express the amount of analyte in moles or as a concentration. For the remainder of this chapter we will limit our treatment to Equation \ref{5.4}. You can extend this treatment to Equation \ref{5.3} by replacing CA with nA.
Single-Point versus Multiple-Point Standardizations
The simplest way to determine the value of kA in Equation \ref{5.4} is to use a single-point standardization in which we measure the signal for a standard, Sstd, that contains a known concentration of analyte, Cstd. Substituting these values into Equation \ref{5.4}
$k_A = \frac {S_{std}} {C_{std}} \label{5.5}$
gives us the value for kA. Having determined kA, we can calculate the concentration of analyte in a sample by measuring its signal, Ssamp, and calculating CA using Equation \ref{5.6}.
$C_A = \frac {S_{samp}} {k_A} \label{5.6}$
A single-point standardization is the least desirable method for standardizing a method. There are two reasons for this. First, any error in our determination of kA carries over into our calculation of CA. Second, our experimental value for kA is based on a single concentration of analyte. To extend this value of kA to other concentrations of analyte requires that we assume a linear relationship between the signal and the analyte’s concentration, an assumption that often is not true [Cardone, M. J.; Palmero, P. J.; Sybrandt, L. B. Anal. Chem. 1980, 52, 1187–1191]. Figure 5.3.1 shows how assuming a constant value of kA leads to a determinate error in CA if kA becomes smaller at higher concentrations of analyte. Despite these limitations, single-point standardizations find routine use when the expected range for the analyte’s concentrations is small. Under these conditions it often is safe to assume that kA is constant (although you should verify this assumption experimentally). This is the case, for example, in clinical labs where many automated analyzers use only a single standard.
The better way to standardize a method is to prepare a series of standards, each of which contains a different concentration of analyte. Standards are chosen such that they bracket the expected range for the analyte’s concentration. A multiple-point standardization should include at least three standards, although more are preferable. A plot of Sstd versus Cstd is called a calibration curve. The exact standardization, or calibration relationship, is determined by an appropriate curve-fitting algorithm.
Linear regression, which also is known as the method of least squares, is one such algorithm. Its use is covered in Section 5.4.
There are two advantages to a multiple-point standardization. First, although a determinate error in one standard introduces a determinate error, its effect is minimized by the remaining standards. Second, because we measure the signal for several concentrations of analyte, we no longer must assume kA is independent of the analyte’s concentration. Instead, we can construct a calibration curve similar to the “actual relationship” in Figure 5.3.1 .
External Standards
The most common method of standardization uses one or more external standards, each of which contains a known concentration of analyte. We call these standards “external” because they are prepared and analyzed separate from the samples.
Appending the adjective “external” to the noun “standard” might strike you as odd at this point, as it seems reasonable to assume that standards and samples are analyzed separately. As we will soon learn, however, we can add standards to our samples and analyze both simultaneously.
Single External Standard
With a single external standard we determine kA using EEquation \ref{5.5} and then calculate the concentration of analyte, CA, using Equation \ref{5.6}.
Example 5.3.1
A spectrophotometric method for the quantitative analysis of Pb2+ in blood yields an Sstd of 0.474 for a single standard for which the concentration of lead is 1.75 ppb. What is the concentration of Pb2+ in a sample of blood for which Ssamp is 0.361?
Solution
Equation \ref{5.5} allows us to calculate the value of kA using the data for the single external standard.
$k_A = \frac {S_{std}} {C_{std}} = \frac {0.474} {1.75 \text{ ppb}} = 0.2709 \text{ ppb}^{-1} \nonumber$
Having determined the value of kA, we calculate the concentration of Pb2+ in the sample of blood is calculated using Equation \ref{5.6}.
$C_A = \frac {S_{samp}} {k_A} = \frac {0.361} {0.2709 \text{ ppb}^{-1}} = 1.33 \text{ ppb} \nonumber$
Multiple External Standards
Figure 5.3.2 shows a typical multiple-point external standardization. The volumetric flask on the left contains a reagent blank and the remaining volumetric flasks contain increasing concentrations of Cu2+. Shown below the volumetric flasks is the resulting calibration curve. Because this is the most common method of standardization, the resulting relationship is called a normal calibration curve.
When a calibration curve is a straight-line, as it is in Figure 5.3.2 , the slope of the line gives the value of kA. This is the most desirable situation because the method’s sensitivity remains constant throughout the analyte’s concentration range. When the calibration curve is not a straight-line, the method’s sensitivity is a function of the analyte’s concentration. In Figure 5.3.1 , for example, the value of kA is greatest when the analyte’s concentration is small and it decreases continuously for higher concentrations of analyte. The value of kA at any point along the calibration curve in Figure 5.3.1 is the slope at that point. In either case, a calibration curve allows to relate Ssamp to the analyte’s concentration.
Example 5.3.2
A second spectrophotometric method for the quantitative analysis of Pb2+ in blood has a normal calibration curve for which
$S_{std} = (0.296 \text{ ppb}^{-1} \times C_{std}) + 0.003 \nonumber$
What is the concentration of Pb2+ in a sample of blood if Ssamp is 0.397?
Solution
To determine the concentration of Pb2+ in the sample of blood, we replace Sstd in the calibration equation with Ssamp and solve for CA.
$C_A = \frac {S_{samp} - 0.003} {0.296 \text{ ppb}^{-1}} = \frac {0.397 - 0.003} {0.296 \text{ ppb}^{-1}} = 1.33 \text{ ppb} \nonumber$
It is worth noting that the calibration equation in this problem includes an extra term that does not appear in Equation \ref{5.6}. Ideally we expect our calibration curve to have a signal of zero when CA is zero. This is the purpose of using a reagent blank to correct the measured signal. The extra term of +0.003 in our calibration equation results from the uncertainty in measuring the signal for the reagent blank and the standards.
Exercise 5.3.1
Figure 5.3.2 shows a normal calibration curve for the quantitative analysis of Cu2+. The equation for the calibration curve is
$S_{std} = 29.59 \text{ M}^{-1} \times C_{std} + 0.015 \nonumber$
What is the concentration of Cu2+ in a sample whose absorbance, Ssamp, is 0.114? Compare your answer to a one-point standardization where a standard of $3.16 \times 10^{-3} \text{ M}$ Cu2+ gives a signal of 0.0931.
Answer
Substituting the sample’s absorbance into the calibration equation and solving for CA give
$S_{samp} = 0.114 = 29.59 \text{ M}^{-1} \times C_{A} + 0.015 \nonumber$
$C_A = 3.35 \times 10^{-3} \text{ M} \nonumber$
For the one-point standardization, we first solve for kA
$k_A = \frac {S_{std}} {C_{std}} = \frac {0.0931} {3.16 \times 10^{-3} \text{ M}} = 29.46 \text{ M}^{-1} \nonumber$
and then use this value of kA to solve for CA.
$C_A = \frac {S_{samp}} {k_A} = \frac {0.114} {29.46 \text{ M}^{-1}} = 3.87 \times 10^{-3} \text{ M} \nonumber$
When using multiple standards, the indeterminate errors that affect the signal for one standard are partially compensated for by the indeterminate errors that affect the other standards. The standard selected for the one-point standardization has a signal that is smaller than that predicted by the regression equation, which underestimates kA and overestimates CA.
An external standardization allows us to analyze a series of samples using a single calibration curve. This is an important advantage when we have many samples to analyze. Not surprisingly, many of the most common quantitative analytical methods use an external standardization.
There is a serious limitation, however, to an external standardization. When we determine the value of kA using Equation \ref{5.5}, the analyte is present in the external standard’s matrix, which usually is a much simpler matrix than that of our samples. When we use an external standardization we assume the matrix does not affect the value of kA. If this is not true, then we introduce a proportional determinate error into our analysis. This is not the case in Figure 5.3.3 , for instance, where we show calibration curves for an analyte in the sample’s matrix and in the standard’s matrix. In this case, using the calibration curve for the external standards leads to a negative determinate error in analyte’s reported concentration. If we expect that matrix effects are important, then we try to match the standard’s matrix to that of the sample, a process known as matrix matching. If we are unsure of the sample’s matrix, then we must show that matrix effects are negligible or use an alternative method of standardization. Both approaches are discussed in the following section.
The matrix for the external standards in Figure 5.3.2 , for example, is dilute ammonia. Because the $\ce{Cu(NH3)4^{2+}}$ complex absorbs more strongly than Cu2+, adding ammonia increases the signal’s magnitude. If we fail to add the same amount of ammonia to our samples, then we will introduce a proportional determinate error into our analysis.
Standard Additions
We can avoid the complication of matching the matrix of the standards to the matrix of the sample if we carry out the standardization in the sample. This is known as the method of standard additions.
Single Standard Addition
The simplest version of a standard addition is shown in Figure 5.3.4 . First we add a portion of the sample, Vo, to a volumetric flask, dilute it to volume, Vf, and measure its signal, Ssamp. Next, we add a second identical portion of sample to an equivalent volumetric flask along with a spike, Vstd, of an external standard whose concentration is Cstd. After we dilute the spiked sample to the same final volume, we measure its signal, Sspike.
The following two equations relate Ssamp and Sspike to the concentration of analyte, CA, in the original sample.
$S_{samp} = k_A C_A \frac {V_o} {V_f} \label{5.7}$
$S_{spike} = k_A \left( C_A \frac {V_o} {V_f} + C_{std} \frac {V_{std}} {V_f} \right) \label{5.8}$
As long as Vstd is small relative to Vo, the effect of the standard’s matrix on the sample’s matrix is insignificant. Under these conditions the value of kA is the same in Equation \ref{5.7} and Equation \ref{5.8}. Solving both equations for kA and equating gives
$\frac {S_{samp}} {C_A \frac {V_o} {V_f}} = \frac {S_{spike}} {C_A \frac {V_o} {V_f} + C_{std} \frac {V_{std}} {V_f}} \label{5.9}$
which we can solve for the concentration of analyte, CA, in the original sample.
Example 5.3.3
A third spectrophotometric method for the quantitative analysis of Pb2+ in blood yields an Ssamp of 0.193 when a 1.00 mL sample of blood is diluted to 5.00 mL. A second 1.00 mL sample of blood is spiked with 1.00 mL of a 1560-ppb Pb2+ external standard and diluted to 5.00 mL, yielding an Sspike of 0.419. What is the concentration of Pb2+ in the original sample of blood?
Solution
We begin by making appropriate substitutions into Equation \ref{5.9} and solving for CA. Note that all volumes must be in the same units; thus, we first convert Vstd from 1.00 mL to $1.00 \times 10^{-3} \text{ mL}$.
$\frac {0.193} {C_A \frac {1.00 \text{ mL}} {5.00 \text{ mL}}} = \frac {0.419} {C_A \frac {1.00 \text{ mL}} {5.00 \text{ mL}} + 1560 \text{ ppb} \frac {1.00 \times 10^{-3} \text{ mL}} {5.00 \text{ mL}}} \nonumber$
$\frac {0.193} {0.200C_A} = \frac {0.419} {0.200C_A + 0.3120 \text{ ppb}} \nonumber$
$0.0386C_A + 0.0602 \text{ ppb} = 0.0838 C_A \nonumber$
$0.0452 C_A = 0.0602 \text{ ppb} \nonumber$
$C_A = 1.33 \text{ ppb} \nonumber$
The concentration of Pb2+ in the original sample of blood is 1.33 ppb.
It also is possible to add the standard addition directly to the sample, measuring the signal both before and after the spike (Figure 5.3.5 ). In this case the final volume after the standard addition is Vo + Vstd and Equation \ref{5.7}, Equation \ref{5.8}, and Equation \ref{5.9} become
$S_{samp} = k_A C_A \nonumber$
$S_{spike} = k_A \left( C_A \frac {V_o} {V_o + V_{std}} + C_{std} \frac {V_{std}} {V_o + V_{std}} \right) \label{5.10}$
$\frac {S_{samp}} {C_A} = \frac {S_{spike}} {C_A \frac {V_o} {V_o + V_{std}} + C_{std} \frac {V_{std}} {V_o + V_{std}}} \label{5.11}$
Example 5.3.4
A fourth spectrophotometric method for the quantitative analysis of Pb2+ in blood yields an Ssamp of 0.712 for a 5.00 mL sample of blood. After spiking the blood sample with 5.00 mL of a 1560-ppb Pb2+ external standard, an Sspike of 1.546 is measured. What is the concentration of Pb2+ in the original sample of blood?
Solution
$\frac {0.712} {C_A} = \frac {1.546} {C_A \frac {5.00 \text{ mL}} {5.005 \text{ mL}} + 1560 \text{ ppb} \frac {5.00 \times 10^{-3} \text{ mL}} {5.005 \text{ mL}}} \nonumber$
$\frac {0.712} {C_A} = \frac {1.546} {0.9990C_A + 1.558 \text{ ppb}} \nonumber$
$0.7113C_A + 1.109 \text{ ppb} = 1.546C_A \nonumber$
$C_A = 1.33 \text{ ppb} \nonumber$
The concentration of Pb2+ in the original sample of blood is 1.33 ppb.
Multiple Standard Additions
We can adapt a single-point standard addition into a multiple-point standard addition by preparing a series of samples that contain increasing amounts of the external standard. Figure 5.3.6 shows two ways to plot a standard addition calibration curve based on Equation \ref{5.8}. In Figure 5.3.6 a we plot Sspike against the volume of the spikes, Vstd. If kA is constant, then the calibration curve is a straight-line. It is easy to show that the x-intercept is equivalent to –CAVo/Cstd.
Example 5.3.5
Beginning with Equation \ref{5.8} show that the equations in Figure 5.3.6 a for the slope, the y-intercept, and the x-intercept are correct.
Solution
We begin by rewriting Equation \ref{5.8} as
$S_{spike} = \frac {k_A C_A V_o} {V_f} + \frac {k_A C_{std}} {V_f} \times V_{std} \nonumber$
which is in the form of the equation for a straight-line
$y = y\text{-intercept} + \text{slope} \times x\text{-intercept} \nonumber$
where y is Sspike and x is Vstd. The slope of the line, therefore, is kACstd/Vf and the y-intercept is kACAVo/Vf. The x-intercept is the value of x when y is zero, or
$0 = \frac {k_A C_A V_o} {V_f} + \frac {k_A C_{std}} {V_f} \times x\text{-intercept} \nonumber$
$x\text{-intercept} = - \frac {k_A C_A V_o / V_f} {K_A C_{std} / V_f} = - \frac {C_A V_o} {C_{std}} \nonumber$
Exercise 5.3.2
Beginning with Equation \ref{5.8} show that the Equations in Figure 5.3.6 b for the slope, the y-intercept, and the x-intercept are correct.
Answer
We begin with Equation \ref{5.8}
$S_{spike} = k_A \left( C_A \frac {V_o} {V_f} + C_{std} \frac {V_{std}} {V_f} \right) \nonumber$
rewriting it as
$S_{spike} = \frac {k_A C_A V_o} {V_f} + k_A \left( C_{std} \frac {V_{std}} {V_f} \right) \nonumber$
which is in the form of the linear equation
$y = y\text{-intercept} + \text{slope} \times x\text{-intercept} \nonumber$
where y is Sspike and x is Cstd $\times$ Vstd/Vf. The slope of the line, therefore, is kA, and the y-intercept is kACAVo/Vf. The x-intercept is the value of x when y is zero, or
$x\text{-intercept} = - \frac {k_A C_A V_o/V_F} {k_A} = - \frac {C_A V_o} {V_f} \nonumber$
Because we know the volume of the original sample, Vo, and the concentration of the external standard, Cstd, we can calculate the analyte’s concentrations from the x-intercept of a multiple-point standard additions.
Example 5.3.6
A fifth spectrophotometric method for the quantitative analysis of Pb2+ in blood uses a multiple-point standard addition based on Equation \ref{5.8}. The original blood sample has a volume of 1.00 mL and the standard used for spiking the sample has a concentration of 1560 ppb Pb2+. All samples were diluted to 5.00 mL before measuring the signal. A calibration curve of Sspike versus Vstd has the following equation
$S_{spike} = 0.266 + 312 \text{ mL}^{-1} \times V_{std} \nonumber$
What is the concentration of Pb2+ in the original sample of blood?
Solution
To find the x-intercept we set Sspike equal to zero.
$S_{spike} = 0.266 + 312 \text{ mL}^{-1} \times V_{std} \nonumber$
Solving for Vstd, we obtain a value of $-8.526 \times 10^{-4} \text{ mL}$ for the x-intercept. Substituting the x-intercept’s value into the equation from Figure 5.3.6 a
$-8.526 \times 10^{-4} \text{ mL} = - \frac {C_A V_o} {C_{std}} = - \frac {C_A \times 1.00 \text{ mL}} {1560 \text{ ppb}} \nonumber$
and solving for CA gives the concentration of Pb2+ in the blood sample as 1.33 ppb.
Exercise 5.3.3
Figure 5.3.6 shows a standard additions calibration curve for the quantitative analysis of Mn2+. Each solution contains 25.00 mL of the original sample and either 0, 1.00, 2.00, 3.00, 4.00, or 5.00 mL of a 100.6 mg/L external standard of Mn2+. All standard addition samples were diluted to 50.00 mL with water before reading the absorbance. The equation for the calibration curve in Figure 5.3.6 a is
$S_{std} = 0.0854 \times V_{std} + 0.1478 \nonumber$
What is the concentration of Mn2+ in this sample? Compare your answer to the data in Figure 5.3.6 b, for which the calibration curve is
$S_{std} = 0.425 \times C_{std}(V_{std}/V_f) + 0.1478 \nonumber$
Answer
Using the calibration equation from Figure 5.3.6 a, we find that the x-intercept is
$x\text{-intercept} = - \frac {0.1478} {0.0854 \text{ mL}^{-1}} = - 1.731 \text{ mL} \nonumber$
If we plug this result into the equation for the x-intercept and solve for CA, we find that the concentration of Mn2+ is
$C_A = - \frac {x\text{-intercept} \times C_{std}} {V_o} = - \frac {-1.731 \text{ mL} \times 100.6 \text{ mg/L}} {25.00 \text{ mL}} = 6.96 \text{ mg/L} \nonumber$
For Figure 5.3.6 b, the x-intercept is
$x\text{-intercept} = - \frac {0.1478} {0.0425 \text{ mL/mg}} = - 3.478 \text{ mg/mL} \nonumber$
and the concentration of Mn2+ is
$C_A = - \frac {x\text{-intercept} \times V_f} {V_o} = - \frac {-3.478 \text{ mg/mL} \times 50.00 \text{ mL}} {25.00 \text{ mL}} = 6.96 \text{ mg/L} \nonumber$
Since we construct a standard additions calibration curve in the sample, we can not use the calibration equation for other samples. Each sample, therefore, requires its own standard additions calibration curve. This is a serious drawback if you have many samples. For example, suppose you need to analyze 10 samples using a five-point calibration curve. For a normal calibration curve you need to analyze only 15 solutions (five standards and ten samples). If you use the method of standard additions, however, you must analyze 50 solutions (each of the ten samples is analyzed five times, once before spiking and after each of four spikes).
Using a Standard Addition to Identify Matrix Effects
We can use the method of standard additions to validate an external standardization when matrix matching is not feasible. First, we prepare a normal calibration curve of Sstd versus Cstd and determine the value of kA from its slope. Next, we prepare a standard additions calibration curve using Equation \ref{5.8}, plotting the data as shown in Figure 5.3.6 b. The slope of this standard additions calibration curve provides an independent determination of kA. If there is no significant difference between the two values of kA, then we can ignore the difference between the sample’s matrix and that of the external standards. When the values of kA are significantly different, then using a normal calibration curve introduces a proportional determinate error.
Internal Standards
To use an external standardization or the method of standard additions, we must be able to treat identically all samples and standards. When this is not possible, the accuracy and precision of our standardization may suffer. For example, if our analyte is in a volatile solvent, then its concentration will increase if we lose solvent to evaporation. Suppose we have a sample and a standard with identical concentrations of analyte and identical signals. If both experience the same proportional loss of solvent, then their respective concentrations of analyte and signals remain identical. In effect, we can ignore evaporation if the samples and the standards experience an equivalent loss of solvent. If an identical standard and sample lose different amounts of solvent, however, then their respective concentrations and signals are no longer equal. In this case a simple external standardization or standard addition is not possible.
We can still complete a standardization if we reference the analyte’s signal to a signal from another species that we add to all samples and standards. The species, which we call an internal standard, must be different than the analyte.
Because the analyte and the internal standard receive the same treatment, the ratio of their signals is unaffected by any lack of reproducibility in the procedure. If a solution contains an analyte of concentration CA and an internal standard of concentration CIS, then the signals due to the analyte, SA, and the internal standard, SIS, are
$S_A = k_A C_A \nonumber$
$S_{IS} = k_{SI} C_{IS} \nonumber$
where $k_A$ and $k_{IS}$ are the sensitivities for the analyte and the internal standard, respectively. Taking the ratio of the two signals gives the fundamental equation for an internal standardization.
$\frac {S_A} {S_{IS}} = \frac {k_A C_A} {k_{IS} C_{IS}} = K \times \frac {C_A} {C_{IS}} \label{5.12}$
Because K is a ratio of the analyte’s sensitivity and the internal standard’s sensitivity, it is not necessary to determine independently values for either kA or kIS.
Single Internal Standard
In a single-point internal standardization, we prepare a single standard that contains the analyte and the internal standard, and use it to determine the value of K in Equation \ref{5.12}.
$K = \left( \frac {C_{IS}} {C_A} \right)_{std} \times \left( \frac {S_A} {S_{IS}} \right)_{std} \label{5.13}$
Having standardized the method, the analyte’s concentration is given by
$C_A = \frac {C_{IS}} {K} \times \left( \frac {S_A} {S_{IS}} \right)_{samp} \nonumber$
Example 5.3.7
A sixth spectrophotometric method for the quantitative analysis of Pb2+ in blood uses Cu2+ as an internal standard. A standard that is 1.75 ppb Pb2+ and 2.25 ppb Cu2+ yields a ratio of (SA/SIS)std of 2.37. A sample of blood spiked with the same concentration of Cu2+ gives a signal ratio, (SA/SIS)samp, of 1.80. What is the concentration of Pb2+ in the sample of blood?
Solution
Equation \ref{5.13} allows us to calculate the value of K using the data for the standard
$K = \left( \frac {C_{IS}} {C_A} \right)_{std} \times \left( \frac {S_A} {S_{IS}} \right)_{std} = \frac {2.25 \text{ ppb } \ce{Cu^{2+}}} {1.75 \text{ ppb } \ce{Pb^{2+}}} \times 2.37 = 3.05 \frac {\text{ppb } \ce{Cu^{2+}}} {\text{ppb } \ce{Pb^{2+}}} \nonumber$
The concentration of Pb2+, therefore, is
$C_A = \frac {C_{IS}} {K} \times \left( \frac {S_A} {S_{IS}} \right)_{samp} = \frac {2.25 \text{ ppb } \ce{Cu^{2+}}} {3.05 \frac {\text{ppb } \ce{Cu^{2+}}} {\text{ppb } \ce{Pb^{2+}}}} \times 1.80 = 1.33 \text{ ppb } \ce{Pb^{2+}} \nonumber$
Multiple Internal Standards
A single-point internal standardization has the same limitations as a single-point normal calibration. To construct an internal standard calibration curve we prepare a series of standards, each of which contains the same concentration of internal standard and a different concentrations of analyte. Under these conditions a calibration curve of (SA/SIS)std versus CA is linear with a slope of K/CIS.
Although the usual practice is to prepare the standards so that each contains an identical amount of the internal standard, this is not a requirement.
Example 5.3.8
A seventh spectrophotometric method for the quantitative analysis of Pb2+ in blood gives a linear internal standards calibration curve for which
$\left( \frac {S_A} {S_{IS}} \right)_{std} = (2.11 \text{ ppb}^{-1} \times C_A) - 0.006 \nonumber$
What is the ppb Pb2+ in a sample of blood if (SA/SIS)samp is 2.80?
Solution
To determine the concentration of Pb2+ in the sample of blood we replace (SA/SIS)std in the calibration equation with (SA/SIS)samp and solve for CA.
$C_A = \frac {\left( \frac {S_A} {S_{IS}} \right)_{samp} + 0.006} {2.11 \text{ ppb}^{-1}} = \frac {2.80 + 0.006} {2.11 \text{ ppb}^{-1}} = 1.33 \text{ ppb } \ce{Pb^{2+}} \nonumber$
The concentration of Pb2+ in the sample of blood is 1.33 ppb.
In some circumstances it is not possible to prepare the standards so that each contains the same concentration of internal standard. This is the case, for example, when we prepare samples by mass instead of volume. We can still prepare a calibration curve, however, by plotting $(S_A / S_{IS})_{std}$ versus CA/CIS, giving a linear calibration curve with a slope of K.
You might wonder if it is possible to include an internal standard in the method of standard additions to correct for both matrix effects and uncontrolled variations between samples; well, the answer is yes as described in the paper “Standard Dilution Analysis,” the full reference for which is Jones, W. B.; Donati, G. L.; Calloway, C. P.; Jones, B. T. Anal. Chem. 2015, 87, 2321-2327. | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/05%3A_Standardizing_Analytical_Methods/5.03%3A_Determining_the_Sensitivity.txt |
In a single-point external standardization we determine the value of kA by measuring the signal for a single standard that contains a known concentration of analyte. Using this value of kA and our sample’s signal, we then calculate the concentration of analyte in our sample (see Example 5.3.1). With only a single determination of kA, a quantitative analysis using a single-point external standardization is straightforward.
A multiple-point standardization presents a more difficult problem. Consider the data in Table 5.4.1 for a multiple-point external standardization. What is our best estimate of the relationship between Sstd and Cstd? It is tempting to treat this data as five separate single-point standardizations, determining kA for each standard, and reporting the mean value for the five trials. Despite it simplicity, this is not an appropriate way to treat a multiple-point standardization.
Table 5.4.1 : Data for a Hypothetical Multiple-Point External Standardization
$C_{std}$ (arbitrary units) $S_{std}$ (arbitrary units) $k_A = S_{std}/C_{std}$
0.000 0.00
0.100 12.36 123.6
0.200 24.83 124.2
0.300 35.91 119.7
0.400 48.79 122.0
0.500 60.42 122.8
mean kA = 122.5
So why is it inappropriate to calculate an average value for kA using the data in Table 5.4.1 ? In a single-point standardization we assume that the reagent blank (the first row in Table 5.4.1 ) corrects for all constant sources of determinate error. If this is not the case, then the value of kA from a single-point standardization has a constant determinate error. Table 5.4.2 demonstrates how an uncorrected constant error affects our determination of kA. The first three columns show the concentration of analyte in a set of standards, Cstd, the signal without any source of constant error, Sstd, and the actual value of kA for five standards. As we expect, the value of kA is the same for each standard. In the fourth column we add a constant determinate error of +0.50 to the signals, (Sstd)e. The last column contains the corresponding apparent values of kA. Note that we obtain a different value of kA for each standard and that each apparent kA is greater than the true value.
Table 5.4.2 : Effect of a Constant Determinate Error on the Value of $k_A$ From a Single-Point Standardization
$C_{std}$
$S_{std}$
(without constant error)
$k_A = S_{std}/C_{std}$
(actual)
$(S_{std})_e$
(with constant error)
$k_A = (S_{std})_e/C_{std}$
(apparent)
1.00 1.00 1.00 1.50 1.50
2.00 2.00 1.00 2.50 1.25
3.00 3.00 1.00 3.50 1.17
4.00 4.00 1.00 4.50 1.13
5.00 5.00 1.00 5.50 1.10
mean kA (true) = 1.00 mean kA (apparent) = 1.23
How do we find the best estimate for the relationship between the signal and the concentration of analyte in a multiple-point standardization? Figure 5.4.1 shows the data in Table 5.4.1 plotted as a normal calibration curve. Although the data certainly appear to fall along a straight line, the actual calibration curve is not intuitively obvious. The process of determining the best equation for the calibration curve is called linear regression.
Linear Regression of Straight Line Calibration Curves
When a calibration curve is a straight-line, we represent it using the following mathematical equation
$y = \beta_0 + \beta_1 x \label{5.1}$
where y is the analyte’s signal, Sstd, and x is the analyte’s concentration, Cstd. The constants $\beta_0$ and $\beta_1$ are, respectively, the calibration curve’s expected y-intercept and its expected slope. Because of uncertainty in our measurements, the best we can do is to estimate values for $\beta_0$ and $\beta_1$, which we represent as b0 and b1. The goal of a linear regression analysis is to determine the best estimates for b0 and b1. How we do this depends on the uncertainty in our measurements.
Unweighted Linear Regression with Errors in y
The most common method for completing the linear regression for Equation \ref{5.1} makes three assumptions:
1. that the difference between our experimental data and the calculated regression line is the result of indeterminate errors that affect y
2. that indeterminate errors that affect y are normally distributed
3. that the indeterminate errors in y are independent of the value of x
Because we assume that the indeterminate errors are the same for all standards, each standard contributes equally in our estimate of the slope and the y-intercept. For this reason the result is considered an unweighted linear regression.
The second assumption generally is true because of the central limit theorem, which we considered in Chapter 4. The validity of the two remaining assumptions is less obvious and you should evaluate them before you accept the results of a linear regression. In particular the first assumption always is suspect because there certainly is some indeterminate error in the measurement of x. When we prepare a calibration curve, however, it is not unusual to find that the uncertainty in the signal, Sstd, is significantly larger than the uncertainty in the analyte’s concentration, Cstd. In such circumstances the first assumption is usually reasonable.
How a Linear Regression Works
To understand the logic of a linear regression consider the example shown in Figure 5.4.2 , which shows three data points and two possible straight-lines that might reasonably explain the data. How do we decide how well these straight-lines fit the data, and how do we determine the best straight-line?
Let’s focus on the solid line in Figure 5.4.2 . The equation for this line is
$\hat{y} = b_0 + b_1 x \label{5.2}$
where b0 and b1 are estimates for the y-intercept and the slope, and $\hat{y}$ is the predicted value of y for any value of x. Because we assume that all uncertainty is the result of indeterminate errors in y, the difference between y and $\hat{y}$ for each value of x is the residual error, r, in our mathematical model.
$r_i = (y_i - \hat{y}_i) \nonumber$
Figure 5.4.3 shows the residual errors for the three data points. The smaller the total residual error, R, which we define as
$R = \sum_{i = 1}^{n} (y_i - \hat{y}_i)^2 \label{5.3}$
the better the fit between the straight-line and the data. In a linear regression analysis, we seek values of b0 and b1 that give the smallest total residual error.
The reason for squaring the individual residual errors is to prevent a positive residual error from canceling out a negative residual error. You have seen this before in the equations for the sample and population standard deviations. You also can see from this equation why a linear regression is sometimes called the method of least squares.
Finding the Slope and y-Intercept
Although we will not formally develop the mathematical equations for a linear regression analysis, you can find the derivations in many standard statistical texts [ See, for example, Draper, N. R.; Smith, H. Applied Regression Analysis, 3rd ed.; Wiley: New York, 1998]. The resulting equation for the slope, b1, is
$b_1 = \frac {n \sum_{i = 1}^{n} x_i y_i - \sum_{i = 1}^{n} x_i \sum_{i = 1}^{n} y_i} {n \sum_{i = 1}^{n} x_i^2 - \left( \sum_{i = 1}^{n} x_i \right)^2} \label{5.4}$
and the equation for the y-intercept, b0, is
$b_0 = \frac {\sum_{i = 1}^{n} y_i - b_1 \sum_{i = 1}^{n} x_i} {n} \label{5.5}$
Although Equation \ref{5.4} and Equation \ref{5.5} appear formidable, it is necessary only to evaluate the following four summations
$\sum_{i = 1}^{n} x_i \quad \sum_{i = 1}^{n} y_i \quad \sum_{i = 1}^{n} x_i y_i \quad \sum_{i = 1}^{n} x_i^2 \nonumber$
Many calculators, spreadsheets, and other statistical software packages are capable of performing a linear regression analysis based on this model. To save time and to avoid tedious calculations, learn how to use one of these tools (and see Section 5.6 for details on completing a linear regression analysis using Excel and R.). For illustrative purposes the necessary calculations are shown in detail in the following example.
Equation \ref{5.4} and Equation \ref{5.5} are written in terms of the general variables x and y. As you work through this example, remember that x corresponds to Cstd, and that y corresponds to Sstd.
Example 5.4.1
Using the data from Table 5.4.1 , determine the relationship between Sstd and Cstd using an unweighted linear regression.
Solution
We begin by setting up a table to help us organize the calculation.
$x_i$ $y_i$ $x_i y_i$ $x_i^2$
0.000 0.00 0.000 0.000
0.100 12.36 1.236 0.010
0.200 24.83 4.966 0.040
0.300 35.91 10.773 0.090
0.400 48.79 19.516 0.160
0.500 60.42 30.210 0.250
Adding the values in each column gives
$\sum_{i = 1}^{n} x_i = 1.500 \quad \sum_{i = 1}^{n} y_i = 182.31 \quad \sum_{i = 1}^{n} x_i y_i = 66.701 \quad \sum_{i = 1}^{n} x_i^2 = 0.550 \nonumber$
Substituting these values into Equation \ref{5.4} and Equation \ref{5.5}, we find that the slope and the y-intercept are
$b_1 = \frac {(6 \times 66.701) - (1.500 \times 182.31)} {(6 \times 0.550) - (1.500)^2} = 120.706 \approx 120.71 \nonumber$
$b_0 = \frac {182.31 - (120.706 \times 1.500)} {6} = 0.209 \approx 0.21 \nonumber$
The relationship between the signal and the analyte, therefore, is
$S_{std} = 120.71 \times C_{std} + 0.21 \nonumber$
For now we keep two decimal places to match the number of decimal places in the signal. The resulting calibration curve is shown in Figure 5.4.4 .
Uncertainty in the Regression Analysis
As shown in Figure 5.4.4 , because indeterminate errors in the signal, the regression line may not pass through the exact center of each data point. The cumulative deviation of our data from the regression line—that is, the total residual error—is proportional to the uncertainty in the regression. We call this uncertainty the standard deviation about the regression, sr, which is equal to
$s_r = \sqrt{\frac {\sum_{i = 1}^{n} \left( y_i - \hat{y}_i \right)^2} {n - 2}} \label{5.6}$
where yi is the ith experimental value, and $\hat{y}_i$ is the corresponding value predicted by the regression line in Equation \ref{5.2}. Note that the denominator of Equation \ref{5.6} indicates that our regression analysis has n – 2 degrees of freedom—we lose two degree of freedom because we use two parameters, the slope and the y-intercept, to calculate $\hat{y}_i$.
Did you notice the similarity between the standard deviation about the regression (Equation \ref{5.6}) and the standard deviation for a sample (Equation 4.1.1)?
A more useful representation of the uncertainty in our regression analysis is to consider the effect of indeterminate errors on the slope, b1, and the y-intercept, b0, which we express as standard deviations.
$s_{b_1} = \sqrt{\frac {n s_r^2} {n \sum_{i = 1}^{n} x_i^2 - \left( \sum_{i = 1}^{n} x_i \right)^2}} = \sqrt{\frac {s_r^2} {\sum_{i = 1}^{n} \left( x_i - \overline{x} \right)^2}} \label{5.7}$
$s_{b_0} = \sqrt{\frac {s_r^2 \sum_{i = 1}^{n} x_i^2} {n \sum_{i = 1}^{n} x_i^2 - \left( \sum_{i = 1}^{n} x_i \right)^2}} = \sqrt{\frac {s_r^2 \sum_{i = 1}^{n} x_i^2} {n \sum_{i = 1}^{n} \left( x_i - \overline{x} \right)^2}} \label{5.8}$
We use these standard deviations to establish confidence intervals for the expected slope, $\beta_1$, and the expected y-intercept, $\beta_0$
$\beta_1 = b_1 \pm t s_{b_1} \label{5.9}$
$\beta_0 = b_0 \pm t s_{b_0} \label{5.10}$
where we select t for a significance level of $\alpha$ and for n – 2 degrees of freedom. Note that Equation \ref{5.9} and Equation \ref{5.10} do not contain a factor of $(\sqrt{n})^{-1}$ because the confidence interval is based on a single regression line.
Example 5.4.2
Calculate the 95% confidence intervals for the slope and y-intercept from Example 5.4.1 .
Solution
We begin by calculating the standard deviation about the regression. To do this we must calculate the predicted signals, $\hat{y}_i$ , using the slope and y-intercept from Example 5.4.1 , and the squares of the residual error, $(y_i - \hat{y}_i)^2$. Using the last standard as an example, we find that the predicted signal is
$\hat{y}_6 = b_0 + b_1 x_6 = 0.209 + (120.706 \times 0.500) = 60.562 \nonumber$
and that the square of the residual error is
$(y_i - \hat{y}_i)^2 = (60.42 - 60.562)^2 = 0.2016 \approx 0.202 \nonumber$
The following table displays the results for all six solutions.
$x_i$ $y_i$ $\hat{y}_i$
$\left( y_i - \hat{y}_i \right)^2$
0.000 0.00 0.209 0.0437
0.100 12.36 12.280 0.0064
0.200 24.83 24.350 0.2304
0.300 35.91 36.421 0.2611
0.400 48.79 48.491 0.0894
0.500 60.42 60.562 0.0202
Adding together the data in the last column gives the numerator of Equation \ref{5.6} as 0.6512; thus, the standard deviation about the regression is
$s_r = \sqrt{\frac {0.6512} {6 - 2}} = 0.4035 \nonumber$
Next we calculate the standard deviations for the slope and the y-intercept using Equation \ref{5.7} and Equation \ref{5.8}. The values for the summation terms are from Example 5.4.1 .
$s_{b_1} = \sqrt{\frac {6 \times (0.4035)^2} {(6 \times 0.550) - (1.500)^2}} = 0.965 \nonumber$
$s_{b_0} = \sqrt{\frac {(0.4035)^2 \times 0.550} {(6 \times 0.550) - (1.500)^2}} = 0.292 \nonumber$
Finally, the 95% confidence intervals ($\alpha = 0.05$, 4 degrees of freedom) for the slope and y-intercept are
$\beta_1 = b_1 \pm ts_{b_1} = 120.706 \pm (2.78 \times 0.965) = 120.7 \pm 2.7 \nonumber$
$\beta_0 = b_0 \pm ts_{b_0} = 0.209 \pm (2.78 \times 0.292) = 0.2 \pm 0.80 \nonumber$
where t(0.05, 4) from Appendix 4 is 2.78. The standard deviation about the regression, sr, suggests that the signal, Sstd, is precise to one decimal place. For this reason we report the slope and the y-intercept to a single decimal place.
Minimizing Uncertainty in Calibration Model
To minimize the uncertainty in a calibration curve’s slope and y-intercept, we evenly space our standards over a wide range of analyte concentrations. A close examination of Equation \ref{5.7} and Equation \ref{5.8} help us appreciate why this is true. The denominators of both equations include the term $\sum_{i = 1}^{n} (x_i - \overline{x}_i)^2$. The larger the value of this term—which we accomplish by increasing the range of x around its mean value—the smaller the standard deviations in the slope and the y-intercept. Furthermore, to minimize the uncertainty in the y-intercept, it helps to decrease the value of the term $\sum_{i = 1}^{n} x_i$ in Equation \ref{5.8}, which we accomplish by including standards for lower concentrations of the analyte.
Obtaining the Analyte's Concentration From a Regression Equation
Once we have our regression equation, it is easy to determine the concentration of analyte in a sample. When we use a normal calibration curve, for example, we measure the signal for our sample, Ssamp, and calculate the analyte’s concentration, CA, using the regression equation.
$C_A = \frac {S_{samp} - b_0} {b_1} \label{5.11}$
What is less obvious is how to report a confidence interval for CA that expresses the uncertainty in our analysis. To calculate a confidence interval we need to know the standard deviation in the analyte’s concentration, $s_{C_A}$, which is given by the following equation
$s_{C_A} = \frac {s_r} {b_1} \sqrt{\frac {1} {m} + \frac {1} {n} + \frac {\left( \overline{S}_{samp} - \overline{S}_{std} \right)^2} {(b_1)^2 \sum_{i = 1}^{n} \left( C_{std_i} - \overline{C}_{std} \right)^2}} \label{5.12}$
where m is the number of replicate we use to establish the sample’s average signal, Ssamp, n is the number of calibration standards, Sstd is the average signal for the calibration standards, and $C_{std_1}$ and $\overline{C}_{std}$ are the individual and the mean concentrations for the calibration standards. Knowing the value of $s_{C_A}$, the confidence interval for the analyte’s concentration is
$\mu_{C_A} = C_A \pm t s_{C_A} \nonumber$
where $\mu_{C_A}$ is the expected value of CA in the absence of determinate errors, and with the value of t is based on the desired level of confidence and n – 2 degrees of freedom.
Equation \ref{5.12} is written in terms of a calibration experiment. A more general form of the equation, written in terms of x and y, is given here.
$s_{x} = \frac {s_r} {b_1} \sqrt{\frac {1} {m} + \frac {1} {n} + \frac {\left( \overline{Y} - \overline{y} \right)^2} {(b_1)^2 \sum_{i = 1}^{n} \left( x_i - \overline{x} \right)^2}} \nonumber$
A close examination of Equation \ref{5.12} should convince you that the uncertainty in CA is smallest when the sample’s average signal, $\overline{S}_{samp}$, is equal to the average signal for the standards, $\overline{S}_{std}$. When practical, you should plan your calibration curve so that Ssamp falls in the middle of the calibration curve. For more information about these regression equations see (a) Miller, J. N. Analyst 1991, 116, 3–14; (b) Sharaf, M. A.; Illman, D. L.; Kowalski, B. R. Chemometrics, Wiley-Interscience: New York, 1986, pp. 126-127; (c) Analytical Methods Committee “Uncertainties in concentrations estimated from calibration experiments,” AMC Technical Brief, March 2006.
Example 5.4.3
Three replicate analyses for a sample that contains an unknown concentration of analyte, yield values for Ssamp of 29.32, 29.16 and 29.51 (arbitrary units). Using the results from Example 5.4.1 and Example 5.4.2 , determine the analyte’s concentration, CA, and its 95% confidence interval.
Solution
The average signal, $\overline{S}_{samp}$, is 29.33, which, using Equation \ref{5.11} and the slope and the y-intercept from Example 5.4.1 , gives the analyte’s concentration as
$C_A = \frac {\overline{S}_{samp} - b_0} {b_1} = \frac {29.33 - 0.209} {120.706} = 0.241 \nonumber$
To calculate the standard deviation for the analyte’s concentration we must determine the values for $\overline{S}_{std}$ and for $\sum_{i = 1}^{2} (C_{std_i} - \overline{C}_{std})^2$. The former is just the average signal for the calibration standards, which, using the data in Table 5.4.1 , is 30.385. Calculating $\sum_{i = 1}^{2} (C_{std_i} - \overline{C}_{std})^2$ looks formidable, but we can simplify its calculation by recognizing that this sum-of-squares is the numerator in a standard deviation equation; thus,
$\sum_{i = 1}^{n} (C_{std_i} - \overline{C}_{std})^2 = (s_{C_{std}})^2 \times (n - 1) \nonumber$
where $s_{C_{std}}$ is the standard deviation for the concentration of analyte in the calibration standards. Using the data in Table 5.4.1 we find that $s_{C_{std}}$ is 0.1871 and
$\sum_{i = 1}^{n} (C_{std_i} - \overline{C}_{std})^2 = (0.1872)^2 \times (6 - 1) = 0.175 \nonumber$
Substituting known values into Equation \ref{5.12} gives
$s_{C_A} = \frac {0.4035} {120.706} \sqrt{\frac {1} {3} + \frac {1} {6} + \frac {(29.33 - 30.385)^2} {(120.706)^2 \times 0.175}} = 0.0024 \nonumber$
Finally, the 95% confidence interval for 4 degrees of freedom is
$\mu_{C_A} = C_A \pm ts_{C_A} = 0.241 \pm (2.78 \times 0.0024) = 0.241 \pm 0.007 \nonumber$
Figure 5.4.5 shows the calibration curve with curves showing the 95% confidence interval for CA.
In a standard addition we determine the analyte’s concentration by extrapolating the calibration curve to the x-intercept. In this case the value of CA is
$C_A = x\text{-intercept} = \frac {-b_0} {b_1} \nonumber$
and the standard deviation in CA is
$s_{C_A} = \frac {s_r} {b_1} \sqrt{\frac {1} {n} + \frac {(\overline{S}_{std})^2} {(b_1)^2 \sum_{i = 1}^{n}(C_{std_i} - \overline{C}_{std})^2}} \nonumber$
where n is the number of standard additions (including the sample with no added standard), and $\overline{S}_{std}$ is the average signal for the n standards. Because we determine the analyte’s concentration by extrapolation, rather than by interpolation, $s_{C_A}$ for the method of standard additions generally is larger than for a normal calibration curve.
Exercise 5.4.1
Figure 5.4.2 shows a normal calibration curve for the quantitative analysis of Cu2+. The data for the calibration curve are shown here.
[Cu2+] (M) Absorbance
0 0
$1.55 \times 10^{-3}$ 0.050
$3.16 \times 10^{-3}$ 0.093
$4.74 \times 10^{-3}$ 0.143
$6.34 \times 10^{-3}$ 0.188
$7.92 \times 10^{-3}$ 0.236
Complete a linear regression analysis for this calibration data, reporting the calibration equation and the 95% confidence interval for the slope and the y-intercept. If three replicate samples give an Ssamp of 0.114, what is the concentration of analyte in the sample and its 95% confidence interval?
Answer
We begin by setting up a table to help us organize the calculation
$x_i$ $y_i$ $x_i y_i$ $x_i^2$
0.000 0.000 0.000 0.000
$1.55 \times 10^{-3}$ 0.050 $7.750 \times 10^{-5}$ $2.403 \times 10^{-6}$
$3.16 \times 10^{-3}$ 0.093 $2.939 \times 10^{-4}$ $9.986 \times 10^{-6}$
$4.74 \times 10^{-3}$ 0.143 $6.778 \times 10^{-4}$ $2.247 \times 10^{-5}$
$6.34 \times 10^{-3}$ 0.188 $1.192 \times 10^{-3}$ $4.020 \times 10^{-5}$
$7.92 \times 10^{-3}$ 0.236 $1.869 \times 10^{-3}$ $6.273 \times 10^{-5}$
Adding the values in each column gives
$\sum_{i = 1}^{n} x_i = 2.371 \times 10^{-2} \quad \sum_{i = 1}^{n} y_i = 0.710 \quad \sum_{i = 1}^{n} x_i y_i = 4.110 \times 10^{-3} \quad \sum_{i = 1}^{n} x_i^2 = 1.378 \times 10^{-4} \nonumber$
When we substitute these values into Equation \ref{5.4} and Equation \ref{5.5}, we find that the slope and the y-intercept are
$b_1 = \frac {6 \times (4.110 \times 10^{-3}) - (2.371 \times 10^{-2}) \times 0.710} {6 \times (1.378 \times 10^{-4}) - (2.371 \times 10^{-2})^2}) = 29.57 \nonumber$
$b_0 = \frac {0.710 - 29.57 \times (2.371 \times 10^{-2}} {6} = 0.0015 \nonumber$
and that the regression equation is
$S_{std} = 29.57 \times C_{std} + 0.0015 \nonumber$
To calculate the 95% confidence intervals, we first need to determine the standard deviation about the regression. The following table helps us organize the calculation.
$x_i$ $y_i$ $\hat{y}_i$ $(y_i - \hat{y}_i)^2$
0.000 0.000 0.0015 $2.250 \times 10^{-6}$
$1.55 \times 10^{-3}$ 0.050 0.0473 $7.110 \times 10^{-6}$
$3.16 \times 10^{-3}$ 0.093 0.0949 $3.768 \times 10^{-6}$
$4.74 \times 10^{-3}$ 0.143 0.1417 $1.791 \times 10^{-6}$
$6.34 \times 10^{-3}$ 0.188 0.1890 $9.483 \times 10^{-6}$
$7.92 \times 10^{-3}$ 0.236 0.2357 $9.339 \times 10^{-6}$
Adding together the data in the last column gives the numerator of Equation \ref{5.6} as $1.596 \times 10^{-5}$. The standard deviation about the regression, therefore, is
$s_r = \sqrt{\frac {1.596 \times 10^{-5}} {6 - 2}} = 1.997 \times 10^{-3} \nonumber$
Next, we need to calculate the standard deviations for the slope and the y-intercept using Equation \ref{5.7} and Equation \ref{5.8}.
$s_{b_1} = \sqrt{\frac {6 \times (1.997 \times 10^{-3})^2} {6 \times (1.378 \times 10^{-4}) - (2.371 \times 10^{-2})^2}} = 0.3007 \nonumber$
$s_{b_0} = \sqrt{\frac {(1.997 \times 10^{-3})^2 \times (1.378 \times 10^{-4})} {6 \times (1.378 \times 10^{-4}) - (2.371 \times 10^{-2})^2}} = 1.441 \times 10^{-3} \nonumber$
and use them to calculate the 95% confidence intervals for the slope and the y-intercept
$\beta_1 = b_1 \pm ts_{b_1} = 29.57 \pm (2.78 \times 0.3007) = 29.57 \text{ M}^{-1} \pm 0.84 \text{ M}^{-1} \nonumber$
$\beta_0 = b_0 \pm ts_{b_0} = 0.0015 \pm (2.78 \times 1.441 \times 10^{-3}) = 0.0015 \pm 0.0040 \nonumber$
With an average Ssamp of 0.114, the concentration of analyte, CA, is
$C_A = \frac {S_{samp} - b_0} {b_1} = \frac {0.114 - 0.0015} {29.57 \text{ M}^{-1}} = 3.80 \times 10^{-3} \text{ M} \nonumber$
The standard deviation in CA is
$s_{C_A} = \frac {1.997 \times 10^{-3}} {29.57} \sqrt{\frac {1} {3} + \frac {1} {6} + \frac {(0.114 - 0.1183)^2} {(29.57)^2 \times (4.408 \times 10^{-5})}} = 4.778 \times 10^{-5} \nonumber$
and the 95% confidence interval is
$\mu = C_A \pm t s_{C_A} = 3.80 \times 10^{-3} \pm \{2.78 \times (4.778 \times 10^{-5})\} \nonumber$
$\mu = 3.80 \times 10^{-3} \text{ M} \pm 0.13 \times 10^{-3} \text{ M} \nonumber$
Evaluating a Linear Regression Model
You should never accept the result of a linear regression analysis without evaluating the validity of the model. Perhaps the simplest way to evaluate a regression analysis is to examine the residual errors. As we saw earlier, the residual error for a single calibration standard, ri, is
$r_i = (y_i - \hat{y}_i) \nonumber$
If the regression model is valid, then the residual errors should be distributed randomly about an average residual error of zero, with no apparent trend toward either smaller or larger residual errors (Figure 5.4.6 a). Trends such as those in Figure 5.4.6 b and Figure 5.4.6 c provide evidence that at least one of the model’s assumptions is incorrect. For example, a trend toward larger residual errors at higher concentrations, Figure 5.4.6 b, suggests that the indeterminate errors affecting the signal are not independent of the analyte’s concentration. In Figure 5.4.6 c, the residual errors are not random, which suggests we cannot model the data using a straight-line relationship. Regression methods for the latter two cases are discussed in the following sections.
Exercise 5.4.2
Using your results from Exercise 5.4.1 , construct a residual plot and explain its significance.
Answer
To create a residual plot, we need to calculate the residual error for each standard. The following table contains the relevant information.
$x_i$ $y_i$ $\hat{y}_i$ $y_i - \hat{y}_i$
0.000 0.000 0.0015 –0.0015
$1.55 \times 10^{-3}$ 0.050 0.0473 0.0027
$3.16 \times 10^{-3}$ 0.093 0.0949 –0.0019
$4.74 \times 10^{-3}$ 0.143 0.1417 0.0013
$6.34 \times 10^{-3}$ 0.188 0.1890 –0.0010
$7.92 \times 10^{-3}$ 0.236 0.2357 0.0003
The figure below shows a plot of the resulting residual errors. The residual errors appear random, although they do alternate in sign, and that do not show any significant dependence on the analyte’s concentration. Taken together, these observations suggest that our regression model is appropriate.
Weighted Linear Regression with Errors in y
Our treatment of linear regression to this point assumes that indeterminate errors affecting y are independent of the value of x. If this assumption is false, as is the case for the data in Figure 5.4.6 b, then we must include the variance for each value of y into our determination of the y-intercept, b0, and the slope, b1; thus
$b_0 = \frac {\sum_{i = 1}^{n} w_i y_i - b_1 \sum_{i = 1}^{n} w_i x_i} {n} \label{5.13}$
$b_1 = \frac {n \sum_{i = 1}^{n} w_i x_i y_i - \sum_{i = 1}^{n} w_i x_i \sum_{i = 1}^{n} w_i y_i} {n \sum_{i =1}^{n} w_i x_i^2 - \left( \sum_{i = 1}^{n} w_i x_i \right)^2} \label{5.14}$
where wi is a weighting factor that accounts for the variance in yi
$w_i = \frac {n (s_{y_i})^{-2}} {\sum_{i = 1}^{n} (s_{y_i})^{-2}} \label{5.15}$
and $s_{y_i}$ is the standard deviation for yi. In a weighted linear regression, each xy-pair’s contribution to the regression line is inversely proportional to the precision of yi; that is, the more precise the value of y, the greater its contribution to the regression.
Example 5.4.4
Shown here are data for an external standardization in which sstd is the standard deviation for three replicate determination of the signal. This is the same data used in Example 5.4.1 with additional information about the standard deviations in the signal.
$C_{std}$ (arbitrary units) $S_{std}$ (arbitrary units) $s_{std}$
0.000 0.00 0.02
0.100 12.36 0.02
0.200 24.83 0.07
0.300 35.91 0.13
0.400 48.79 0.22
0.500 60.42 0.33
Determine the calibration curve’s equation using a weighted linear regression. As you work through this example, remember that x corresponds to Cstd, and that y corresponds to Sstd.
Solution
We begin by setting up a table to aid in calculating the weighting factors.
$C_{std}$ (arbitrary units) $S_{std}$ (arbitrary units) $s_{std}$ $(s_{y_i})^{-2}$ $w_i$
0.000 0.00 0.02 2500.00 2.8339
0.100 12.36 0.02 250.00 2.8339
0.200 24.83 0.07 204.08 0.2313
0.300 35.91 0.13 59.17 0.0671
0.400 48.79 0.22 20.66 0.0234
0.500 60.42 0.33 9.18 0.0104
Adding together the values in the fourth column gives
$\sum_{i = 1}^{n} (s_{y_i})^{-2} \nonumber$
which we use to calculate the individual weights in the last column. As a check on your calculations, the sum of the individual weights must equal the number of calibration standards, n. The sum of the entries in the last column is 6.0000, so all is well. After we calculate the individual weights, we use a second table to aid in calculating the four summation terms in Equation \ref{5.13} and Equation \ref{5.14}.
$x_i$ $y_i$ $w_i$ $w_i x_i$ $w_i y_i$ $w_i x_i^2$ $w_i x_i y_i$
0.000 0.00 2.8339 0.0000 0.0000 0.0000 0.0000
0.100 12.36 2.8339 0.2834 35.0270 0.0283 3.5027
0.200 24.83 0.2313 0.0463 5.7432 0.0093 1.1486
0.300 35.91 0.0671 0.0201 2.4096 0.0060 0.7229
0.400 48.79 0.0234 0.0094 1.1417 0.0037 0.4567
0.500 60.42 0.0104 0.0052 0.6284 0.0026 0.3142
Adding the values in the last four columns gives
$\sum_{i = 1}^{n} w_i x_i = 0.3644 \quad \sum_{i = 1}^{n} w_i y_i = 44.9499 \quad \sum_{i = 1}^{n} w_i x_i^2 = 0.0499 \quad \sum_{i = 1}^{n} w_i x_i y_i = 6.1451 \nonumber$
Substituting these values into the Equation \ref{5.13} and Equation \ref{5.14} gives the estimated slope and estimated y-intercept as
$b_1 = \frac {(6 \times 6.1451) - (0.3644 \times 44.9499)} {(6 \times 0.0499) - (0.3644)^2} = 122.985 \nonumber$
$b_0 = \frac{44.9499 - (122.985 \times 0.3644)} {6} = 0.0224 \nonumber$
The calibration equation is
$S_{std} = 122.98 \times C_{std} + 0.2 \nonumber$
Figure 5.4.7 shows the calibration curve for the weighted regression and the calibration curve for the unweighted regression in Example 5.4.1 . Although the two calibration curves are very similar, there are slight differences in the slope and in the y-intercept. Most notably, the y-intercept for the weighted linear regression is closer to the expected value of zero. Because the standard deviation for the signal, Sstd, is smaller for smaller concentrations of analyte, Cstd, a weighted linear regression gives more emphasis to these standards, allowing for a better estimate of the y-intercept.
Equations for calculating confidence intervals for the slope, the y-intercept, and the concentration of analyte when using a weighted linear regression are not as easy to define as for an unweighted linear regression [Bonate, P. J. Anal. Chem. 1993, 65, 1367–1372]. The confidence interval for the analyte’s concentration, however, is at its optimum value when the analyte’s signal is near the weighted centroid, yc , of the calibration curve.
$y_c = \frac {1} {n} \sum_{i = 1}^{n} w_i x_i \nonumber$
Weighted Linear Regression with Errors in Both x and y
If we remove our assumption that indeterminate errors affecting a calibration curve are present only in the signal (y), then we also must factor into the regression model the indeterminate errors that affect the analyte’s concentration in the calibration standards (x). The solution for the resulting regression line is computationally more involved than that for either the unweighted or weighted regression lines. Although we will not consider the details in this textbook, you should be aware that neglecting the presence of indeterminate errors in x can bias the results of a linear regression.
See, for example, Analytical Methods Committee, “Fitting a linear functional relationship to data with error on both variable,” AMC Technical Brief, March, 2002), as well as this chapter’s Additional Resources.
Curvilinear and Multivariate Regression
A straight-line regression model, despite its apparent complexity, is the simplest functional relationship between two variables. What do we do if our calibration curve is curvilinear—that is, if it is a curved-line instead of a straight-line? One approach is to try transforming the data into a straight-line. Logarithms, exponentials, reciprocals, square roots, and trigonometric functions have been used in this way. A plot of log(y) versus x is a typical example. Such transformations are not without complications, of which the most obvious is that data with a uniform variance in y will not maintain that uniform variance after it is transformed.
It is worth noting that the term “linear” does not mean a straight-line. A linear function may contain more than one additive term, but each such term has one and only one adjustable multiplicative parameter. The function
$y = ax + bx^2 \nonumber$
is an example of a linear function because the terms x and x2 each include a single multiplicative parameter, a and b, respectively. The function
$y = x^b \nonumber$
is nonlinear because b is not a multiplicative parameter; it is, instead, a power. This is why you can use linear regression to fit a polynomial equation to your data.
Sometimes it is possible to transform a nonlinear function into a linear function. For example, taking the log of both sides of the nonlinear function above gives a linear function.
$\log(y) = b \log(x) \nonumber$
Another approach to developing a linear regression model is to fit a polynomial equation to the data, such as $y = a + b x + c x^2$. You can use linear regression to calculate the parameters a, b, and c, although the equations are different than those for the linear regression of a straight-line. If you cannot fit your data using a single polynomial equation, it may be possible to fit separate polynomial equations to short segments of the calibration curve. The result is a single continuous calibration curve known as a spline function.
For details about curvilinear regression, see (a) Sharaf, M. A.; Illman, D. L.; Kowalski, B. R. Chemometrics, Wiley-Interscience: New York, 1986; (b) Deming, S. N.; Morgan, S. L. Experimental Design: A Chemometric Approach, Elsevier: Amsterdam, 1987.
The regression models in this chapter apply only to functions that contain a single independent variable, such as a signal that depends upon the analyte’s concentration. In the presence of an interferent, however, the signal may depend on the concentrations of both the analyte and the interferent
$S = k_A C_A + k_I CI + S_{reag} \nonumber$
where kI is the interferent’s sensitivity and CI is the interferent’s concentration. Multivariate calibration curves are prepared using standards that contain known amounts of both the analyte and the interferent, and modeled using multivariate regression.
See Beebe, K. R.; Kowalski, B. R. Anal. Chem. 1987, 59, 1007A–1017A. for additional details, and check out this chapter’s Additional Resources for more information about linear regression with errors in both variables, curvilinear regression, and multivariate regression. | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/05%3A_Standardizing_Analytical_Methods/5.04%3A_Linear_Regression_and_Calibration_Curves.txt |
Thus far in our discussion of strategies for standardizing analytical methods, we have assumed that a suitable reagent blank is available to correct for signals arising from sources other than the analyte. We did not, however ask an important question: “What constitutes an appropriate reagent blank?” Surprisingly, the answer is not immediately obvious.
In one study, approximately 200 analytical chemists were asked to evaluate a data set consisting of a normal calibration curve, a separate analyte-free blank, and three samples with different sizes, but drawn from the same source [Cardone, M. J. Anal. Chem. 1986, 58, 433–438]. The first two columns in Table 5.5.1 shows a series of external standards and their corresponding signals. The normal calibration curve for the data is
$S_{std} = 0.0750 \times W_{std} + 0.1250 \nonumber$
where the y-intercept of 0.1250 is the calibration blank. A separate reagent blank gives the signal for an analyte-free sample.
Table 5.5.1 : Data Used to Study the Blank in an Analytical Method
$W_{std}$ $S_{std}$ $\text{Sample Number}$ $W_{samp}$ $S_{samp}$
1.6667 0.2500 1 62.4746 0.8000
5.0000 0.5000 2 82.7915 1.0000
8.3333 0.7500 3 103.1085 1.2000
11.6667 0.8413
18.1600 1.4870 $\text{reagent blank}$ 0.1000
19.9333 1.6200
$S_{std} = 0.0750 \times W_{std} + 0.1250$
$W_{std} \text{: weight of analyte used to prepare the external standard; diluted to a volume, } V$
$W_{samp} \text{: weight of sample used to prepare sample as analyzed; diluted to a volume, } V$
In working up this data, the analytical chemists used at least four different approaches to correct the signals: (a) ignoring both the calibration blank, CB, and the reagent blank, RB, which clearly is incorrect; (b) using the calibration blank only; (c) using the reagent blank only; and (d) using both the calibration blank and the reagent blank. The first four rows of Table 5.5.2 shows the equations for calculating the analyte’s concentration using each approach, along with the reported concentrations for the analyte in each sample.
Table 5.5.2 : Equations and Resulting Concentrations of Analyte for Different Approaches to Correcting for the Blank
Concentration of Analyte in...
Approach for Correcting the Signal Equation Sample 1 Sample 2 Sample 3
ignore calibration and reagent blanks $C_A = \frac {W_A} {W_{samp}} = \frac {S_{samp}} {k_A W_{samp}}$ 0.1707 0.1610 0.1552
use calibration blank only $C_A = \frac {W_A} {W_{samp}} = \frac {S_{samp} -CB} {k_A W_{samp}}$ 0.1441 0.1409 0.1390
use reagent blank only $C_A = \frac {W_A} {W_{samp}} = \frac {S_{samp} - RB} {k_A W_{samp}}$ 0.1494 0.1449 0.1422
use both calibration and reagent blanks $C_A = \frac {W_A} {W_{samp}} = \frac {S_{samp} -CB -RB} {k_A W_{samp}}$ 0.1227 0.1248 0.1266
use total Youden blank $C_A = \frac {W_A} {W_{samp}} = \frac {S_{samp} -TYB} {k_A W_{samp}}$ 0.1313 0.1313 0.1313
$C_A = \text{ concentration of analyte; } W_A = \text{ weight of analyte; } W_{samp} \text{ weight of sample; }$
$k_A = \text{ slope of calibration curve (0.0750; slope of calibration equation); } CB = \text{ calibration blank (0.125; intercept of calibration equation); }$
$RB = \text{ reagent blank (0.100); } TYB = \text{ total Youden blank (0.185; see text)}$
That all four methods give a different result for the analyte’s concentration underscores the importance of choosing a proper blank, but does not tell us which blank is correct. Because all four methods fail to predict the same concentration of analyte for each sample, none of these blank corrections properly accounts for an underlying constant source of determinate error.
To correct for a constant method error, a blank must account for signals from any reagents and solvents used in the analysis and any bias that results from interactions between the analyte and the sample’s matrix. Both the calibration blank and the reagent blank compensate for signals from reagents and solvents. Any difference in their values is due to indeterminate errors in preparing and analyzing the standards.
Because we are considering a matrix effect of sorts, you might think that the method of standard additions is one way to overcome this problem. Although the method of standard additions can compensate for proportional determinate errors, it cannot correct for a constant determinate error; see Ellison, S. L. R.; Thompson, M. T. “Standard additions: myth and reality,” Analyst, 2008, 133, 992–997.
Unfortunately, neither a calibration blank nor a reagent blank can correct for a bias that results from an interaction between the analyte and the sample’s matrix. To be effective, the blank must include both the sample’s matrix and the analyte and, consequently, it must be determined using the sample itself. One approach is to measure the signal for samples of different size, and to determine the regression line for a plot of Ssamp versus the amount of sample. The resulting y-intercept gives the signal in the absence of sample, and is known as the total Youden blank [Cardone, M. J. Anal. Chem. 1986, 58, 438–445]. This is the true blank correction. The regression line for the three samples in Table 5.5.1 is
$S_{samp} = 0.009844 \times W_{samp} + 0.185 \nonumber$
giving a true blank correction of 0.185. As shown in Table 5.5.2 , using this value to correct Ssamp gives identical values for the concentration of analyte in all three samples.
The use of the total Youden blank is not common in analytical work, with most chemists relying on a calibration blank when using a calibration curve and a reagent blank when using a single-point standardization. As long we can ignore any constant bias due to interactions between the analyte and the sample’s matrix, which is often the case, the accuracy of an analytical method will not suffer. It is a good idea, however, to check for constant sources of error before relying on either a calibration blank or a reagent blank. | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/05%3A_Standardizing_Analytical_Methods/5.05%3A_Compensating_for_the_Reagent_Blank.txt |
Although the calculations in this chapter are relatively straightforward—consisting, as they do, mostly of summations—it is tedious to work through problems using nothing more than a calculator. Both Excel and R include functions for completing a linear regression analysis and for visually evaluating the resulting model.
Excel
Let’s use Excel to fit the following straight-line model to the data in Example 5.4.1.
$y = \beta_0 + \beta_1 x \nonumber$
Enter the data into a spreadsheet, as shown in Figure 5.6.1 . Depending upon your needs, there are many ways that you can use Excel to complete a linear regression analysis. We will consider three approaches here.
Using Excel's Built-In Functions
If all you need are values for the slope, $\beta_1$, and the y-intercept, $\beta_0$, you can use the following functions:
= intercept(known_y's, known_x's)
= slope(known_y's, known_x's)
where known_y’s is the range of cells that contain the signals (y), and known_x’s is the range of cells that contain the concentrations (x). For example, if you click on an empty cell and enter
= slope(B2:B7, A2:A7)
Excel returns exact calculation for the slope (120.705 714 3).
Using Excel's Data Analysis Tools
To obtain the slope and the y-intercept, along with additional statistical details, you can use the data analysis tools in the Data Analysis ToolPak. The ToolPak is not a standard part of Excel’s instillation. To see if you have access to the Analysis ToolPak on your computer, select Tools from the menu bar and look for the Data Analysis... option. If you do not see Data Analysis..., select Add-ins... from the Tools menu. Check the box for the Analysis ToolPak and click on OK to install them.
Select Data Analysis... from the Tools menu, which opens the Data Analysis window. Scroll through the window, select Regression from the available options, and press OK. Place the cursor in the box for Input Y range and then click and drag over cells B1:B7. Place the cursor in the box for Input X range and click and drag over cells A1:A7. Because cells A1 and B1 contain labels, check the box for Labels.
Including labels is a good idea. Excel’s summary output uses the x-axis label to identify the slope.
Select the radio button for Output range and click on any empty cell; this is where Excel will place the results. Clicking OK generates the information shown in Figure 5.6.2 .
There are three parts to Excel’s summary of a regression analysis. At the top of Figure 5.6.2 is a table of Regression Statistics. The standard error is the standard deviation about the regression, sr. Also of interest is the value for Multiple R, which is the model’s correlation coefficient, r, a term with which you may already be familiar. The correlation coefficient is a measure of the extent to which the regression model explains the variation in y. Values of r range from –1 to +1. The closer the correlation coefficient is to ±1, the better the model is at explaining the data. A correlation coefficient of 0 means there is no relationship between x and y. In developing the calculations for linear regression, we did not consider the correlation coefficient. There is a reason for this. For most straight-line calibration curves the correlation coefficient is very close to +1, typically 0.99 or better. There is a tendency, however, to put too much faith in the correlation coefficient’s significance, and to assume that an r greater than 0.99 means the linear regression model is appropriate. Figure 5.6.3 provides a useful counterexample. Although the regression line has a correlation coefficient of 0.993, the data clearly is curvilinear. The take-home lesson here is simple: do not fall in love with the correlation coefficient!
The second table in Figure 5.6.2 is entitled ANOVA, which stands for analysis of variance. We will take a closer look at ANOVA in Chapter 14. For now, it is sufficient to understand that this part of Excel’s summary provides information on whether the linear regression model explains a significant portion of the variation in the values of y. The value for F is the result of an F-test of the following null and alternative hypotheses.
H0: the regression model does not explain the variation in y
HA: the regression model does explain the variation in y
The value in the column for Significance F is the probability for retaining the null hypothesis. In this example, the probability is $2.5 \times 10^{-6}\%$, which is strong evidence for accepting the regression model. As is the case with the correlation coefficient, a small value for the probability is a likely outcome for any calibration curve, even when the model is inappropriate. The probability for retaining the null hypothesis for the data in Figure 5.6.3 , for example, is $9.0 \times 10^{-7}\%$.
See Chapter 4.6 for a review of the F-test.
The third table in Figure 5.6.2 provides a summary of the model itself. The values for the model’s coefficients—the slope,$\beta_1$, and the y-intercept, $\beta_0$—are identified as intercept and with your label for the x-axis data, which in this example is Cstd. The standard deviations for the coefficients, $s_{b_0}$ and $s_{b_1}$, are in the column labeled Standard error. The column t Stat and the column P-value are for the following t-tests.
slope: $H_0 \text{: } \beta_1 = 0 \quad H_A \text{: } \beta_1 \neq 0$
y-intercept: $H_0 \text{: } \beta_0 = 0 \quad H_A \text{: } \beta_0 \neq 0$
The results of these t-tests provide convincing evidence that the slope is not zero, but there is no evidence that the y-intercept differs significantly from zero. Also shown are the 95% confidence intervals for the slope and the y-intercept (lower 95% and upper 95%).
See Chapter 4.6 for a review of the t-test.
Programming the Formulas Yourself
A third approach to completing a regression analysis is to program a spreadsheet using Excel’s built-in formula for a summation
=sum(first cell:last cell)
and its ability to parse mathematical equations. The resulting spreadsheet is shown in Figure 5.6.4 .
Using Excel to Visualize the Regression Model
You can use Excel to examine your data and the regression line. Begin by plotting the data. Organize your data in two columns, placing the x values in the left-most column. Click and drag over the data and select Charts from the ribbon. Select Scatter, choosing the option without lines that connect the points. To add a regression line to the chart, click on the chart’s data and select Chart: Add Trendline... from the main men. Pick the straight-line model and click OK to add the line to your chart. By default, Excel displays the regression line from your first point to your last point. Figure 5.6.5 shows the result for the data in Figure 5.6.1 .
Excel also will create a plot of the regression model’s residual errors. To create the plot, build the regression model using the Analysis ToolPak, as described earlier. Clicking on the option for Residual plots creates the plot shown in Figure 5.6.6 .
Limitations to Using Excel for a Regression Analysis
Excel’s biggest limitation for a regression analysis is that it does not provide a function to calculate the uncertainty when predicting values of x. In terms of this chapter, Excel can not calculate the uncertainty for the analyte’s concentration, CA, given the signal for a sample, Ssamp. Another limitation is that Excel does not have a built-in function for a weighted linear regression. You can, however, program a spreadsheet to handle these calculations.
Exercise 5.6.1
Use Excel to complete the regression analysis in Exercise 5.4.1.
Answer
Begin by entering the data into an Excel spreadsheet, following the format shown in Figure 5.6.1 . Because Excel’s Data Analysis tools provide most of the information we need, we will use it here. The resulting output, which is shown below, provides the slope and the y-intercept, along with their respective 95% confidence intervals.
Excel does not provide a function for calculating the uncertainty in the analyte’s concentration, CA, given the signal for a sample, Ssamp. You must complete these calculations by hand. With an Ssamp of 0.114, we find that CA is
$C_A = \frac {S_{samp} - b_0} {b_1} = \frac {0.114 - 0.0014} {29.59 \text{ M}^{-1}} = 3.80 \times 10^{-3} \text{ M} \nonumber$
The standard deviation in CA is
$s_{C_A} = \frac {1.996 \times 10^{-3}} {29.59} \sqrt{\frac {1} {3} + \frac {1} {6} + \frac {(0.114 - 0.1183)^2} {(29.59)^2 \times 4.408 \times 10^{-5})}} = 4.772 \times 10^{-5} \nonumber$
and the 95% confidence interval is
$\mu = C_A \pm ts_{C_A} = 3.80 \times 10^{-3} \pm \{2.78 \times (4.772 \times 10^{-5}) \} \nonumber$
$\mu = 3.80 \times 10^{-3} \text{ M} \pm 0.13 \times 10^{-3} \text{ M} \nonumber$
R
Let’s use R to fit the following straight-line model to the data in Example 5.4.1.
$y = \beta_0 + \beta_1 x \nonumber$
Entering Data and Creating the Regression Model
To begin, create objects that contain the concentration of the standards and their corresponding signals.
> conc = c(0, 0.1, 0.2, 0.3, 0.4, 0.5)
> signal = c(0, 12.36, 24.83, 35.91, 48.79, 60.42)
The command for a straight-line linear regression model is
lm(y ~ x)
where y and x are the objects the objects our data. To access the results of the regression analysis, we assign them to an object using the following command
> model = lm(signal ~ conc)
where model is the name we assign to the object.
As you might guess, lm is short for linear model.
You can choose any name for the object that contains the results of the regression analysis.
Evaluating the Linear Regression Model
To evaluate the results of a linear regression we need to examine the data and the regression line, and to review a statistical summary of the model. To examine our data and the regression line, we use the plot command, which takes the following general form
plot(x, y, optional arguments to control style)
where x and y are the objects that contain our data, and the abline command
abline(object, optional arguments to control style)
where object is the object that contains the results of the linear regression. Entering the commands
> plot(conc, signal, pch = 19, col = “blue”, cex = 2)
> abline(model, col = “red”)
creates the plot shown in Figure 5.6.7 .
To review a statistical summary of the regression model, we use the summary command.
> summary(model)
The resulting output, shown in Figure 5.6.8 , contains three sections.
The first section of R’s summary of the regression model lists the residual errors. To examine a plot of the residual errors, use the command
> plot(model, which = 1)
which produces the result shown in Figure 5.6.9 . Note that R plots the residuals against the predicted (fitted) values of y instead of against the known values of x. The choice of how to plot the residuals is not critical, as you can see by comparing Figure 5.6.9 to Figure 5.6.6 . The line in Figure 5.6.9 is a smoothed fit of the residuals.
The reason for including the argument which = 1 is not immediately obvious. When you use R’s plot command on an object created by the lm command, the default is to create four charts summarizing the model’s suitability. The first of these charts is the residual plot; thus, which = 1 limits the output to this plot.
The second section of Figure 5.6.8 provides the model’s coefficients—the slope, $\beta_1$, and the y-intercept, $\beta_0$—along with their respective standard deviations (Std. Error). The column t value and the column Pr(>|t|) are for the following t-tests.
slope: $H_0 \text{: } \beta_1 = 0 \quad H_A \text{: } \beta_1 \neq 0$
y-intercept: $H_0 \text{: } \beta_0 = 0 \quad H_A \text{: } \beta_0 \neq 0$
The results of these t-tests provide convincing evidence that the slope is not zero, but no evidence that the y-intercept differs significantly from zero.
The last section of the regression summary provides the standard deviation about the regression (residual standard error), the square of the correlation coefficient (multiple R-squared), and the result of an F-test on the model’s ability to explain the variation in the y values. For a discussion of the correlation coefficient and the F-test of a regression model, as well as their limitations, refer to the section on using Excel’s data analysis tools.
Predicting the Uncertainty in $C_A$ Given $S_{samp}$
Unlike Excel, R includes a command for predicting the uncertainty in an analyte’s concentration, CA, given the signal for a sample, Ssamp. This command is not part of R’s standard installation. To use the command you need to install the “chemCal” package by entering the following command (note: you will need an internet connection to download the package).
> install.packages(“chemCal”)
After installing the package, you need to load the functions into R using the following command. (note: you will need to do this step each time you begin a new R session as the package does not automatically load when you start R).
> library(“chemCal”)
You need to install a package once, but you need to load the package each time you plan to use it. There are ways to configure R so that it automatically loads certain packages; see An Introduction to R for more information (click here to view a PDF version of this document).
The command for predicting the uncertainty in CA is inverse.predict, which takes the following form for an unweighted linear regression
inverse.predict(object, newdata, alpha = value)
where object is the object that contains the regression model’s results, new-data is an object that contains values for Ssamp, and value is the numerical value for the significance level. Let’s use this command to complete Example 5.4.3. First, we create an object that contains the values of Ssamp
> sample = c(29.32, 29.16, 29.51)
and then we complete the computation using the following command
> inverse.predict(model, sample, alpha = 0.05)
producing the result shown in Figure 5.6.10 . The analyte’s concentration, CA, is given by the value $Prediction, and its standard deviation, $s_{C_A}$, is shown as$Standard Error. The value for $Confidence is the confidence interval, $\pm t s_{C_A}$, for the analyte’s concentration, and$Confidence Limits provides the lower limit and upper limit for the confidence interval for CA.
Using R for a Weighted Linear Regression
R’s command for an unweighted linear regression also allows for a weighted linear regression if we include an additional argument, weights, whose value is an object that contains the weights.
lm(y ~ x, weights = object)
Let’s use this command to complete Example 5.4.4. First, we need to create an object that contains the weights, which in R are the reciprocals of the standard deviations in y, $(s_{y_i})^{-2}$. Using the data from Example 5.4.4, we enter
> syi=c(0.02, 0.02, 0.07, 0.13, 0.22, 0.33)
> w=1/syi^2
to create the object that contains the weights. The commands
> modelw= lm(signal ~ conc, weights = w)
> summary(modelw)
generate the output shown in Figure 5.6.11 . Any difference between the results shown here and the results shown in Example 5.4.4 are the result of round-off errors in our earlier calculations.
You may have noticed that this way of defining weights is different than that shown in Equation 5.4.15. In deriving equations for a weighted linear regression, you can choose to normalize the sum of the weights to equal the number of points, or you can choose not to—the algorithm in R does not normalize the weights.
Exercise 5.6.2
Use R to complete the regression analysis in Exercise 5.4.1.
Answer
The figure below shows the R session for this problem, including loading the chemCal package, creating objects to hold the values for Cstd, Sstd, and Ssamp. Note that for Ssamp, we do not have the actual values for the three replicate measurements. In place of the actual measurements, we just enter the average signal three times. This is okay because the calculation depends on the average signal and the number of replicates, and not on the individual measurements. | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/05%3A_Standardizing_Analytical_Methods/5.06%3A_Using_Excel_and_R_for_a_Linear_Regression.txt |
1. Suppose you use a serial dilution to prepare 100 mL each of a series of standards with concentrations of $1.00 \times10^{-5}$, $1.00 \times10^{-4}$, $1.00 \times10^{-3}$, and $1.00 \times10^{-2}$ M from a 0.100 M stock solution. Calculate the uncertainty for each solution using a propagation of uncertainty, and compare to the uncertainty if you prepare each solution as a single dilution of the stock solution. You will find tolerances for different types of volumetric glassware and digital pipets in Table 4.2.1 and Table 4.2.2. Assume that the uncertainty in the stock solution’s molarity is ±0.0002.
2. Three replicate determinations of Stotal for a standard solution that is 10.0 ppm in analyte give values of 0.163, 0.157, and 0.161 (arbitrary units). The signal for the reagent blank is 0.002. Calculate the concentration of analyte in a sample with a signal of 0.118.
3. A 10.00-g sample that contains an analyte is transferred to a 250-mL volumetric flask and diluted to volume. When a 10.00 mL aliquot of the resulting solution is diluted to 25.00 mL it gives a signal of 0.235 (arbitrary units). A second 10.00-mL portion of the solution is spiked with 10.00 mL of a 1.00-ppm standard solution of the analyte and diluted to 25.00 mL. The signal for the spiked sample is 0.502. Calculate the weight percent of analyte in the original sample.
4. A 50.00 mL sample that contains an analyte gives a signal of 11.5 (arbitrary units). A second 50 mL aliquot of the sample, which is spiked with 1.00 mL of a 10.0-ppm standard solution of the analyte, gives a signal of 23.1. What is the analyte’s concentration in the original sample?
5. A standard additions calibration curve based on Equation 5.3.10 places $S_{spike} \times (V_o + V_{std})$ on the y-axis and $C_{std} \times V_{std}$ on the x-axis. Derive equations for the slope and the y-intercept and explain how you can determine the amount of analyte in a sample from the calibration curve. In addition, clearly explain why you cannot plot Sspike on the y-axis and $C_{std} \times \{V_{std}/(V_o + V_{std})\}$ on the x-axis.
6. A standard sample contains 10.0 mg/L of analyte and 15.0 mg/L of internal standard. Analysis of the sample gives signals for the analyte and the internal standard of 0.155 and 0.233 (arbitrary units), respectively. Sufficient internal standard is added to a sample to make its concentration 15.0 mg/L. Analysis of the sample yields signals for the analyte and the internal standard of 0.274 and 0.198, respectively. Report the analyte’s concentration in the sample.
7. For each of the pair of calibration curves shown ibelow, select the calibration curve that uses the more appropriate set of standards. Briefly explain the reasons for your selections. The scales for the x-axis and the y-axis are the same for each pair.
8. The following data are for a series of external standards of Cd2+ buffered to a pH of 4.6.
[Cd2+] (nM) 15.4 30.4 44.9 59.0 72.7 86.0
$S_{spike}$ (nA) 4.8 11.4 18.2 26.6 32.3 37.7
(a) Use a linear regression analysis to determine the equation for the calibration curve and report confidence intervals for the slope and the y-intercept.
(b) Construct a plot of the residuals and comment on their significance.
At a pH of 3.7 the following data were recorded for the same set of external standards.
[Cd2+] (nM) 15.4 30.4 44.9 59.0 72.7 86.0
$S_{spike}$ (nA) 15.0 42.7 58.5 77.0 101 118
(c) How much more or less sensitive is this method at the lower pH?
(d) A single sample is buffered to a pH of 3.7 and analyzed for cadmium, yielding a signal of 66.3 nA. Report the concentration of Cd2+ in the sample and its 95% confidence interval.
The data in this problem are from Wojciechowski, M.; Balcerzak, J. Anal. Chim. Acta 1991, 249, 433–445.
9. To determine the concentration of analyte in a sample, a standard addition is performed. A 5.00-mL portion of sample is analyzed and then successive 0.10-mL spikes of a 600.0 ppb standard of the analyte are added, analyzing after each spike. The following table shows the results of this analysis.
$V_{spike}$ (mL) 0.00 0.10 0.20 0.30
$S_{total}$ (arbitrary units) 15.0 42.7 58.5 77.0
Construct an appropriate standard additions calibration curve and use a linear regression analysis to determine the concentration of analyte in the original sample and its 95% confidence interval.
10. Troost and Olavsesn investigated the application of an internal standardization to the quantitative analysis of polynuclear aromatic hydrocarbons. The following results were obtained for the analysis of phenanthrene using isotopically labeled phenanthrene as an internal standard. Each solution was analyzed twice.
$C_A/C_{IS}$
0.50 1.25 2.00 3.00 4.00
$S_A/S_{IS}$
0.514
0.522
0.993
1.024
1.486
1.471
2.044
2.080
2.342
2.550
(a) Determine the equation for the calibration curve using a linear regression, and report confidence intervals for the slope and the y-intercept. Average the replicate signals for each standard before you complete the linear regression analysis.
(b) Based on your results explain why the authors concluded that the internal standardization was inappropriate.
The data in this problem are from Troost, J. R.; Olavesen, E. Y. Anal. Chem. 1996, 68, 708–711.
11. In Chapter 4.6. we used a paired t-test to compare two analytical methods that were used to analyze independently a series of samples of variable composition. An alternative approach is to plot the results for one method versus the results for the other method. If the two methods yield identical results, then the plot should have an expected slope, $\beta_1$, of 1.00 and an expected y-intercept, $\beta_0$, of 0.0. We can use a t-test to compare the slope and the y-intercept from a linear regression to the expected values. The appropriate test statistic for the y-intercept is found by rearranging Equation 5.4.10.
$t_{exp} = \frac {|\beta_0 - b_0|} {s_{b_0}} = \frac {|b_0|} {s_{b_0}} \nonumber$
Rearranging Equation 5.4.9 gives the test statistic for the slope.
$t_{exp} = \frac {|\beta_1 - b_1} {s_{b_1}} = \frac {|b_1|} {s_{b_1}} \nonumber$
Reevaluate the data in Problem 25 from Chapter 4 using the same significance level as in the original problem.
Although this is a common approach for comparing two analytical methods, it does violate one of the requirements for an unweighted linear regression—that indeterminate errors affect y only. Because indeterminate errors affect both analytical methods, the result of an unweighted linear regression is biased. More specifically, the regression underestimates the slope, b1, and overestimates the y-intercept, b0. We can minimize the effect of this bias by placing the more precise analytical method on the x-axis, by using more samples to increase the degrees of freedom, and by using samples that uniformly cover the range of concentrations.
For more information, see Miller, J. C.; Miller, J. N. Statistics for Analytical Chemistry, 3rd ed. Ellis Horwood PTR Prentice-Hall: New York, 1993. Alternative approaches are found in Hartman, C.; Smeyers-Verbeke, J.; Penninckx, W.; Massart, D. L. Anal. Chim. Acta 1997, 338, 19–40, and Zwanziger, H. W.; Sârbu, C. Anal. Chem. 1998, 70, 1277–1280.
12. Consider the following three data sets, each of which gives values of y for the same values of x.
x y1 y2 y3
10.00 8.04 9.14 7.46
8.00 6.95 8.14 6.77
13.00 7.58 8.74 12.74
9.00 8.81 8.77 7.11
11.00 8.33 9.26 7.81
14.00 9.96 8.10 8.84
6.00 7.24 6.13 6.08
4.00 4.26 3.10 5.39
12.00 10.84 9.13 8.15
7.00 4.82 7.26 6.42
5.00 5.68 4.74 5.73
(a) An unweighted linear regression analysis for the three data sets gives nearly identical results. To three significant figures, each data set has a slope of 0.500 and a y-intercept of 3.00. The standard deviations in the slope and the y-intercept are 0.118 and 1.125 for each data set. All three standard deviations about the regression are 1.24. Based on these results for a linear regression analysis, comment on the similarity of the data sets.
(b) Complete a linear regression analysis for each data set and verify that the results from part (a) are correct. Construct a residual plot for each data set. Do these plots change your conclusion from part (a)? Explain.
(c) Plot each data set along with the regression line and comment on your results.
(d) Data set 3 appears to contain an outlier. Remove the apparent outlier and reanalyze the data using a linear regression. Comment on your result.
(e) Briefly comment on the importance of visually examining your data.
These three data sets are taken from Anscombe, F. J. “Graphs in Statistical Analysis,” Amer. Statis. 1973, 27, 17-21.
13. Fanke and co-workers evaluated a standard additions method for a voltammetric determination of Tl. A summary of their results is tabulated in the following table.
ppm Tl added Instrument Response ($\mu$A)
0.000 2.53 2.50 2.70 2.63 2.70 2.80 2.52
0.387 8.42 7.96 8.54 8.18 7.70 8.34 7.98
1.851 29.65 28.70 29.05 28.30 29.20 29.95 28.95
5.734 84.8 85.6 86.0 85.2 84.2 86.4 87.8
Use a weighted linear regression to determine the standardization relationship for this data.
The data in this problem are from Franke, J. P.; de Zeeuw, R. A.; Hakkert, R. Anal. Chem. 1978, 50, 1374–1380. | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/05%3A_Standardizing_Analytical_Methods/5.07%3A_Problems.txt |
Although there are many experiments in the literature that incorporate external standards, the method of standard additions, or internal standards, the issue of choosing a method standardization is not the experiment’s focus. One experiment designed to consider the issue of selecting a method of standardization is given here.
• Harvey, D. T. “External Standards or Standard Additions? Selecting and Validating a Method of Standardization,” J. Chem. Educ. 2002, 79, 613–615.
In addition to the texts listed as suggested readings in Chapter 4, the following text provide additional details on linear regression.
• Draper, N. R.; Smith, H. Applied Regression Analysis, 2nd. ed.; Wiley: New York, 1981.
The following articles providing more details about linear regression.
• Analytical Methods Committee “Is my calibration linear?” AMC Technical Brief, December 2005.
• Analytical Methods Committee “Robust regression: an introduction, “AMCTB 50, 2012.
• Badertscher, M.; Pretsch, E. “Bad results from good data,” Trends Anal. Chem. 2006, 25, 1131–1138.
• Boqué, R.; Rius, F. X.; Massart, D. L. “Straight Line Calibration: Something More Than Slopes, Intercepts, and Correlation Coefficients,” J. Chem. Educ. 1993, 70, 230–232.
• Danzer, K.; Currie, L. A. “Guidelines for Calibration in Analytical Chemistry. Part 1. Fundamentals and Single Component Calibration,” Pure Appl. Chem. 1998, 70, 993–1014.
• Henderson, G. “Lecture Graphic Aids for Least-Squares Analysis,” J. Chem. Educ. 1988, 65, 1001–1003.
• Logan, S. R. “How to Determine the Best Straight Line,” J. Chem. Educ. 1995, 72, 896–898.
• Mashkina, E.; Oldman, K. B. “Linear Regressions to Which the Standard Formulas do not Apply,” ChemTexts, 2015, 1, 1–11.
• Miller, J. N. “Basic Statistical Methods for Analytical Chemistry. Part 2. Calibration and Regression Methods,” Analyst 1991, 116, 3–14.
• Raposo, F. “Evaluation of analytical calibration based on least-squares linear regression for instrumental techniques: A tutorial review,” Trends Anal. Chem. 2016, 77, 167–185.
• Renman, L., Jagner, D. “Asymmetric Distribution of Results in Calibration Curve and Standard Addition Evaluations,” Anal. Chim. Acta 1997, 357, 157–166.
• Rodriguez, L. C.; Gamiz-Gracia; Almansa-Lopez, E. M.; Bosque-Sendra, J. M. “Calibration in chemical measurement processes. II. A methodological approach,” Trends Anal. Chem. 2001, 20, 620–636.
Useful papers providing additional details on the method of standard additions are gathered here.
• Bader, M. “A Systematic Approach to Standard Addition Methods in Instrumental Analysis,” J. Chem. Educ. 1980, 57, 703–706.
• Brown, R. J. C.; Roberts, M. R.; Milton, M. J. T. “Systematic error arising form ‘Sequential’ Standard Addition Calibrations. 2. Determination of Analyte Mass Fraction in Blank Solutions,” Anal. Chim. Acta 2009, 648, 153–156.
• Brown, R. J. C.; Roberts, M. R.; Milton, M. J. T. “Systematic error arising form ‘Sequential’ Standard Addition Calibrations: Quantification and correction,” Anal. Chim. Acta 2007, 587, 158–163.
• Bruce, G. R.; Gill, P. S. “Estimates of Precision in a Standard Additions Analysis,” J. Chem. Educ. 1999, 76, 805–807.
• Kelly, W. R.; MacDonald, B. S.; Guthrie “Gravimetric Approach to the Standard Addition Method in Instrumental Analysis. 1.” Anal. Chem. 2008, 80, 6154–6158.
• Meija, J.; Pagliano, E.; Mester, Z. “Coordinate Swapping in Standard Addition Graphs for Analytical Chemistry: A Simplified Path for Uncertainty Calculation in Linear and Nonlinear Plots,” Anal. Chem. 2014, 86, 8563–8567.
• Nimura, Y.; Carr, M. R. “Reduction of the Relative Error in the Standard Additions Method,” Analyst 1990, 115, 1589–1595.
Approaches that combine a standard addition with an internal standard are described in the following paper.
• Jones, W. B.; Donati, G. L.; Calloway, C. P.; Jones, B. T. “Standard Dilution Analysis,” Anal. Chem. 2015, 87, 2321–2327.
The following papers discusses the importance of weighting experimental data when use linear regression.
• Analytical Methods Committee “Why are we weighting?” AMC Technical Brief, June 2007.
• Karolczak, M. “To Weight or Not to Weight? An Analyst’s Dilemma,” Current Separations 1995, 13, 98–104.
Algorithms for performing a linear regression with errors in both X and Y are discussed in the following papers. Also included here are papers that address the difficulty of using linear regression to compare two analytical methods.
• Irvin, J. A.; Quickenden, T. L. “Linear Least Squares Treatment When There are Errors in Both x and y,” J. Chem. Educ. 1983, 60, 711–712.
• Kalantar, A. H. “Kerrich’s Method for y = ax Data When Both y and x Are Uncertain,” J. Chem. Educ. 1991, 68, 368–370.
• Macdonald, J. R.; Thompson, W. J. “Least-Squares Fitting When Both Variables Contain Errors: Pitfalls and Possibilities,” Am. J. Phys. 1992, 60, 66–73.
• Martin, R. F. “General Deming Regression for Estimating Systematic Bias and Its Confidence Interval in Method-Comparison Studies,” Clin. Chem. 2000, 46, 100–104.
• Ogren, P. J.; Norton, J. R. “Applying a Simple Linear Least-Squares Algorithm to Data with Uncertain- ties in Both Variables,” J. Chem. Educ. 1992, 69, A130–A131.
• Ripley, B. D.; Thompson, M. “Regression Techniques for the Detection of Analytical Bias,” Analyst 1987, 112, 377–383.
• Tellinghuisen, J. “Least Squares in Calibration: Dealing with Uncertainty in x,” Analyst, 2010, 135, 1961–1969.
Outliers present a problem for a linear regression analysis. The following papers discuss the use of robust linear regression techniques.
• Glaister, P. “Robust Linear Regression Using Thiel’s Method,” J. Chem. Educ. 2005, 82, 1472–1473.
• Glasser, L. “Dealing with Outliers: Robust, Resistant Regression,” J. Chem. Educ. 2007, 84, 533–534.
• Ortiz, M. C.; Sarabia, L. A.; Herrero, A. “Robust regression techniques. A useful alternative for the detection of outlier data in chemical analysis,” Talanta 2006, 70, 499–512.
The following papers discusses some of the problems with using linear regression to analyze data that has been mathematically transformed into a linear form, as well as alternative methods of evaluating curvilinear data.
• Chong, D. P. “On the Use of Least Squares to Fit Data in Linear Form,” J. Chem. Educ. 1994, 71, 489–490.
• Hinshaw, J. V. “Nonlinear Calibration,” LCGC 2002, 20, 350–355.
• Lieb, S. G. “Simplex Method of Nonlinear Least-Squares - A Logical Complementary Method to Linear Least-Squares Analysis of Data,” J. Chem. Educ. 1997, 74, 1008–1011.
• Zielinski, T. J.; Allendoerfer, R. D. “Least Squares Fitting of Nonlinear Data in the Undergraduate Laboratory,” J. Chem. Educ. 1997, 74, 1001–1007.
More information on multivariate and multiple regression can be found in the following papers.
• Danzer, K.; Otto, M.; Currie, L. A. “Guidelines for Calibration in Analytical Chemistry. Part 2. Multispecies Calibration,” Pure Appl. Chem. 2004, 76, 1215–1225.
• Escandar, G. M.; Faber, N. M.; Goicoechea, H. C.; de la Pena, A. M.; Olivieri, A.; Poppi, R. J. “Second- and third-order multivariate calibration: data, algorithms and applications,” Trends Anal. Chem. 2007, 26, 752–765.
• Kowalski, B. R.; Seasholtz, M. B. “Recent Developments in Multivariate Calibration,” J. Chemometrics 1991, 5, 129–145.
• Lang, P. M.; Kalivas, J. H. “A Global Perspective on Multivariate Calibration Methods,” J. Chemomet- rics 1993, 7, 153–164.
• Madden, S. P.; Wilson, W.; Dong, A.; Geiger, L.; Mecklin, C. J. “Multiple Linear Regression Using a Graphing Calculator,” J. Chem. Educ. 2004, 81, 903–907.
• Olivieri, A. C.; Faber, N. M.; Ferré, J.; Boqué, R.; Kalivas, J. H.; Mark, H. “Uncertainty Estimation and Figures of Merit for Multivariate Calibration,” Pure Appl. Chem. 2006, 78, 633–661.
An additional discussion on method blanks, including the use of the total Youden blank, is found in the fol- lowing papers.
• Cardone, M. J. “Detection and Determination of Error in Analytical Methodology. Part II. Correc- tion for Corrigible Systematic Error in the Course of Real Sample Analysis,” J. Assoc. Off. Anal. Chem. 1983, 66, 1283–1294.
• Cardone, M. J. “Detection and Determination of Error in Analytical Methodology. Part IIB. Direct Calculational Technique for Making Corrigible Systematic Error Corrections,” J. Assoc. Off. Anal. Chem. 1985, 68, 199–202.
• Ferrus, R.; Torrades, F. “Bias-Free Adjustment of Analytical Methods to Laboratory Samples in Routine Analytical Procedures,” Anal. Chem. 1988, 60, 1281–1285.
• Vitha, M. F.; Carr, P. W.; Mabbott, G. A. “Appropriate Use of Blanks, Standards, and Controls in Chemical Measurements,” J. Chem. Educ. 2005, 82, 901–902.
There are a variety of computational packages for completing linear regression analyses. These papers provide details on there use in a variety of contexts.
• Espinosa-Mansilla, A.; de la Peña, A. M.; González-Gómez, D. “Using Univariate Linear Regression Calibration Software in the MATLAB Environment. Application to Chemistry Laboratory Practices,” Chem. Educator 2005, 10, 1–9.
• Harris, D. C. “Nonlinear Least-Squares Curve Fitting with Microsoft Excel Solver,” J. Chem. Educ. 1998, 75, 119–121.
• Kim, M. S.; Bukart, M.; Kim, M. H. “A Method Visual Interactive Regression,” J. Chem. Educ. 2006, 83, 1884.
• Machuca-Herrera, J. G. “Nonlinear Curve Fitting with Spreadsheets,” J. Chem. Educ. 1997, 74, 448–449.
• Smith, E. T.; Belogay, E. A.; Hõim “Linear Regression and Error Analysis for Calibration Curves and Standard Additions: An Excel Spreadsheet Exercise for Undergraduates,” Chem. Educator 2010, 15, 100–102.
• Smith, E. T.; Belogay, E. A.; Hõim “Using Multiple Linear Regression to Analyze Mixtures: An Excel Spreadsheet Exercise for Undergraduates,” Chem. Educator 2010, 15, 103–107.
• Young, S. H.; Wierzbicki, A. “Mathcad in the Chemistry Curriculum. Linear Least-Squares Regres- sion,” J. Chem. Educ. 2000, 77, 669.
• Young, S. H.; Wierzbicki, A. “Mathcad in the Chemistry Curriculum. Non-Linear Least-Squares Re- gression,” J. Chem. Educ. 2000, 77, 669.
5.09: Chapter Summary and Key Terms
Summary
In a quantitative analysis we measure a signal, Stotal, and calculate the amount of analyte, nA or CA, using one of the following equations.
Stotal = kAnA + Sreag
Stotal = kACA + Sreag
To obtain an accurate result we must eliminate determinate errors that affect the signal, Stotal, the method’s sensitivity, kA, and the signal due to the reagents, Sreag.
To ensure that we accurately measure Stotal, we calibrate our equipment and instruments. To calibrate a balance, for example, we use a standard weight of known mass. The manufacturer of an instrument usually suggests appropriate calibration standards and calibration methods.
To standardize an analytical method we determine its sensitivity. There are several standardization strategies available to us, including external standards, the method of standard addition, and internal standards. The most common strategy is a multiple-point external standardization and a normal calibration curve. We use the method of standard additions, in which we add known amounts of analyte to the sample, when the sample’s matrix complicates the analysis. When it is difficult to reproducibly handle samples and standards, we may choose to add an internal standard.
Single-point standardizations are common, but are subject to greater uncertainty. Whenever possible, a multiple-point standardization is preferred, with results displayed as a calibration curve. A linear regression analysis provides an equation for the standardization.
A reagent blank corrects for any contribution to the signal from the reagents used in the analysis. The most common reagent blank is one in which an analyte-free sample is taken through the analysis. When a simple reagent blank does not compensate for all constant sources of determinate error, other types of blanks, such as the total Youden blank, are used.
Key Terms
calibration curve
linear regression
multiple-point standardization
reagent grade
serial dilution
total Youden blank
external standard
matrix matching
normal calibration curve
residual error
single-point standardization
unweighted linear regression
internal standard
method of standard additions
primary standard
secondary standard
standard deviation about the regression
weighted linear regression | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/05%3A_Standardizing_Analytical_Methods/5.08%3A_Additional_Resources.txt |
Regardless of the problem on which an analytical chemist is working, its solution requires a knowledge of chemistry and the ability to apply that knowledge to solve a problem. For example, an analytical chemist who is studying the effect of pollution on spruce trees needs to know, or know where to find, the chemical differences between p‐hydroxybenzoic acid and p‐hydroxyacetophenone, two common phenols found in the needles of spruce trees.
The ability to “think as a chemist” is a product of your experience in the classroom and in the laboratory. For the most part, the material in this text assumes you are familiar with topics covered in earlier courses; however, because of its importance to analytical chemistry, this chapter provides a review of equilibrium chemistry. Much of the material in this chapter should be familiar to you, although some topics—ladder diagrams and activity, for example—likely afford you with new ways to look at equilibrium chemistry.
• 6.1: Reversible Reactions and Chemical Equilibria
Although a system at equilibrium appears static on a macroscopic level, it is important to remember that the forward and the reverse reactions continue to occur. A reaction at equilibrium exists in a steady‐state, in which the rate at which a species forms equals the rate at which it is consumed.
• 6.2: Thermodynamics and Equilibrium Chemistry
Thermodynamics is the study of thermal, electrical, chemical, and mechanical forms of energy. The study of thermodynamics crosses many disciplines, including physics, engineering, and chemistry. Of the various branches of thermodynamics, the most important to chemistry is the study of how energy changes during a chemical reaction.
• 6.3: Manipulating Equilibrium Constants
We can take advantage of two useful relationships when we work with equilibrium constants. First, if we reverse a reaction’s direction, the equilibrium constant for the new reaction is the inverse of that for the original reaction. Second, if we add together two reactions to form a new reaction, the equilibrium constant for the new reaction is the product of the equilibrium constants for the original reactions.
• 6.4: Equilibrium Constants for Chemical Reactions
Several types of chemical reactions are important in analytical chemistry, either in preparing a sample for analysis or during the analysis. The most significant of these are precipitation reactions, acid–base reactions, complexation reactions, and oxidation–reduction reactions. In this section we review these reactions and their equilibrium constant expressions.
• 6.5: Le Châtelier’s Principle
The observation that a system at equilibrium responds to an external action by reequilibrating itself in a manner that diminishes that action, is formalized as Le Châtelier’s principle.
• 6.6: Ladder Diagrams
In this section we introduce the ladder diagram as a simple graphical tool for visualizing equilibrium chemistry. We will use ladder diagrams to determine what reactions occur when we combine several reagents, to estimate the approximate composition of a system at equilibrium, and to evaluate how a change to solution conditions might affect an analytical method.
• 6.7: Solving Equilibrium Problems
Ladder diagrams are a useful tool for evaluating chemical reactivity and for providing a reasonable estimate of a chemical system’s composition at equilibrium. If we need a more exact quantitative description of the equilibrium condition, then a ladder diagram is insufficient; instead, we need to find an algebraic solution. In this section we will learn how to set‐up and solve equilibrium problems.
• 6.8: Buffer Solutions
Adding as little as 0.1 mL of concentrated HCl to a liter of $\text{H}_2\text{O}$ shifts the pH from 7.0 to 3.0. Adding the same amount of HCl to a liter of a solution that 0.1 M in acetic acid and 0.1 M in sodium acetate, however, results in a negligible change in pH. Why do these two solutions respond so differently to the addition of HCl? A mixture of acetic acid and sodium acetate is one example of an acid–base buffer.
• 6.9: Activity Effects
Careful measurements on the metal–ligand complex $\text{Fe(SCN)}^{2+}$ suggest its stability, and thus its equilibrium constant, decreases in the presence of inert ions. Understanding why this is so is critical to developing a complete understanding of equilibrium chemistry.
• 6.10: Using Excel and R to Solve Equilibrium Problems
In solving equilibrium problems we typically make assumptions to simplify the algebra. These assumptions are important because they allow us to reduce the problem to an equation in $x$ that we can solve by simply taking a square‐root, a cube‐root, or by using the quadratic equation. Without these assumptions, most equilibrium problems result in a higher‐order equation that is more challenging to solve. Both Excel and R are useful tools for solving such equations.
• 6.11: Some Final Thoughts on Equilibrium Calculations
In this chapter we developed several tools to evaluate the composition of a system at equilibrium. These tools differ in how precisely they allow us to answer questions involving equilibrium chemistry. They also differ in how easy they are to use. An important part of having several tools to choose from is knowing when to each is most useful.
• 6.12: Problems
End-of-chapter problems to test your understanding of topics in this chapter.
• 6.13: Additional Resources
A compendium of resources to accompany topics in this chapter.
• 6.14: Chapter Summary and Key Terms
Summary of chapter's main topics and a list of key terms introduced in this chapter.
06: Equilibrium Chemistry
In 1798, the chemist Claude Berthollet accompanied Napoleon’s military expedition to Egypt. While visiting the Natron Lakes, a series of salt water lakes carved from limestone, Berthollet made an observation that led him to an important discovery. When exploring the lake’s shore, Berthollet found deposits of Na2CO3, a result he found surprising. Why did Berthollet find this result surprising and how did it contribute to an important discovery? Answering these questions provides us with an example of chemical reasoning and introduces us to the topic of this chapter.
Napoleon’s expedition to Egypt was the first to include a significant scientific presence. The Commission of Sciences and Arts, which included Claude Berthollet, began with 151 members, and operated in Egypt for three years. In addition to Berthollet’s work, other results included a publication on mirages and a detailed catalogs of plant and animal life, mineralogy, and archeology. For a review of the Commission’s contributions, see Gillispie, C. G. “Scientific Aspects of the French Egyptian Expedition, 1798‐1801,” Proc. Am. Phil. Soc. 1989, 133, 447–474.
At the end of the 18th century, chemical reactivity was explained in terms of elective affinities [Quilez, J. Chem. Educ. Res. Pract. 2004, 5, 69–87]. If, for example, substance A reacts with substance BC to form AB
$\text{A}+\text{BC} \rightarrow \text{AB}+\text{C} \nonumber$
then A and B were said to have an elective affinity for each other. With elective affinity as the driving force for chemical reactivity, reactions were understood to proceed to completion and to proceed in one direction. Once formed, the compound AB could not revert to A and BC.
$\text{A}+\text{BC} \nrightarrow \text{AB}+\text{C} \nonumber$
From his experience in the laboratory, Berthollet knew that adding solid Na2CO3 to a solution of CaCl2 produces a precipitate of CaCO3.
$\mathrm{Na}_{2} \mathrm{CO}_{3}(s)+\mathrm{CaCl}_{2}(a q) \rightarrow 2 \mathrm{NaCl}(a q)+\mathrm{CaCO}_{3}(s) \nonumber$
Understanding this, Berthollet was surprised to find solid Na2CO3 forming on the edges of the lake, particularly since the deposits formed only when the lake’s salt water, NaCl(aq), was in contact with solid limestone, CaCO3(s). Where the lake was in contact with clay soils, there was little or no Na2CO3.
Natron is another name for the mineral sodium carbonate, Na2CO3•10H2O. In nature, it usually contains impurities of NaHCO3 and NaCl. In ancient Egypt, natron was mined and used for a variety of purposes, including as a cleaning agent and in mummification.
Berthollet’s important insight was recognizing that the chemistry leading to the formation of Na2CO3 is the reverse of that seen in the laboratory.
$2 \mathrm{NaCl}(a q)+\mathrm{CaCO}_{3}(s) \rightarrow \mathrm{Na}_{2} \mathrm{CO}_{3}(s)+\mathrm{CaCl}_{2}(a q) \nonumber$
Using this insight Berthollet reasoned that the reaction is reversible, and that the relative amounts of NaCl, CaCO3, Na2CO3, and CaCl2 determine the direction in which the reaction occurs and the final composition of the reaction mixture. We recognize a reaction’s ability to move in both directions by using a double arrow when we write the reaction.
$\mathrm{Na}_{2} \mathrm{CO}_{3}(s)+\mathrm{CaCl}_{2}(a q) \rightleftharpoons 2 \mathrm{NaCl}(a q)+\mathrm{CaCO}_{3}(s) \nonumber$
For obvious reasons, we call the double arrow, $\rightleftharpoons$, an equilibrium arrow.
Berthollet’s reasoning that reactions are reversible was an important step in understanding chemical reactivity. When we mix together solutions of Na2CO3 and CaCl2 they react to produce NaCl and CaCO3. As the reaction takes place, if we monitor the mass of Ca2+ that remains in solution and the mass of CaCO3 that precipitates, the result looks something like Figure 6.1.1 . At the start of the reaction the mass of Ca2+ decreases and the mass of CaCO3 increases. Eventually the reaction reaches a point after which there is no further change in the amounts of these species. Such a condition is called a state of equilibrium.
Although a system at equilibrium appears static on a macroscopic level, it is important to remember that the forward and the reverse reactions continue to occur. A reaction at equilibrium exists in a steady‐state, in which the rate at which a species forms equals the rate at which it is consumed. | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/06%3A_Equilibrium_Chemistry/6.01%3A_Reversible_Reactions_and_Chemical_Equilibria.txt |
Thermodynamics is the study of thermal, electrical, chemical, and mechanical forms of energy. The study of thermodynamics crosses many disciplines, including physics, engineering, and chemistry. Of the various branches of thermodynamics, the most important to chemistry is the study of how energy changes during a chemical reaction.
Consider, for example, the general equilibrium reaction shown in Equation \ref{6.1}, which involves the species A, B, C, and D, with stoichiometric coefficients of a, b, c, and d.
$a A+b B \rightleftharpoons c C+d D \label{6.1}$
By convention, we identify the species on the left side of the equilibrium arrow as reactants and those on the right side of the equilibrium arrow as products. As Berthollet discovered, writing a reaction in this fashion does not guarantee that the reaction of A and B to produce C and D is favorable. Depending on initial conditions the reaction may move to the left, it may move to the right, or it may exist in a state of equilibrium. Understanding the factors that determine the reaction’s final equilibrium position is one of the goals of chemical thermodynamics.
The direction of a reaction is that which lowers the overall free energy. At a constant temperature and pressure, which is typical of many benchtop chemical reactions, a reaction’s free energy is given by the Gibb’s free energy function
$\Delta G=\Delta H-T \Delta S \label{6.2}$
where T is the temperature in kelvin, and ∆G, ∆H, and ∆S are the differences in the Gibb's free energy, the enthalpy, and the entropy between the products and the reactants.
Enthalpy is a measure of the flow of energy, as heat, during a chemical reaction. A reaction that releases heat has a negative ∆H and is called exothermic. An endothermic reaction absorbs heat from its surroundings and has a positive ∆H. Entropy is a measure of energy that is unavailable for useful, chemical work. The entropy of an individual species is always positive and generally is larger for gases than for solids, and for more complex molecules than for simpler molecules. Reactions that produce a large number of simple, gaseous products usually have a positive ∆S.
For many students, entropy is the most difficult topic in thermodynamics to understand. For a rich resource on entropy, visit the following web site: http://entropysite.oxy.edu/.
The sign of ∆G indicates the direction in which a reaction moves to reach its equilibrium position. A reaction is thermodynamically favorable when its enthalpy, ∆H, decreases and its entropy, ∆S, increases. Substituting the inequalities ∆H < 0 and ∆S > 0 into Equation \ref{6.2} shows that a reaction is thermodynamically favorable when ∆G is negative. When ∆G is positive the reaction is unfavorable as written (although the reverse reaction is favorable). A reaction at equilibrium has a ∆G of zero.
Equation \ref{6.2} shows that the sign of ∆G depends on the signs of ∆H and of ∆S, and the temperature, T. The following table summarizes the possibilities.
$\Delta H$ $\Delta S$ $\Delta G$
$–$ $+$ $\Delta G < 0$ at all temperatures
$-$ $-$ $\Delta G < 0$ at low temperatures only
$+$ $+$ $\Delta G < 0$ at high temperatures only
$+$ $-$
$\Delta G > 0$ at all temperatures
Note that the what constitutes "low temperatures" or "high temperatures" depends on the reaction.
As a reaction moves from its initial, non‐equilibrium condition to its equilibrium position, its value of ∆G approaches zero. At the same time, the chemical species in the reaction experience a change in their concentrations. The Gibb's free energy, therefore, must be a function of the concentrations of reactants and products.
As shown in Equation \ref{6.3}, we can divide the Gibb’s free energy, ∆G, into two terms.
$\triangle G=\Delta G^{\circ}+R T \ln Q_r \label{6.3}$
The first term, ∆Go, is the change in the Gibb’s free energy when each species in the reaction is in its standard state, which we define as follows: gases with unit partial pressures, solutes with unit concentrations, and pure solids and pure liquids. The second term includes the reaction quotient, $Q_r$, which accounts for non‐standard state pressures and concentrations. For reaction \ref{6.1} the reaction quotient is
$Q_r = \frac{[\mathrm{C}]^{c}[\mathrm{D}]^{d}}{[\mathrm{A}]^{a}[\mathrm{B}]^{b}} \label{6.4}$
where the terms in brackets are the concentrations of the reactants and products. Note that we define the reaction quotient with the products in the numerator and the reactants in the denominator. In addition, we raise the concentration of each species to a power equivalent to its stoichiometry in the balanced chemical reaction. For a gas, we use partial pressure in place of concentration. Pure solids and pure liquids do not appear in the reaction quotient.
Although not shown here, each concentration term in Equation \ref{6.4} is divided by the corresponding standard state concentration; thus, the term [C]c really means
$\left\{\frac{[\mathrm{C}]}{[\mathrm{C}]^{\circ}}\right\} \nonumber$
where [C]o is the standard state concentration for C. There are two important consequences of this: (1) the value of Q is unitless; and (2) the ratio has a value of 1 for a pure solid or a pure liquid. This is the reason that pure solids and pure liquids do not appear in the reaction quotient.
At equilibrium the Gibb’s free energy is zero, and Equation \ref{6.3} simplifies to
$\triangle G^{\circ}=-R T \ln K \nonumber$
where K is an equilibrium constant that defines the reaction’s equilibrium position. The equilibrium constant is just the reaction quotient’s numerical value when we substitute equilibrium concentrations into Equation \ref{6.4}.
$K = \frac{[\mathrm{C}]_{\mathrm{eq}}^{c}[\mathrm{D}]_{\mathrm{eq}}^{d}}{[\mathrm{A}]_{\mathrm{eq}}^{a}[\mathrm{B}]_{\mathrm{eq}}^{b}} \label{6.5}$
Here we include the subscript “eq” to indicate a concentration at equilibrium. Although generally we will omit the “eq” when we write an equilibrium constant expressions, it is important to remember that the value of K is determined by equilibrium concentrations.
As written, Equation \ref{6.5} is a limiting law that applies only to infinitely dilute solutions where the chemical behavior of one species is unaffected by the presence of other species. Strictly speaking, Equation \ref{6.5} is written in terms of activities instead of concentrations. We will return to this point in Chapter 6.9. For now, we will stick with concentrations as this convention already is familiar to you.
6.03: Manipulating Equilibrium Constants
We will take advantage of two useful relationships when we work with equilibrium constants. First, if we reverse a reaction’s direction, the equilibrium constant for the new reaction is the inverse of that for the original reaction. For example, the equilibrium constant for the reaction
$\mathrm{A}+2 \mathrm{B}\rightleftharpoons \mathrm{AB}_{2} \quad \quad K_{1}=\frac{\left[\mathrm{AB}_{2}\right]}{[\mathrm{A}][\mathrm{B}]^{2}} \nonumber$
is the inverse of that for the reaction
$\mathrm{AB}_{2}\rightleftharpoons \mathrm{A}+2 \mathrm{B} \quad \quad K_{2}=\left(K_{1}\right)^{-1}=\frac{[\mathrm{A}][\mathrm{B}]^{2}}{\left[\mathrm{AB}_{2}\right]} \nonumber$
Second, if we add together two reactions to form a new reaction, the equilibrium constant for the new reaction is the product of the equilibrium constants for the original reactions.
$A+C\rightleftharpoons A C \quad \quad K_{3}=\frac{[A C]}{[A][C]} \nonumber$
$\mathrm{AC}+\mathrm{C}\rightleftharpoons\mathrm{AC}_{2} \quad \quad K_{4}=\frac{\left[\mathrm{AC}_{2}\right]}{[\mathrm{AC}][\mathrm{C}]} \nonumber$
$\mathrm{A}+2 \mathrm{C}\rightleftharpoons \mathrm{AC}_{2} \quad \quad K_{5}=K_{3} \times K_{4}=\frac{[\mathrm{AC}]}{[\mathrm{A}][\mathrm{C}]} \times \frac{\left[\mathrm{AC}_{2}\right]}{[\mathrm{AC}][\mathrm{C}]}=\frac{\left[\mathrm{AC}_{2}\right]}{[\mathrm{A}][\mathrm{C}]^{2}} \nonumber$
Example 6.3.1
Calculate the equilibrium constant for the reaction
$2 \mathrm{A}+\mathrm{B}\rightleftharpoons \mathrm{C}+3 \mathrm{D} \nonumber$
given the following information
$\begin{array}{ll}{\text{Rxn} \ 1 : A+B\rightleftharpoons D} & {K_{1}=0.40} \ {\text{Rxn} \ 2 : A+E\rightleftharpoons C+D+F} & {K_{2}=0.10} \ {\text{Rxn} \ 3 : C+E\rightleftharpoons B} & {K_{3}=2.0} \ {\text{Rxn} \ 4 : F+C\rightleftharpoons D+B} & {K_{4}=5.0}\end{array} \nonumber$
Solution
The overall reaction is equivalent to
$\text{Rxn} \ 1+\text{Rxn} \ 2-\text{Rxn} \ 3+\text{Rxn} \ 4 \nonumber$
Subtracting a reaction is equivalent to adding the reverse reaction; thus, the overall equilibrium constant is
$K=\frac{K_{1} \times K_{2} \times K_{4}}{K_{3}}=\frac{0.40 \times 0.10 \times 5.0}{2.0}=0.10 \nonumber$
Exercise 6.3.1
Calculate the equilibrium constant for the reaction
$C+D+F \rightleftharpoons 2 A+3 B \nonumber$
using the equilibrium constants from Example 6.3.1 .
Answer
The overall reaction is equivalent to
$\operatorname{Rxn} 4-2 \times \operatorname{Rxn} 1 \nonumber$
Subtracting a reaction is equivalent to adding the reverse reaction; thus, the overall equilibrium constant is
$K=\frac{K_{4}}{\left(K_{1}\right)^{2}}=\frac{(5.0)}{(0.40)^{2}}=31.25 \approx 31 \nonumber$ | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/06%3A_Equilibrium_Chemistry/6.02%3A_Thermodynamics_and_Equilibrium_Chemistry.txt |
Several types of chemical reactions are important in analytical chemistry, either in preparing a sample for analysis or during the analysis. The most significant of these are precipitation reactions, acid–base reactions, complexation reactions, and oxidation–reduction reactions. In this section we review these reactions and their equilibrium constant expressions.
Another common name for an oxidation–reduction reaction is a redox reaction, where “red” is short for reduction and “ox” is short for oxidation.
Precipitation Reactions
In a precipitation reaction, two or more soluble species combine to form an insoluble precipitate. The most common precipitation reaction is a metathesis reaction in which two soluble ionic compounds exchange parts. For example, if we add a solution of lead nitrate, Pb(NO3)2, to a solution of potassium chloride, KCl, a precipitate of lead chloride, PbCl2, forms. We usually write a precipitation reaction as a net ionic equation, which shows only the precipitate and those ions that form the precipitate; thus, the precipitation reaction for PbCl2 is
$\mathrm{Pb}^{2+}(a q)+2 \mathrm{Cl}^{-}(a q) \rightleftharpoons \mathrm{PbCl}_{2}(s) \nonumber$
When we write the equilibrium constant for a precipitation reaction, we focus on the precipitate’s solubility; thus, for PbCl2, the solubility reaction is
$\mathrm{PbCl}_{2}(s)\rightleftharpoons \mathrm{Pb}^{2+}(a q)+2 \mathrm{Cl}^{-}(a q) \nonumber$
and its equilibrium constant, or solubility product, Ksp, is
$K_{\mathrm{sp}}=\left[\mathrm{Pb}^{2+}\right]\left[\mathrm{Cl}^{-}\right]^{2} \label{6.1}$
Even though it does not appear in the Ksp expression, it is important to remember that Equation \ref{6.1} is valid only if PbCl2(s) is present and in equilibrium with Pb2+ and Cl. You will find values for selected solubility products in Appendix 10.
Acid–Base Reactions
A useful definition of acids and bases is that independently introduced in 1923 by Johannes Brønsted and Thomas Lowry. In the Brønsted‐Lowry definition, an acid is a proton donor and a base is a proton acceptor. Note the connection between these definitions—defining a base as a proton acceptor implies there is an acid available to donate the proton. For example, in reaction \ref{6.2} acetic acid, CH3COOH, donates a proton to ammonia, NH3, which serves as the base.
$\mathrm{CH}_{3} \mathrm{COOH}(aq)+\mathrm{NH}_{3}(aq) \rightleftharpoons \mathrm{NH}_{4}^{+}(aq)+\mathrm{CH}_{3} \mathrm{COO}^{-}(aq) \label{6.2}$
When an acid and a base react, the products are a new acid and a new base. For example, the acetate ion, CH3COO, in reaction \ref{6.2} is a base that can accept a proton from the acidic ammonium ion, $\text{NH}_4^+$, forming acetic acid and ammonia. We call the acetate ion the conjugate base of acetic acid, and we call the ammonium ion the conjugate acid of ammonia.
Strong and Weak Acids
The reaction of an acid with its solvent (typically water) is an acid dissociation reaction. We divide acids into two categories—strong and weak—based on their ability to donate a proton to the solvent. A strong acid, such as HCl, almost completely transfers its proton to the solvent, which acts as the base.
$\mathrm{HCl}(a q)+\mathrm{H}_{2} \mathrm{O}(l) \rightarrow \mathrm{H}_{3} \mathrm{O}^{+}(a q)+\mathrm{Cl}^{-}(a q) \nonumber$
We use a single arrow ($\rightarrow$) in place of the equilibrium arrow ($\rightleftharpoons$) because we treat HCl as if it dissociates completely in an aqueous solution. In water, the common strong acids are hydrochloric acid (HCl), hydroiodic acid (HI), hydrobromic acid (HBr), nitric acid (HNO3), perchloric acid (HClO4), and the first proton of sulfuric acid (H2SO4).
The strength of an acid is a function of the acid and the solvent. For example, HCl does not act as a strong acid in methanol. In this case we use the equilibrium arrow when writing the acid–base reaction.
$\mathrm{HCl}+\mathrm{CH}_{3} \mathrm{OH}\rightleftharpoons \mathrm{CH}_{3} \mathrm{OH}_{2}^{+}+\mathrm{Cl}^{-} \nonumber$
A weak acid, of which aqueous acetic acid is one example, does not completely donate its acidic proton to the solvent. Instead, most of the acid remains undissociated with only a small fraction present as the conjugate base.
$\mathrm{CH}_{3} \mathrm{COOH}(a q)+\mathrm{H}_{2} \mathrm{O}(l)\rightleftharpoons \mathrm{H}_{3} \mathrm{O}^{+}(a q)+\mathrm{CH}_{3} \mathrm{COO}^{-}(a q) \nonumber$
The equilibrium constant for this reaction is an acid dissociation constant, Ka, which we write as
$K_{a}=\frac{\left[\mathrm{CH}_{3} \mathrm{COO}^{-}\right]\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]}{\left[\mathrm{CH}_{3} \mathrm{COOH}\right]}=1.75 \times 10^{-5} \nonumber$
The magnitude of K provides information about a weak acid's relative strength, with a smaller Ka corresponding to a weaker acid. The ammonium ion, $\text{NH}_4^+$, for example, has a Ka of $5.702 \times 10^{-10}$ and is a weaker acid than acetic acid.
Earlier we noted that we omit pure solids and pure liquids from equilibrium constant expressions. Because the solvent, H2O, is not pure, you might wonder why we have not included it in acetic acid’s Ka expression. Recall that we divide each term in an equilibrium constant expression by its standard state value. Because the concentration of H2O is so large—it is approximately 55.5 mol/L—its concentration as a pure liquid and as a solvent are virtually identical. The ratio
$\frac{\left[\mathrm{H}_{2} \mathrm{O}\right]}{\left[\mathrm{H}_{2} \mathrm{O}\right]^{\circ}} \nonumber$
is essentially 1.00.
A monoprotic weak acid, such as acetic acid, has only a single acidic proton and a single acid dissociation constant. Other acids, such as phosphoric acid, have multiple acidic protons, each characterized by an acid dissociation constant. We call such acids polyprotic. Phosphoric acid, for example, has three acid dissociation reactions and three acid dissociation constants.
$\mathrm{H}_{3} \mathrm{PO}_{4}(a q)+\mathrm{H}_{2} \mathrm{O}(l) \rightleftharpoons \mathrm{H}_{3} \mathrm{O}^{+}(a q)+\mathrm{H}_{2} \mathrm{PO}_{4}^{-}(a q) \nonumber$
$K_{\mathrm{al}}=\frac{\left[\mathrm{H}_{2} \mathrm{PO}_{4}^{-}\right]\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]}{\left[\mathrm{H}_{3} \mathrm{PO}_{4}\right]}=7.11 \times 10^{-3} \nonumber$
$\mathrm{H}_{2} \mathrm{PO}_{4}^-(a q)+\mathrm{H}_{2} \mathrm{O}(l)\rightleftharpoons \mathrm{H}_{3} \mathrm{O}^{+}(a q)+\mathrm{HPO}_{4}^{2-}(a q) \nonumber$
$K_{a 2}=\frac{\left[\mathrm{HPO}_{4}^{2-}\right]\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]}{\left[\mathrm{H}_{2} \mathrm{PO}_{4}^-\right]}=6.32 \times 10^{-8} \nonumber$
$\mathrm{HPO}_{4}^{2-}(a q)+\mathrm{H}_{2} \mathrm{O}({l})\rightleftharpoons \mathrm{H}_{3} \mathrm{O}^{+}(a q)+\mathrm{PO}_{4}^{3-}(a q) \nonumber$
$K_{\mathrm{a} 3}=\frac{\left[\mathrm{PO}_{4}^{3-}\right]\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]}{\left[\mathrm{HPO}_{4}^{2-}\right]}=4.5 \times 10^{-13} \nonumber$
The decrease in the acid dissociation constants from Ka1 to Ka3 tells us that each successive proton is harder to remove. Consequently, H3PO4 is a stronger acid than $\text{H}_2\text{PO}_4^-$, and $\text{H}_2\text{PO}_4^-$ is a stronger acid than $\text{HPO}_4^{2-}$.
Strong and Weak Bases
The most common example of a strong base is an alkali metal hydroxide, such as sodium hydroxide, NaOH, which completely dissociates to produce hydroxide ion.
$\mathrm{NaOH}(s) \rightarrow \mathrm{Na}^{+}(a q)+\mathrm{OH}^{-}(a q) \nonumber$
A weak base, such as the acetate ion, CH3COO, only partially accepts a proton from the solvent, and is characterized by a base dissociation constant, Kb. For example, the base dissociation reaction and the base dissociation constant for the acetate ion are
$\mathrm{CH}_{3} \mathrm{COO}^{-}(a q)+\mathrm{H}_{2} \mathrm{O}(l)\rightleftharpoons \mathrm{OH}^{-}(a q)+\mathrm{CH}_{3} \mathrm{COOH}(a q) \nonumber$
$K_{\mathrm{b}}=\frac{\left[\mathrm{CH}_{3} \mathrm{COOH}\right]\left[\mathrm{OH}^{-}\right]}{\left[\mathrm{CH}_{3} \mathrm{COO}^{-}\right]}=5.71 \times 10^{-10} \nonumber$
A polyprotic weak base, like a polyprotic acid, has more than one base dissociation reaction and more than one base dissociation constant.
Amphiprotic Species
Some species can behave as either a weak acid or as a weak base. For example, the following two reactions show the chemical reactivity of the bicarbonate ion, $\text{HCO}_3^-$, in water.
$\mathrm{HCO}_{3}^{-}(a q)+\mathrm{H}_{2} \mathrm{O}(l)\rightleftharpoons \mathrm{H}_{3} \mathrm{O}^{+}(a q)+\mathrm{CO}_{3}^{2-}(a q) \label{6.3}$
$\mathrm{HCO}_{3}^{-}(a q)+\mathrm{H}_{2} \mathrm{O}(l)\rightleftharpoons \mathrm{OH}^{-}(a q)+\mathrm{H}_{2} \mathrm{CO}_{3}(a q) \label{6.4}$
A species that is both a proton donor and a proton acceptor is called amphiprotic. Whether an amphiprotic species behaves as an acid or as a base depends on the equilibrium constants for the competing reactions. For bicarbonate, the acid dissociation constant for reaction \ref{6.3}
$K_{a 2}=\frac{\left[\mathrm{CO}_{3}^{2-}\right]\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]}{\left[\mathrm{HCO}_{3}^{-}\right]}=4.69 \times 10^{-11} \nonumber$
is smaller than the base dissociation constant for reaction \ref{6.4}.
$K_{\mathrm{b} 2}=\frac{\left[\mathrm{H}_{2} \mathrm{CO}_{3}\right]\left[\mathrm{OH}^{-}\right]}{\left[\mathrm{HCO}_{3}^{-}\right]}=2.25 \times 10^{-8} \nonumber$
Because bicarbonate is a stronger base than it is an acid, we expect that an aqueous solution of $\text{HCO}_3^-$ is basic.
Dissociation of Water
Water is an amphiprotic solvent because it can serve as an acid or as a base. An interesting feature of an amphiprotic solvent is that it is capable of reacting with itself in an acid–base reaction.
$2 \mathrm{H}_{2} \mathrm{O}(l)\rightleftharpoons \mathrm{H}_{3} \mathrm{O}^{+}(a q)+\mathrm{OH}^{-}(a q) \label{6.5}$
We identify the equilibrium constant for this reaction as water’s dissociation constant, Kw,
$K_{w}=\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\left[\mathrm{OH}^{-}\right]=1.00 \times 10^{-14} \label{6.6}$
at a temperature of 24oC. The value of Kw varies substantially with temperature. For example, at 20oC Kw is $6.809 \times 10^{-15}$, while at 30oC Kw is $1.469 \times 10^{-14}$. At 25oC, Kw is $1.008 \times 10^{-14}$, which is sufficiently close to $1.00 \times 10^{-14}$ that we can use the latter value with negligible error.
An important consequence of Equation \ref{6.6} is that the concentration of H3O+ and the concentration of OH are related. If we know [H3O+] for a solution, then we can calculate [OH] using Equation \ref{6.6}.
Example 6.4.1
What is the [OH] if the [H3O+] is $6.12 \times 10^{-5}$ M?
Solution
$\left[\mathrm{OH}^{-}\right]=\frac{K_{w}}{\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]}=\frac{1.00 \times 10^{-14}}{6.12 \times 10^{-5}}=1.63 \times 10^{-10} \nonumber$
The pH Scale
Equation \ref{6.6} allows us to develop a pH scale ($\text{pH} = - \log [\text{H}_3\text{O}^+]$) that indicates a solution’s acidity. When the concentrations of H3O+ and OH are equal a solution is neither acidic nor basic; that is, the solution is neutral. Letting
$\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=\left[\mathrm{OH}^{-}\right] \nonumber$
substituting into Equation \ref{6.6}
$K_{w}=\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]^{2}=1.00 \times 10^{-14} \nonumber$
and solving for [H3O+] gives
$\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=\sqrt{1.00 \times 10^{-14}}=1.00 \times 10^{-7} \nonumber$
A neutral solution of water at 25oC has a hydronium ion concentration of $1.00 \times 10^{-7}$ M and a pH of 7.00. In an acidic solution the concentration of H3O+ is greater than that for OH, which means that
$\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]>1.00 \times 10^{-7} \mathrm{M} \nonumber$
The pH of an acidic solution, therefore, is less than 7.00. A basic solution, on the other hand, has a pH greater than 7.00. Figure 6.4.1 shows the pH scale and pH values for some representative solutions.
Tabulating Values for Ka and Kb
A useful observation about weak acids and weak bases is that the strength of a weak base is inversely proportional to the strength of its conjugate weak acid. Consider, for example, the dissociation reactions of acetic acid and acetate.
$\mathrm{CH}_{3} \mathrm{COOH}(a q)+\mathrm{H}_{2} \mathrm{O}(l)\rightleftharpoons \ \mathrm{H}_{3} \mathrm{O}^{+}(a q)+\mathrm{CH}_{3} \mathrm{COO}^{-}(a q) \label{6.7}$
$\mathrm{CH}_{3} \mathrm{COO}^{-}(a q)+\mathrm{H}_{2} \mathrm{O}(l)\rightleftharpoons \mathrm{OH}^{-}(a q)+\mathrm{CH}_{3} \mathrm{COOH}(a q) \label{6.8}$
Adding together these two reactions gives the reaction
$2 \mathrm{H}_{2} \mathrm{O}(l)\rightleftharpoons \mathrm{H}_{3} \mathrm{O}^{+}(a q)+\mathrm{OH}^{-}(a q) \nonumber$
for which the equilibrium constant is Kw. Because adding together two reactions is equivalent to multiplying their respective equilibrium constants, we may express Kw as the product of Ka for CH3COOH and Kb for CH3COO–.
$K_{\mathrm{w}}=K_{\mathrm{a}, \mathrm{CH}_{3} \mathrm{COOH}} \times K_{\mathrm{b}, \mathrm{CH}_{3} \mathrm{COO}^{-}} \nonumber$
For any weak acid, HA, and its conjugate weak base, A, we can generalize this to the following equation
$K_{\mathrm{w}}=K_{\mathrm{a}, \mathrm{HA}} \times K_{\mathrm{b}, \mathrm{A}^{-}} \label{6.9}$
where HA and A are a conjugate acid–base pair. The relationship between Ka and Kb for a conjugate acid–base pair simplifies our tabulation of acid and base dissociation constants. Appendix 11 includes acid dissociation constants for a variety of weak acids. To find the value of Kb for a weak base, use Equation \ref{6.9} and the Ka value for its corresponding weak acid.
A common mistake when using Equation \ref{6.9} is to forget that it applies to a conjugate acid–base pair only.
Example 6.4.2
Using Appendix 11, calculate values for the following equilibrium constants.
1. Kb for pyridine, C5H5N
2. Kb for dihydrogen phosphate, $\text{H}_2\text{PO}_4^-$
Solution
$\text { (a) } K_{\mathrm{b}, \mathrm{C}_5 \mathrm{H}_{5} \mathrm{N}}=\frac{K_{\mathrm{w}}}{K_{\mathrm{a}, \mathrm{C}_{\mathrm{5}} \mathrm{H}_{5} \mathrm{NH}^{+}}}=\frac{1.00 \times 10^{-14}}{5.90 \times 10^{-6}}=1.69 \times 10^{-9} \nonumber$
$\text { (b) } K_{\mathrm{b}, \mathrm{H}_2 \mathrm{PO}_{4}^- }=\frac{K_{\mathrm{w}}}{K_{\mathrm{a}, \mathrm{H}_{\mathrm{3}} \mathrm{PO}_{4} }}=\frac{1.00 \times 10^{-14}}{7.11 \times 10^{-3}}=1.41 \times 10^{-12} \nonumber$
When finding the Kb value for a polyprotic weak base, be careful to choose the correct Ka value. Remember that Equation \ref{6.9} applies to a conjugate acid–base pair only. The conjugate acid of $\text{H}_2\text{PO}_4^-$ is H3PO4, not $\text{HPO}_4^{2-}$.
Exercise 6.4.1
Using Appendix 11, calculate Kb values for hydrogen oxalate, $\text{HC}_2\text{O}_4^-$, and for oxalate, $\text{C}_2\text{O}_4^{2-}$.
Answer
The Kb for hydrogen oxalate is
$K_{\mathrm{b}, \mathrm{HC}_{2} \mathrm{O}_{4}^-}=\frac{K_{\mathrm{w}}}{K_{\mathrm{a}, \mathrm{H}_{2} \mathrm{C}_{2} \mathrm{O}_{4}}}=\frac{1.00 \times 10^{-14}}{5.60 \times 10^{-2}}=1.79 \times 10^{-13} \nonumber$
and the Kb for oxalate is
$K_{\mathrm{b}, \mathrm{C}_{2} \mathrm{O}_{4}^{2-}}=\frac{K_{\mathrm{w}}}{K_{\mathrm{a}, \mathrm{HC}_{2} \mathrm{O}_{\mathrm{4}}^-}}=\frac{1.00 \times 10^{-14}}{5.42 \times 10^{-5}}=1.85 \times 10^{-10} \nonumber$
As we expect, the Kb value for $\text{C}_2\text{O}_4^{2-}$ is larger than that for $\text{HC}_2\text{O}_4^-$.
Complexation Reactions
A more general definition of acids and bases was proposed in 1923 by G. N. Lewis. The Brønsted‐Lowry definition of acids and bases focuses on an acid’s proton‐donating ability and a base’s proton‐accepting ability. Lewis theory, on the other hand, uses the breaking and the forming of covalent bonds to describe acids and bases. In this treatment, an acid is an electron pair acceptor and a base in an electron pair donor. Although we can apply Lewis theory to the treatment of acid–base reactions, it is more useful for treating complexation reactions between metal ions and ligands.
The following reaction between the metal ion Cd2+ and the ligand NH3 is typical of a complexation reaction.
$\mathrm{Cd}^{2+}(a q)+4: \mathrm{NH}_{3}(a q)\rightleftharpoons \mathrm{Cd}\left( : \mathrm{NH}_{3}\right)_{4}^{2+}(a q) \label{6.10}$
The product of this reaction is a metal–ligand complex. In writing this reaction we show ammonia as :NH3, using a pair of dots to emphasize the pair of electrons that it donates to Cd2+. In subsequent reactions we will omit this notation.
Metal-Ligand Formation Constants
We characterize the formation of a metal–ligand complex by a formation constant, Kf. For example, the complexation reaction between Cd2+ and NH3, reaction \ref{6.10}, has the following equilibrium constant.
$K_{f}=\frac{\left[\mathrm{Cd}\left(\mathrm{NH}_{3}\right)_{4}^{2+}\right]}{\left[\mathrm{Cd}^{2+}\right]\left[\mathrm{NH}_{3}\right]^{4}}=5.5 \times 10^{7} \label{6.11}$
The reverse of reaction \ref{6.10} is a dissociation reaction, which we characterize by a dissociation constant, Kd, that is the reciprocal of Kf.
Many complexation reactions occur in a stepwise fashion. For example, the reaction between Cd2+ and NH3 involves four successive reactions.
$\mathrm{Cd}^{2+}(a q)+\mathrm{NH}_{3}(a q) \rightleftharpoons \mathrm{Cd}\left(\mathrm{NH}_{3}\right)^{2+}(a q) \label{6.12}$
$\mathrm{Cd}\left(\mathrm{NH}_{3}\right)^{2+}(a q)+\mathrm{NH}_{3}(a q)\rightleftharpoons \mathrm{Cd}\left(\mathrm{NH}_{3}\right)_{2}^{2+}(a q) \label{6.13}$
$\mathrm{Cd}\left(\mathrm{NH}_{3}\right)_{2}^{2+}(a q)+\mathrm{NH}_{3}(a q)\rightleftharpoons \mathrm{Cd}\left(\mathrm{NH}_{3}\right)_{3}^{2+}(a q) \label{6.14}$
$\mathrm{Cd}\left(\mathrm{NH}_{3}\right)_{3}^{2+}(a q)+\mathrm{NH}_{3}(a q)\rightleftharpoons \mathrm{Cd}\left(\mathrm{NH}_{3}\right)_{4}^{2+}(a q) \label{6.15}$
To avoid ambiguity, we divide formation constants into two categories. A stepwise formation constant, which we designate as Ki for the ith step, describes the successive addition of one ligand to the metal–ligand complex from the previous step. Thus, the equilibrium constants for reactions \ref{6.12}–\ref{6.15} are, respectively, K1, K2, K3, and K4. An overall, or cumulative formation constant, which we designate as $\beta_i$, describes the addition of i ligands to the free metal ion. The equilibrium constant in Equation \ref{6.11} is correctly identified as $\beta_4$, where
$\beta_{4}=K_{1} \times K_{2} \times K_{3} \times K_{4} \nonumber$
In general
$\beta_{n}=K_{1} \times K_{2} \times \cdots \times K_{n}=\prod_{i=1}^{n} K_{i} \nonumber$
Stepwise and overall formation constants for selected metal–ligand complexes are in Appendix 12.
Metal-Ligand Complexation and Solubility
A formation constant describes the addition of one or more ligands to a free metal ion. To find the equilibrium constant for a complexation reaction that includes a solid, we combine appropriate Ksp and Kf expressions. For example, the solubility of AgCl increases in the presence of excess chloride ions as the result of the following complexation reaction.
$\operatorname{AgCl}(s)+\mathrm{Cl}^{-}(a q)\rightleftharpoons\operatorname{Ag}(\mathrm{Cl})_{2}^{-}(a q) \label{6.16}$
We can write this reaction as the sum of three other equilibrium reactions with known equilibrium constants—the solubility of AgCl, which is described by its Ksp reaction
$\mathrm{AgCl}(s) \rightleftharpoons \mathrm{Ag}^{+}(a q)+\mathrm{Cl}^{-}(a q) \nonumber$
and the stepwise formation of $\text{AgCl}_2^-$, which is described by K1and K 2 reactions.
$\mathrm{Ag}^{+}(a q)+\mathrm{Cl}^{-}(a q) \rightleftharpoons \operatorname{Ag} \mathrm{Cl}(a q) \nonumber$
$\operatorname{AgCl}(a q)+\mathrm{Cl}^{-}(a q) \rightleftharpoons \operatorname{AgCl}_{2}^{-}(a q) \nonumber$
The equilibrium constant for reaction \ref{6.16}, therefore, is $K_\text{sp} \times K_1 \times K_2$.
Example 6.4.3
Determine the value of the equilibrium constant for the reaction
$\mathrm{PbCl}_{2}(s)\rightleftharpoons \mathrm{PbCl}_{2}(a q) \nonumber$
Solution
We can write this reaction as the sum of three other reactions. The first of these reactions is the solubility of PbCl2(s), which is described by its Ksp reaction.
$\mathrm{PbCl}_{2}(s)\rightleftharpoons \mathrm{Pb}^{2+}(a q)+2 \mathrm{Cl}^{-}(a q) \nonumber$
The remaining two reactions are the stepwise formation of PbCl2(aq), which are described by K1 and K2.
$\mathrm{Pb}^{2+}(a q)+\mathrm{Cl}^{-}(a q)\rightleftharpoons \mathrm{PbCl}^{+}(a q) \nonumber$
$\mathrm{PbCl}^{+}(a q)+\mathrm{Cl}^{-}(a q)\rightleftharpoons \mathrm{PbCl}_{2}(a q) \nonumber$
Using values for Ksp, K1, and K2 from Appendix 10 and Appendix 12, we find that the equilibrium constant is
$K=K_{\mathrm{sp}} \times K_{1} \times K_{2}=\left(1.7 \times 10^{-5}\right) \times 38.9 \times 1.62=1.1 \times 10^{-3} \nonumber$
Exercise 6.4.2
What is the equilibrium constant for the following reaction? You will find appropriate equilibrium constants in Appendix 10 and Appendix 12.
$\operatorname{Ag} \mathrm{Br}(s)+2 \mathrm{S}_{2} \mathrm{O}_{3}^{2-}(a q)\rightleftharpoons\operatorname{Ag}\left(\mathrm{S}_{2} \mathrm{O}_{3}\right)_2^{3-}(a q)+\mathrm{Br}^{-}(a q) \nonumber$
Answer
We can write the reaction as a sum of three other reactions. The first reaction is the solubility of AgBr(s), which we characterize by its Ksp.
$\operatorname{AgBr}(s)\rightleftharpoons\operatorname{Ag}^{+}(a q)+\mathrm{Br}^{-}(a q) \nonumber$
The remaining two reactions are the stepwise formation of $\text{Ag(S}_2\text{O}_3)_2^{3-}$, which we characterize by K1 and K2.
$\mathrm{Ag}^{+}(a q)+\mathrm{S}_{2} \mathrm{O}_{3}^{2-}(a q)\rightleftharpoons\operatorname{Ag}\left(\mathrm{S}_{2} \mathrm{O}_{3}\right)^{-}(a q) \nonumber$
$\operatorname{Ag}\left(\mathrm{S}_{2} \mathrm{O}_{3}\right)^{-}(a q)+\mathrm{S}_{2} \mathrm{O}_{3}^{2-}(a q)\rightleftharpoons\operatorname{Ag}\left(\mathrm{S}_{2} \mathrm{O}_{3}\right)_{2}^{3-}(a q) \nonumber$
Using values for Ksp, K1, and K2 from Appendix 10 and Appendix 12, we find that the equilibrium constant for our reaction is
$K=K_{sp} \times K_{1} \times K_{2}=\left(5.0 \times 10^{-13}\right)\left(6.6 \times 10^{8}\right)\left(7.1 \times 10^{4}\right)=23 \nonumber$
Oxidation–Reduction (Redox) Reactions
An oxidation–reduction reaction occurs when electrons move from one reactant to another reactant. As a result of this transfer of electrons, the reactants undergo a change in oxidation state. Those reactant that increases its oxidation state undergoes oxidation, and the reactant that decreases its oxidation state undergoes reduction. For example, in the following redox reaction between Fe3+ and oxalic acid, H2C2O4, iron is reduced because its oxidation state changes from +3 to +2.
$2 \mathrm{Fe}^{3+}(a q)+\mathrm{H}_{2} \mathrm{C}_{2} \mathrm{O}_{4}(a q)+2 \mathrm{H}_{2} \mathrm{O}(l)\rightleftharpoons \ {2 \mathrm{Fe}^{2+}(a q)+2 \mathrm{CO}_{2}(g)+2 \mathrm{H}_{3} \mathrm{O}^{+}(a q)} \label{6.17}$
Oxalic acid, on the other hand, is oxidized because the oxidation state for carbon increases from +3 in H2C2O4 to +4 in CO2.
We can divide a redox reaction, such as reaction \ref{6.17}, into separate half‐reactions that show the oxidation and the reduction processes.
$\mathrm{H}_{2} \mathrm{C}_{2} \mathrm{O}_{4}(a q)+2 \mathrm{H}_{2} \mathrm{O}(l)\rightleftharpoons 2 \mathrm{CO}_{2}(g)+2 \mathrm{H}_{3} \mathrm{O}^{+}(a q)+2 e^{-} \nonumber$
$\mathrm{Fe}^{3+}(a q)+e^{-} \rightleftharpoons \mathrm{Fe}^{2+}(a q) \nonumber$
It is important to remember, however, that an oxidation reaction and a reduction reaction always occur as a pair. We formalize this relationship by identifying as a reducing agent the reactant that is oxidized, because it provides the electrons for the reduction half‐reaction. Conversely, the reactant that is reduced is an oxidizing agent. In reaction \ref{6.17}, Fe3+ is the oxidizing agent and H2C2O4 is the reducing agent.
The products of a redox reaction also have redox properties. For example, the Fe2+ in reaction \ref{6.17} is oxidized to Fe3+ when CO2 is reduced to H2C2O4. Borrowing some terminology from acid–base chemistry, Fe2+ is the conjugate reducing agent of the oxidizing agent Fe3+, and CO2 is the conjugate oxidizing agent of the reducing agent H2C2O4.
Thermodynamics of Redox Reactions
Unlike precipitation reactions, acid–base reactions, and complexation reactions, we rarely express the equilibrium position of a redox reaction with an equilibrium constant. Because a redox reaction involves a transfer of electrons from a reducing agent to an oxidizing agent, it is convenient to consider the reaction’s thermodynamics in terms of the electron.
For a reaction in which one mole of a reactant undergoes oxidation or reduction, the net transfer of charge, Q, in coulombs is
$Q=n F \nonumber$
where n is the moles of electrons per mole of reactant, and F is Faraday’s constant (96485 C/mol). The free energy, ∆G, to move this charge, Q, over a change in potential, E, is
$\triangle G=E Q \nonumber$
The change in free energy (in kJ/mole) for a redox reaction, therefore, is
$\Delta G=-n F E \label{6.18}$
where ∆G has units of kJ/mol. The minus sign in Equation \ref{6.18} is the result of a different convention for assigning a reaction’s favorable direction. In thermodynamics, a reaction is favored when ∆G is negative, but an oxidation‐reduction reaction is favored when E is positive. Substituting Equation \ref{6.18} into equation 6.2.3
$-n F E=-n F E^{\circ}+R T \ln Q_r \nonumber$
and dividing by –nF, leads to the well‐known Nernst equation
$E=E^{\circ}-\frac{R T}{n F} \ln Q_r \nonumber$
where Eo is the potential under standard‐state conditions. Substituting appropriate values for R and F, assuming a temperature of 25 oC (298 K), and switching from ln to log gives the potential in volts as
$E=E^{\mathrm{o}}-\frac{0.05916}{n} \log Q_r \label{6.19}$
Standard Potentials
A redox reaction’s standard potential, Eo, provides an alternative way of expressing its equilibrium constant and, therefore, its equilibrium position. Because a reaction at equilibrium has a ∆G of zero, the potential, E, also is zero at equilibrium. Substituting these values into Equation \ref{6.19} and rearranging provides a relationship between E o and K
$E^{\circ}=\frac{0.05916}{n} \log K \label{6.20}$
A standard potential is the potential when all species are in their standard states. You may recall that we define standard state conditions as follows: all gases have unit partial pressures, all solutes have unit concentrations, and all solids and liquids are pure.
We generally do not tabulate standard potentials for redox reactions. Instead, we calculate Eo using the standard potentials for the corresponding oxidation half‐reaction and reduction half‐reaction. By convention, standard potentials are provided for reduction half‐reactions. The standard potential for a redox reaction, Eo, is
$E^{\circ}=E_{red}^{\circ}-E_{ox}^{\circ} \nonumber$
where $E_{red}^{\circ}$ and $E_{ox}^{\circ}$ are the standard reduction potentials for the reduction half‐reaction and the oxidation half‐reaction.
Because we cannot measure the potential for a single half‐reaction, we arbitrarily assign a standard reduction potential of zero to a reference half‐reaction
$2 \mathrm{H}_{3} \mathrm{O}^{+}(a q)+2 e^{-}\rightleftharpoons 2 \mathrm{H}_{2} \mathrm{O}(l)+\mathrm{H}_{2}(g) \nonumber$
and report all other reduction potentials relative to this reference. Appendix 13 contains a list of selected standard reduction potentials. The more positive the standard reduction potential, the more favorable the reduction reaction is under standard state conditions. For example, under standard state conditions the reduction of Cu2+ to Cu (Eo = +0.3419 V) is more favorable than the reduction of Zn2+ to Zn (Eo = –0.7618 V).
Example 6.4.4
Calculate (a) the standard potential, (b) the equilibrium constant, and (c) the potential when [Ag+] = 0.020 M and [Cd2+] = 0.050 M, for the following reaction at 25oC.
$\mathrm{Cd}(s)+2 \mathrm{Ag}^{+}(a q)\rightleftharpoons2 \mathrm{Ag}(s)+\mathrm{Cd}^{2+}(a q) \nonumber$
Solution
(a) In this reaction Cd is oxidized and Ag+ is reduced. The standard cell potential, therefore, is
$E^{\circ} = E^{\circ}_{\text{Ag}^+/ \text{Ag}} - E^{\circ}_{\text{Cd}^{2+}/ \text{Cd}} = 0.7996 - (-0.4030) = 1.2026 \ \text{V} \nonumber$
(b) To calculate the equilibrium constant we substitute appropriate values into Equation \ref{6.20}.
$E^{\circ}=1.2026 \ \mathrm{V}=\frac{0.05916 \ \mathrm{V}}{2} \log K \nonumber$
Solving for K gives the equilibrium constant as
$\begin{array}{l}{\log K=40.6558} \ {K=4.527 \times 10^{40}}\end{array} \nonumber$
(c) To calculate the potential when [Ag+] is 0.020 M and [Cd2+] is 0.050M, we use the appropriate relationship for the reaction quotient, Qr, in Equation \ref{6.19}.
$\begin{array}{c}{E=E^{\circ}-\frac{0.05916 \ \mathrm{V}}{n} \log \frac{\left[\mathrm{Cd}^{2+}\right]}{\left[\mathrm{Ag}^{+}\right]^{2}}} \ {E=1.2026 \ \mathrm{V}-\frac{0.05916 \ \mathrm{V}}{2} \log \frac{0.050}{(0.020)^{2}}=1.14 \ \mathrm{V}}\end{array} \nonumber$
Exercise 6.4.3
For the following reaction at 25oC
$5 \mathrm{Fe}^{2+}(a q)+\mathrm{MnO}_{4}^{-}(a q)+8 \mathrm{H}^{+}(a q) \rightleftharpoons 5 \mathrm{Fe}^{3+}(a q)+\mathrm{Mn}^{2+}(a q)+4 \mathrm{H}_{2} \mathrm{O}(l) \nonumber$
calculate (a) the standard potential, (b) the equilibrium constant, and (c) the potential under these conditions: [Fe2+] = 0.50 M, [Fe3+] = 0.10 M, [$\text{MnO}_4^{-}$] = 0.025 M, [Mn2+] = 0.015 M, and a pH of 7.00. See Appendix 13 for standard state reduction potentials.
Answer
The two half‐reactions are the oxidation of Fe2+ and the reduction of $\text{MnO}_4^-$.
$\mathrm{Fe}^{2+}(a q) \rightleftharpoons \mathrm{Fe}^{3+}(a q)+e^{-} \nonumber$
$\mathrm{MnO}_{4}^{-}(a q)+8 \mathrm{H}^{+}(a q)+5 e^{-} \rightleftharpoons \mathrm{Mn}^{2+}(a q)+4 \mathrm{H}_{2} \mathrm{O}(l) \nonumber$
From Appendix 13, the standard state reduction potentials for these half‐reactions are
$E_{\text{Fe}^{3+}/\text{Fe}^{2+}}^{\circ} = 0.771 \ \text{V and } E_{\text{MnO}_4^-/\text{Mn}^{2+}}^{\circ} = 1.51 \ \text{V} \nonumber$
(a) The standard state potential for the reaction is
$E^{\circ} = E_{\text{MnO}_4^-/\text{Mn}^{2+}}^{\circ} - E_{\text{Fe}^{3+}/\text{Fe}^{2+}}^{\circ} = 1.51 \ \text{V} - 0.771 \ \text{V } = 0.74 \ \text{V} \nonumber$
(b) To calculate the equilibrium constant we substitute appropriate values into Equation \ref{6.20}.
$E^{\circ}=0.74 \ \mathrm{V}=\frac{0.05916}{5} \log K \nonumber$
Solving for K gives its value as $3.5 \times 10^{62}$.
(c) To calculate the potential under these non‐standard state conditions, we make appropriate substitutions into the Nernst equation.
$E=E^{\circ}-\frac{R T}{n F} \ln \frac{\left[\mathrm{Mn}^{2+}\right]\left[\mathrm{Fe}^{3+}\right]^{5}}{\left[\mathrm{MnO}_{4}^{-}\right]\left[\mathrm{Fe}^{2+}\right]^{5}\left[\mathrm{H}^{+}\right]^{8}} \nonumber$
$E=0.74-\frac{0.05916}{5} \log \frac{(0.015)(0.10)^{5}}{(0.025)(0.50)^{5}\left(1 \times 10^{-7}\right)^{8}}=0.12 \ \mathrm{V} \nonumber$
When writing precipitation, acid–base, and metal–ligand complexation reactions, we represent acidity as H3O+. Redox reactions more commonly are written using H+ instead of H3O+. For the reaction in Exercise 6.4.3 , we could replace H+ with H3O+ and increase the stoichiometric coefficient for H2O from 4 to 12. | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/06%3A_Equilibrium_Chemistry/6.04%3A_Equilibrium_Constants_for_Chemical_Reactions.txt |
At a temperature of 25oC, acetic acid’s dissociation reaction
$\mathrm{CH}_{3} \mathrm{COOH}(a q)+\mathrm{H}_{2} \mathrm{O}(l)\rightleftharpoons \mathrm{H}_{3} \mathrm{O}^{+}(a q)+\mathrm{CH}_{3} \mathrm{COO}^{-}(a q) \nonumber$
has an equilibrium constant of
$K_{\mathrm{a}}=\frac{\left[\mathrm{CH}_{3} \mathrm{COO}^{-}\right]\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]}{\left[\mathrm{CH}_{3} \mathrm{COOH}\right]}=1.75 \times 10^{-5} \label{6.1}$
Because Equation \ref{6.1} has three variables—[CH3COOH], [CH3COO], and [H3O+]—it does not have a unique mathematical solution. Nevertheless, although two solutions of acetic acid may have different values for [CH3COOH], [CH3COO], and [H3O+], each solution has the same value of Ka.
If we add sodium acetate to a solution of acetic acid, the concentration of CH3COO increases, which suggests there is an increase in the value of Ka; however, because Ka must remain constant, the concentration of all three species in Equation \ref{6.1} must change to restore Ka to its original value. In this case, a partial reaction of CH3COOand H3O+ decreases their concentrations, increases the concentration of CH3COOH, and reestablishes the equilibrium.
The observation that a system at equilibrium responds to an external action by reequilibrating itself in a manner that diminishes that action, is formalized as Le Châtelier’s principle. One common action is to change the concentration of a reactant or product for a system at equilibrium. As noted above for a solution of acetic acid, if we add a product to a reaction at equilibrium the system responds by converting some of the products into reactants. Adding a reactant has the opposite effect, resulting in the conversion of reactants to products.
When we add sodium acetate to a solution of acetic acid, we directly apply the action to the system. It is also possible to apply a change concentration indirectly. Consider, for example, the solubility of AgCl.
$\mathrm{AgCl}(s) \rightleftharpoons \mathrm{Ag}^{+}(a q)+\mathrm{Cl}^{-}(a q) \label{6.2}$
The effect on the solubility of AgCl of adding AgNO3 is obvious, but what is the effect if we add a ligand that forms a stable, soluble complex with Ag+? Ammonia, for example, reacts with Ag+ as shown here
$\mathrm{Ag}^{+}(a q)+2 \mathrm{NH}_{3}(a q) \rightleftharpoons \mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}^{+}(a q) \label{6.3}$
Adding ammonia decreases the concentration of Ag+ as the $\text{Ag(NH}_3)_2^+$ complex forms. In turn, a decrease in the concentration of Ag+ increases the solubility of AgCl as reaction \ref{6.2} reestablishes its equilibrium position. Adding together reaction \ref{6.2} and reaction \ref{6.3} clarifies the effect of ammonia on the solubility of AgCl, by showing ammonia as a reactant.
$\mathrm{AgCl}(s)+2 \mathrm{NH}_{3}(a q) \rightleftharpoons \mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}^{+}(a q)+\mathrm{Cl}^{-}(a q) \label{6.4}$
So what is the effect on the solubility of AgCl of adding AgNO3? Adding AgNO3 increases the concentration of Ag+ in solution. To reestablish equilibrium, some of the Ag+ and Cl react to form additional AgCl; thus, the solubility of AgCl decreases. The solubility product, Ksp, of course, remains unchanged.
Example 6.5.1
What happens to the solubility of AgCl if we add HNO3 to the equilibrium solution defined by reaction \ref{6.4}?
Solution
Nitric acid is a strong acid, which reacts with ammonia as shown here
$\mathrm{HNO}_{3}(a q)+\mathrm{NH}_{3}(a q)\rightleftharpoons \mathrm{NH}_{4}^{+}(a q)+\mathrm{NO}_{3}^{-}(a q) \nonumber$
Adding nitric acid lowers the concentration of ammonia. Decreasing ammonia’s concentration causes reaction \ref{6.4} to move from products to reactants, decreasing the solubility of AgCl.
Increasing or decreasing the partial pressure of a gas is the same as increasing or decreasing its concentration. Because the concentration of a gas depends on its partial pressure, and not on the total pressure of the system, adding or removing an inert gas has no effect on a reaction’s equilibrium position.
We can use the ideal gas law to deduce the relationship between pressure and concentration. Starting with PV = nRT, we solve for the molar concentration
$M=\frac{n}{V}=\frac{P}{R T} \nonumber$
Of course, this assumes that the gas is behaving ideally, which usually is a reasonable assumption under normal laboratory conditions.
Most reactions involve reactants and products dispersed in a solvent. If we change the amount of solvent by diluting the solution, then the concentrations of all reactants and products must increase; conversely, if we allow the solvent to evaporate partially, then the concentration of the solutes must increase. The effect of simultaneously changing the concentrations of all reactants and products is not intuitively as obvious as when we change the concentration of a single reactant or product. As an example, let’s consider how diluting a solution affects the equilibrium position for the formation of the aqueous silver‐amine complex (reaction \ref{6.3}). The equilibrium constant for this reaction is
$\beta_{2}=\frac{\left[\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}^{+}\right]_{\mathrm{eq}}}{\left[\mathrm{Ag}^{+}\right]_{\mathrm{eq}}\left[\mathrm{NH}_{3}\right]_{\mathrm{eq}}^{2}} \label{6.5}$
where we include the subscript “eq” for clarification. If we dilute a portion of this solution with an equal volume of water, each of the concentration terms in Equation \ref{6.5} is cut in half. The reaction quotient, Qr, becomes
$Q_r=\frac{0.5\left[\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}^{+}\right]_{\mathrm{eq}}}{0.5\left[\mathrm{Ag}^{+}\right]_{\mathrm{eq}}(0.5)^{2}\left[\mathrm{NH}_{3}\right]_{\mathrm{eq}}^{2}}=\frac{0.5}{(0.5)^{3}} \times \frac{\left[\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}^{+}\right]_{\mathrm{eq}}}{\left[\mathrm{Ag}^{+}\right]_{\mathrm{eq}}\left[\mathrm{NH}_{3}\right]_{\mathrm{eq}}^{2}}=4 \beta_{2} \label{6.6}$
Because Qr is greater than $\beta_2$, equilibrium is reestablished by shifting the reaction to the left, decreasing the concentration of $\text{Ag(NH}_3)_2^+$. Note that the new equilibrium position lies toward the side of the equilibrium reaction that has the greatest number of solute particles (one Ag+ ion and two molecules of NH3 versus a single metal‐ligand complex). If we concentrate the solution of $\text{Ag(NH}_3)_2^+$ by evaporating some of the solvent, equilibrium is reestablished in the opposite direction. This is a general conclusion that we can apply to any reaction. Increasing volume always favors the direction that produces the greatest number of particles, and decreasing volume always favors the direction that produces the fewest particles. If the number of particles is the same on both sides of the reaction, then the equilibrium position is unaffected by a change in volume. | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/06%3A_Equilibrium_Chemistry/6.05%3A_Le_Chateliers_Principle.txt |
When we develop or evaluate an analytical method, we often need to understand how the chemistry that takes place affects our results. Suppose we wish to isolate Ag+ by precipitating it as AgCl. If we also need to control pH, then we must use a reagent that does not adversely affect the solubility of AgCl. It is a mistake to use NH3 to adjust the pH, for example, because it increases the solubility of AgCl (see reaction 6.5.4). One of the primary sources of determinate errors in many analytical methods is failing to account for potential chemical interferences.
In this section we introduce the ladder diagram as a simple graphical tool for visualizing equilibrium chemistry. We will use ladder diagrams to determine what reactions occur when we combine several reagents, to estimate the approximate composition of a system at equilibrium, and to evaluate how a change to solution conditions might affect an analytical method.
Although not specifically on the topic of ladder diagrams as developed in this section, the following papers provide appropriate background information: (a) Runo, J. R.; Peters, D. G. J. Chem. Educ. 1993, 70, 708–713; (b) Vale, J.; Fernández‐Pereira, C.; Alcalde, M. J. Chem. Educ. 1993, 70, 790–795; (c) Fernández‐Pereira, C.; Vale, J. Chem. Educator 1996, 6, 1–18; (d) Fernández‐ Pereira, C.; Vale, J.; Alcalde, M. Chem. Educator 2003, 8, 15–21; (e) Fernández‐Pereira, C.; Alcalde, M.; Villegas, R.; Vale, J. J. Chem. Educ. 2007, 84, 520–525. Ladder diagrams are a great tool for helping you to think intuitively about analytical chemistry. We will make frequent use of them in the chapters to follow.
Ladder Diagrams for Acid–Base Equilibria
Let’s use acetic acid, CH3COOH, to illustrate the process we will use to draw and to interpret an acid–base ladder diagram. Before we draw the diagram, however, let’s consider the equilibrium reaction in more detail. Acetic acid's acid dissociation reaction and equilibrium constant expression are
$\mathrm{CH}_{3} \mathrm{COOH}(a q)+\mathrm{H}_{2} \mathrm{O}(l)\rightleftharpoons \mathrm{H}_{3} \mathrm{O}^{+}(a q)+\mathrm{CH}_{3} \mathrm{COO}^{-}(a q) \nonumber$
$K_{\mathrm{a}}=\frac{\left[\mathrm{CH}_{3} \mathrm{COO}^{-}\right]\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]}{\left[\mathrm{CH}_{3} \mathrm{COOH}\right]}=1.75 \times 10^{-5} \nonumber$
First, let’s take the logarithm of each term in this equation and multiply through by –1
$-\log K_{a}=4.76=-\log \left[\mathrm{H}_{3} \mathrm{O}^{+}\right]-\log \frac{\left[\mathrm{CH}_{3} \mathrm{COO}^{-}\right]}{\left[\mathrm{CH}_{3} \mathrm{COOH}\right]} \nonumber$
Now, let’s replace –log[H3O+] with pH and rearrange the equation to obtain the result shown here.
$\mathrm{pH}=4.76+\log \frac{\left[\mathrm{CH}_{3} \mathrm{COO}^{-}\right]}{\left[\mathrm{CH}_{3} \mathrm{COOH}\right]} \label{6.1}$
Equation \ref{6.1} tells us a great deal about the relationship between pH and the relative amounts of acetic acid and acetate at equilibrium. If the concentrations of CH3COOH and CH3COO are equal, then Equation \ref{6.1} reduces to
$\mathrm{pH}=4.76+\log (1)=4.76+0=4.76 \nonumber$
If the concentration of CH3COO is greater than that of CH3COOH, then the log term in Equation \ref{6.1} is positive and the pH is greater than 4.76. This is a reasonable result because we expect the concentration of the conjugate base, CH3COO, to increase as the pH increases. Similar reasoning will convince you that the pH is less than 4.76 when the concentration of CH3COOH exceeds that of CH3COO.
Now we are ready to construct acetic acid’s ladder diagram (Figure 6.6.1 ). First, we draw a vertical arrow that represents the solution’s pH, with smaller (more acidic) pH levels at the bottom and larger (more basic) pH levels at the top. Second, we draw a horizontal line at a pH equal to acetic acid’s pKa value. This line, or step on the ladder, divides the pH axis into regions where either CH3COOH or CH3COO is the predominate species. This completes the ladder diagram.
Using the ladder diagram, it is easy to identify the predominate form of acetic acid at any pH. At a pH of 3.5, for example, acetic acid exists primarily as CH3COOH. If we add sufficient base to the solution such that the pH increases to 6.5, the predominate form of acetic acid is CH3COO.
Example 6.6.1
Draw a ladder diagram for the weak base p‐nitrophenolate and identify its predominate form at a pH of 6.00.
Solution
To draw a ladder diagram for a weak base, we simply draw the ladder diagram for its conjugate weak acid. From Appendix 11, the pKa for p‐nitrophenol is 7.15. The resulting ladder diagram is shown in Figure 6.6.2 . At a pH of 6.00, p‐nitrophenolate is present primarily in its weak acid form.
Exercise 6.6.1
Draw a ladder diagram for carbonic acid, H2CO3. Because H2CO3 is a diprotic weak acid, your ladder diagram will have two steps. What is the predominate form of carbonic acid when the pH is 7.00? Relevant equilibrium constants are in Appendix 11.
Answer
From Appendix 11, the pKa values for H2CO3 are 6.352 and 10.329. The ladder diagram for H2CO3 is shown below. The predominate form at a pH of 7.00 is $\text{HCO}_3^-$.
A ladder diagram is particularly useful for evaluating the reactivity between a weak acid and a weak base. Figure 6.6.3 , for example, shows a single ladder diagram for acetic acid/acetate and for p‐nitrophenol/p‐nitrophenolate. An acid and a base can not co‐exist if their respective areas of predominance do not overlap. If we mix together solutions of acetic acid and sodium p‐nitrophenolate, the reaction
$\mathrm{C}_{6} \mathrm{H}_{4} \mathrm{NO}_{2}^{-}(a q)+\mathrm{CH}_{3} \mathrm{COOH}(a q)\rightleftharpoons \text{CH}_3\text{COO}^-(aq) + \text{C}_6\text{H}_4\text{NO}_2\text{H}(aq) \label{6.2}$
occurs because the areas of predominance for acetic acid and p‐nitrophenolate do not overlap. The solution’s final composition depends on which species is the limiting reagent. The following example shows how we can use the ladder diagram in Figure 6.6.3 to evaluate the result of mixing together solutions of acetic acid and p‐nitrophenolate.
Example 6.6.2
Predict the approximate pH and the final composition after mixing together 0.090 moles of acetic acid and 0.040 moles of p‐nitrophenolate.
Solution
The ladder diagram in Figure 6.6.3 indicates that the reaction between acetic acid and p‐nitrophenolate is favorable. Because acetic acid is in excess, we assume the reaction of p‐nitrophenolate to p‐nitrophenol is complete. At equilibrium essentially no p‐nitrophenolate remains and there are 0.040 mol of p‐nitrophenol. Converting p‐nitrophenolate to p‐nitrophenol consumes 0.040 moles of acetic acid; thus
$\begin{array}{c}{\text { moles } \mathrm{CH}_{3} \mathrm{COOH}=0.090-0.040=0.050 \ \mathrm{mol}} \ {\text { moles } \mathrm{CH}_{3} \mathrm{COO}^{-}=0.040 \ \mathrm{mol}}\end{array} \nonumber$
According to the ladder diagram, the pH is 4.74 when there are equal amounts of CH3COOH and CH3COO. Because we have slightly more CH3COOH than CH3COO, the pH is slightly less than 4.74.
Exercise 6.6.2
Using Figure 6.6.3 , predict the approximate pH and the composition of the solution formed by mixing together 0.090 moles of p‐nitrophenolate and 0.040 moles of acetic acid.
Answer
The ladder diagram in Figure 6.6.3 indicates that the reaction between acetic acid and p‐nitrophenolate is favorable. Because p‐nitrophenolate is in excess, we assume the reaction of acetic acid to acetate is complete. At equilibrium essentially no acetic acid remains and there are 0.040 moles of acetate. Converting acetic acid to acetate consumes 0.040 moles of p‐nitrophenolate; thus
$\text { moles } p \text {-nitrophenolate }=0.090-0.040=0.050 \text { mol } \nonumber$
$\text { moles } p\text{-nitrophenol }=0.040 \ \mathrm{mol} \nonumber$
According to the ladder diagram for this system, the pH is 7.15 when there are equal concentrations of p‐nitrophenol and p‐nitrophenolate. Because we have slightly more p‐nitrophenolate than we have p‐nitrophenol, the pH is slightly greater than 7.15.
If the areas of predominance for an acid and a base overlap, then we do not expect that much of a reaction will occur. For example, if we mix together solutions of CH3COO and p‐nitrophenol, we do not expect a significant change in the moles of either reagent. Furthermore, the pH of the mixture must be between 4.76 and 7.15, with the exact pH depending upon the relative amounts of CH3COO and p‐nitrophenol.
We also can use an acid–base ladder diagram to evaluate the effect of pH on other equilibria. For example, the solubility of CaF2
$\mathrm{CaF}_{2}(s) \rightleftharpoons \mathrm{Ca}^{2+}(a q)+2 \mathrm{F}^{-}(a q) \nonumber$
is affected by pH because F is a weak base. From Le Châtelier’s principle, we know that converting F to HF will increase the solubility of CaF2. To minimize the solubility of CaF2 we need to maintain the solution’s pH so that F is the predominate species. The ladder diagram for HF (Figure 6.6.4 ) shows us that maintaining a pH of more than 3.17 will minimize solubility losses.
Ladder Diagrams for Complexation Equilibria
We can apply the same principles for constructing and interpreting an acid–base ladder diagram to equilibria that involve metal–ligand complexes. For a complexation reaction we define the ladder diagram’s scale using the concentration of uncomplexed, or free ligand, pL. Using the formation of $\text{Cd(NH}_3)^{2+}$ as an example
$\mathrm{Cd}^{2+}(a q)+\mathrm{NH}_{3}(a q) \rightleftharpoons \mathrm{Cd}\left(\mathrm{NH}_{3}\right)^{2+}(a q) \nonumber$
we can show that log K1 is the dividing line between the areas of predominance for Cd2+ and for $\text{Cd(NH}_3)^{2+}$.
$K_{1}=3.55 \times 10^{2}=\frac{\left[\mathrm{Cd}\left(\mathrm{NH}_{3}\right)^{2+}\right]}{\left[\mathrm{Cd}^{2+}\right]\left[\mathrm{NH}_{3}\right]} \nonumber$
$\log K_{1}=\log \left(3.55 \times 10^{2}\right)=\log \frac{\left[\mathrm{Cd}\left(\mathrm{NH}_{3}\right)^{2+}\right]}{\left[\mathrm{Cd}^{2+}\right]}-\log \left[\mathrm{NH}_{3}\right] \nonumber$
$\log K_{1}=2.55=\log \frac{\left[\mathrm{Cd}\left(\mathrm{NH}_{3}\right)^{2+}\right]}{\left[\mathrm{Cd}^{2+}\right]}+\mathrm{p} \mathrm{NH}_{3} \nonumber$
$\mathrm{p} \mathrm{NH}_{3}=\log K_{1}+\log \frac{\left[\mathrm{Cd}^{2+}\right]}{\left[\mathrm{Cd}\left(\mathrm{NH}_{3}\right)^{2+}\right]}=2.55+\log \frac{\left[\mathrm{Cd}^{2+}\right]}{\left[\mathrm{Cd}\left(\mathrm{NH}_{3}\right)^{2+}\right]} \nonumber$
Thus, Cd2+ is the predominate species when pNH3 is greater than 2.55 (a concentration of NH3 smaller than $2.82 \times 10^{-3}$ M) and for a pNH3 value less than 2.55, $\text{Cd(NH}_3)^{2+}$ is the predominate species. Figure 6.6.5 shows a complete metal–ligand ladder diagram for Cd2+ and NH3 that includes additional Cd–NH3 complexes.
Example 6.6.3
Draw a single ladder diagram for the Ca(EDTA)2– and the Mg(EDTA)2– metal–ligand complexes. Use your ladder diagram to predict the result of adding 0.080 moles of Ca2+ to 0.060 moles of Mg(EDTA)2–. EDTA is an abbreviation for the ligand ethylenediaminetetraacetic acid.
Solution
Figure 6.6.6 shows the ladder diagram for this system of metal–ligand complexes. Because the predominance regions for Ca2+ and Mg(EDTA)2‐ do not overlap, the reaction
$\mathrm{Ca}^{2+}(a q)+\mathrm{Mg}(\mathrm{EDTA})^{2-}(a q) \rightleftharpoons \mathrm{Ca}(\mathrm{EDTA})^{2-}(a q)+\mathrm{Mg}^{2+}(a q) \nonumber$
proceeds essentially to completion. Because Ca2+ is the excess reagent, the composition of the final solution is approximately
$\text { moles } \mathrm{Ca}^{2+}=0.080-0.060=0.020 \ \mathrm{mol} \nonumber$
$\text { moles } \mathrm{Ca}(\mathrm{EDTA})^{2-}=0.060 \ \mathrm{mol} \nonumber$
$\text { moles } \mathrm{Mg}^{2+}=0.060 \ \mathrm{mol} \nonumber$
$\text { moles } \mathrm{Mg}(\mathrm{EDTA})^{2-}=0 \ \mathrm{mol} \nonumber$
The metal–ligand ladder diagram in Figure 6.6.5 uses stepwise formation constants. We also can construct a ladder diagram using cumulative formation constants. For example, the first three stepwise formation constants for the reaction of Zn2+ with NH3
$\mathrm{Zn}^{2+}(a q)+\mathrm{NH}_{3}(a q) \rightleftharpoons \mathrm{Zn}\left(\mathrm{NH}_{3}\right)^{2+}(a q) \quad K_{1}=1.6 \times 10^{2} \nonumber$
$\mathrm{Zn}\left(\mathrm{NH}_{3}\right)^{2+}(a q)+\mathrm{NH}_{3}(a q)\rightleftharpoons\mathrm{Zn}\left(\mathrm{NH}_{3}\right)_{2}^{2+}(a q) \quad K_{2}=1.95 \times 10^{2} \nonumber$
$\mathrm{Zn}\left(\mathrm{NH}_{3}\right)_{2}^{2+}(a q)+\mathrm{NH}_{3}(a q)=\mathrm{Zn}\left(\mathrm{NH}_{3}\right)_{3}^{2+}(a q) \quad K_{3}=2.3 \times 10^{2} \nonumber$
suggests that the formation of $\text{Zn(NH}_3)_3^{2+}$ is more favorable than the formation of $\text{Zn(NH}_3)^{2+}$ or $\text{Zn(NH}_3)_2^{2+}$. For this reason, the equilibrium is best represented by the cumulative formation reaction shown here.
$\mathrm{Zn}^{2+}(a q)+3 \mathrm{NH}_{3}(a q)\rightleftharpoons \mathrm{Zn}\left(\mathrm{NH}_{3}\right)_{3}^{2+}(a q) \quad \beta_{3}=7.2 \times 10^{6} \nonumber$
Because K3 is greater than K2, which is greater than K1, the formation of the metal‐ligand complex $\text{Zn(NH}_3)_3^{2+}$ is more favorable than the formation of the other metal ligand complexes. For this reason, at lower values of pNH3 the concentration of $\text{Zn(NH}_3)_3^{2+}$ is larger than the concentrations of $\text{Zn(NH}_3)^{2+}$ or $\text{Zn(NH}_3)_2^{2+}$. The value of $\beta_3$ is
$\beta_{3}=K_{1} \times K_{2} \times K_{3} \nonumber$
To see how we incorporate this cumulative formation constant into a ladder diagram, we begin with the reaction’s equilibrium constant expression.
$\beta_{3}=\frac{\left[\mathrm{Zn}\left(\mathrm{NH}_{3}\right)_{3}^{2+}\right]}{\left[\mathrm{Zn}^{2+}\right]\left[\mathrm{NH}_{3}\right]^{3}} \nonumber$
Taking the log of each side
$\log \beta_{3}=\log \frac{\left[\mathrm{Zn}\left(\mathrm{NH}_{3}\right)_{3}^{2+}\right]}{\left[\mathrm{Zn}^{2+}\right]}-3 \log \left[\mathrm{NH}_{3}\right] \nonumber$
and rearranging gives
$\mathrm{pNH}_{3}=\frac{1}{3} \log \beta_{3}+\frac{1}{3} \log \frac{\left[\mathrm{Zn}^{2+}\right]}{\left[\mathrm{Zn}\left(\mathrm{NH}_{3}\right)_{3}^{2+}\right]} \nonumber$
When the concentrations of Zn and $\text{Zn(NH}_3)_3^{2+}$ are equal, then
$\mathrm{p} \mathrm{NH}_{3}=\frac{1}{3} \log \beta_{3}=2.29 \nonumber$
In general, for the metal–ligand complex MLn, the step for a cumulative formation constant is
$\mathrm{pL}=\frac{1}{n} \log \beta_{n} \nonumber$
Figure 6.6.7 shows the complete ladder diagram for the Zn2+–NH3 system.
Ladder Diagrams for Oxidation/Reduction Equilibria
We also can construct ladder diagrams to help us evaluate redox equilibria. Figure 6.6.8 shows a typical ladder diagram for two half‐reactions in which the scale is the potential, E.
The Nernst equation defines the areas of predominance. Using the Fe3+/Fe2+ half‐reaction as an example, we write
$E=E^{\circ}-\frac{R T}{n F} \ln \frac{\left[\mathrm{Fe}^{2+}\right]}{\left[\mathrm{Fe}^{3+}\right]}=0.771-0.05916 \log \frac{\left[\mathrm{Fe}^{2+}\right]}{\left[\mathrm{Fe}^{3+}\right]} \nonumber$
At a potential more positive than the standard state potential, the predominate species is Fe3+, whereas Fe2+ predominates at potentials more negative than Eo. When coupled with the step for the Sn4+/Sn2+ half‐reaction we see that Sn2+ is a useful reducing agent for Fe3+. If Sn2+ is in excess, the potential of the resulting solution is near +0.154 V.
Because the steps on a redox ladder diagram are standard state potentials, a complication arises if solutes other than the oxidizing agent and reducing agent are present at non‐standard state concentrations. For example, the potential for the half‐reaction
$\mathrm{UO}_{2}^{2+}(a q)+4 \mathrm{H}_{3} \mathrm{O}^{+}(a q)+2 e^{-} \rightleftharpoons \mathrm{U}^{4+}(a q)+6 \mathrm{H}_{2} \mathrm{O}(l) \nonumber$
depends on the solution’s pH. To define areas of predominance in this case we begin with the Nernst equation
$E=+0.327-\frac{0.05916}{2} \log \frac{\left[\mathrm{U}^{4+}\right]}{\left[\mathrm{UO}_{2}^{2+}\right]\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]^{4}} \nonumber$
and factor out the concentration of H3O+.
$E=+0.327+\frac{0.05916}{2} \log \left[\mathrm{H}_{3} \mathrm{O}^{+}\right]^{4}-\frac{0.05916}{2} \log \frac{\left[\mathrm{U}^{4+}\right]}{\left[\mathrm{UO}_{2}^{2+}\right]}\nonumber$
From this equation we see that the area of predominance for $\text{UO}_2^{2+}$ and U4+ is defined by a step at a potential where [U4+] = [$\text{UO}_2^{2+}$].
$E=+0.327+\frac{0.05916}{2} \log \left[\mathrm{H}_{3} \mathrm{O}^{+}\right]^{4}=+0.327-0.1183 \mathrm{pH} \nonumber$
Figure 6.6.9 shows how pH affects the step for the $\text{UO}_2^{2+}$ /U4+ half‐reaction. | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/06%3A_Equilibrium_Chemistry/6.06%3A_Ladder_Diagrams.txt |
Ladder diagrams are a useful tool for evaluating chemical reactivity and for providing a reasonable estimate of a chemical system’s composition at equilibrium. If we need a more exact quantitative description of the equilibrium condition, then a ladder diagram is insufficient; instead, we need to find an algebraic solution. In this section we will learn how to set‐up and solve equilibrium problems. We will start with a simple problem and work toward more complex problems.
A Simple Problem: The Solubility of Pb(IO3)2
If we place an insoluble compound such as Pb(IO3)2 in deionized water, the solid dissolves until the concentrations of Pb2+ and $\text{IO}_3^-$ satisfy the solubility product for Pb(IO3)2. At equilibrium the solution is saturated with Pb(IO3)2, which means simply that no more solid can dissolve. How do we determine the equilibrium concentrations of Pb2+ and $\text{IO}_3^-$, and what is the molar solubility of Pb(IO3)2 in this saturated solution?
When we first add solid Pb(IO3)2 to water, the concentrations of Pb2+ and $\text{IO}_3^-$ are zero and the reaction quotient, Qr, is
$Q_r = \left[\mathrm{Pb}^{2+}\right]\left[\mathrm{IO}_{3}^{-}\right]^{2}=0 \nonumber$
As the solid dissolves, the concentrations of these ions increase, but Qr remains smaller than Ksp. We reach equilibrium and “satisfy the solubility product” when Qr = Ksp.
We begin by writing the equilibrium reaction and the solubility product expression for Pb(IO3)2.
$\mathrm{Pb}\left(\mathrm{IO}_{3}\right)_{2}(s)\rightleftharpoons \mathrm{Pb}^{2+}(a q)+2 \mathrm{IO}_{3}^{-}(a q) \nonumber$
e$K_{\mathrm{sp}}=\left[\mathrm{Pb}^{2+}\right]\left[\mathrm{IO}_{3}^{-}\right]^{2}=2.5 \times 10^{-13} \label{6.1}$
As Pb(IO3)2 dissolves, two $\text{IO}_3^-$ ions form for each ion of Pb2+. If we assume that the change in the molar concentration of Pb2+ at equilibrium is x, then the change in the molar concentration of $\text{IO}_3^-$ is 2x. The following table helps us keep track of the initial concentrations, the change in con‐ centrations, and the equilibrium concentrations of Pb2+ and $\text{IO}_3^-$.
concentrations Pb(IO3)2 (s) $\rightleftharpoons$ Pb2+ (aq) + 2$\text{IO}_3^-$ (aq)
initial solid 0 0
change solid +x + 2x
equilibrium solid x 2x
Because a solid, such as Pb(IO3)2 , does not appear in the solubility product expression, we do not need to keep track of its concentration. Remember, however, that the Ksp value applies only if there is some solid Pb(IO3)2 present at equilibrium.
Substituting the equilibrium concentrations into Equation \ref{6.1} and solving gives
$(x)(2 x)^{2}=4 x^{3}=2.5 \times 10^{-13} \nonumber$
$x=3.97 \times 10^{-5} \nonumber$
Substituting this value of x back into the equilibrium concentration expressions for Pb2+ and $\text{IO}_3^-$ gives their concentrations as
$\left[\mathrm{Pb}^{2+}\right]=x=4.0 \times 10^{-5} \mathrm{M} \text { and }\left[\mathrm{IO}_{3}^{-}\right]=2 x=7.9 \times 10^{-5} \nonumber$
Because one mole of Pb(IO3)2 contains one mole of Pb2+, the molar solubility of Pb(IO3)2 is equal to the concentration of Pb2+, or $4.0 \times 10^{-5}$ M.
We can express a compound’s solubility in two ways: as its molar solubility (mol/L) or as its mass solubility (g/L). Be sure to express your answer clearly.
Exercise 6.7.1
Calculate the molar solubility and the mass solubility for Hg2Cl2, given the following solubility reaction and Ksp value.
$\mathrm{Hg}_{2} \mathrm{Cl}_{2}(s)\rightleftharpoons \mathrm{Hg}_{2}^{2+}(a q)+2 \mathrm{Cl}^{-}(a q) \quad K_{\mathrm{sp}}=1.2 \times 10^{-8} \nonumber$
Answer
When Hg2Cl2 dissolves, two Cl are produced for each ion of $\text{Hg}_2^{2+}$. If we assume x is the change in the molar concentration of $\text{Hg}_2^{2+}$, then the change in the molar concentration of Cl is 2x. The following table helps us keep track of our solution to this problem.
concentration Hg2Cl2 (s) $\rightleftharpoons$ $\text{Hg}_2^{2+}$ (aq) + Cl(aq)
initial solid 0 0
change solid +x +2x
equilibrium solid x 2x
Substituting the equilibrium concentrations into the Ksp expression forHg2Cl2 gives
$K_{\mathrm{sp}}=\left[\mathrm{Hg}_{2}^{2+}\right]\left[\mathrm{Cl}^{-}\right]^{2}=(x)(2 \mathrm{x})^{2}=4 x^{3}=1.2 \times 10^{-18} \nonumber$
$x=6.69 \times 10^{-7} \nonumber$
Substituting x back into the equilibrium expressions for $\text{Hg}_2^{2+}$ and Cl gives their concentrations as
$\left[\mathrm{Hg}_{2}^{2+}\right]=x=6.7 \times 10^{-7} \ \mathrm{M} \quad\left[\mathrm{Cl}^{-}\right]=2 x=1.3 \times 10^{-6} \ \mathrm{M} \nonumber$
The molar solubility is equal to [$\text{Hg}_2^{2+}$], or $6.7 \times 10^{-7}$ mol/L.
A More Complex Problem: The Common Ion Effect
Calculating the solubility of Pb(IO3)2 in deionized water is a straightforward problem because the solid’s dissolution is the only source of Pb2+ and $\text{IO}_3^-$. But what if we add Pb(IO3)2 to a solution of 0.10 M Pb(NO3)2? Before we set‐up and solve this problem algebraically, think about the system’s chemistry and decide whether the solubility of Pb(IO3)2 will increase, decrease, or remain the same. Beginning a problem by thinking about the likely answer is a good habit to develop. Knowing what answers are reasonable will help you spot errors in your calculations and give you more confidence that your solution to a problem is correct. Because the solution already contains a source of Pb2+, we can use Le Châtelier’s principle to predict that the solubility of Pb(IO3)2 is smaller than that in our previous problem.
We begin by setting up a table to help us keep track of the concentrations of Pb2+ and $\text{IO}_3^-$ as this system moves toward and reaches equilibrium.
concentrations Pb(IO3)2 (s) $\rightleftharpoons$ Pb2+ (aq) + 2$\text{IO}_3^-$ (aq)
initial solid 0.10 0
change solid +x + 2x
equilibrium solid 0.10 + x 2x
Substituting the equilibrium concentrations into Equation \ref{6.1}
$(0.10+x)(2 x)^{2}=2.5 \times 10^{-13} \nonumber$
and multiplying out the terms on the equation’s left side leaves us with
$4 x^{3}+0.40 x^{2}=2.5 \times 10^{-13} \label{6.2}$
This is a more difficult equation to solve than that for the solubility of Pb(IO3)2 in deionized water, and its solution is not immediately obvious. We can find a rigorous solution to Equation \ref{6.2} using computational software packages and spreadsheets, some of which are described in Chapter 6.10.
There are several approaches to solving cubic equations, but none are computationally easy using just paper and pencil.
How might we solve Equation \ref{6.2} if we do not have access to a computer? One approach is to use our understanding of chemistry to simplify the problem. From Le Châtelier’s principle we know that a large initial concentration of Pb2+ will decrease significantly the solubility of Pb(IO3)2. One reasonable assumption is that the initial concentration of Pb2+ is very close to its equilibrium concentration. If this assumption is correct, then the following approximation is reasonable
$\left[\mathrm{Pb}^{2+}\right]=0.10+x \approx 0.10 \nonumber$
Substituting this approximation into Equation \ref{6.1} and solving for x gives
$(0.10)(2 x)^{2}=0.4 x^{2}=2.5 \times 10^{-13} \nonumber$
$x=7.91 \times 10^{-7} \nonumber$
Before we accept this answer, we must verify that our approximation is reasonable. The difference between the actual concentration of Pb2+, which is 0.10 + x M, and our assumption that the concentration of Pb2+ is 0.10 M is $7.9 \times 10^{-7}$, or $7.9 \times 10^{-4}$ % of the assumed concentration. This is a negligible error. If we accept the result of our calculation, we find that the equilibrium concentrations of Pb2+ and $\text{IO}_3^-$ are
$\left[\mathrm{Pb}^{2+}\right]=0.10+x \approx 0.10 \ \mathrm{M} \text { and }\left[\mathrm{IO}_{3}^{-}\right]=2 x=1.6 \times 10^{-6} \ \mathrm{M} \nonumber$
\begin{aligned} \% \text { error } &=\frac{\text { actual }-\text { assumed }}{\text { assumed }} \times 100 \ &=\frac{(0.10+x)-0.10}{0.10} \times 100 \ &=\frac{7.91 \times 10^{-7}}{0.10} \times 100 \ &=7.91 \times 10^{-4} \% \end{aligned} \nonumber
The molar solubility of Pb(IO3)2 is equal to the additional concentration of Pb2+ in solution, or $7.9 \times 10^{-4}$ mol/L. As expected, we find that Pb(IO3)2 is less soluble in the presence of a solution that already contains one of its ions. This is known as the common ion effect.
As outlined in the following example, if an approximation leads to an error that is unacceptably large, then we can extend the process of making and evaluating approximations.
One “rule of thumb” when making an approximation is that it should not introduce an error of more than ±5%. Although this is not an unreasonable choice, what matters is that the error makes sense within the context of the problem you are solving.
Example 6.7.1
Calculate the solubility of Pb(IO3)2 in $1.0 \times 10^{-4}$ M Pb(NO3)2.
Solution
If we let x equal the change in the concentration of Pb2+, then the equilibrium concentrations of Pb2+ and $\text{IO}_3^-$ are
$\left[\mathrm{P} \mathrm{b}^{2+}\right]=1.0 \times 10^{-4}+ \ x \text { and }\left[\mathrm{IO}_{3}^-\right]=2 x \nonumber$
Substituting these concentrations into Equation \ref{6.1} leaves us with
$\left(1.0 \times 10^{-4}+ \ x\right)(2 x)^{2}=2.5 \times 10^{-13} \nonumber$
To solve this equation for x, let’s make the following assumption
$\left[\mathrm{Pb}^{2+}\right]=1.0 \times 10^{-4}+ \ x \approx 1.0 \times 10^{-4} \ \mathrm{M} \nonumber$
Solving for x gives its value as $2.50 \times 10^{-5}$; however, when we substitute this value for x back, we find that the calculated concentration of Pb2+ at equilibrium
$\left[\mathrm{Pb}^{2+}\right]=1.0 \times 10^{-4}+ \ x=1.0 \times 10^{-4}+ \ 2.50 \times 10^{-5}=1.25 \times 10^{-4} \ \mathrm{M} \nonumber$
is 25% greater than our assumption of $1.0 \times 10^{-4}$ M. This error is unreasonably large.
Rather than shouting in frustration, let’s make a new assumption. Our first assumption—that the concentration of Pb2+ is $1.0 \times 10^{-4}$ M—was too small. The calculated concentration of $1.25 \times 10^{-4}$ M, therefore, probably is a too large, but closer to the correct concentration than was our first assumption. For our second approximation, let’s assume that
$\left[\mathrm{Pb}^{2+}\right]=1.0 \times 10^{-4}+ \ x=1.25 \times 10^{-4} \mathrm{M} \nonumber$
Substituting into Equation \ref{6.1} and solving for x gives its value as $2.24 \times 10^{-5}$. The resulting concentration of Pb2+ is
$\left[\mathrm{Pb}^{2+}\right]=1.0 \times 10^{-4}+ \ 2.24 \times 10^{-5}=1.22 \times 10^{-4} \ \mathrm{M} \nonumber$
which differs from our assumption of $1.25 \times 10^{-4}$ M by 2.4%. Because the original concentration of Pb2+ is given to two significant figure, this is a more reasonable error. Our final solution, to two significant figures, is
$\left[\mathrm{Pb}^{2+}\right]=1.2 \times 10^{-4} \ \mathrm{M} \text { and }\left[\mathrm{IO}_{3}\right]=4.5 \times 10^{-5} \ \mathrm{M} \nonumber$
and the molar solubility of Pb(IO3)2 is $2.2 \times 10^{-5}$ mol/L. This iterative approach to solving the problems is known as the method of successive approximations.
Exercise 6.7.2
Calculate the molar solubility for Hg2Cl2 in 0.10 M NaCl and compare your answer to its molar solubility in deionized water (see Exercise 6.7.1 ).
Answer
We begin by setting up a table to help us keep track of the concentrations $\text{Hg}_2^{2+}$ and Cl as this system moves toward and reaches equilibrium.
concentration Hg2Cl2 (s) $\rightleftharpoons$ $\text{Hg}_2^{2+}$ (aq) + Cl(aq)
initial solid 0 0.10
change solid +x +2x
equilibrium solid x 0.1 + 2x
Substituting the equilibrium concentrations into the Ksp expression for Hg2Cl2 leaves us with a difficult to solve cubic equation.
$K_{\mathrm{sp}}=\left[\mathrm{Hg}_{2}^{2+}\right]\left[\mathrm{Cl}^{-}\right]^{2}=(x)(0.10+2 x)^{2}=4 x^{3}+0.40 x^{2}+0.010 x \nonumber$
Let’s make an assumption to simplify this problem. Because we expect the value of x to be small, let’s assume that
$\left[\mathrm{Cl}^{-}\right]=0.10+2 x \approx 0.10 \nonumber$
This simplifies our problem to
$K_{\mathrm{sp}}=\left[\mathrm{Hg}_{2}^{2+}\right]\left[\mathrm{Cl}^{-}\right]^{2}=(x)(0.10)^{2}=0.010 x=1.2 \times 10^{-18} \nonumber$
which gives the value of x as $1.2 \times 10^{-16}$ M. The difference between the actual concentration of Cl, which is (0.10 + 2x) M, and our assumption that it is 0.10 M introduces an error of $2.4 \times 10^{-13}$ %. This is a negligible error. The molar solubility of Hg2Cl2 is the same as the concentration of $\text{Hg}_2^{2+}$, or $1.2 \times 10^{-16}$ M. As expected, the molar solubility in 0.10 M NaCl is less than $6.7 \times 10^{-7}$ mol/L, which is its solubility in water (see solution to Exercise 6.7.1 ).
A Systematic Approach to Solving Equilibrium Problems
Calculating the solubility of Pb(IO3)2 in a solution of Pb(NO3)2 is more complicated than calculating its solubility in deionized water. The calculation, however, is still relatively easy to organize and the simplifying assumptions are fairly obvious. This problem is reasonably straightforward because it involves only one equilibrium reaction and one equilibrium constant.
Determining the equilibrium composition of a system with multiple equilibrium reactions is more complicated. In this section we introduce a systematic approach to setting‐up and solving equilibrium problems. As shown in Table 6.7.1 , this approach involves four steps.
Table 6.7.1 . Systematic Approach to Solving Equilibrium Problems
Step 1
Write all relevant equilibrium reactions and equilibrium constant expressions.
Step 2
Count the unique species that appear in the equilibrium constant expressions; these are your unknowns. You have enough information to solve the problem if the number of unknowns equals the number of equilibrium constant expressions. If not, add a mass balance equation and/or a charge balance equation. Continue adding equations until the number of equations equals the number of unknowns.
Step 3
Combine your equations and solve for one unknown. Whenever possible, simplify the algebra by making appropriate assumptions. If you make an assumption, set a limit for its error. This decision influences your evaluation of the assumption.
Step 4
Check your assumptions. If any assumption proves invalid, return to the previous step and continue solving. The problem is complete when you have an answer that does not violate any of your assumptions.
In addition to equilibrium constant expressions, two other equations are important to this systematic approach to solving an equilibrium problem. The first of these equations is a mass balance equation, which simply is a statement that matter is conserved during a chemical reaction. In a solution of acetic acid, for example, the combined concentrations of the conjugate weak acid, CH3COOH, and the conjugate weak base, CH3COO, must equal acetic acid’s initial concentration, $C_{\text{CH}_3\text{COOH}}$.
$C_{\mathrm{CH}_{\mathrm{3}} \mathrm{COOH}}=\left[\mathrm{CH}_{3} \mathrm{COOH}\right]+\left[\mathrm{CH}_{3} \mathrm{COO}^{-}\right] \nonumber$
You may recall from Chapter 2 that this is the difference between a formal concentration and a molar concentration. The variable C represents a formal concentration.
The second equation is a charge balance equation, which requires that the total positive charge from the cations equal the total negative charge from the anions. Mathematically, the charge balance equation is
$\sum_{i=1}^{n}\left(z^{+}\right)_{i}\left[{C^{z}}^+\right]_{i} = -\sum_{j=1}^{m}(z^-)_{j}\left[{A^{z}}^-\right]_{j} \nonumber$
where [Cz+]i and [Az-]j are, respectively, the concentrations of the ith cation and the jth anion, and (z+)i and (z)j are the charges for the ith cation and the jth anion. Every ion in solution, even if it does not appear in an equilibrium reaction, must appear in the charge balance equation. For example, the charge balance equation for an aqueous solution of Ca(NO3)2 is
$2 \times\left[\mathrm{Ca}^{2+}\right]+\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=\left[\mathrm{OH}^{+}\right]+\left[\mathrm{NO}_{3}^-\right] \nonumber$
Note that we multiply the concentration of Ca2+ by two and that we include the concentrations of H3O+ and OH.
A charge balance is a conservation of a charge. The minus sign in front of the summation term on the right side of the charge balance equation ensures that both summations are positive. There are situations where it is impossible to write a charge balance equation because we do not have enough information about the solution’s composition. For example, suppose we fix a solution’s pH using a buffer. If the buffer’s composition is not specified, then we cannot write a charge balance equation.
Example 6.7.2
Write mass balance equations and a charge balance equation for a 0.10 M solution of NaHCO3.
Solution
It is easier to keep track of the species in solution if we write down the reactions that define the solution’s composition. These reactions are the dissolution of a soluble salt
$\mathrm{NaHCO}_{3}(s) \rightarrow \mathrm{Na}^{+}(a q)+\mathrm{HCO}_{3}^{-}(a q) \nonumber$
and the acid–base dissociation reactions of $\text{HCO}_3^-$ and H2O
$\mathrm{HCO}_{3}^{-}(a q)+\mathrm{H}_{2} \mathrm{O}(l)\rightleftharpoons \mathrm{H}_{3} \mathrm{O}^{+}(a q)+\mathrm{CO}_{3}^{2-}(a q) \nonumber$
$\mathrm{HCO}_{3}^{-}(a q)+\mathrm{H}_{2} \mathrm{O}(l)\rightleftharpoons \mathrm{OH}^{-}(a q)+\mathrm{H}_{2} \mathrm{CO}_{3}(a q) \nonumber$
$2 \mathrm{H}_{2} \mathrm{O}(l)\rightleftharpoons \mathrm{H}_{3} \mathrm{O}^{+}(a q)+\mathrm{OH}^{-}(a q) \nonumber$
The mass balance equations are
$0.10 \mathrm{M}=\left[\mathrm{H}_{2} \mathrm{CO}_{3}\right]+\left[\mathrm{HCO}_{3}^{-}\right]+\left[\mathrm{CO}_{3}^{2-}\right] \nonumber$
$0.10 \ \mathrm{M}=\left[\mathrm{Na}^{+}\right] \nonumber$
and the charge balance equation is
$\left[\mathrm{Na}^{+}\right]+\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=\left[\mathrm{OH}^{-}\right]+\left[\mathrm{HCO}_{3}^-\right]+2 \times\left[\mathrm{CO}_{3}^{2-}\right] \nonumber$
Exercise 6.7.3
Write appropriate mass balance and charge balance equations for a solution containing 0.10 M KH2PO4 and 0.050 M Na2HPO4.
Answer
To help us determine what ions are in solution, let’s write down all the reaction needed to prepare the solutions and the equilibrium reactions that take place within these solutions. These reactions are the dissolution of two soluble salts
$\mathrm{KH}_{2} \mathrm{PO}_{4}(s) \longrightarrow \mathrm{K}^{+}(a q)+\mathrm{H}_{2} \mathrm{PO}_{4}^{-}(a q) \nonumber$
$\mathrm{NaHPO}_{4}(s) \longrightarrow \mathrm{Na}^{+}(a q)+\mathrm{HPO}_{4}^{2-}(a q) \nonumber$
and the acid–base dissociation reactions for $\text{H}_2\text{PO}_4^-$, $\text{HPO}_4^{2-}$. and H2O.
$\mathrm{H}_{2} \mathrm{PO}_{4}^{-}(a q)+\mathrm{H}_{2} \mathrm{O}(l)\rightleftharpoons\mathrm{H}_{3} \mathrm{O}^{+}(a q)+\mathrm{HPO}_{4}^{2-}(a q) \nonumber$
$\mathrm{H}_{2} \mathrm{PO}_{4}^{-}(a q)+\mathrm{H}_{2} \mathrm{O}(l)\rightleftharpoons \mathrm{OH}^{-}(a q)+\mathrm{H}_{3} \mathrm{PO}_{4}(a q) \nonumber$
$\mathrm{HPO}_{4}^{2-}(a q)+\mathrm{H}_{2} \mathrm{O}(l)\rightleftharpoons \mathrm{H}_{3} \mathrm{O}^{+}(a q)+\mathrm{PO}_{4}^{3-}(a q) \nonumber$
$2 \mathrm{H}_{2} \mathrm{O}(l)\rightleftharpoons \mathrm{H}_{3} \mathrm{O}^{+}(a q)+\mathrm{OH}^{-}(a q) \nonumber$
Note that we did not include the base dissociation reaction for $\text{HPO}_4^{2-}$ because we already accounted for its product, $\text{H}_2\text{PO}_4^-$, in another reaction. The mass balance equations for K+ and Na+ are straightforward
$\left[\mathrm{K}^{+}\right]=0.10 \ \mathrm{M} \text { and }\left[\mathrm{Na}^{+}\right]=0.10 \ \mathrm{M} \nonumber$
but the mass balance equation for phosphate takes a bit more thought. Both $\text{H}_2\text{PO}_4^-$ and $\text{HPO}_4^{2-}$ produce the same ions in solution. We can, therefore, imagine that the solution initially contains 0.15 M KH2PO4, which gives the following mass balance equation.
$\left[\mathrm{H}_{3} \mathrm{PO}_{4}\right]+\left[\mathrm{H}_{2} \mathrm{PO}_{4}^-\right]+\left[\mathrm{HPO}_{4}^{2-}\right]+\left[\mathrm{PO}_{4}^{3-}\right]=0.15 \ \mathrm{M} \nonumber$
The charge balance equation is
$\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]+\left[\mathrm{K}^{+}\right]+\left[\mathrm{Na}^{+}\right] =\left[\mathrm{H}_{2} \mathrm{PO}_{4}^{-}\right]+2 \times\left[\mathrm{HPO}_{4}^{2-}\right] +3 \times\left[\mathrm{PO}_{4}^{3-}\right]+\left[\mathrm{OH}^{-}\right] \nonumber$
pH of a Monoprotic Weak Acid
To illustrate the systematic approach to solving equilibrium problems, let’s calculate the pH of 1.0 M HF. Two equilibrium reactions affect the pH. The first, and most obvious, is the acid dissociation reaction for HF
$\mathrm{HF}(a q)+\mathrm{H}_{2} \mathrm{O}(l) \rightleftharpoons \mathrm{H}_{3} \mathrm{O}^{+}(a q)+\mathrm{F}^{-}(a q) \nonumber$
for which the equilibrium constant expression is
$K_{\mathrm{a}}=\frac{\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\left[\mathrm{F}^{-}\right]}{[\mathrm{HF}]}=6.8 \times 10^{-4} \label{6.3}$
The second equilibrium reaction is the dissociation of water, which is an obvious yet easily neglected reaction
$2 \mathrm{H}_{2} \mathrm{O}(l) \rightleftharpoons \mathrm{H}_{3} \mathrm{O}^{+}(a q)+\mathrm{OH}^{-}(a q) \nonumber$
$K_{w}=\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\left[\mathrm{OH}^{-}\right]=1.00 \times 10^{-14} \label{6.4}$
Counting unknowns, we find four: [HF], [F], [H3O+], and [OH]. To solve this problem we need two additional equations. These equations are a mass balance equation on hydrofluoric acid
$C_{\mathrm{HF}}=[\mathrm{HF}]+\left[\mathrm{F}^{-}\right]=1.0 \mathrm{M} \label{6.5}$
and a charge balance equation
$\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=\left[\mathrm{OH}^{-}\right]+\left[\mathrm{F}^{-}\right] \label{6.6}$
With four equations and four unknowns, we are ready to solve the problem. Before doing so, let’s simplify the algebra by making two assumptions.
Assumption One. Because HF is a weak acid, we know that the solution is acidic. For an acidic solution it is reasonable to assume that
$\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]>>\left[\mathrm{OH}^{-}\right] \nonumber$
which simplifies the charge balance equation to
$\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=\left[\mathrm{F}^{-}\right] \label{6.7}$
Assumption Two. Because HF is a weak acid, very little of it dissociates to form F. Most of the HF remains in its conjugate weak acid form and it is reasonable to assume that
$[\mathrm{HF}]>>\left[\mathrm{F}^{-}\right] \nonumber$
which simplifies the mass balance equation to
$C_{\mathrm{HF}}=[\mathrm{HF}]=1.0 \ \mathrm{M} \label{6.8}$
For this exercise let’s accept an assumption if it introduces an error of less than ±5%.
Substituting Equation \ref{6.7} and Equation \ref{6.8} into Equation \ref{6.3}, and solving for the concentration of H3O+ gives us
$\mathrm{K}_{\mathrm{a}}=\frac{\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\left[\mathrm{F}^{-}\right]}{[\mathrm{HF}]}=\frac{\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]}{\mathrm{C}_{\mathrm{HF}}}=\frac{\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]^{2}}{\mathrm{C}_{\mathrm{HF}}}=6.8 \times 10^{-4} \nonumber$
$\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=\sqrt{K_{\mathrm{a}} C_{\mathrm{HF}}}=\sqrt{\left(6.8 \times 10^{-4}\right)(1.0)}=2.6 \times 10^{-2} \nonumber$
Before accepting this answer, we must verify our assumptions. The first assumption is that [OH] is significantly smaller than [H3O+]. Using Equation \ref{6.4}, we find that
$\left[\mathrm{OH}^{-}\right]=\frac{K_{\mathrm{w}}}{\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]}=\frac{1.00 \times 10^{-14}}{2.6 \times 10^{-2}}=3.8 \times 10^{-13} \nonumber$
Clearly this assumption is acceptable. The second assumption is that [F] is significantly smaller than [HF]. From Equation \ref{6.7} we have
$\left[\mathrm{F}^{-}\right]=2.6 \times 10^{-2} \ \mathrm{M} \nonumber$
Because [F] is 2.60% of CHF, this assumption also is acceptable. Given that [H3O+] is $2.6 \times 10^{-2}$ M, the pH of 1.0 M HF is 1.59.
How does the calculation change if we require that the error introduced in our assumptions be less than ±1%? In this case we no longer can assume that [HF] >> [F] and we cannot simplify the mass balance equation. Solving the mass balance equation for [HF]
$[\mathrm{HF}]=C_{\mathrm{HF}}-\left[\mathrm{F}^{-}\right]=C_{\mathrm{HF}}-\left[\mathrm{H}_{3} \mathrm{O}^{+}\right] \nonumber$
and substituting into the Ka expression along with Equation \ref{6.7} gives
$K_{\mathrm{a}}=\frac{\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]^{2}}{C_{\mathrm{HF}}-\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]} \nonumber$
Rearranging this equation leaves us with a quadratic equation
$\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]^{2}+K_{\mathrm{a}}\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]-K_{\mathrm{a}} C_{\mathrm{HF}}=0 \nonumber$
which we solve using the quadratic formula
$x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a} \nonumber$
where a, b, and c are the coefficients in the quadratic equation
$a x^{2}+b x+c=0 \nonumber$
Solving a quadratic equation gives two roots, only one of which has chemical significance. For our problem, the equation’s roots are
$x=\frac{-6.8 \times 10^{-4} \pm \sqrt{\left(6.8 \times 10^{-4}\right)^{2}-(4)(1)\left(-6.8 \times 10^{-4}\right)}}{(2)(1)} \nonumber$
$x=\frac{-6.8 \times 10^{-4} \pm 5.22 \times 10^{-2}}{2} \nonumber$
$x=2.57 \times 10^{-2} \text { or }-2.64 \times 10^{-2} \nonumber$
Only the positive root is chemically significant because the negative root gives a negative concentration for H3O+. Thus, [H3O+] is $2.57 \times 10^{-2}$ M and the pH is 1.59.
You can extend this approach to calculating the pH of a monoprotic weak base by replacing Ka with Kb, replacing CHF with the weak base’s concentration, and solving for [OH] in place of [H3O+].
Exercise 6.7.4
Calculate the pH of 0.050 M NH3. State any assumptions you make in solving the problem, limiting the error for any assumption to ±5%. The Kb value for NH3 is $1.75 \times 10^{-5}$.
Answer
To determine the pH of 0.050 M NH3, we need to consider two equilibrium reactions: the base dissociation reaction for NH3
$\mathrm{NH}_{3}(a q)+\mathrm{H}_{2} \mathrm{O}(l) \rightleftharpoons \mathrm{OH}^{-}(a q)+\mathrm{NH}_{4}^{+}(a q) \nonumber$
and water’s dissociation reaction.
$2 \mathrm{H}_{2} \mathrm{O}(l)\rightleftharpoons \mathrm{H}_{3} \mathrm{O}^{+}(a q)+\mathrm{OH}^{-}(a q) \nonumber$
These two reactions contain four species whose concentrations we need to consider: NH3, $\text{NH}_4^+$, H3O+, and OH. We need four equations to solve the problem—these equations are the Kb equation for NH3
$K_{\mathrm{b}}=\frac{\left[\mathrm{NH}_{4}^{+}\right]\left[\mathrm{OH}^{-}\right]}{\left[\mathrm{NH}_{3}\right]}=1.75 \times 10^{-5} \nonumber$
the Kw equation for H2O
$K_{w}=\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\left[\mathrm{OH}^{-}\right] \nonumber$
a mass balance equation on ammonia
$C_{\mathrm{NH}_{3}}=0.050 \ \mathrm{M}=\left[\mathrm{NH}_{3}\right]+\left[\mathrm{NH}_{4}^{+}\right] \nonumber$
and a charge balance equation
$\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]+\left[\mathrm{NH}_{4}^{+}\right]=\left[\mathrm{OH}^{-}\right] \nonumber$
To solve this problem, we will make two assumptions. Because NH3 is a base, our first assumption is
$\left[\mathrm{OH}^{-}\right]>>\left[\mathrm{H}_{3} \mathrm{O}^{+}\right] \nonumber$
which simplifies the charge balance equation to
$\left[\mathrm{NH}_{4}^{+}\right]=\left[\mathrm{OH}^{-}\right] \nonumber$
Because NH3 is a weak base, our second assumption is
$\left[\mathrm{NH}_{3}\right]>>\left[\mathrm{NH}_{4}^{+}\right] \nonumber$
which simplifies the mass balance equation to
$C_{\mathrm{NH}_{3}}=0.050 \ \mathrm{M}=\left[\mathrm{NH}_{3}\right] \nonumber$
Substituting the simplified charge balance equation and mass balance equation into the Kb equation leave us with
$K_{\mathrm{b}}=\frac{\left[\mathrm{NH}_{4}^{+}\right]\left[\mathrm{OH}^{-}\right]}{\left[\mathrm{NH}_{3}\right]}=\frac{\left[\mathrm{OH}^{-}\right]\left[\mathrm{OH}^{-}\right]}{C_{\mathrm{NH}_3}}=\frac{\left[\mathrm{OH}^{-}\right]^{2}}{C_{\mathrm{NH_3}}}=1.75 \times 10^{-5} \nonumber$
$\left[\mathrm{OH}^{-}\right]=\sqrt{K_{\mathrm{b}} C_{\mathrm{NH_3}}}=\sqrt{\left(1.75 \times 10^{-5}\right)(0.050)}=9.35 \times 10^{-4} \nonumber$
Before we accept this answer, we must verify our two assumptions. The first assumption is that the concentration of OH is significantly greater than the concentration of H3O+. Using Kw, we find that
$\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=\frac{K_{\mathrm{w}}}{\left[\mathrm{OH}^{-}\right]}=\frac{1.00 \times 10^{-14}}{9.35 \times 10^{-4}}=1.07 \times 10^{-11} \nonumber$
Clearly this assumption is acceptable. Our second assumption is that the concentration of NH3 is significantly greater than the concentration of $\text{NH}_4^+$. Using our simplified charge balance equation, we find that
$\left[\mathrm{NH}_{4}^{+}\right]=\left[\mathrm{OH}^{-}\right]=9.35 \times 10^{-4} \nonumber$
Because the concentration of $\text{NH}_4^+$ is 1.9% of $C_{\text{NH}_3}$, our second assumption also is reasonable. Given that [H3O+] is $1.07 \times 10^{-11}$, the pH is 10.97.
pH of a Polyprotic Acid or Base
A more challenging problem is to find the pH of a solution that contains a polyprotic weak acid or one of its conjugate species. As an example, consider the amino acid alanine, whose structure is shown in Figure 6.7.1 . The ladder diagram in Figure 6.7.2 shows alanine’s three acid–base forms and their respective areas of predominance. For simplicity, we identify these species as H2L+, HL, and L.
Figure 6.7.2 . Ladder diagram for alanine.
pH of 0.10 M Alanine Hydrochloride (H2L+)
Alanine hydrochloride is the salt of the diprotic weak acid H2L+ and Cl. Because H2L+ has two acid dissociation reactions, a complete systematic solution to this problem is more complicated than that for a monoprotic weak acid. The ladder diagram in Figure 6.7.2 helps us simplify the problem. Because the areas of predominance for H2L+ and L are so far apart, we can assume that a solution of H2L+ will not contain a significant amount of L. As a result, we can treat H2L+ as though it is a monoprotic weak acid. Calculating the pH of 0.10 M alanine hydrochloride, which is 1.72, is left to the reader as an exercise.
pH of 0.10 M Sodium Alaninate (L–)
The alaninate ion is a diprotic weak base. Because L has two base dissociation reactions, a complete systematic solution to this problem is more complicated than that for a monoprotic weak base. Once again, the ladder diagram in Figure 6.7.2 helps us simplify the problem. Because the areas of predominance for H2L+ and L are so far apart, we can assume that a solution of L will not contain a significant amount of H2L+. As a result, we can treat L as though it is a monoprotic weak base. Calculating the pH of 0.10 M sodium alaninate, which is 11.42, is left to the reader as an exercise.
pH of 0.10 M Alanine (HL)
Finding the pH of a solution of alanine is more complicated than our previous two examples because we cannot ignore the presence of either H2L+ or L. To calculate the solution’s pH we must consider alanine’s acid dissociation reaction
$\mathrm{HL}(a q)+\mathrm{H}_{2} \mathrm{O}(l)\rightleftharpoons \mathrm{H}_{3} \mathrm{O}^{+}(a q)+\mathrm{L}^{-}(a q) \nonumber$
and its base dissociation reaction
$\mathrm{HL}(a q)+\mathrm{H}_{2} \mathrm{O}(l)\rightleftharpoons \mathrm{OH}^{-}(a q)+\mathrm{H}_{2} \mathrm{L}^{+}(a q) \nonumber$
and, as always, we must also consider the dissociation of water
$2 \mathrm{H}_{2} \mathrm{O}(l)\rightleftharpoons \mathrm{H}_{3} \mathrm{O}^{+}(a q)+\mathrm{OH}^{-}(a q) \nonumber$
This leaves us with five unknowns—[H2L+], [HL], [L], [H3O+], and [OH]—for which we need five equations. These equations are Ka2 and Kb2 for alanine
$K_{\mathrm{a} 2}=\frac{\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\left[\mathrm{L}^{-}\right]}{[\mathrm{HL}]} \nonumber$
$K_{\mathrm{b} 2}=\frac{K_{\mathrm{w}}}{K_{\mathrm{a1}}}=\frac{\left[\mathrm{OH}^{-}\right]\left[\mathrm{H}_{2} \mathrm{L}^{+}\right]}{[\mathrm{HL}]} \nonumber$
the Kw equation
$K_{\mathrm{w}}=\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\left[\mathrm{OH}^{-}\right] \nonumber$
a mass balance equation for alanine
$C_{\mathrm{HL}}=\left[\mathrm{H}_{2} \mathrm{L}^{+}\right]+[\mathrm{HL}]+[\mathrm{L}^{-}] \nonumber$
and a charge balance equation
$\left[\mathrm{H}_{2} \mathrm{L}^{+}\right]+\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=[\mathrm{OH}^-]+[\mathrm{L^-}] \nonumber$
Because HL is a weak acid and a weak base, it seems reasonable to assume that little of it will dissociate and that
$[\mathrm{HL}]>>\left[\mathrm{H}_{2} \mathrm{L}^{+}\right]+[\mathrm{L}^-] \nonumber$
which allows us to simplify the mass balance equation to
$C_{\mathrm{HL}}=[\mathrm{HL}] \nonumber$
Next we solve Kb2 for [H2L+]
$\left[\mathrm{H}_{2} \mathrm{L}^{+}\right]=\frac{K_{\mathrm{w}}[\mathrm{HL}]}{K_{\mathrm{a1}}\left[\mathrm{OH}^{-}\right]}=\frac{\left[\mathrm{H}_{3} \mathrm{O}^{+}\right][\mathrm{HL}]}{K_{\mathrm{a1}}}=\frac{C_{\mathrm{HL}}\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]}{K_{\mathrm{a1}}} \nonumber$
and solve Ka2 for [L]
$[\mathrm{L^-}]=\frac{K_{a2}[\mathrm{HL}]}{\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]}=\frac{K_{a2} C_{\mathrm{HL}}}{\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]} \nonumber$
Substituting these equations for [H2L+] and [L], and the equation for Kw, into the charge balance equation give us
$\frac{C_{\mathrm{HL}}\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]}{K_{\mathrm{a1}}}+\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=\frac{K_{\mathrm{w}}}{\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]}+\frac{K_{a2} C_{\mathrm{HL}}}{\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]} \nonumber$
which we simplify to
$\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\left(\frac{C_{\mathrm{HL}}}{K_{\mathrm{a1}}}+1\right)=\frac{1}{\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]}\left(K_{\mathrm{w}}+K_{a2} C_{\mathrm{HL}}\right) \nonumber$
$\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]^{2}=\frac{\left(K_{\mathrm{a} 2} C_{\mathrm{HL}}+K_{\mathrm{w}}\right)}{\frac{C_{\mathrm{HL}}}{K_{\mathrm{a1}}}+1}=\frac{K_{\mathrm{a1}}\left(K_{\mathrm{a2}} C_{\mathrm{HL}}+K_{\mathrm{w}}\right)}{C_{\mathrm{HL}}+K_{\mathrm{a1}}} \nonumber$
$\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=\sqrt{\frac{\left(K_{\mathrm{a1}} K_{a2} C_{\mathrm{HL}}+K_{\mathrm{a1}} K_{\mathrm{w}}\right)}{C_{\mathrm{HL}}+K_{\mathrm{a1}}}} \nonumber$
We can further simplify this equation if Ka1Kw << Ka1Ka2CHL, and if Ka1 << CHL, leaving us with
$\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=\sqrt{K_{\mathrm{a1}} K_{\mathrm{a} 2}} \nonumber$
For a solution of 0.10 M alanine the [H3O+] is
$\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=\sqrt{\left(4.487 \times 10^{-3}\right)\left(1.358 \times 10^{-10}\right)}=7.806 \times 10^{-7} \ \mathrm{M} \nonumber$
or a pH of 6.11.
Exercise 6.7.5
Verify that each assumption in our solution for the pH of 0.10 M alanine is reasonable, using ±5% as the limit for the acceptable error.
Answer
In solving for the pH of 0.10 M alanine, we made the following three assumptions: (a) [HL] >> [H2L+] + [L]; (b) Ka1Kw << Ka1Ka2CHL; and (c) Ka1 << CHL. Assumptions (b) and (c) are easy to check. The value of Ka1 ($4.487 \times 10^{-3}$) is 4.5% of CHL (0.10), and Ka1Kw ($4.487 \times 10^{-17}$) is 0.074% of Ka1Ka2CHL ($6.093 \times 10^{-14}$). Each of these assumptions introduces an error of less than ±5%.
To test assumption (a) we need to calculate the concentrations of H2L+ and L, which we accomplish using the equations for Ka1 and Ka2.
$\left[\mathrm{H}_{2} \mathrm{L}^{+}\right]=\frac{\left[\mathrm{H}_{3} \mathrm{O}^{+}\right][\mathrm{HL}]}{K_{\mathrm{a1}}}=\frac{\left(7.807 \times 10^{-7}\right)(0.10)}{4.487 \times 10^{-3}}=1.74 \times 10^{-5} \nonumber$
$\left[\mathrm{L}^{-}\right]=\frac{K_{a 2}[\mathrm{HL}]}{\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]}=\frac{\left(1.358 \times 10^{-10}\right)(0.10)}{7.807 \times 10^{-7}}=1.74 \times 10^{-5} \nonumber$
Because these concentrations are less than ±5% of CHL, the first assumption also is acceptable.
Effect of Complexation on Solubility
One method for increasing a precipitate’s solubility is to add a ligand that forms soluble complexes with one of the precipitate’s ions. For example, the solubility of AgI increases in the presence of NH3 due to the formation of the soluble $\text{Ag(NH}_3)_2^+$ complex. As a final illustration of the systematic approach to solving equilibrium problems, let’s calculate the molar solubility of AgI in 0.10 M NH3.
We begin by writing the relevant equilibrium reactions, which includes the solubility of AgI, the acid–base chemistry of NH3 and H2O, and the metal‐ligand complexation chemistry between Ag+ and NH3.
$\begin{array}{c}{\operatorname{AgI}(s)\rightleftharpoons\operatorname{Ag}^{+}(a q)+\mathrm{I}^{-}(a q)} \ {\mathrm{NH}_{3}(a q)+\mathrm{H}_{2} \mathrm{O}(l)\rightleftharpoons \mathrm{OH}^{-}(a q)+\mathrm{NH}_{4}^{+}(a q)} \ {2 \mathrm{H}_{2} \mathrm{O}(l)\rightleftharpoons \mathrm{H}_{3} \mathrm{O}^{+}(a q)+\mathrm{OH}^{-}(a q)} \ {\mathrm{Ag}^{+}(a q)+2 \mathrm{NH}_{3}(a q) \rightleftharpoons \mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}^{+}(a q)}\end{array} \nonumber$
This leaves us with seven unknowns—[Ag+], [I], [NH3], [$\text{NH}_4^+$ ], [OH], [H3O+], and [$\text{Ag(NH}_3)_2^+$]—and a need for seven equations. Four of the equations we need to solve this problem are the equilibrium constant expressions
$K_{\mathrm{sp}}=\left[\mathrm{Ag}^{+}\right]\left[\mathrm{I}^{-}\right]=8.3 \times 10^{-17} \label{6.9}$
$K_{\mathrm{b}}=\frac{\left[\mathrm{NH}_{4}^{+}\right]\left[\mathrm{OH}^{-}\right]}{\left[\mathrm{NH}_{3}\right]}=1.75 \times 10^{-5} \label{6.10}$
$K_{\mathrm{w}}=\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\left[\mathrm{OH}^{-}\right]=1.00 \times 10^{-14} \label{6.11}$
$\beta_{2}=\frac{\left[\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}^{+}\right]}{\left[\mathrm{Ag}^{+}\right]\left[\mathrm{NH}_{3}\right]^{2}}=1.7 \times 10^{7} \label{6.12}$
We still need three additional equations. The first of these equations is a mass balance for NH3.
$C_{\mathrm{NH}_{3}}=\left[\mathrm{NH}_{3}\right]+\left[\mathrm{NH}_{4}^{+}\right]+2 \times\left[\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}^{+}\right] \label{6.13}$
In writing this mass balance equation we multiply the concentration of $\text{Ag(NH}_3)_2^+$ by two since there are two moles of NH3 per mole of $\text{Ag(NH}_3)_2^+$. The second additional equation is a mass balance between iodide and silver. Because AgI is the only source of I and Ag+, each iodide in solution must have an associated silver ion, which may be Ag+ or $\text{Ag(NH}_3)_2^+$ ; thus
$\left[\mathrm{I}^{-}\right]=\left[\mathrm{Ag}^{+}\right]+\left[\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}^{+}\right] \label{6.14}$
Finally, we include a charge balance equation.
$\left[\mathrm{Ag}^{+}\right]+\left[\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}^{+}\right]+\left[\mathrm{NH}_{4}^{+}\right]+\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=[\mathrm{OH}^-]+[\mathrm{I}^-] \label{6.15}$
Although the problem looks challenging, three assumptions greatly simplify the algebra.
Assumption One. Because the formation of the $\text{Ag(NH}_3)_2^+$ complex is so favorable ($\beta_2$ is $1.7 \times 10^7$), there is very little free Ag+ in solution and it is reasonable to assume that
$\left[\mathrm{Ag}^{+}\right]<<\left[\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}^{+}\right] \nonumber$
Assumption Two. Because NH3 is a weak base we may reasonably assume that most uncomplexed ammonia remains as NH3; thus
$\left[\mathrm{NH}_{4}^{+}\right]<<\left[\mathrm{NH}_{3}\right] \nonumber$
Assumption Three. Because Ksp for AgI is significantly smaller than $\beta_2$ for $\text{Ag(NH}_3)_2^+$, the solubility of AgI probably is small enough that very little ammonia is needed to form the metal–ligand complex; thus
$\left[\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}^{+}\right]<<\left[\mathrm{NH}_{3}\right] \nonumber$
As we use these assumptions to simplify the algebra, let’s set ±5% as the limit for error.
Assumption two and assumption three suggest that the concentration of NH3 is much larger than the concentrations of either $\text{NH}_4^+$ or $\text{Ag(NH}_3)_2^+$, which allows us to simplify the mass balance equation for NH3 to
$C_{\mathrm{NH}_{3}}=\left[\mathrm{NH}_{3}\right] \label{6.16}$
Finally, using assumption one, which suggests that the concentration of $\text{Ag(NH}_3)_2^+$ is much larger than the concentration of Ag+, we simplify the mass balance equation for I to
$\left[\mathrm{I}^{-}\right]=\left[\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}^{+}\right] \label{6.17}$
Now we are ready to combine equations and to solve the problem. We begin by solving Equation \ref{6.9} for [Ag+] and substitute it into $\beta_2$ (Equation \ref{6.12}), which leaves us with
$\beta_{2}=\frac{\left[\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}^{+}\right][\mathrm{I}^-]}{K_{\mathrm{sp}}\left[\mathrm{NH}_{3}\right]^{2}} \label{6.18}$
Next we substitute Equation \ref{6.16} and Equation \ref{6.17} into Equation \ref{6.18}, obtaining
$\beta_{2}=\frac{\left[\mathrm{I}^{-}\right]^{2}}{K_{\mathrm{sp}}\left(C_{\mathrm{NH}_3}\right)^{2}} \label{6.19}$
Solving Equation \ref{6.19} for [I] gives
$\left[\mathrm{I}^{-}\right]=C_{\mathrm{NH}_3} \sqrt{\beta_{2} K_{s p}} = \ (0.10) \sqrt{\left(1.7 \times 10^{7}\right)\left(8.3 \times 10^{-17}\right)}=3.76 \times 10^{-6} \ \mathrm{M} \nonumber$
Because one mole of AgI produces one mole of I, the molar solubility of AgI is the same as the [I], or $3.8 \times 10^{-6}$ mol/L.
Before we accept this answer we need to check our assumptions. Substituting [I] into Equation \ref{6.9}, we find that the concentration of Ag+ is
$\left[\mathrm{Ag}^{+}\right]=\frac{K_{\mathrm{p}}}{[\mathrm{I}^-]}=\frac{8.3 \times 10^{-17}}{3.76 \times 10^{-6}}=2.2 \times 10^{-11} \ \mathrm{M} \nonumber$
Substituting the concentrations of I and Ag+ into the mass balance equation for iodide (Equation \ref{6.14}), gives the concentration of $\text{Ag(NH}_3)_2^+$ as
$\left[\operatorname{Ag}\left(\mathrm{NH}_{3}\right)_{2}^{+}\right]=[\mathrm{I}^-]-\left[\mathrm{Ag}^{+}\right]=3.76 \times 10^{-6}-2.2 \times 10^{-11}=3.76 \times 10^{-6} \ \mathrm{M} \nonumber$
Our first assumption that [Ag+] is significantly smaller than the [$\text{Ag(NH}_3)_2^+$] is reasonable.
Substituting the concentrations of Ag+ and $\text{Ag(NH}_3)_2^+$ into Equation \ref{6.12} and solving for [NH3], gives
$\left[\mathrm{NH}_{3}\right]=\sqrt{\frac{\left[\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}^{+}\right]}{\left[\mathrm{Ag}^{+}\right] \beta_{2}}}=\sqrt{\frac{3.76 \times 10^{-6}}{\left(2.2 \times 10^{-11}\right)\left(1.7 \times 10^{7}\right)}}=0.10 \ \mathrm{M} \nonumber$
From the mass balance equation for NH3 (Equation \ref{6.12}) we see that [$\text{NH}_4^+$] is negligible, verifying our second assumption that $[\text{NH}_4^+]$ is significantly smaller than [NH3]. Our third assumption that [$\text{Ag(NH}_3)_2^+$] is significantly smaller than [NH3] also is reasonable.
Did you notice that our solution to this problem did not make use of Equation \ref{6.15}, the charge balance equation? The reason for this is that we did not try to solve for the concentration of all seven species. If we need to know the reaction mixture’s complete composition at equilibrium, then we will need to incorporate the charge balance equation into our solution. | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/06%3A_Equilibrium_Chemistry/6.07%3A_Solving_Equilibrium_Problems.txt |
Adding as little as 0.1 mL of concentrated HCl to a liter of H2O shifts the pH from 7.0 to 3.0. Adding the same amount of HCl to a liter of a solution that 0.1 M in acetic acid and 0.1 M in sodium acetate, however, results in a negligible change in pH. Why do these two solutions respond so differently to the addition of HCl?
A mixture of acetic acid and sodium acetate is one example of an acid–base buffer. To understand how this buffer works to limit the change in pH, we need to consider its acid dissociation reaction
$\mathrm{CH}_{3} \mathrm{COOH}(a q)+\mathrm{H}_{2} \mathrm{O}(l)\rightleftharpoons\mathrm{H}_{3} \mathrm{O}^{+}(a q)+\mathrm{CH}_{3} \mathrm{COO}^{-}(a q) \nonumber$
and its corresponding acid dissociation constant
$K_{a}=\frac{\left[\mathrm{CH}_{3} \mathrm{COO}^{-}\right]\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]}{\left[\mathrm{CH}_{3} \mathrm{COOH}\right]}=1.75 \times 10^{-5} \label{6.1}$
Taking the negative log of the terms in Equation \ref{6.1} and solving for pH leaves us with the result shown here.
$\mathrm{pH}=\mathrm{p} K_{\mathrm{a}}+\log \frac{\left[\mathrm{CH}_{3} \mathrm{COO}^{-}\right]}{\left[\mathrm{CH}_{3} \mathrm{COOH}\right]} \nonumber$
$\mathrm{pH}=4.76+\log \frac{\left[\mathrm{CH}_{3} \mathrm{COO}^{-}\right]}{\left[\mathrm{CH}_{3} \mathrm{COOH}\right]} \label{6.2}$
You may recall that we developed these same equations in Chapter 6.6 when we introduced ladder diagrams.
Buffering occurs because of the logarithmic relationship between pH and the concentration ratio of acetate and acetic acid. Here is an example to illustrate this point. If the concentrations of acetic acid and acetate are equal, the buffer’s pH is 4.76. If we convert 10% of the acetate to acetic acid, by adding a strong acid, the ratio [CH3COO]/[CH3COOH] changes from 1.00 to 0.818, and the pH decreases from 4.76 to 4.67—a decrease of only 0.09 pH units.
The ratio [CH3COO]/[CH3COOH] becomes 0.9/1.1 = 0.818 and the pH becomes
$\mathrm{pH}=4.76+\log (0.818)=4.67 \nonumber$
Systematic Solution to Buffer Problems
Equation \ref{6.2} is written in terms of the equilibrium concentrations of CH3COOH and of CH3COO. A more useful relationship relates a buffer’s pH to the initial concentrations of the weak acid and the weak base. We can derive a general buffer equation by considering the following reactions for a weak acid, HA, and the soluble salt of its conjugate weak base, NaA.
$\begin{array}{c}{\mathrm{NaA}(s) \rightarrow \mathrm{Na}^{+}(a q)+\mathrm{A}^{-}(a q)} \ {\mathrm{HA}(a q)+\mathrm{H}_{2} \mathrm{O}(l)\rightleftharpoons \mathrm{H}_{3} \mathrm{O}^{+}(a q)+\mathrm{A}^{-}(a q)} \ {2 \mathrm{H}_{2} \mathrm{O}(l)\rightleftharpoons \mathrm{H}_{3} \mathrm{O}^{+}(a q)+\mathrm{OH}^{-}(a q)}\end{array} \nonumber$
Because the concentrations of Na+, A, HA, H3O+, and OH are unknown, we need five equations to define the solution’s composition. Two of these equations are the equilibrium constant expressions for HA and H2O.
$K_{a}=\frac{\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\left[\mathrm{A}^{-}\right]}{[\mathrm{HA}]} \label{6.3}$
$K_{w}=\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\left[\mathrm{OH}^{-}\right] \nonumber$
The remaining three equations are mass balance equations for HA and Na+
$C_{\mathrm{HA}}+C_{\mathrm{NaA}}=[\mathrm{HA}]+\left[\mathrm{A}^{-}\right] \label{6.4}$
$C_{\mathrm{NaA}}=\left[\mathrm{Na}^{+}\right] \label{6.5}$
and a charge balance equation
$\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]+\left[\mathrm{Na}^{+}\right]=\left[\mathrm{OH}^{-}\right]+\left[\mathrm{A}^{-}\right] \label{6.6}$
Substituting Equation \ref{6.5} into Equation \ref{6.6} and solving for [A] gives
$\left[\mathrm{A}^{-}\right]=C_{\mathrm{NaA} }-\left[\mathrm{OH}^{-}\right]+\left[\mathrm{H}_{3} \mathrm{O}^{+}\right] \label{6.7}$
Next, we substitute Equation \ref{6.7} into Equation \ref{6.4}, which gives the concentration of HA as
$[\mathrm{HA}]=C_{\mathrm{HA}}+\left[\mathrm{OH}^{-}\right]-\left[\mathrm{H}_{3} \mathrm{O}^{+}\right] \label{6.8}$
Finally, we substitute Equation \ref{6.7} and Equation \ref{6.8} into Equation \ref{6.3} and solve for pH to arrive at a general equation for a buffer’s pH.
$\mathrm{pH}=\mathrm{p} K_{\mathrm{a}}+\log \frac{C_{\mathrm{NaA} }-\left[\mathrm{OH}^{-}\right]+\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]}{C_{\mathrm{HA}}+\left[\mathrm{OH}^{-}\right]-\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]} \nonumber$
If the initial concentrations of the weak acid, CHA, and the weak base, CNaA. are significantly greater than [H3O+] and [OH], then we can simplify the general equation to the Henderson–Hasselbalch equation.
$\mathrm{pH}=\mathrm{p} K_{\mathrm{a}}+\log \frac{C_{\mathrm{NaA}}}{C_{\mathrm{HA}}} \label{6.9}$
As outlined below, the Henderson–Hasselbalch equation provides a simple way to calculate the pH of a buffer, and to determine the change in pH upon adding a strong acid or strong base.
Lawrence Henderson (1878‐1942) first developed a relationship between [H3O+], [HA], and [A] while studying the buffering of blood. Kurt Hasselbalch (1874‐1962) modified Henderson’s equation by transforming it to the logarithmic form shown in Equation \ref{6.9}. The assumptions that lead to Equation \ref{6.9} result in a minimal error in pH (<±5%) for larger concentrations of HA and A, for concentrations of HA and A that are similar in magnitude, and for weak acid’s with pKa values closer to 7. For most problems in this textbook, Equation \ref{6.9} provides acceptable results. Be sure, however, to test your assumptions. For a discussion of the Henderson–Hasselbalch equation, including the error inherent in Equation \ref{6.9}, see Po, H. N.; Senozan, N. M. “The Henderson–Hasselbalch Equation: Its History and Limitations,” J. Chem. Educ. 2001, 78, 1499–1503.
Example 6.8.1
Calculate the pH of a buffer that is 0.020 M in NH3 and 0.030 M in NH4Cl. What is the pH after we add 1.0 mL of 0.10 M NaOH to 0.10 L of this buffer?
Solution
The acid dissociation constant for $\text{NH}_4^+$ is $5.70 \times 10^{-10}$, which is a pKa of 9.24. Substituting the initial concentrations of NH3 and NH4Cl into Equation \ref{6.9} and solving, we find that the buffer’s pH is
$\mathrm{pH}=9.24+\log \frac{0.020}{0.030}=9.06 \nonumber$
With a pH of 9.06, the concentration of H3O+ is $8.71 \times 10^{-10}$ and the concentration of OH is $1.15 \times 10^{-5}$. Because both of these concentrations are much smaller than either $C_{\text{NH}_3}$ or $C_{\text{NH}_4\text{Cl}}$, the approximations used to derive Equation \ref{6.9} are reasonable.
Adding NaOH converts a portion of the $\text{NH}_4^+$ to NH3 following reaction
$\mathrm{NH}_{4}^{+}(a q)+\mathrm{OH}^{-}(a q) \rightleftharpoons \mathrm{H}_{2} \mathrm{O}(l)+\mathrm{NH}_{3}(a q) \nonumber$
Because this reaction’s equilibrium constant is so large (it is equal to (Kb)-1 or $5.7 \times 10^4$), we may treat the reaction as if it goes to completion. The new concentrations of $\text{NH}_4^+$ and NH3 are
$C_{\mathrm{NH}_{4}^{+}}=\frac{\operatorname{mol} \ \mathrm{NH}_{4}^{+}- \ \mathrm{mol} \mathrm{OH}^{-}}{V_{\mathrm{total}}} \nonumber$
$C_{\mathrm{NH}_4^+}=\frac{(0.030 \ \mathrm{M})(0.10 \ \mathrm{L})-(0.10 \ \mathrm{M})\left(1.0 \times 10^{-3} \ \mathrm{L}\right)}{0.10 \ \mathrm{L}+1.0 \times 10^{-3} \ \mathrm{L}}=0.029 \ \mathrm{M} \nonumber$
$C_{\mathrm{NH}_{3}}=\frac{\mathrm{mol} \ \mathrm{NH}_{3}+\mathrm{mol} \ \mathrm{OH}^{-}}{V_{\mathrm{total}}} \nonumber$
$C_{\mathrm{NH}_3}=\frac{(0.020 \ \mathrm{M})(0.10 \ \mathrm{L})+(0.10 \ \mathrm{M})\left(1.0 \times 10^{-3} \ \mathrm{L}\right)}{0.10 \ \mathrm{L}+1.0 \times 10^{-3} \ \mathrm{L}}=0.021 \ \mathrm{M} \nonumber$
Substituting these concentrations into the equation 6.60 gives a pH of
$\mathrm{pH}=9.24+\log \frac{0.021}{0.029}=9.10 \nonumber$
Note that adding NaOH increases the pH from 9.06 to 9.10. As we expect, adding a base makes the pH more basic. Checking to see that the pH changes in the right direction is one way to catch a calculation error.
Exercise 6.8.1
Calculate the pH of a buffer that is 0.10 M in KH2PO4 and 0.050 M in Na2HPO4. What is the pH after we add 5.0 mL of 0.20 M HCl to 0.10 L of this buffer. Use Appendix 11 to find the appropriate Ka value.
Answer
The acid dissociation constant for $\text{H}_2\text{PO}_4^-$ is $6.32 \times 10^{-8}$, or a pKa of 7.199. Substituting the initial concentrations of $\text{H}_2\text{PO}_4^-$ and $\text{HPO}_4^{2-}$ into Equation \ref{6.9} and solving gives the buffer’s pH as
$\mathrm{pH}=7.199+\log \frac{\left[\mathrm{HPO}_{4}^{2-}\right]}{\left[\mathrm{H}_{2} \mathrm{PO}_{4}^{-}\right]}=7.199+\log \frac{0.050}{0.10}=6.898 \approx 6.90\nonumber$
Adding HCl converts a portion of $\text{HPO}_4^{2-}$ to $\text{H}_2\text{PO}_4^-$ as a result of the following reaction
$\mathrm{HPO}_{4}^{2-}(a q)+\mathrm{H}_{3} \mathrm{O}^{+}(a q)\rightleftharpoons \mathrm{H}_{2} \mathrm{O}(l)+\mathrm{H}_{2} \mathrm{PO}_{4}^{-}(a q) \nonumber$
Because this reaction’s equilibrium constant is so large (it is $1.59 \times 10^7$), we may treat the reaction as if it goes to completion. The new concentrations of $\text{H}_2\text{PO}_4^-$ and $\text{HPO}_4^{2-}$ are
$C_{\mathrm{H}_{2} \mathrm{PO}_{4}^{4-}}=\frac{\mathrm{mol} \ \mathrm{H}_{2} \mathrm{PO}_{4}^{-}+\mathrm{mol} \ \mathrm{HCl}}{V_{\mathrm{total}}} \nonumber$
$C_{\mathrm{H}_{2} \mathrm{PO}_{4}^{4-}}=\frac{(0.10 \ \mathrm{M})(0.10 \ \mathrm{L})+(0.20 \ \mathrm{M})\left(5.0 \times 10^{-3} \ \mathrm{L}\right)}{0.10 \ \mathrm{L}+5.0 \times 10^{-3} \ \mathrm{L}}=0.105 \ \mathrm{M} \nonumber$
$C_{\mathrm{HPO}_{4}^{2-}}=\frac{\mathrm{mol} \ \mathrm{HPO}_{4}^{2-}-\mathrm{mol} \ \mathrm{HCl}}{V_{\mathrm{total}}} \nonumber$
$C_{\mathrm{HPO}_{4}^{2-}}=\frac{(0.05 \ \mathrm{M})(0.10 \ \mathrm{L})-(0.20 \ \mathrm{M})\left(5.0 \times 10^{-3} \ \mathrm{L}\right)}{0.10 \ \mathrm{L}+5.0 \times 10^{-3} \ \mathrm{L}}=0.0381 \ \mathrm{M} \nonumber$
Substituting these concentrations into Equation \ref{6.9} gives a pH of
$\mathrm{pH}=7.199+\log \frac{\left[\mathrm{HPO}_{4}^{2-}\right]}{\left[\mathrm{H}_{2} \mathrm{PO}_{4}^-\right]}=7.199+\log \frac{0.0381}{0.105}=6.759 \approx 6.76 \nonumber$
As we expect, adding HCl decreases the buffer’s pH by a small amount, dropping from 6.90 to 6.76.
We can use a multiprotic weak acid to prepare buffers at as many different pH’s as there are acidic protons, with the Henderson–Hasselbalch equation applying in each case. For example, for malonic acid (pKa1 = 2.85 and pKa2 = 5.70) we can prepare buffers with pH values of
$\begin{array}{l}{\mathrm{pH}=2.85+\log \frac{C_{\mathrm{HM}^{-}}}{C_{\mathrm{H}_{2} \mathrm{M}}}} \ {\mathrm{pH}=5.70+\log \frac{C_{\mathrm{M}^{2-}}}{C_{\mathrm{HM}^-}}}\end{array} \nonumber$
where H2M, HM and M2– are malonic acid’s different acid–base forms.
Although our treatment of buffers is based on acid–base chemistry, we can extend buffers to equilibria that involve complexation or redox reactions. For example, the Nernst equation for a solution that contains Fe2+ and Fe3+ is similar in form to the Henderson‐Hasselbalch equation.
$E=E_{\mathrm{Fe}^{3+} / \mathrm{Fe}^{2+}}^{\circ}-0.05916 \log \frac{\left[\mathrm{Fe}^{2+}\right]}{\left[\mathrm{Fe}^{3+}\right]} \nonumber$
A solution that contains similar concentrations of Fe2+ and Fe3+ is buffered to a potential near the standard state reduction potential for Fe3+. We call such solutions redox buffers. Adding a strong oxidizing agent or a strong reducing agent to a redox buffer results in a small change in potential.
Representing Buffer Solutions with Ladder Diagrams
A ladder diagram provides a simple way to visualize a solution’s predominate species as a function of solution conditions. It also provides a convenient way to show the range of solution conditions over which a buffer is effective. For example, an acid–base buffer exists when the concentrations of the weak acid and its conjugate weak base are similar. For convenience, let’s assume that an acid–base buffer exists when
$\frac{1}{10} \leq \frac{\left[\mathrm{CH}_{3} \mathrm{COO}^{-}\right]}{\left[\mathrm{CH}_{3} \mathrm{COOH}\right]} \leq \frac{10}{1} \nonumber$
Substituting these ratios into the Henderson–Hasselbalch equation
\begin{aligned} \mathrm{pH} &=\mathrm{p} K_{\mathrm{a}}+\log \frac{1}{10}=\mathrm{p} K_{\mathrm{a}}-1 \ \mathrm{pH} &=\mathrm{p} K_{\mathrm{a}}+\log \frac{10}{1}=\mathrm{p} K_{\mathrm{a}}+1 \end{aligned} \nonumber
shows that an acid–base buffer works over a pH range of pKa ± 1.
Using the same approach, it is easy to show that a metal‐ligand complexation buffer for MLn exists when
$\mathrm{pL}=\log K_{n} \pm 1 \text { or } \mathrm{pL}=\log \beta_{n} \pm \frac{1}{n} \nonumber$
where Kn or $\beta_n$ is the relevant stepwise or overall formation constant. For an oxidizing agent and its conjugate reducing agent, a redox buffer exists when
$E=E^{\circ} \pm \frac{1}{n} \times \frac{R T}{F}=E^{\circ} \pm \frac{0.05916}{n}\left(\text { at } 25^{\circ} \mathrm{C}\right) \nonumber$
Figure 6.8.1 shows ladder diagrams with buffer regions for several equilibrium systems.
Preparing a Buffer
Buffer capacity is the ability of a buffer to resist a change in pH when we add to it a strong acid or a strong base. A buffer’s capacity to resist a change in pH is a function of the concentrations of the weak acid and the weak base, as well as their relative proportions. The importance of the weak acid’s concentration and the weak base’s concentration is obvious. The more moles of weak acid and weak base a buffer has, the more strong base or strong acid it can neutralize without a significant change in its pH.
Although a higher concentration of buffering agents provides greater buffer capacity, there are reasons for using smaller concentrations, including the formation of unwanted precipitates and the tolerance of biological systems for high concentrations of dissolved salts.
The relative proportions of a weak acid and a weak base also affects how much the pH changes when we add a strong acid or a strong base. A buffer that is equimolar in weak acid and weak base requires a greater amount of strong acid or strong base to bring about a one unit change in pH. Consequently, a buffer is most effective against the addition of strong acids or strong bases when its pH is near the weak acid’s pKa value.
Buffer solutions are often prepared using standard “recipes” found in the chemical literature [see, for example, (a) Bower, V. E.; Bates, R. G. J. Res. Natl. Bur. Stand. (U. S.) 1955, 55, 197– 200; (b) Bates, R. G. Ann. N. Y. Acad. Sci. 1961, 92, 341–356; (c) Bates, R. G. Determination of pH, 2nd ed.; Wiley‐Interscience: New York, 1973]. In addition, there are computer programs and on‐line calculators to aid in preparing buffers [(a) Lambert, W. J. J. Chem. Educ. 1990, 67, 150–153; (b) http://www.bioinformatics.org/JaMBW/5/4/index.html.]. Perhaps the simplest way to make a buffer, however, is to prepare a solution that contains an appropriate conjugate weak acid and weak base, measure its pH, and then adjust the pH to the desired value by adding small portions of either a strong acid or a strong base.
A good “rule of thumb” when choosing a buffer is to select one whose reagents have a pKa value close to your desired pH. | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/06%3A_Equilibrium_Chemistry/6.08%3A_Buffer_Solutions.txt |
Careful measurements on the metal–ligand complex Fe(SCN)2+ suggest its stability decreases in the presence of inert ions [Lister, M. W.; Rivington, D. E. Can. J. Chem. 1995, 33, 1572–1590]. We can demonstrate this by adding an inert salt to an equilibrium mixture of Fe3+ and SCN. Figure 6.9.1 a shows the result of mixing together equal volumes of 1.0 mM FeCl3 and 1.5 mM KSCN, both of which are colorless. The solution’s reddish–orange color is due to the formation of Fe(SCN)2+.
$\mathrm{Fe}^{3+}(a q)+\mathrm{SCN}^{-}(a q) \rightleftharpoons \mathrm{Fe}(\mathrm{SCN})^{2+}(a q) \label{6.1}$
Adding 10 g of KNO3 to the solution and stirring to dissolve the solid, produces the result shown in Figure 6.9.1 b. The solution’s lighter color suggests that adding KNO3 shifts reaction \ref{6.1} to the left, decreasing the concentration of Fe(SCN)2+ and increasing the concentrations of Fe3+ and SCN. The result is a decrease in the complex’s formation constant, K1.
$K_{1}=\frac{\left[\mathrm{Fe}(\mathrm{SCN})^{2+}\right]}{\left[\mathrm{Fe}^{3+}\right]\left[\mathrm{SCN}^{-}\right]} \label{6.2}$
Why should adding an inert electrolyte affect a reaction’s equilibrium position? We can explain the effect of KNO3 on the formation of Fe(SCN)2+ if we consider the reaction on a microscopic scale. The solution in Figure 6.9.1 b contains a variety of cations and anions: Fe3+, SCN, K+, $\text{NO}_3^-$, H3O+, and OH. Although the solution is homogeneous, on average, there are slightly more anions in regions near the Fe3+ ions, and slightly more cations in regions near the SCN ions. As shown in Figure 6.9.2 , each Fe3+ ion and each SCN ion is surrounded by an ionic atmosphere of opposite charge ($\delta^–$ and $\delta^+$) that partially screen the ions from each other. Because each ion’s apparent charge at the edge of its ionic atmosphere is less than its actual charge, the force of attraction between the two ions is smaller. As a result, the formation of Fe(SCN)2+ is slightly less favorable and the formation constant in Equation \ref{6.2} is slightly smaller. Higher concentrations of KNO3 increase $\delta^–$ and $\delta^+$, resulting in even smaller values for the formation constant.
Ionic Strength
To factor the concentration of ions into the formation constant for Fe(SCN)2+, we need a way to express that concentration in a meaningful way. Because both an ion’s concentration and its charge are important, we define the solution’s ionic strength, $\mu$ as
$\mu=\frac{1}{2} \sum_{i=1}^{n} c_{i} z_{i}^{2} \nonumber$
where ci and zi are the concentration and charge of the ith ion.
Example 6.9.1
Calculate the ionic strength of a solution of 0.10 M NaCl. Repeat the calculation for a solution of 0.10 M Na2SO4.
Solution
The ionic strength for 0.10 M NaCl is
$\begin{array}{c}{\mu=\frac{1}{2}\left\{\left[\mathrm{Na}^{+}\right] \times(+1)^{2}+\left[\mathrm{Cl}^{-}\right] \times(-1)^{2}\right\}} \ {\mu=\frac{1}{2}\left\{(0.10) \times(+1)^{2}+(0.10) \times(-1)^{2}\right\}=0.10 \ \mathrm{M}}\end{array} \nonumber$
For 0.10 M Na2SO4 the ionic strength is
$\begin{array}{c}{\mu=\frac{1}{2}\left\{\left[\mathrm{Na}^{+}\right] \times(+1)^{2}+\left[\mathrm{SO}_{4}^{2-}\right] \times(-2)^{2}\right\}} \ {\mu=\frac{1}{2}\left\{(0.20) \times(+1)^{2}+(0.10) \times(-2)^{2}\right\}=0.30 \ \mathrm{M}}\end{array} \nonumber$
In calculating the ionic strengths of these solutions we are ignoring the presence of H3O+ and OH, and, in the case of Na2SO4, the presence of $\text{HSO}_4^-$ from the base dissociation reaction of $\text{SO}_4^{2-}$. In the case of 0.10 M NaCl, the concentrations of H3O+ and OH are $1.0 \times 10^{-7}$, which is significantly smaller than the concentrations of Na+ and Cl. Because $\text{SO}_4^{2-}$ is a very weak base (Kb = $1.0 \times 10^{-12}$), the solution is only slightly basic (pH = 7.5), and the concentrations of H3O+, OH, and $\text{HSO}_4^-$ are negligible. Although we can ignore the presence of H3O+, OH, and $\text{HSO}_4^-$ when we calculate the ionic strength of these two solutions, be aware that an equilibrium reaction can generate ions that might affect the solution’s ionic strength.
Note that the unit for ionic strength is molarity, but that a salt’s ionic strength need not match its molar concentration. For a 1:1 salt, such as NaCl, ionic strength and molar concentration are identical. The ionic strength of a 2:1 electrolyte, such as Na2SO4, is three times larger than the electrolyte’s molar concentration.
Activity and Activity Coefficients
Figure 6.9.1 shows that adding KNO3 to a mixture of Fe3+ and SCN decreases the formation constant for Fe(SCN)2+. This creates a contradiction. Earlier in this chapter we showed that there is a relationship between a reaction’s standard‐state free energy, ∆Go, and its equilibrium constant, K.
$\triangle G^{\circ}=-R T \ln K \nonumber$
Because a reaction has only one standard‐state, its equilibrium constant must be independent of solution conditions. Although ionic strength affects the apparent formation constant for Fe(SCN)2+, reaction \ref{6.1} must have an underlying thermodynamic formation constant that is independent of ionic strength.
The apparent formation constant for Fe(SCN)2+, as shown in Equation \ref{6.2}, is a function of concentrations. In place of concentrations, we define the true thermodynamic equilibrium constant using activities. The activity of species A, aA, is the product of its concentration, [A], and a solution‐dependent activity coefficient, $\gamma_A$
$a_{A}=[A] \gamma_{A} \nonumber$
The true thermodynamic formation constant for Fe(SCN)2+, therefore, is
$K_{1}=\frac{a_{\mathrm{Fe}(S \mathrm{CN})^{2+}}}{a_{\mathrm{Fe}^{3+}} \times a_{\mathrm{SCN}^-}}=\frac{\left[\mathrm{Fe}(\mathrm{SCN})^{2+}\right] \gamma_{\mathrm{Fe}(\mathrm{SCN})^{2+}}}{\left[\mathrm{Fe}^{3+}\right] \gamma_{\mathrm{Fe}^{3+}}\left[\mathrm{SCN}^{-}\right] \gamma_{\mathrm{SCN}^{-}}} \nonumber$
Unless otherwise specified, the equilibrium constants in the appendices are thermodynamic equilibrium constants.
A species’ activity coefficient corrects for any deviation between its physical concentration and its ideal value. For a gas, a pure solid, a pure liquid, or a non‐ionic solute, the activity coefficient is approximately one under most reasonable experimental conditions.
For a gas the proper terms are fugacity and fugacity coefficient, instead of activity and activity coefficient.
For a reaction that involves only these species, the difference between activity and concentration is negligible. The activity coefficient for an ion, however, depends on the solution’s ionic strength, the ion’s charge, and the ion’s size. It is possible to estimate activity coefficients using the extended Debye‐Hückel equation
$\log \gamma_{A}=\frac{-0.51 \times z_{A}^{2} \times \sqrt{\mu}}{1+3.3 \times \alpha_{A} \times \sqrt{\mu}} \label{6.3}$
where zA is the ion’s charge, $\alpha_A$ is the hydrated ion’s effective diameter in nanometers (Table 6.2), $\mu$ is the solution’s ionic strength, and 0.51 and 3.3 are constants appropriate for an aqueous solution at 25oC. A hydrated ion’s effective radius is the radius of the ion plus those water molecules closely bound to the ion. The effective radius is greater for smaller, more highly charged ions than it is for larger, less highly charged ions.
Table 6.9.1 . Effective Diameters ($\alpha$) for Selected Ions
ion effective diameter (nm)
H3O+
0.9
Li+
0.6
Na+, $\text{IO}_3^-$, $\text{HSO}_3^-$, $\text{HCO}_3^-$, $\text{H}_2\text{PO}_4^-$
0.45
OH, F, SCN, HS, $\text{ClO}_3^-$, $\text{ClO}_4^-$, $\text{MnO}_4^-$
0.35
K+, Cl, Br, I, CN, $\text{NO}_2^-$, $\text{NO}_3^-$
0.3
Cs+, Tl+, Ag+, $\text{NH}_4^+$
0.25
Mg2+, Be2+
0.8
Ca2+, Cu2+, Zn2+, Sn2+, Mn2+, Fe2+, Ni2+, Co2+
0.6
Sr2+, Ba2+, Cd2+, Hg2+, S2–
0.5
Pb2+, $\text{SO}_4^{2-}$, $\text{SO}_3^{2-}$
0.45
$\text{Hg}_2^{2+}$, $\text{SO}_4^{2-}$, $\text{S}_22\text{O}_3^{2-}$, $\text{CrO}_4^{2-}$, $\text{HPO}_4^{2-}$
0.40
Al3+, Fe3+, Cr3+
0.9
$\text{PO}_4^{3-}$, $\text{Fe(CN)}_6^{3-}$
0.4
Zr4+, Ce4+, Sn4+
1.1
$\text{Fe(CN)}_6^{4-}$
0.5
Source: Kielland, J. J. Am. Chem. Soc. 1937, 59, 1675–1678.
Several features of Equation \ref{6.3} deserve our attention. First, as the ionic strength approaches zero an ion’s activity coefficient approaches a value of one. In a solution where $\mu = 0$, an ion’s activity and its concentration are identical. We can take advantage of this fact to determine a reaction’s thermodynamic equilibrium constant by measuring the apparent equilibrium constant for several increasingly smaller ionic strengths and extrapolating back to an ionic strength of zero. Second, an activity coefficient is smaller, and the effect of activity is more important, for an ion with a higher charge and a smaller effective radius. Finally, the extended Debye‐Hückel equation provides a reasonable estimate of an ion’s activity coefficient when the ionic strength is less than 0.1. Modifications to Equation \ref{6.3} extend the calculation of activity coefficients to higher ionic strengths [Davies, C. W. Ion Association, Butterworth: London, 1962].
Including Activity Coefficients When Solving Equilibrium Problems
Earlier in this chapter we calculated the solubility of Pb(IO3)2 in deionized water, obtaining a result of $4.0 \times 10^{-5}$ mol/L. Because the only significant source of ions is from the solubility reaction, the ionic strength is very low and we can assume that $\gamma \approx 1$ for both Pb2+ and $\text{IO}_3^-$. In calculating the solubility of Pb(IO3)2 in deionized water, we do not need to account for ionic strength. But what if we need to know the solubility of Pb(IO3)2 in a solution that contains other, inert ions? In this case we need to include activity coefficients in our calculation.
Example 6.9.2
Calculate the solubility of Pb(IO3)2 in a matrix of 0.020 M Mg(NO3)2.
Solution
We begin by calculating the solution’s ionic strength. Since Pb(IO3)2 is only sparingly soluble, we will assume we can ignore its contribution to the ionic strength; thus
$\mu=\frac{1}{2}\left\{(0.020)(+2)^{2}+(0.040)(-1)^{2}\right\}=0.060 \ \mathrm{M} \nonumber$
Next, we use Equation \ref{6.3} to calculate the activity coefficients for Pb2+ and $\text{IO}_3^-$.
$\log \gamma_{\mathrm{Pb}^{2+}}=\frac{-0.51 \times(+2)^{2} \times \sqrt{0.060}}{1+3.3 \times 0.45 \times \sqrt{0.060}}=-0.366 \nonumber$
$\gamma_{\mathrm{Pb}^{2+}}=0.431 \nonumber$
$\log \gamma_{\mathrm{IO}_{3}^{-}}=\frac{-0.51 \times(-1)^{2} \times \sqrt{0.060}}{1+3.3 \times 0.45 \times \sqrt{0.060}}=-0.0916 \nonumber$
$\gamma_{\mathrm{IO}_{3}^-}=0.810 \nonumber$
Defining the equilibrium concentrations of Pb2+ and $\text{IO}_3^-$ in terms of the variable x
Concentrations
Pb(IO3)2 (s) $\rightleftharpoons$ Pb2+ (aq) + 2$\text{IO}_3^-$ (aq)
initial solid 0 0
change solid +x +2x
equilibrium solid x 2x
and substituting into the thermodynamic solubility product for Pb(IO3)2 leaves us with
$K_{\mathrm{sp}}=a_{\mathrm{Pb}^{2+}} \times a_{\mathrm{IO}_{3}^-}^{2}=\gamma_{\mathrm{Pb}^{2+}}\left[\mathrm{Pb}^{2+}\right] \times \gamma_{\mathrm{IO}_3^-}^{2}\left[\mathrm{IO}_{3}^{-}\right]^{2}=2.5 \times 10^{-13} \nonumber$
$K_{\mathrm{sp}}=(0.431)(x)(0.810)^{2}(2 x)^{2}=2.5 \times 10^{-13} \nonumber$
$K_{\mathrm{sp}}=1.131 x^{3}=2.5 \times 10^{-13} \nonumber$
Solving for x gives $6.0 \times 10^{-5}$ and a molar solubility of $6.0 \times 10^{-5}$ mol/L for Pb(IO3)2. If we ignore activity, as we did in our earlier calculation, we report the molar solubility as $4.0 \times 10^{-5}$ mol/L. Failing to account for activity in this case underestimates the molar solubility of Pb(IO3)2 by 33%.
The solution’s equilibrium composition is
$\begin{array}{c}{\left[\mathrm{Pb}^{2+}\right]=6.0 \times 10^{-5} \ \mathrm{M}} \ {\left[\mathrm{IO}_{3}^{-}\right]=1.2 \times 10^{-4} \ \mathrm{M}} \ {\left[\mathrm{Mg}^{2+}\right]=0.020 \ \mathrm{M}} \ {\left[\mathrm{NO}_{3}^{-}\right]=0.040 \ \mathrm{M}}\end{array} \nonumber$
Because the concentrations of both Pb2+ and $\text{IO}_3^-$ are much smaller than the concentrations of Mg2+ and $\text{NO}_3^-$ our decision to ignore the contribution of Pb2+ and $\text{IO}_3^-$ to the ionic strength is reasonable.
How do we handle the calculation if we can not ignore the concentrations of Pb2+ and $\text{IO}_3^-$ when calculating the ionic strength. One approach is to use the method of successive approximations. First, we recalculate the ionic strength using the concentrations of all ions, including Pb2+ and $\text{IO}_3^-$. Next, we recalculate the activity coefficients for Pb2+ and $\text{IO}_3^-$ using this new ionic strength and then recalculate the molar solubility. We continue this cycle until two successive calculations yield the same molar solubility within an acceptable margin of error.
Exercise 6.9.1
Calculate the molar solubility of Hg2Cl2 in 0.10 M NaCl, taking into account the effect of ionic strength. Compare your answer to that from Exercise 6.7.2 in which you ignored the effect of ionic strength.
Answer
We begin by calculating the solution’s ionic strength. Because NaCl is a 1:1 ionic salt, the ionic strength is the same as the concentration of NaCl; thus $\mu$ = 0.10 M. This assumes, of course, that we can ignore the contributions of $\text{Hg}_2^{2+}$ and Cl from the solubility of Hg2Cl2.
Next we use Equation \ref{6.3} to calculate the activity coefficients for $\text{Hg}_2^{2+}$ and Cl.
$\log \gamma_{\mathrm{Hg}_{2}^{2+}}=\frac{-0.51 \times(+2)^{2} \times \sqrt{0.10}}{1+3.3 \times 0.40 \times \sqrt{0.10}}=-0.455 \nonumber$
$\gamma_{\mathrm{H} \mathrm{g}_{2}^{2+}}=0.351 \nonumber$
$\log \gamma_{\mathrm{Cl}^{-}}=\frac{-0.51 \times(-1)^{2} \times \sqrt{0.10}}{1+3.3 \times 0.3 \times \sqrt{0.10}}=-0.12 \nonumber$
$\gamma_{\mathrm{Cl}^-}=0.75 \nonumber$
Defining the equilibrium concentrations of $\text{Hg}_2^{2+}$ and Cl in terms of the variable x
concentrations Hg2Cl2 (s) $\rightleftharpoons$ $\text{Hg}_2^{2+}$ (aq) + 2Cl (aq)
initial solid 0 0.10
change solid +x +2x
equilibrium solid x 0.10 + 2x
and substituting into the thermodynamic solubility product for Hg2Cl2, leave us with
$K_{\mathrm{sp}}=a_{\mathrm{Hg}_{2}^{2+}}\left(a_{\mathrm{Cl}^-}\right)^{2} = \gamma_{\mathrm{Hg}_{2}^{2+}}\left[\mathrm{Hg}_{2}^{2+}\right]\left(\gamma_{\mathrm{Cl}^{-}}\right)^{2}\left[\mathrm{Cl}^{-}\right]^{2}=1.2 \times 10^{-18} \nonumber$
Because the value of x likely is small, let’s simplify this equation to
$(0.351)(x)(0.75)^{2}(0.1)^{2}=1.2 \times 10^{-18} \nonumber$
Solving for x gives its value as $6.1 \times 10^{-16}$. Because x is the concentration of $\text{Hg}_2^{2+}$ and 2x is the concentration of Cl, our decision to ignore their contributions to the ionic strength is reasonable. The molar solubility of Hg2Cl2 in 0.10 M NaCl is $6.1 \times 10^{-16}$ mol/L. In Exercise 6.7.2, where we ignored ionic strength, we determined that the molar solubility of Hg2Cl2 is $1.2 \times 10^{-16}$ mol/L, a result that is $5 \times$ smaller than the its actual value.
As Example 6.9.2 and Exercise 6.9.1 show, failing to correct for the effect of ionic strength can lead to a significant error in an equilibrium calculation. Nevertheless, it is not unusual to ignore activities and to assume that the equilibrium constant is expressed in terms of concentrations. There is a practical reason for this—in an analysis we rarely know the exact composition, much less the ionic strength of aqueous samples or of solid samples brought into solution. Equilibrium calculations are a useful guide when we develop an analytical method; however, it only is when we complete an analysis and evaluate the results that can we judge whether our theory matches reality. In the end, work in the laboratory is the most critical step in developing a reliable analytical method.
This is a good place to revisit the meaning of pH. In Chapter 2 we defined pH as
$\mathrm{pH}=-\log \left[\mathrm{H}_{3} \mathrm{O}^{+}\right] \nonumber$
Now we see that the correct definition is
$\begin{array}{c}{\mathrm{pH}=-\log a_{\mathrm{H}_{3} \mathrm{O}^{+}}} \ {\mathrm{pH}=-\log \gamma_{\mathrm{H}_{3} \mathrm{O}^{+}}\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]}\end{array} \nonumber$
Failing to account for the effect of ionic strength can lead to a significant error in the reported concentration of H3O+. For example, if the pH of a solution is 7.00 and the activity coefficient for H3O+ is 0.90, then the concentration of H3O+ is $1.11 \times 10^{-7}$ M, not $1.00 \times 10^{-7}$ M, an error of +11%. Fortunately, when we develop and carry out an analytical method, we are more interested in controlling pH than in calculating [H3O+]. As a result, the difference between the two definitions of pH rarely is of significant concern. | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/06%3A_Equilibrium_Chemistry/6.09%3A_Activity_Effects.txt |
In solving equilibrium problems we typically make one or more assumptions to simplify the algebra. These assumptions are important because they allow us to reduce the problem to an equation in x that we can solve by simply taking a square‐root, a cube‐root, or by using the quadratic equation. Without these assumptions, most equilibrium problems result in a cubic equation (or a higher‐order equation) that is more challenging to solve. Both Excel and R are useful tools for solving such equations.
Although we focus here on the use of Excel and R to solve equilibrium problems, you also can use WolframAlpha; for details, see Cleary, D. A. “Use of WolframAlpha in Equilibrium Calculations,” Chem. Educator, 2014, 19, 182–186.
Excel
Excel offers a useful tool—the Solver function—for finding the chemically significant root of a polynomial equation. In addition, it is easy to solve a system of simultaneous equations by constructing a spreadsheet that allows you to test and evaluate multiple solutions. Let’s work through two examples.
Example 1: Solubility of Pb(IO3)2 in 0.10 M Pb(NO3)2
In our earlier treatment of this problem we arrived at the following cubic equation
$4 x^{3}+0.40 x^{2}=2.5 \times 10^{-13} \nonumber$
where x is the equilibrium concentration of Pb2+. Although there are several approaches for solving cubic equations with paper and pencil, none are computationally easy. One approach is to iterate in on the answer by finding two values of x, one that leads to a result larger than $2.5 \times 10^{-13}$ and one that gives a result smaller than $2.5 \times 10^{-13}$. With boundaries established for the value of x, we shift the upper limit and the lower limit until the precision of our answer is satisfactory. Without going into details, this is how Excel’s Solver function works.
To solve this problem, we first rewrite the cubic equation so that its right‐side equals zero.
$4 x^{3}+0.40 x^{2}-2.5 \times 10^{-13}=0 \nonumber$
Next, we set up the spreadsheet shown in Figure 6.10.1 a, placing the formula for the cubic equation in cell B2, and entering our initial guess for x in cell B1. Because Pb(IO3)2 is not very soluble, we expect that x is small and set our initial guess to 0. Finally, we access the Solver function by selecting Solver... from the Tools menu, which opens the Solver Parameters window.
To define the problem, place the cursor in the box for Set Target Cell and then click on cell B2. Select the Value of: radio button and enter 0 in the box. Place the cursor in the box for By Changing Cells: and click on cell B1. Together, these actions instruct the Solver function to change the value of x, which is in cell B1, until the cubic equation in cell B2 equals zero. Before we actually solve the function, we need to consider whether there are any limitations for an acceptable result. For example, we know that x cannot be smaller than 0 because a negative concentration is not possible. We also want to ensure that the solution’s precision is acceptable. Click on the button labeled Options... to open the Solver Options window. Checking the option for Assume Non-Negative forces the Solver to maintain a positive value for the contents of cell B1, meeting one of our criteria. Setting the precision requires a bit more thought. The Solver function uses the precision to decide when to stop its search, doing so when
$|\text { expected value }-\text { calculated value } | \times 100=\text { precision }(\%) \nonumber$
where expected value is the target cell’s desired value (0 in this case), calculated value is the function’s current value (cell B1 in this case), and precision is the value we enter in the box for Precision. Because our initial guess of x = 0 gives a calculated result of $2.5 \times 10^{-13}$, accepting the Solver’s default precision of $1 \times 10^{-6}$ will stop the search after one cycle. To be safe, let’s set the precision to $1 \times 10^{-18}$. Click OK and then Solve. When the Solver function finds a solution, the results appear in your spreadsheet (see Figure 6.10.1 b). Click OK to keep the result, or Cancel to return to the original values. Note that the answer here agrees with our earlier result of $7.91 \times 10^{-7}$ M for the solubility of Pb(IO3)2.
Be sure to evaluate the reasonableness of Solver’s answer. If necessary, repeat the process using a smaller value for the precision.
Example 2: pH of 1.0 M HF
In developing our earlier solution to this problem we began by identifying four unknowns and writing out the following four equations.
$K_{\mathrm{a}}=\frac{\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\left[\mathrm{F}^{-}\right]}{[\mathrm{HF}]}=6.8 \times 10^{-4} \nonumber$
$K_{w}=\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\left[\mathrm{OH}^{-}\right]=1.00 \times 10^{-14} \nonumber$
$C_{\mathrm{HF}}=[\mathrm{HF}]+\left[\mathrm{F}^{-}\right] \nonumber$
$\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=\left[\mathrm{OH}^{-}\right]+\left[\mathrm{F}^{-}\right] \nonumber$
Next, we made two assumptions that allowed us to simplify the problem to an equation that is easy to solve.
$\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=\sqrt{K_{\mathrm{a}} C_{\mathrm{HF}}}=\sqrt{\left(6.8 \times 10^{-4}\right)(1.0)}=2.6 \times 10^{-2} \nonumber$
Although we did not note this at the time, without making assumptions the solution to our problem is a cubic equation
$\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]^{3}+K_{\mathrm{a}}\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]^{2}- \left(K_{a} C_{\mathrm{HF}}+K_{\mathrm{w}}\right)\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]-K_{\mathrm{a}} K_{\mathrm{w}}=0 \label{6.1}$
that we can solve using Excel’s Solver function. Of course, this assumes that we successfully complete the derivation!
Another option is to use Excel to solve the four equations simultaneously by iterating in on values for [HF], [F], [H3O+], and [OH]. Figure 6.10.2 a shows a spreadsheet for this purpose. The cells in the first row contain initial guesses for the equilibrium pH. Using the ladder diagram in Figure 6.8.1, pH values between 1 and 3 seems reasonable. You can add additional columns if you wish to include more pH values. The formulas in rows 2–5 use the definition of pH to calculate [H3O+], Kw to calculate [OH], the charge balance equation to calculate [F], and Ka to calculate [HF]. To evaluate the initial guesses, we use the mass balance expression for HF, rewriting it as
$[\mathrm{HF}]+\left[\mathrm{F}^{-}\right]-C_{\mathrm{HF}}=[\mathrm{HF}]+[\mathrm{F}]-1.0=0 \nonumber$
and entering it in the last row; the values in these cells gives the calculation’s error for each pH.
Figure 6.10.2 b shows the actual values for the spreadsheet in Figure 6.10.2 a. The negative value in cells B6 and C6 means that the combined concentrations of HF and F are too small, and the positive value in cell D6 means that their combined concentrations are too large. The actual pH, therefore, is between 1.00 and 2.00. Using these pH values as new limits for the spreadsheet’s first row, we continue to narrow the range for the actual pH. Figure 6.10.2 c shows a final set of guesses, with the actual pH falling between 1.59 and 1.58. Because the error for 1.59 is smaller than that for 1.58, we accept a pH of 1.59 as the answer. Note that this is an agreement with our earlier result.
You also can solve this set of simultaneous equations using Excel’s Solver function. To do so, create the spreadsheet in Figure 6.10.2 a, but omit all columns other than A and B. Select Solver... from the Tools menu and define the problem by using B6 for Set Target Cell, setting its desired value to 0, and selecting B1 for By Changing Cells:. You may need to play with the Solver’s options to find a suitable solution to the problem, and it is wise to try several different initial guesses. The Solver function works well for relatively simple problems, such as finding the pH of 1.0 M HF. As problems become more complex and include more unknowns, the Solver function becomes a less reliable tool for solving equilibrium problems.
Exercise 6.10.1
Using Excel, calculate the solubility of AgI in 0.10 M NH3 without making any assumptions. See our earlier treatment of this problem for the relevant equilibrium reactions and constants.
Answer
For a list of the relevant equilibrium reactions and equilibrium constants, see our earlier treatment of this problem. To solve this problem using Excel, let’s set up the following spreadsheet
copying the contents of cells B1‐B9 into several additional columns. The initial guess for pI in cell B1 gives the concentration of I in cell B2. Cells B3–B8 calculate the remaining concentrations, using the Ksp to obtain [Ag+], using the mass balance on iodide and silver to obtain [$\text{Ag(NH}_3)_2^+$], using b2 to calculate [NH3], using the mass balance on ammonia to find [$\text{NH}_4^+$], using Kb to calculate [OH], and using Kw to calculate [H3O+]. The system’s charge balance equation provides a means for determining the calculation’s error.
$\left[\mathrm{Ag}^{+}\right]+\left[\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}^{+}\right]+\left[\mathrm{NH}_{4}^{+}\right]+\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]-\left[\mathrm{I}^{-}\right]+\left[\mathrm{OH}^{-}\right]=0 \nonumber$
The largest possible value for pI, which corresponds to the smallest concentration of I and the lowest possible solubility, occurs for a simple, saturated solution of AgI. When [Ag+] = [I], the concentration of iodide is
$\left[\mathrm{I}^{-}\right]=\sqrt{K_{\mathrm{sp}}}=\sqrt{8.3 \times 10^{-17}}=9.1 \times 10^{-9} \nonumber$
which corresponds to a pI of 8.04. Entering initial guesses for pI of 4, 5, 6, 7, and 8 shows that the error changes sign between a pI of 5 and 6. Continuing in this way to narrow down the range for pI, we find that the error function is closest to zero at a pI of 5.42. The concentration of I at equilibrium, and the molar solubility of AgI, is $3.8 \times 10^{-6}$ mol/L, which agrees with our earlier solution to this problem.
R
R has a simple command—uniroot—for finding the chemically significant root of a polynomial equation. In addition, it is easy to write a function to solve a set of simultaneous equations by iterating in on a solution. Let’s work through two examples.
Example 1: Solubility of Pb(IO3)2 in 0.10 M Pb(NO3)2
In our earlier treatment of this problem we arrived at the following cubic equation
$4 x^{3}+0.40 x^{2}=2.5 \times 10^{-13} \nonumber$
where x is the equilibrium concentration of Pb2+. Although there are several approaches for solving cubic equations with paper and pencil, none are computationally easy. One approach to solving the problem is to iterate in on the answer by finding two values of x, one that leads to a result larger than $2.5 \times 10^{-13}$ and one that gives a result smaller than $2.5 \times 10^{-13}$. Having established boundaries for the value of x, we then shift the upper limit and the lower limit until the precision of our answer is satisfactory. Without going into details, this is how the uniroot command works.
The general form of the uniroot command is
uniroot(function, lower, upper, tol)
where function is an object that contains the equation whose root we seek, lower and upper are boundaries for the root, and tol is the desired precision for the root. To create an object that contains the equation, we rewrite it so that its right‐side equals zero.
$4 x^{3}+0.40 x^{2}-2.5 \times 10^{-13} = 0 \nonumber$
Next, we enter the following code, which defines our cubic equation as a function with the name eqn.
> eqn = function(x) {4*x^3 + 0.4*x^2 – 2.5e–13}
Because our equation is a function, the uniroot command can send a value of x to eqn and receive back the equation’s corresponding value.
For example, entering
> eqn(2)
passes the value x = 2 to the function and returns an answer of 33.6.
Finally, we use the uniroot command to find the root.
> uniroot(eqn, lower = 0, upper = 0.1, tol = 1e–18)
Because Pb(IO3)2 is not very soluble, we expect that x is small and set the lower limit to 0. The choice for the upper limit is less critical. To ensure that the solution has sufficient precision, we set the tolerance to a value that is smaller than the expected root. Figure 6.10.3 shows the resulting output. The value \$root is the equation’s root, which is in good agreement with our earlier result of $7.91 \times 10^{-7}$ for the molar solubility of Pb(IO3)2. The other results are the equation’s value for the root, the number of iterations needed to find the root, and the root’s estimated precision.
Example 2: pH of 1.0 M HF
In developing our earlier solution to this problem we began by identifying four unknowns and writing out the following four equations.
$K_{\mathrm{a}}=\frac{\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\left[\mathrm{F}^{-}\right]}{[\mathrm{HF}]}=6.8 \times 10^{-4} \nonumber$
$K_{w}=\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\left[\mathrm{OH}^{-}\right]=1.00 \times 10^{-14} \nonumber$
$C_{\mathrm{HF}}=[\mathrm{HF}]+\left[\mathrm{F}^{-}\right] \nonumber$
$\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=\left[\mathrm{OH}^{-}\right]+\left[\mathrm{F}^{-}\right] \nonumber$
Next, we made two assumptions that allowed us to simplify the problem to an equation that is easy to solve.
$\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=\sqrt{K_{\mathrm{a}} C_{\mathrm{HF}}}=\sqrt{\left(6.8 \times 10^{-4}\right)(1.0)}=2.6 \times 10^{-2} \nonumber$
Although we did not note this at the time, without making assumptions the solution to our problem is a cubic equation
$\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]^{3}+K_{\mathrm{a}}\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]^{2}- \left(K_{a} C_{\mathrm{HF}}+K_{\mathrm{w}}\right)\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]-K_{\mathrm{a}} K_{\mathrm{w}}=0 \nonumber$
that we can solve using the uniroot command. Of course, this assumes that we successfully complete the derivation!
Another option is to write a function to solve the four equations simultaneously. Here is the code for this function, which we will call eval.
> eval = function(pH){
+ h3o = 10^–pH
+ oh = 1e–14/h3o
+ hf = (h3o * f )/6.8e–4
+ error= hf + f – 1
+ output= data.frame(pH, error)
+ print(output)
+}
The open curly braces, {, tells R that we intend to enter our function over several lines. When we press enter at the end of a line, R changes its prompt from > to +, indicating that we are continuing to enter the same command. The closed curly brace, }, on the last line indicates that we have completed the function. The command data.frame combines two or more objects into a table, which we then print out so that we can view the results of the calculations. You can adapt this function to other problems by changing the variable you pass to the function and the equations you include within the function.
Let’s examine more closely how this function works. The function accepts a guess for the pH and uses the definition of pH to calculate [H3O+], Kw to calculate [OH], the charge balance equation to calculate [F], and Ka to calculate [HF]. The function then evaluates the solution using the mass balance expression for HF, rewriting it as
$[\mathrm{HF}]+\left[\mathrm{F}^{-}\right]-C_{\mathrm{HF}}=[\mathrm{HF}]+\left[\mathrm{F}^{-}\right]-1.0=0 \nonumber$
The function then gathers together the initial guess for the pH and the error and prints them as a table.
The beauty of this function is that the object we pass to it, pH, can contain many values, which makes it easy to search for a solution. Because HF is an acid, we know that the solution is acidic. This sets an upper limit of 7 for the pH. We also know that the pH of 1.0 M HF is no smaller than 0 as this is the pH if HF was a strong acid. For our first pass, let’s enter the following code
> pH= c(7, 6, 5, 4, 3, 2, 1)
> eval(pH)
which varies the pH within these limits. The result, which is shown in Figure 6.10.4 a, indicates that the pH is less than 2 and greater than 1 because it is in this interval that the error changes sign.
For our second pass, let’s explore pH values between 2.0 and 1.0 to further narrow down the problem’s solution.
> pH = c(2.0, 1.9, 1.8, 1.7, 1.6, 1.5, 1.4, 1.3, 1.2, 1.1, 1.0)
> eval(pH)
The result in Figure 6.10.4 b show that the pH must be less than 1.6 and greater than 1.5. A third pass between these limits gives the result shown in Figure 6.10.4 c, which is consistent with our earlier result of a pH 1.59.
Exercise 6.10.2
Using R, calculate the solubility of AgI in 0.10 M NH3 without making any assumptions. See our earlier treatment of this problem for the relevant equilibrium reactions and constants.
Answer
To solve this problem, let’s use the following function
> eval = function(pI){
+ I = 10^–pI
+ Ag = 8.3e–17/I
+ AgNH3 = Ag – I
+ NH3 = (AgNH3/(1.7e7*Ag))^0.5
+ NH4 = 0.10‐NH3 – 2 * AgNH3
+ OH = 1.75e–5 * NH3/NH4
+ H3O = 1e–14/OH
+ error = Ag + AgNH3 + NH4 + H3O – OH – I
+ output = data.frame(pI, error)
+ print(output)
+}
The function accepts an initial guess for pI and calculates the concentrations of each species in solution using the definition of pI to calculate [I], using the Ksp to obtain [Ag+], using the mass balance on iodide and silver to obtain [$\text{Ag(NH}_3)_2^+$], using $\beta_2$ to calculate [NH3], using the mass balance on ammonia to find [$\text{NH}_4^+$], using Kb to calculate [OH], and using Kw to calculate [H3O+]. The system’s charge balance equation provides a means for determining the calculation’s error.
$\left[\mathrm{Ag}^{+}\right]+\left[\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}^{+}\right]+\left[\mathrm{NH}_{4}^{+}\right]+\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]-\left[\mathrm{I}^{-}\right]+\left[\mathrm{OH}^{-}\right]=0 \nonumber$
The largest possible value for pI—corresponding to the smallest concentration of I and the lowest possible solubility—occurs for a simple, saturated solution of AgI. When [Ag+] = [I], the concentration of iodide is
$\left[\mathrm{I}^{-}\right]=\sqrt{K_{\mathrm{sp}}}=\sqrt{8.3 \times 10^{-17}}=9.1 \times 10^{-9} \nonumber$
corresponding to a pI of 8.04. The following session shows the function in action.
> pI=c(4, 5, 6, 7, 8)
> eval(pI)
pI error
1 4 ‐2.56235615
2 5 ‐0.16620930
3 6 0.07337101
4 7 0.09734824
5 8 0.09989073
> pI =c(5.1, 5.2, 5.3, 5.4, 5.5, 5.6, 5.7, 5.8, 5.9, 6.0)
> eval(pI)
pI error
1 5.1 ‐0.11144658
2 5.2 ‐0.06794105
3 5.3 ‐0.03336475
4 5.4 ‐0.00568116
5 5.5 0.01571549
6 5.6 0.03308929
7 5.7 0.04685937
8 5.8 0.05779214
9 5.9 0.06647475
10 6.0 0.07337101
> pI =c(5.40, 5.41, 5.42, 5.43, 5.44, 5.45, 5.46, 5.47, 5.48, 5.49, 5.50)
> eval(pI)
pI error
1 5.40 ‐0.0056811605
2 5.41 ‐0.0030715484
3 5.42 0.0002310369
4 5.43 ‐0.0005134898
5 5.44 0.0028281878
6 5.45 0.0052370980
7 5.46 0.0074758181
8 5.47 0.0096260370
9 5.48 0.0117105498
10 5.49 0.0137387291
11 5.50 0.0157154889
The error function is closest to zero at a pI of 5.42. The concentration of I at equilibrium, and the molar solubility of AgI, is $3.8 \times 10^{-6}$ mol/L, which agrees with our earlier solution to this problem. | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/06%3A_Equilibrium_Chemistry/6.10%3A_Using_Excel_and_R_to_Solve_Equilibrium_Problems.txt |
In this chapter we developed several tools to evaluate the composition of a system at equilibrium. These tools differ in how precisely they allow us to answer questions involving equilibrium chemistry. They also differ in how easy they are to use. An important part of having several tools to choose from is knowing when to each is most useful. If you need to know whether a reaction if favorable or you need to estimate a solution’s pH, then a ladder diagram usually will meet your needs. On the other hand, if you require a more accurate or more precise estimate of a compound’s solubility, then a rigorous calculation that includes activity coefficients is necessary.
A critical part of solving an equilibrium problem is to know what equilibrium reactions to include. The need to include all relevant reactions is obvious, and at first glance this does not appear to be a significant problem—it is, however, a potential source of significant errors. The tables of equilibrium constants in this textbook, although extensive, are a small subset of all known equilibrium constants, which makes it easy to overlook an important equilibrium reaction. Commercial and freeware computational programs with extensive databases are available for equilibrium modeling, two examples of which are Visual Minteq (Windows only) and CurTiPot (for Excel); Visual Minteq can model acid–base, solubility, complexation, and redox equilibria; CurTiPot is limited to acid–base equilibria. Both programs account for the effect of activity. The R package CHNOSZ is used to model aqueous geochemistry systems and the properities of proteins.
An integrated set of tools for thermodynamic calculations in aqueous geochemistry and geobiochemistry. Functions are provided for writing balanced reactions to form species from user-selected basis species and for calculating the standard molal properties of species and reactions, including the standard Gibbs energy and equilibrium constant. Calculations of the non-equilibrium chemical affinity and equilibrium chemical activity of species can be portrayed on diagrams as a function of temperature, pressure, or activity of basis species; in two dimensions, this gives a maximum affinity or predominance diagram. The diagrams have formatted chemical formulas and axis labels, and water stability limits can be added to Eh-pH, oxygen fugacity- temperature, and other diagrams with a redox variable. The package has been developed to handle common calculations in aqueous geochemistry, such as solubility due to complexation of metal ions, mineral buffers of redox or pH, and changing the basis species across a diagram ("mosaic diagrams"). CHNOSZ also has unique capabilities for comparing the compositional and thermodynamic properties of different proteins.
Finally, a consideration of equilibrium chemistry can only help us decide if a reaction is favorable; however, it does not guarantee that the reaction occurs. How fast a reaction approaches its equilibrium position does not depend on the reaction’s equilibrium constant because the rate of a chemical reaction is a kinetic, not a thermodynamic, phenomenon. We will consider kinetic effects and their application in analytical chemistry in Chapter 13.
6.12: Problems
1. Write equilibrium constant expressions for the following reactions. What is the value for each reaction’s equilibrium constant?
(a) $\mathrm{NH}_{3}(a q)+\mathrm{H}_{3} \mathrm{O}^{+}(a q) \rightleftharpoons \mathrm{N} \mathrm{H}_{4}^{+}(a q)$
(b) $\operatorname{PbI}_{2}(s)+\mathrm{S}^{2-}(a q) \rightleftharpoons \operatorname{PbS}(s)+2 \mathrm{I}^{-}(a q)$
(c) $\operatorname{CdY}^{2-}(a q)+4 \mathrm{CN}^{-}(a q) \rightleftharpoons \mathrm{Cd}(\mathrm{CN})_{4}^{2-}(a q)+\mathrm{Y}^{4-}(a q)$; note: Y is the shorthand symbol for EDTA
(d) $\mathrm{AgCl}(s)+2 \mathrm{NH}_{3}(a q)\rightleftharpoons\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}^{+}(a q)+\mathrm{Cl}^{-}(a q)$
(e) $\mathrm{BaCO}_{3}(s)+2 \mathrm{H}_{3} \mathrm{O}^{+}(a q)\rightleftharpoons \mathrm{Ba}^{2+}(a q)+\mathrm{H}_{2} \mathrm{CO}_{3}(a q)+2 \mathrm{H}_{2} \mathrm{O}(l)$
2. Use a ladder diagram to explain why the first reaction is favorable and why the second reaction is unfavorable.
$\mathrm{H}_{3} \mathrm{PO}_{4}(a q)+\mathrm{F}^{-}(a q)\rightleftharpoons\mathrm{HF}(a q)+\mathrm{H}_{2} \mathrm{PO}_{4}^{-}(a q) \nonumber$
$\mathrm{H}_{3} \mathrm{PO}_{4}(a q)+2 \mathrm{F}^{-}(a q)\rightleftharpoons2 \mathrm{HF}(a q)+\mathrm{HPO}_{4}^{2-}(a q) \nonumber$
Determine the equilibrium constant for these reactions and verify that they are consistent with your ladder diagram.
3. Calculate the potential for the following redox reaction for a solution in which [Fe3+] = 0.050 M, [Fe2+] = 0.030 M, [Sn2+] = 0.015 M and [Sn4+] = 0.020 M.
$2 \mathrm{Fe}^{3+}(a q)+\mathrm{Sn}^{2+}(a q)\rightleftharpoons\mathrm{Sn}^{4+}(a q)+2 \mathrm{Fe}^{2+}(a q) \nonumber$
4. Calculate the standard state potential and the equilibrium constant for each of the following redox reactions. Assume that [H3O+] is 1.0 M for an acidic solution and that [OH] is 1.0 M for a basic solution. Note that these reactions are not balanced. Reactions (a) and (b) are in acidic solution; reaction (c) is in a basic solution.
(a) $\mathrm{MnO}_{4}^{-}(a q)+\mathrm{H}_{2} \mathrm{SO}_{3}(a q)\rightleftharpoons \mathrm{Mn}^{2+}(a q)+\mathrm{SO}_{4}^{2-}(a q)$
(b) $\mathrm{IO}_{3}^{-}(a q)+\mathrm{I}^{-}(a q) \rightleftharpoons \mathrm{I}_{2}(a q)$
(c) $\mathrm{ClO}^{-}(a q)+\mathrm{I}^{-}(a q) \rightleftharpoons \mathrm{IO}_{3}^{-}(a q)+\mathrm{Cl}^{-}(a q)$
5. One analytical method for determining the concentration of sulfur is to oxidize it to $\text{SO}_4^{2-}$ and then precipitate it as BaSO4 by adding BaCl2. The mass of the resulting precipitate is proportional to the amount of sulfur in the original sample. The accuracy of this method depends on the solubility of BaSO4, the reaction for which is shown here.
$\mathrm{BaSO}_{4}(s) \rightleftharpoons \mathrm{Ba}^{2+}(a q)+\mathrm{SO}_{4}^{2-}(a q) \nonumber$
For each of the following, predict the affect on the solubility of BaSO4: (a) decreasing the solution’s pH; (b) adding more BaCl2; and (c) increasing the solution’s volume by adding H2O.
6. Write a charge balance equation and one or more mass balance equations for the following solutions.
(a) 0.10 M NaCl
(b) 0.10 M HCl
(c) 0.10 M HF
(d) 0.10 M NaH2PO4
(e) MgCO3 (saturated solution)
(f) 0.10 M $\text{Ag(CN)}_2^-$ (prepared using AgNO3 and KCN)
(g) 0.10 M HCl and 0.050 M NaNO2
7. Use the systematic approach to equilibrium problems to calculate the pH of the following solutions. Be sure to state and justify any assumptions you make in solving the problems.
(a) 0.050 M HClO4
(b) $1.00 \times 10^{-7}$ M HCl
(c) 0.025 M HClO
(d) 0.010 M HCOOH
(e) 0.050 M Ba(OH)2
(f) 0.010 M C5H5N
8. Construct ladder diagrams for the following diprotic weak acids (H2A) and estimate the pH of 0.10 M solutions of H2A, NaHA, and Na2A.
(a) maleic acid
(b) malonic acid
(c) succinic acid
9. Use the systematic approach to solving equilibrium problems to calculate the pH of (a) malonic acid, H2A; (b) sodium hydrogenmalonate, NaHA; and (c) sodium malonate, Na2A. Be sure to state and justify any assumptions you make in solving the problems.
10. Ignoring activity effects, calculate the molar solubility of Hg2Br2 in the following solutions. Be sure to state and justify any assumption you make in solving the problems.
(a) a saturated solution of Hg2Br2
(b) 0.025 M Hg2(NO3)2 saturated with Hg2Br2
(c) 0.050 M NaBr saturated with Hg2Br2
11. The solubility of CaF2 is controlled by the following two reactions
$\mathrm{CaF}_{2}(s) \rightleftharpoons \mathrm{Ca}^{2+}(a q)+2 \mathrm{F}^{-}(a q) \nonumber$
$\mathrm{HF}(a q)+\mathrm{H}_{2} \mathrm{O}(l)\rightleftharpoons\mathrm{H}_{3} \mathrm{O}^{+}(a q)+\mathrm{F}^{-}(a q) \nonumber$
Calculate the molar solubility of CaF2 in a solution that is buffered to a pH of 7.00. Use a ladder diagram to help simplify the calculations. How would your approach to this problem change if the pH is buffered to 2.00? What is the solubility of CaF2 at this pH? Be sure to state and justify any assumptions you make in solving the problems.
12. Calculate the molar solubility of Mg(OH)2 in a solution buffered to a pH of 7.00. How does this compare to its solubility in unbuffered deionized water with an initial pH of 7.00? Be sure to state and justify any assumptions you make in solving the problem.
13. Calculate the solubility of Ag3PO4 in a solution buffered to a pH of 9.00. Be sure to state and justify any assumptions you make in solving the problem.
14. Determine the equilibrium composition of saturated solution of AgCl. Assume that the solubility of AgCl is influenced by the following reactions
$\mathrm{AgCl}(s) \rightleftharpoons \mathrm{Ag}^{+}(a q)+\mathrm{Cl}^{-}(a q) \nonumber$
$\operatorname{Ag}^{+}(a q)+\mathrm{Cl}^{-}(a q) \rightleftharpoons \operatorname{AgCl}(a q) \nonumber$
$\operatorname{AgCl}(a q)+\mathrm{Cl}^{-}(a q) \rightleftharpoons \operatorname{AgCl}_{2}^-(a q) \nonumber$
Be sure to state and justify any assumptions you make in solving the problem.
15. Calculate the ionic strength of the following solutions
(a) 0.050 M NaCl
(b) 0.025 M CuCl2
(c) 0.10 M Na2SO4
16. Repeat the calculations in Problem 10, this time correcting for the effect of ionic strength. Be sure to state and justify any assumptions you make in solving the problems.
17. Over what pH range do you expect Ca3(PO4)2 to have its minimum solubility?
18. Construct ladder diagrams for the following systems, each of which consists of two or three equilibrium reactions. Using your ladder diagrams, identify all reactions that are likely to occur in each system?
(a) HF and H3PO4
(b) $\text{Ag(CN)}_2^-$, $\text{Ni(CN)}_4^{2-}$, and $\text{Fe(CN)}_6^{3-}$
(c) $\text{Cr}_2\text{O}_7^{2-}/\text{Cr}^{3+}$ and Fe3+/Fe2+
19. Calculate the pH of the following acid–base buffers. Be sure to state and justify any assumptions you make in solving the problems.
(a) 100.0 mL of 0.025 M formic acid and 0.015 M sodium formate
(b) 50.00 mL of 0.12 M NH3 and 3.50 mL of 1.0 M HCl
(c) 5.00 g of Na2CO3 and 5.00 g of NaHCO3 diluted to 0.100 L
20. Calculate the pH of the buffers in Problem 19 after adding 5.0 mL of 0.10 M HCl. Be sure to state and justify any assumptions you make in solving the problems.
21. Calculate the pH of the buffers in Problem 19 after adding 5.0 mL of 0.10 M NaOH. Be sure to state and justify any assumptions you make in solving the problems.
22. Consider the following hypothetical complexation reaction between a metal, M, and a ligand, L
$\mathrm{M}(a q)+\mathrm{L}(a q) \rightleftharpoons \mathrm{ML}(a q) \nonumber$
for which the formation constant is $1.5 \times 10^8$. (a) Derive an equation similar to the Henderson–Hasselbalch equation that relates pM to the concentrations of L and ML. (b) What is the pM for a solution that contains 0.010 mol of M and 0.020 mol of L? (c) What is pM if you add 0.002 mol of M to this solution? Be sure to state and justify any assumptions you make in solving the problem.
23. A redox buffer contains an oxidizing agent and its conjugate reducing agent. Calculate the potential of a solution that contains 0.010 mol of Fe3+ and 0.015 mol of Fe2+. What is the potential if you add sufficient oxidizing agent to convert 0.002 mol of Fe2+ to Fe3+? Be sure to state and justify any assumptions you make in solving the problem.
24. Use either Excel or R to solve the following problems. For these problems, make no simplifying assumptions.
(a) the solubility of CaF2 in deionized water
(b) the solubility of AgCl in deionized water
(c) the pH of 0.10 M fumaric acid
25. Derive equation 6.10.1 for the rigorous solution to the pH of 0.1 M HF. | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/06%3A_Equilibrium_Chemistry/6.11%3A_Some_Final_Thoughts_on_Equilibrium_Calculations.txt |
The following experiments involve the experimental determination of equilibrium constants, the characterization of buffers, and, in some cases, demonstrations of the importance of activity effects.
• “The Effect of Ionic Strength on an Equilibrium Constant (A Class Study)” in Chemical Principles in Practice, J. A. Bell, Ed., Addison-Wesley: Reading, MA, 1967.
• “Equilibrium Constants for Calcium Iodate Solubility and Iodic Acid Dissociation” in Chemical Principles in Practice, J. A. Bell, Ed., Addison-Wesley: Reading, MA, 1967.
• “The Solubility of Silver Acetate” in Chemical Principles in Practice, J. A. Bell, Ed., Addison-Wesley: Reading, MA, 1967.
• Cobb, C. L.; Love, G. A. “Iron(III) Thiocyanate Revisited: A Physical Chemistry Equilibrium Lab Incorporating Ionic Strength Effects,” J. Chem. Educ. 1998, 75, 90–92.
• Green, D. B.; Rechtsteiner, G.; Honodel, A. “Determination of the Thermodynamic Solubility Product, Ksp, of PbI2 Assuming Nonideal Behavior,” J. Chem. Educ. 1996, 73, 789–792.
• Russo, S. O.; Hanania, I. H. “Buffer Capacity,” J. Chem. Educ. 1987, 64, 817–819.
• Stolzberg, R. J. “Discovering a Change in Equilibrium Constant with Change in Ionic Strength,” J. Chem. Educ. 1999, 76, 640–641.
• Wiley, J. D. “The Effect of Ionic Strength on the Solubility of an Electrolyte,” J. Chem. Educ. 2004, 81, 1644–1646.
A nice discussion of Berthollet’s discovery of the reversibility of reactions is found in
• Roots-Bernstein, R. S. Discovering, Harvard University Press: Cambridge, MA, 1989.
The following texts provide additional coverage of equilibrium chemistry.
• Butler, J. N. Ionic Equilibria: A Mathematical Approach; Addison-Wesley: Reading, MA, 1964.
• Butler, J. N. Solubility and pH Calculations; Addison-Wesley: Reading, MA, 1973.
• Fernando, Q.; Ryan, M. D. Calculations in Analytical Chemistry, Harcourt Brace Jovanovich: New York, 1982.
• Freiser, H.; Fernando, Q. Ionic Equilibria in Analytical Chemistry, Wiley: New York, 1963.
• Freiser, H. Concepts and Calculations in Analytical Chemistry, CRC Press: Boca Raton, 1992.
• Gordus, A. A. Schaum’s Outline of Analytical Chemistry; McGraw-Hill: New York, 1985.
• Ramette, R. W. Chemical Equilibrium and Analysis, Addison-Wesley: Reading, MA, 1981.
The following papers discuss a variety of general aspects of equilibrium chemistry.
• Cepría, G.; Salvatella, L. “General Procedure for the Easy Calculation of pH in an Introductory Course of General or Analytical Chemistry,” J. Chem. Educ. 2014, 91, 524–530.
• Gordus, A. A. “Chemical Equilibrium I. The Thermodynamic Equilibrium Concept,” J. Chem. Educ. 1991, 68, 138–140.
• Gordus, A. A. “Chemical Equilibrium II. Deriving an Exact Equilibrium Equation,” J. Chem. Educ. 1991, 68, 215–217.
• Gordus, A. A. “Chemical Equilibrium III. A Few Math Tricks,” J. Chem. Educ. 1991, 68, 291–293.
• Gordus, A. A. “Chemical Equilibrium IV. Weak Acids and Bases,” J. Chem. Educ. 1991, 68, 397–399.
• Gordus, A. A. “Chemical Equilibrium VI. Buffer Solutions,” J. Chem. Educ. 1991, 68, 656–658.
• Gordus, A. A. “Chemical Equilibrium VII. Precipitates, “J. Chem. Educ. 1991, 68, 927–930.
• Reijenga, J.; Van Hoof, A.; van Loon, A.; Teunissen, B. “Development of Methods for the Determination of pKa Values,” Analytical Chemistry Insights, 2013, 8, 53–71.
• Thomson, B. M.; Kessick, M. A. “On the Preparation of Buffer Solutions,” J. Chem. Educ. 1981, 58, 743–746.
• Weltin, E. “Are the Equilibrium Concentrations for a Chemical Reaction Always Uniquely Determined by the Initial Concentrations?” J. Chem. Educ. 1990, 67, 548.
• Weltin, E. “Are the Equilibrium Compositions Uniquely Determined by the Initial Compositions? Properties of the Gibbs Free Energy Function,” J. Chem. Educ. 1995, 72, 508–511.
Collected here are a papers that discuss a variety of approaches to solving equilibrium problems.
• Ault, A. “Do pH in Your Head,” J. Chem. Educ. 1999, 76, 936–938.
• Chaston, S. “Calculating Complex Equilibrium Concentrations by a Next Guess Factor Method,” J. Chem. Educ. 1993, 70, 622–624.
• Donato, H. “Graphing Calculator Strategies for Solving Chemical Equilibrium Problems,” J. Chem. Educ. 1999, 76, 632–634.
• Glaser, R. E. Delarosa, M. A.; Salau, A. O.; Chicone, C. “Dynamical Approach to Multiequilibria Problems for Mixtures of Acids and Their Conjugate Bases,” J. Chem. Educ. 2014, 91, 1009–1016.
• Olivieri, A. C. “Solution of Acid-Base Equilibria by Successive Approximations,” J. Chem. Educ. 1990, 67, 229–231.
• Weltin, E. “A Numerical Method to Calculate Equilibrium Concentrations for Single-Equation Systems,” J. Chem. Educ. 1991, 68, 486–487.
• Weltin, E. “Calculating Equilibrium Concentrations,” J. Chem. Educ. 1992, 69, 393–396.
• Weltin, E. “Calculating Equilibrium Concentrations for Stepwise Binding of Ligands and Polyprotic Acid-Base Systems,” J. Chem. Educ. 1993, 70, 568–571.
• Weltin, E. “Equilibrium Calculations are Easier Than You Think - But You do Have to Think!” J. Chem. Educ. 1993, 70, 571–573.
• Weltin, E. “Calculating Equilibrium Concentrations by Iteration: Recycle Your Approximations,” J. Chem. Educ. 1995, 72, 36–38.
Additional historical background on the development of the Henderson-Hasselbalch equation is provided by the following papers.
• de Levie, R. “The Henderson Approximation and the Mass Action Law of Guldberg and Waage,” Chem. Educator 2002, 7, 132–135.
• de Levie, R. “The Henderson-Hasselbalch Equation: Its History and Limitations,” J. Chem. Educ. 2003, 80, 146.
A simulation is a useful tool for helping students gain an intuitive understanding of a topic. Gathered here are some simulations for teaching equilibrium chemistry.
• Edmonson, L. J.; Lewis, D. L. “Equilibrium Principles: A Game for Students,” J. Chem. Educ. 1999, 76, 502.
• Huddle, P. A.; White, M. W.; Rogers, F. “Simulations for Teaching Chemical Equilibrium,” J. Chem. Educ. 2000, 77, 920–926.
The following papers provide additional resources on ionic strength, activity, and the effect of ionic strength and activity on equilibrium reactions and pH.
• Clark, R. W.; Bonicamp, J. M. “The Ksp-Solubility Conundrum,” J. Chem. Educ. 1998, 75, 1182– 1185.
• de Levie, R. “On Teaching Ionic Activity Effects: What, When, and Where?” J. Chem. Educ. 2005, 82, 878–884.
• McCarty, C. G.; Vitz, E. “pH Paradoxes: Demonstrating That It Is Not True That pH = –log[H+],”J. Chem. Educ. 2006, 83, 752–757.
• Ramshaw, J. D. “Fugacity and Activity in a Nutshell,” J. Chem. Educ. 1995, 72, 601–603.
• Sastre de Vicente, M. E. “The Concept of Ionic Strength Eighty Years After Its Introduction,” J. Chem. Educ. 2004, 81, 750–753.
• Solomon, T. “The Definition and Unit of Ionic Strength,” J. Chem. Educ. 2001, 78, 1691–1692.
For a contrarian’s view of equilibrium chemistry, please see the following papers.
• Hawkes, S. J. “Buffer Calculations Deceive and Obscure,” Chem. Educator, 1996, 1, 1–8.
• Hawkes, S. J. “What Should We Teach Beginners About Solubility and Solubility Products?” J. Chem. Educ. 1998, 75, 1179–1181.
• Hawkes, S. J. “Complexation Calculations are Worse Than Useless,” J. Chem. Educ. 1999, 76, 1099–1100.
• Hawkes, S. J. “Easy Deviation of pH ≈ (pKa1 + pKa2)/2 Using Autoprotolysis of HA: Doubtful Value of the Supposedly More Rigorous Equation,” J. Chem. Educ. 2000, 77, 1183–1184. See, also, an exchange of letters between J. J. Roberts and S. J. Hawkes, J. Chem. Educ. 2002, 79, 161–162. | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/06%3A_Equilibrium_Chemistry/6.13%3A_Additional_Resources.txt |
Chapter Summary
Analytical chemistry is more than a collection of techniques; it is the application of chemistry to the analysis of samples. As we will see in later chapters, almost all analytical methods use chemical reactivity to accomplish one or more of the following: dissolve a sample, separate analytes from interferents, transform an analyte into a more useful form, or provide a signal. Equilibrium chemistry and thermodynamics provide us with a means for predicting which reactions are likely to be favorable.
The most important types of reactions are precipitation reactions, acid–base reactions, metal‐ligand complexation reactions, and oxidation–reduction reactions. In a precipitation reaction two or more soluble species combine to produce an insoluble precipitate, which we characterize using a solubility product.
An acid–base reaction occurs when an acid donates a proton to a base. The reaction’s equilibrium position is described using either an acid dissociation constant, Ka, or a base dissociation constant, Kb. The product of Ka and Kb for an acid and its conjugate base is the dissociation constant for water, Kw.
When a ligand donates one or more pairs of electron to a metal ion, the result is a metal–ligand complex. Two types of equilibrium constants are used to describe metal–ligand complexation: stepwise formation constants and overall formation constants. There are two stepwise formation constants for the metal–ligand complex ML2, each of which describes the addition of one ligand; thus, K1 represents the addition of the first ligand to M, and K2 represents the addition of the second ligand to ML. Alternatively, we can use a cumulative, or overall formation constant, $\beta_2$, for the metal–ligand complex ML2, in which both ligands are added to M.
In an oxidation–reduction reaction, one of the reactants is oxidized and another reactant is reduced. Instead of using an equilibrium constants to characterize an oxidation–reduction reactions, we use the potential, positive values of which indicate a favorable reaction. The Nernst equation relates this potential to the concentrations of reactants and products.
Le Châtelier’s principle provides a means for predicting how a system at equilibrium responds to a change in conditions. If we apply a stress to a system at equilibrium—by adding a reactant or product, by adding a reagent that reacts with a reactant or product, or by changing the volume—the system will respond by moving in the direction that relieves the stress.
You should be able to describe a system at equilibrium both qualitatively and quantitatively. You can develop a rigorous solution to an equilibrium problem by combining equilibrium constant expressions with appropriate mass balance and charge balance equations. Using this systematic approach, you can solve some quite complicated equilibrium problems. If a less rigorous answer is acceptable, then a ladder diagram may help you estimate the equilibrium system’s composition.
Solutions that contain relatively similar amounts of a weak acid and its conjugate base experience only a small change in pH upon the addition of a small amount of strong acid or of strong base. We call these solutions buffers. A buffer can also be formed using a metal and its metal–ligand complex, or an oxidizing agent and its conjugate reducing agent. Both the systematic approach to solving equilibrium problems and ladder diagrams are useful tools for characterizing buffers.
A quantitative solution to an equilibrium problem may give an answer that does not agree with experimental results if we do not consider the effect of ionic strength. The true, thermodynamic equilibrium constant is a function of activities, a, not concentrations. A species’ activity is related to its molar concentration by an activity coefficient, $\gamma$. Activity coefficients are estimated using the extended Debye‐Hückel equation, making possible a more rigorous treatment of equilibria.
Key Terms
acid
activity coefficient
base dissociation constant
charge balance equation
dissociation constant
equilibrium
formation constant
Henderson–Hasselbalch equation
Le Châtelier’s principle
metal–ligand complex
Nernst equation
pH scale
precipitate
reduction
steady state
acid dissociation constant
amphiprotic
buffer
common ion effect
enthalpy
equilibrium constant
Gibb’s free energy
ionic strength
ligand
method of successive approximations
oxidation
polyprotic
redox reaction
standard‐state
stepwise formation constant
activity
base
buffer capacity
cumulative formation constant
entropy
extended Debye‐Hückel equation
half‐reaction
ladder diagram
mass balance equation
monoprotic
oxidizing agent
potential
reducing agent
standard potential
solubility product | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/06%3A_Equilibrium_Chemistry/6.14%3A_Chapter_Summary_and_Key_Terms.txt |
When we use an analytical method to solve a problem, there is no guarantee that will obtain accurate or precise results. In designing an analytical method we consider potential sources of determinate error and indeterminate error, and we take appropriate steps—such as reagent blanks and the calibration of instruments—to minimize their effect. Why might a carefully designed analytical method give poor results? One possible reason is that we may have failed to account for errors associated with the sample. If we collect the wrong sample, or if we lose analyte when we prepare the sample for analysis, then we introduce a determinate source of error. If we fail to collect enough samples, or if we collect samples of the wrong size, then the precision of our analysis may suffer. In this chapter we consider how to collect samples and how to prepare them for analysis.
• 7.1: The Importance of Sampling
If the individual samples do not represent accurately the population from which they are drawn—a population that we call the target population—then even a careful analysis will yield an inaccurate result. Extrapolating a result from a sample to its target population always introduces a determinate sampling error. To minimize this determinate sampling error, we must collect the right sample.
• 7.2: Designing a Sampling Plan
A sampling plan must support the goals of an analysis. A material scientist interested in characterizing a metal’s surface chemistry is more likely to choose a freshly exposed surface, created by cleaving the sample under vacuum, than a surface previously exposed to the atmosphere. In a qualitative analysis, a sample need not be identical to the original substance if there is sufficient analyte present to ensure its detection.
• 7.3: Implementing the Sampling Plan
Implementing a sampling plan usually involves three steps: physically removing the sample from its target population, preserving the sample, and preparing the sample for analysis.
• 7.4: Separating the Analyte From Interferents
When an analytical method is selective for the analyte, analyzing a sample is a relatively simple task. For example, a quantitative analysis for glucose in honey is relatively easy to accomplish if the method is selective for glucose, even in the presence of other reducing sugars, such as fructose. Unfortunately, few analytical methods are selective toward a single species; thus, we must separate analytes from interferents.
• 7.5: General Theory of Separation Effiiciency
The goal of an analytical separation is to remove either the analyte or the interferent from the sample’s matrix. To achieve this separation we must identify at least one significant difference between the analyte’s and the interferent’s chemical or physical properties. A significant difference in properties, however, is not sufficient to effect a separation if the conditions that favor the extraction of interferent from the sample also removes a small amount of analyte.
• 7.6: Classifying Separation Techniques
We can separate an analyte and an interferent if there is a significant difference in at least one of their chemical or physical properties, such as size, mass or density, the ability to form complexes, a change in physical state, a change in chemical state, or the ability to partition between phases.
• 7.7: Liquid-Liquid Extractions
A liquid–liquid extraction is an important separation technique for environmental, clinical, and industrial laboratories. In a simple liquid–liquid extraction the solute partitions itself between two immiscible phases. One phase usually is an aqueous solvent and the other phase is an organic solvent, such as the pentane used to extract trihalomethanes from water.
• 7.8: Separation Versus Preconcentration
Two common analytical problems are matrix components that interfere with an analyte’s analysis and an analyte with a concentration that is too small to analyze accurately. As we have learned in this chapter, we can use a separation to solve the first problem. Interestingly, we often can use a separation to solve the second problem as well.
• 7.9: Problems
End-of-chapter problems to test your understanding ot topics in this chapter.
• 7.10: Additional Resources
A compendium of resources to accompany topics in this chapter.
• 7.11: Chapter Summary and Key Terms
Summary of chapter's main topics and a list of keyterms introduced in this chapter.
07: Obtaining and Preparing Samples for Analysis
When a manufacturer lists a chemical as ACS Reagent Grade, they must demonstrate that it conforms to specifications set by the American Chemical Society (ACS). For example, the ACS specifications for commercial NaBr require that the concentration of iron is less than 5 ppm. To verify that a production lot meets this standard, the manufacturer collects and analyzes several samples, reporting the average result on the product’s label (Figure 7.1.1 ).
If the individual samples do not represent accurately the population from which they are drawn—a population that we call the target population—then even a careful analysis will yield an inaccurate result. Extrapolating a result from a sample to its target population always introduces a determinate sampling error. To minimize this determinate sampling error, we must collect the right sample.
Even if we collect the right sample, indeterminate sampling errors may limit the usefulness of our analysis. Equation \ref{7.1} shows that a confidence interval about the mean, $\overline{X}$ , is proportional to the standard deviation, s, of the analysis
$\mu=\overline{X} \pm \frac{t s}{\sqrt{n}} \label{7.1}$
where n is the number of samples and t is a statistical factor that accounts for the probability that the confidence interval contains the true value, $\mu$.
Equation \ref{7.1} should be familiar to you. See Chapter 4 to review confidence intervals and see Appendix 4 for values of t.
Each step of an analysis contributes random error that affects the overall standard deviation. For convenience, let’s divide an analysis into two steps—collecting the samples and analyzing the samples—each of which is characterized by a variance. Using a propagation of uncertainty, the relationship between the overall variance, s2, and the variances due to sampling, $s_{samp}^2$, and the variance due to the analytical method, $s_{meth}^2$, is
$s^{2}=s_{samp}^{2}+s_{meth}^{2} \label{7.2}$
Although Equation \ref{7.1} is written in terms of a standard deviation, s, a propagation of uncertainty is written in terms of variances, s2. In this section, and those that follow, we will use both standard deviations and variances to discuss sampling uncertainty. For a review of the propagation of uncertainty, see Chapter 4.3 and Appendix 2.
Equation \ref{7.2} shows that the overall variance for an analysis is limited by either the analytical method or sampling, or by both. Unfortunately, analysts often try to minimize the overall variance by improving only the method’s precision. This is a futile effort, however, if the standard deviation for sampling is more than three times greater than that for the method [Youden, Y. J. J. Assoc. Off. Anal. Chem. 1981, 50, 1007–1013]. Figure 7.1.2 shows how the ratio ssamp/smeth affects the method’s contribution to the overall variance. As shown by the dashed line, if the sample’s standard deviation is $3 \times$ the method’s standard deviation, then indeterminate method errors explain only 10% of the overall variance. If indeterminate sampling errors are significant, decreasing smeth provides only limited improvement in the overall precision.
Example 7.1.1
A quantitative analysis gives a mean concentration of 12.6 ppm for an analyte. The method’s standard deviation is 1.1 ppm and the standard deviation for sampling is 2.1 ppm. (a) What is the overall variance for the analysis? (b) By how much does the overall variance change if we improve smeth by 10% to 0.99 ppm? (c) By how much does the overall variance change if we improve ssamp by 10% to 1.9 ppm?
Solution
(a) The overall variance is
$s^{2}=s_{samp}^{2}+s_{meth}^{2}=(2.1 \ \mathrm{ppm})^{2}+(1.1 \ \mathrm{ppm})^{2}=5.6 \ \mathrm{ppm}^{2} \nonumber$
(b) Improving the method’s standard deviation changes the overall variance to
$s^{2}=(2.1 \ \mathrm{ppm})^{2}+(0.99 \ \mathrm{ppm})^{2}=5.4 \ \mathrm{ppm}^{2} \nonumber$
Improving the method’s standard deviation by 10% improves the overall variance by approximately 4%.
(c) Changing the standard deviation for sampling
$s^{2}=(1.9 \ \mathrm{ppm})^{2}+(1.1 \ \mathrm{ppm})^{2}=4.8 \ \mathrm{ppm}^{2} \nonumber$
improves the overall variance by almost 15%. As expected, because ssamp is larger than smeth, we achieve a bigger improvement in the overall variance when we focus our attention on sampling problems.
Exercise 7.1.1
Suppose you wish to reduce the overall variance in Example 7.1.1 to 5.0 ppm2. If you focus on the method, by what percentage do you need to reduce smeth? If you focus on the sampling, by what percentage do you need to reduce ssamp?
Answer
To reduce the overall variance by improving the method’s standard deviation requires that
$s^{2}=5.00 \ \mathrm{ppm}^{2} = s_{samp}^{2}+s_{m e t h}^{2} = (2.1 \mathrm{ppm})^{2}+s_{m e t h}^{2} \nonumber$
Solving for smeth gives its value as 0.768 ppm. Relative to its original value of 1.1 ppm, this is a reduction of $3.0 \times 10^1$%. To reduce the overall variance by improving the standard deviation for sampling requires that
$s^{2}=5.00 \ \mathrm{ppm}^{2} = s_{samp}^{2}+s_{meth}^{2} = s_{samp}^{2}+(1.1 \ \mathrm{ppm})^{2} \nonumber$
Solving for ssamp gives its value as 1.95 ppm. Relative to its original value of 2.1 ppm, this is reduction of 7.1%.
To determine which step has the greatest effect on the overall variance, we need to measure both ssamp and smeth. The analysis of replicate samples provides an estimate of the overall variance. To determine the method’s variance we must analyze samples under conditions where we can assume that the sampling variance is negligible; the sampling variance is then determined by difference.
There are several ways to minimize the standard deviation for sampling. Here are two examples. One approach is to use a standard reference material (SRM) that has been carefully prepared to minimize indeterminate sampling errors. When the sample is homogeneous—as is the case, for example, with an aqueous sample—then another useful approach is to conduct replicate analyses on a single sample.
Example 7.1.2
The following data were collected as part of a study to determine the effect of sampling variance on the analysis of drug-animal feed formulations [Fricke, G. H.; Mischler, P. G.; Staffieri, F. P.; Houmyer, C. L. Anal. Chem. 1987, 59, 1213– 1217].
% drug (w/w) % drug (w/w)
0.0114 0.0099 0.0105 0.0105 0.0109 0.0107
0.0102 0.0106 0.0087 0.0103 0.0103 0.0104
0.0100 0.0095 0.0098 0.0101 0.0101 0.013
0.0105 0.0095 0.0097
The data on the left were obtained under conditions where both ssamp and smeth contribute to the overall variance. The data on the right were obtained under conditions where ssamp is insignificant. Determine the overall variance, and the standard deviations due to sampling and the analytical method. To which source of indeterminate error—sampling or the method—should we turn our attention if we want to improve the precision of the analysis?
Solution
Using the data on the left, the overall variance, s2, is $4.71 \times 10^{-7}$. To find the method’s contribution to the overall variance, $s_{meth}^2$, we use the data on the right, obtaining a value of $7.00 \times 10^{-8}$. The variance due to sampling, $s_{samp}^2$, is
$s_{samp}^{2}=s^{2}-s_{meth}^{2} = 4.71 \times 10^{-7}-7.00 \times 10^{-8}=4.01 \times 10^{-7} \nonumber$
Converting variances to standard deviations gives ssamp as $6.33 \times 10^{-4}$ and smeth as $2.65 \times 10^{-4}$. Because ssamp is more than twice as large as smeth, improving the precision of the sampling process will have the greatest impact on the overall precision.
Exercise 7.1.2
A polymer’s density provides a measure of its crystallinity. The standard deviation for the determination of density using a single sample of a polymer is $1.96 \times 10^{-3}$ g/cm3. The standard deviation when using different samples of the polymer is $3.65 \times 10^{-2}$ g/cm3. Determine the standard deviations due to sampling and to the analytical method.
Answer
The analytical method’s standard deviation is $1.96 \times 10^{-3}$ g/cm3 as this is the standard deviation for the analysis of a single sample of the polymer. The sampling variance is
$s_{sa m p}^{2}=s^{2}-s_{meth}^{2}= \left(3.65 \times 10^{-2}\right)^{2}-\left(1.96 \times 10^{-3}\right)^{2}=1.33 \times 10^{-3} \nonumber$
Converting the variance to a standard deviation gives smeth as $3.64 \times 10^{-2}$ g/cm3. | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/07%3A_Obtaining_and_Preparing_Samples_for_Analysis/7.01%3A_The_Importance_of_Sampling.txt |
A sampling plan must support the goals of an analysis. For example, a material scientist interested in characterizing a metal’s surface chemistry is more likely to choose a freshly exposed surface, created by cleaving the sample under vacuum, than a surface previously exposed to the atmosphere. In a qualitative analysis, a sample need not be identical to the original substance provided there is sufficient analyte present to ensure its detection. In fact, if the goal of an analysis is to identify a trace-level component, it may be desirable to discriminate against major components when collecting samples.
For an interesting discussion of the importance of a sampling plan, see Buger, J. et al. “Do Scientists and Fishermen Collect the Same Size Fish? Possible Implications for Exposure Assessment,” Environ. Res. 2006, 101, 34–41.
For a quantitative analysis, the sample’s composition must represent accurately the target population, a requirement that necessitates a careful sampling plan. Among the issues we need to consider are these five questions.
1. From where within the target population should we collect samples?
2. What type of samples should we collect?
3. What is the minimum amount of sample needed for each analysis?
4. How many samples should we analyze?
5. How can we minimize the overall variance for the analysis?
Where to Sample the Target Population
A sampling error occurs whenever a sample’s composition is not identical to its target population. If the target population is homogeneous, then we can collect individual samples without giving consideration to where we collect sample. Unfortunately, in most situations the target population is heterogeneous and attention to where we collect samples is important. For example, due to settling a medication available as an oral suspension may have a higher concentration of its active ingredients at the bottom of the container. The composition of a clinical sample, such as blood or urine, may depend on when it is collected. A patient’s blood glucose level, for instance, will change in response to eating and exercise. Other target populations show both a spatial and a temporal heterogeneity. The concentration of dissolved O2 in a lake is heterogeneous due both to a change in seasons and to point sources of pollution.
The composition of a homogeneous target population is the same regardless of where we sample, when we sample, or the size of our sample. For a heterogeneous target population, the composition is not the same at different locations, at different times, or for different sample sizes.
If the analyte’s distribution within the target population is a concern, then our sampling plan must take this into account. When feasible, homogenizing the target population is a simple solution, although this often is impracticable. In addition, homogenizing a sample destroys information about the analyte’s spatial or temporal distribution within the target population, information that may be of importance.
Random Sampling
The ideal sampling plan provides an unbiased estimate of the target population’s properties. A random sampling is the easiest way to satisfy this requirement [Cohen, R. D. J. Chem. Educ. 1991, 68, 902–903]. Despite its apparent simplicity, a truly random sample is difficult to collect. Haphazard sampling, in which samples are collected without a sampling plan, is not random and may reflect an analyst’s unintentional biases.
Here is a simple method to ensure that we collect random samples. First, we divide the target population into equal units and assign to each unit a unique number. Then, we use a random number table to select the units to sample. Example 7.2.1 provides an illustrative example. Appendix 14 provides a random number table that you can use to design a sampling plan.
Example 7.2.1
To analyze a polymer’s tensile strength, individual samples of the polymer are held between two clamps and stretched. To evaluate a production lot, the manufacturer’s sampling plan calls for collecting ten 1 cm $\times$ 1 cm samples from a 100 cm $\times$ 100 cm polymer sheet. Explain how we can use a random number table to ensure that we collect these samples at random.
Solution
As shown by the grid below, we divide the polymer sheet into 10 000 1 cm $\times$ 1 cm squares, each identified by its row number and its column number, with numbers running from 0 to 99.
For example, the blue square is in row 98 and in column 1. To select ten squares at random, we enter the random number table in Appendix 14 at an arbitrary point and let the entry’s last four digits represent the row number and the column number for the first sample. We then move through the table in a predetermined fashion, selecting random numbers until we have 10 samples. For our first sample, let’s use the second entry in the third column of Appendix 14 , which is 76831. The first sample, therefore, is row 68 and column 31. If we proceed by moving down the third column, then the 10 samples are as follows:
sample number row column sample number roq column
1 76831 68 31 6 41701 17 01
2
66558
65 58 7 38605 86 05
3 33266 32 66 8 64516 45 16
4 12032 20 32 9 13015 30 15
5 14063 40 63 10 12138 21 38
When we collect a random sample we make no assumptions about the target population, which makes this the least biased approach to sampling. On the other hand, a random sample often requires more time and expense than other sampling strategies because we need to collect a greater number of samples to ensure that we adequately sample the target population, particularly when that population is heterogenous [Borgman, L. E.; Quimby, W. F. in Keith, L. H., ed. Principles of Environmental Sampling, American Chemical Society: Washington, D. C., 1988, 25–43].
Judgmental Sampling
The opposite of random sampling is selective, or judgmental sampling in which we use prior information about the target population to help guide our selection of samples. Judgmental sampling is more biased than random sampling, but requires fewer samples. Judgmental sampling is useful if we wish to limit the number of independent variables that might affect our results. For example, if we are studying the bioaccumulation of PCB’s in fish, we may choose to exclude fish that are too small, too young, or that appear diseased.
Systematic Sampling
Random sampling and judgmental sampling represent extremes in bias and in the number of samples needed to characterize the target population. Systematic sampling falls in between these extremes. In systematic sampling we sample the target population at regular intervals in space or time. Figure 7.2.1 shows an aerial photo of the Great Salt Lake in Utah. A railroad line divides the lake into two sections that have different chemical compositions. To compare the lake’s two sections—and to evaluate spatial variations within each section—we use a two-dimensional grid to define sampling locations, collecting samples at the center of each location. When a population’s is heterogeneous in time, as is common in clinical and environmental studies, then we might choose to collect samples at regular intervals in time.
If a target population’s properties have a periodic trend, a systematic sampling will lead to a significant bias if our sampling frequency is too small. This is a common problem when sampling electronic signals where the problem is known as aliasing. Consider, for example, a signal that is a simple sign wave. Figure 7.2.2 a shows how an insufficient sampling frequency underestimates the signal’s true frequency. The apparent signal, shown by the dashed red line that passes through the five data points, is significantly different from the true signal shown by the solid blue line.
According to the Nyquist theorem, to determine accurately the frequency of a periodic signal, we must sample the signal at least twice during each cycle or period. If we collect samples at an interval of $\Delta t$, then the highest frequency we can monitor accurately is $(2 \Delta t)^{-1}$. For example, if we collect one sample each hour, then the highest frequency we can monitor is (2 $\times$ 1 hr)–1 or 0.5 hr–1, a period of less than 2 hr. If our signal’s period is less than 2 hours (a frequency of more than 0.5 hr–1), then we must use a faster sampling rate. Ideally, we use a sampling rate that is at least 3–4 times greater than the highest frequency signal of interest. If our signal has a period of one hour, then we should collect a new sample every 15-20 minutes.
Systematic–Judgmental Sampling
Combinations of the three primary approaches to sampling also are possible [Keith, L. H. Environ. Sci. Technol. 1990, 24, 610–617]. One such combination is systematic–judgmental sampling, in which we use prior knowledge about a system to guide a systematic sampling plan. For example, when monitoring waste leaching from a landfill, we expect the plume to move in the same direction as the flow of groundwater—this helps focus our sampling, saving money and time. The systematic–judgmental sampling plan in Figure 7.2.3 includes a rectangular grid for most of the samples and linear transects to explore the plume’s limits [Flatman, G. T.; Englund, E. J.; Yfantis, A. A. in Keith, L. H., ed. Principles of Environmental Sampling, American Chemical Society: Washington, D. C., 1988, 73–84].
Stratified Sampling
Another combination of the three primary approaches to sampling is judgmental–random, or stratified sampling. Many target populations consist of distinct units, or strata. For example, suppose we are studying particulate Pb in urban air. Because particulates come in a range of sizes—some visible and some microscopic—and come from many sources—such as road dust, diesel soot, and fly ash to name a few—we can subdivide the target population by size or by source. If we choose a random sampling plan, then we collect samples without considering the different strata, which may bias the sample toward larger particulates. In a stratified sampling we divide the target population into strata and collect random samples from within each stratum. After we analyze the samples from each stratum, we pool their respective means to give an overall mean for the target population. The advantage of stratified sampling is that individual strata usually are more homogeneous than the target population. The overall sampling variance for stratified sampling always is at least as good, and often is better than that obtained by simple random sampling. Because a stratified sampling requires that we collect and analyze samples from several strata, it often requires more time and money.
Convenience Sampling
One additional method of sampling deserves mention. In convenience sampling we select sample sites using criteria other than minimizing sampling error and sampling variance. In a survey of rural groundwater quality, for example, we can choose to drill wells at sites selected at random or we can choose to take advantage of existing wells; the latter usually is the preferred choice. In this case cost, expedience, and accessibility are more important than ensuring a random sample
What Type of Sample to Collect
Having determined from where to collect samples, the next step in designing a sampling plan is to decide on the type of sample to collect. There are three common methods for obtaining samples: grab sampling, composite sampling, and in situ sampling.
The most common type of sample is a grab sample in which we collect a portion of the target population at a specific time or location, providing a “snapshot” of the target population. If our target population is homogeneous, a series of random grab samples allows us to establish its properties. For a heterogeneous target population, systematic grab sampling allows us to characterize how its properties change over time and/or space.
A composite sample is a set of grab samples that we combine into a single sample before analysis. Because information is lost when we combine individual samples, normally we analyze separately each grab sample. In some situations, however, there are advantages to working with a composite sample.
One situation where composite sampling is appropriate is when our interest is in the target population’s average composition over time or space. For example, wastewater treatment plants must monitor and report the average daily composition of the treated water they release to the environment. The analyst can collect and analyze a set of individual grab samples and report the average result, or she can save time and money by combining the grab samples into a single composite sample and report the result of her analysis of the composite sample.
Composite sampling also is useful when a single sample does not supply sufficient material for the analysis. For example, analytical methods for the quantitative analysis of PCB’s in fish often require as much as 50 g of tissue, an amount that may be difficult to obtain from a single fish. Combining and homogenizing tissue samples from several fish makes it easy to obtain the necessary 50-g sample.
A significant disadvantage of grab samples and composite samples is that we cannot use them to monitor continuously a time-dependent change in the target population. In situ sampling, in which we insert an analytical sensor into the target population, allows us to monitor the target population without removing individual grab samples. For example, we can monitor the pH of a solution in an industrial production line by immersing a pH electrode in the solution’s flow.
Example 7.2.2
A study of the relationship between traffic density and the concentrations of Pb, Cd, and Zn in roadside soils uses the following sampling plan [Nabulo, G.; Oryem-Origa, H.; Diamond, M. Environ. Res. 2006, 101, 42–52]. Samples of surface soil (0–10 cm) are collected at distances of 1, 5, 10, 20, and 30 m from the road. At each distance, 10 samples are taken from different locations and mixed to form a single sample. What type of sampling plan is this? Explain why this is an appropriate sampling plan.
Solution
This is a systematic–judgemental sampling plan using composite samples. These are good choices given the goals of the study. Automobile emissions release particulates that contain elevated concentrations of Pb, Cd, and Zn—this study was conducted in Uganda where leaded gasoline was still in use—which settle out on the surrounding roadside soils as “dry rain.” Samples collected near the road and samples collected at fixed distances from the road provide sufficient data for the study, while minimizing the total number of samples. Combining samples from the same distance into a single, composite sample has the advantage of decreasing sampling uncertainty.
How Much Sample to Collect
To minimize sampling errors, samples must be of an appropriate size. If a sample is too small its composition may differ substantially from that of the target population, which introduces a sampling error. Samples that are too large, however, require more time and money to collect and analyze, without providing a significant improvement in the sampling error.
Let’s assume our target population is a homogeneous mixture of two types of particles. Particles of type A contain a fixed concentration of analyte, and particles of type B are analyte-free. Samples from this target population follow a binomial distribution. If we collect a sample of n particles, then the expected number of particles that contains analyte, nA, is
$n_{A}=n p \nonumber$
where p is the probability of selecting a particle of type A. The standard deviation for sampling is
$s_{samp}=\sqrt{n p(1-p)} \label{7.1}$
To calculate the relative standard deviation for sampling, $\left( s_{samp} \right)_{rel}$, we divide Equation \ref{7.1} by nA, obtaining
$\left(s_{samp}\right)_{r e l}=\frac{\sqrt{n p(1-p)}}{n p} \nonumber$
Solving for n allows us to calculate the number of particles we need to provide a desired relative sampling variance.
$n=\frac{1-p}{p} \times \frac{1}{\left(s_{s a m p}\right)_{rel}^{2}} \label{7.2}$
Example 7.2.3
Suppose we are analyzing a soil where the particles that contain analyte represent only $1 \times 10^{-7}$% of the population. How many particles must we collect to give a percent relative standard deviation for sampling of 1%?
Solution
Since the particles of interest account for $1 \times 10^{-7}$% of all particles, the probability, p, of selecting one of these particles is $1 \times 10^{-9}$. Substituting into Equation \ref{7.2} gives
$n=\frac{1-\left(1 \times 10^{-9}\right)}{1 \times 10^{-9}} \times \frac{1}{(0.01)^{2}}=1 \times 10^{13} \nonumber$
To obtain a relative standard deviation for sampling of 1%, we need to collect $1 \times 10^{13}$ particles.
Depending on the particle size, a sample of 1013 particles may be fairly large. Suppose this is equivalent to a mass of 80 g. Working with a sample this large clearly is not practical. Does this mean we must work with a smaller sample and accept a larger relative standard deviation for sampling? Fortunately the answer is no. An important feature of Equation \ref{7.2} is that the relative standard deviation for sampling is a function of the number of particles instead of their combined mass. If we crush and grind the particles to make them smaller, then a sample of 1013 particles will have a smaller mass. If we assume that a particle is spherical, then its mass is proportional to the cube of its radius.
$\operatorname{mass} \propto r^{3} \nonumber$
If we decrease a particle’s radius by a factor of 2, for example, then we decrease its mass by a factor of 23, or 8. This assumes, of course, that the process of crushing and grinding particles does not change the composition of the particles.
Example 7.2.4
Assume that a sample of 1013 particles from Example 7.2.3 weighs 80 g and that the particles are spherical. By how much must we reduce a particle’s radius if we wish to work with 0.6-g samples?
Solution
To reduce the sample’s mass from 80 g to 0.6 g, we must change its mass by a factor of
$\frac{80}{0.6}=133 \times \nonumber$
To accomplish this we must decrease a particle’s radius by a factor of
\begin{aligned} r^{3} &=133 \times \ r &=5.1 \times \end{aligned} \nonumber
Decreasing the radius by a factor of approximately 5 allows us to decrease the sample’s mass from 80 g to 0.6 g.
Treating a population as though it contains only two types of particles is a useful exercise because it shows us that we can improve the relative standard deviation for sampling by collecting more particles. Of course, a real population likely contains more than two types of particles, with the analyte present at several levels of concentration. Nevertheless, the sampling of many well-mixed populations approximate binomial sampling statistics because they are homogeneous on the scale at which they are sampled. Under these conditions the following relationship between the mass of a random grab sample, m, and the percent relative standard deviation for sampling, R, often is valid
$m R^{2}=K_{s} \label{7.3}$
where Ks is a sampling constant equal to the mass of a sample that produces a percent relative standard deviation for sampling of ±1% [Ingamells, C. O.; Switzer, P. Talanta 1973, 20, 547–568].
Example 7.2.5
The following data were obtained in a preliminary determination of the amount of inorganic ash in a breakfast cereal.
mass of cereal (g) 0.9956 0.9981 1.0036 0.9994 1.0067
%w/w ash 1.34 1.29 1.32 1.26 1.28
What is the value of Ks and what size sample is needed to give a percent relative standard deviation for sampling of ±2.0%. Predict the percent relative standard deviation and the absolute standard deviation if we collect 5.00-g samples.
Solution
To determine the sampling constant, Ks, we need to know the average mass of the cereal samples and the relative standard deviation for the amount of ash in those samples. The average mass of the cereal samples is 1.0007 g. The average %w/w ash and its absolute standard deviation are, respectively, 1.298 %w/w and 0.03194 %w/w. The percent relative standard deviation, R, therefore, is
$R=\frac{s_{\text { samp }}}{\overline{X}}=\frac{0.03194 \% \ \mathrm{w} / \mathrm{w}}{1.298 \% \ \mathrm{w} / \mathrm{w}} \times 100=2.46 \% \nonumber$
Solving for Ks gives its value as
$K_{s}=m R^{2}=(1.0007 \mathrm{g})(2.46)^{2}=6.06 \ \mathrm{g} \nonumber$
To obtain a percent relative standard deviation of ±2%, samples must have a mass of at least
$m=\frac{K_{s}}{R^{2}}=\frac{6.06 \mathrm{g}}{(2.0)^{2}}=1.5 \ \mathrm{g} \nonumber$
If we use 5.00-g samples, then the expected percent relative standard deviation is
$R=\sqrt{\frac{K_{s}}{m}}=\sqrt{\frac{6.06 \mathrm{g}}{5.00 \mathrm{g}}}=1.10 \% \nonumber$
and the expected absolute standard deviation is
$s_{\text { samp }}=\frac{R \overline{X}}{100}=\frac{(1.10)(1.298 \% \mathrm{w} / \mathrm{w})}{100}=0.0143 \% \mathrm{w} / \mathrm{w} \nonumber$
Exercise 7.2.1
Olaquindox is a synthetic growth promoter in medicated feeds for pigs. In an analysis of a production lot of feed, five samples with nominal masses of 0.95 g were collected and analyzed, with the results shown in the following table.
mass (g) 0.9530 0.9728 0.9660 0.9402 0.9576
mg olaquindox/kg feed 23.0 23.8 21.0 26.5 21.4
What is the value of Ks and what size samples are needed to obtain a percent relative deviation for sampling of 5.0%? By how much do you need to reduce the average particle size if samples must weigh no more than 1 g?
Answer
To determine the sampling constant, Ks, we need to know the average mass of the samples and the percent relative standard deviation for the concentration of olaquindox in the feed. The average mass for the five samples is 0.95792 g. The average concentration of olaquindox in the samples is 23.14 mg/kg with a standard deviation of 2.200 mg/kg. The percent relative standard deviation, R, is
$R=\frac{s_{\text { samp }}}{\overline{X}} \times 100=\frac{2.200 \ \mathrm{mg} / \mathrm{kg}}{23.14 \ \mathrm{mg} / \mathrm{kg}} \times 100=9.507 \approx 9.51 \nonumber$
Solving for Ks gives its value as
$K_{s}=m R^{2}=(0.95792 \mathrm{g})(9.507)^{2}=86.58 \ \mathrm{g} \approx 86.6 \ \mathrm{g} \nonumber$
To obtain a percent relative standard deviation of 5.0%, individual samples need to have a mass of at least
$m=\frac{K_{s}}{R^{2}}=\frac{86.58 \ \mathrm{g}}{(5.0)^{2}}=3.5 \ \mathrm{g} \nonumber$
To reduce the sample’s mass from 3.5 g to 1 g, we must change the mass by a factor of
$\frac{3.5 \ \mathrm{g}}{1 \ \mathrm{g}}=3.5 \times \nonumber$
If we assume that the sample’s particles are spherical, then we must reduce a particle’s radius by a factor of
\begin{aligned} r^{3} &=3.5 \times \ r &=1.5 \times \end{aligned} \nonumber
How Many Samples to Collect
In the previous section we considered how much sample we need to minimize the standard deviation due to sampling. Another important consideration is the number of samples to collect. If the results from our analysis of the samples are normally distributed, then the confidence interval for the sampling error is
$\mu=\overline{X} \pm \frac{t s_{samp}}{\sqrt{n_{samp}}} \label{7.4}$
where nsamp is the number of samples and ssamp is the standard deviation for sampling. Rearranging Equation \ref{7.4} and substituting e for the quantity $\overline{X} - \mu$, gives the number of samples as
$n_{samp}=\frac{t^{2} s_{samp}^{2}}{e^{2}} \label{7.5}$
Because the value of t depends on nsamp, the solution to Equation \ref{7.5} is found iteratively.
When we use Equation \ref{7.5}, we must express the standard deviation for sampling, ssamp, and the error, e, in the same way. If ssamp is reported as a percent relative standard deviation, then the error, e, is reported as a percent relative error. When you use Equation \ref{7.5}, be sure to check that you are expressing ssamp and e in the same way.
Example 7.2.6
In Example 7.2.5 we determined that we need 1.5-g samples to establish an ssamp of ±2.0% for the amount of inorganic ash in cereal. How many 1.5-g samples do we need to collect to obtain a percent relative sampling error of ±0.80% at the 95% confidence level?
Solution
Because the value of t depends on the number of samples—a result we have yet to calculate—we begin by letting nsamp = $\infty$ and using t(0.05, $\infty$) for t. From Appendix 4, the value for t(0.05, $\infty$) is 1.960. Substituting known values into Equation \ref{7.5} gives the number of samples as
$n_{samp}=\frac{(1.960)^{2}(2.0)^{2}}{(0.80)^{2}}=24.0 \approx 24 \nonumber$
Letting nsamp = 24, the value of t(0.05, 23) from Appendix 4 is 2.073. Recalculating nsamp gives
$n_{samp}=\frac{(2.073)^{2}(2.0)^{2}}{(0.80)^{2}}=26.9 \approx 27 \nonumber$
When nsamp = 27, the value of t(0.05, 26) from Appendix 4 is 2.060. Recalculating nsamp gives
$n_{samp}=\frac{(2.060)^{2}(2.0)^{2}}{(0.80)^{2}}=26.52 \approx 27 \nonumber$
Because two successive calculations give the same value for nsamp, we have an iterative solution to the problem. We need 27 samples to achieve a percent relative sampling error of ±0.80% at the 95% confidence level.
Exercise 7.2.2
Assuming that the percent relative standard deviation for sampling in the determination of olaquindox in medicated feed is 5.0% (see Exercise 7.2.1 ), how many samples do we need to analyze to obtain a percent relative sampling error of ±2.5% at $\alpha$ = 0.05?
Answer
Because the value of t depends on the number of samples—a result we have yet to calculate—we begin by letting nsamp = $\infty$ and using t(0.05, $\infty$) for the value of t. From Appendix 4, the value for t(0.05, $\infty$) is 1.960. Our first estimate for nsamp is
$n_{samp}=\frac{t^{2} s_{s a m p}^{2}}{e^{2}} = \frac{(1.96)^{2}(5.0)^{2}}{(2.5)^{2}}=15.4 \approx 15 \nonumber$
Letting nsamp = 15, the value of t(0.05,14) from Appendix 4 is 2.145. Recalculating nsamp gives
$n_{samp}=\frac{t^{2} s_{samp}^{2}}{e^{2}}=\frac{(2.145)^{2}(5.0)^{2}}{(2.5)^{2}}=18.4 \approx 18 \nonumber$
Letting nsamp = 18, the value of t(0.05,17) from Appendix 4 is 2.103. Recalculating nsamp gives
$n_{samp}=\frac{t^{2} s_{samp}^{2}}{e^{2}}=\frac{(2.103)^{2}(5.0)^{2}}{(2.5)^{2}}=17.7 \approx 18 \nonumber$
Because two successive calculations give the same value for nsamp, we need 18 samples to achieve a sampling error of ±2.5% at the 95% confidence interval.
Equation \ref{7.5} provides an estimate for the smallest number of samples that will produce the desired sampling error. The actual sampling error may be substantially larger if ssamp for the samples we collect during the subsequent analysis is greater than ssamp used to calculate nsamp. This is not an uncommon problem. For a target population with a relative sampling variance of 50 and a desired relative sampling error of ±5%, Equation \ref{7.5} predicts that 10 samples are sufficient. In a simulation using 1000 samples of size 10, however, only 57% of the trials resulted in a sampling error of less than ±5% [Blackwood, L. G. Environ. Sci. Technol. 1991, 25, 1366–1367]. Increasing the number of samples to 17 was sufficient to ensure that the desired sampling error was achieved 95% of the time.
For an interesting discussion of why the number of samples is important, see Kaplan, D.; Lacetera, N.; Kaplan, C. “Sample Size and Precision in NIH Peer Review,” Plos One, 2008, 3(7), 1–3. When reviewing grants, individual reviewers report a score between 1.0 and 5.0 (two significant figures). NIH reports the average score to three significant figures, implying that a difference of 0.01 is significant. If the individual scores have a standard deviation of 0.1, then a difference of 0.01 is significant at $\alpha = 0.05$ only if there are 384 reviews. The authors conclude that NIH review panels are too small to provide a statistically meaningful separation between proposals receiving similar scores.
Minimizing the Overall Variance
A final consideration when we develop a sampling plan is how we can minimize the overall variance for the analysis. Equation 7.1.2 shows that the overall variance is a function of the variance due to the method, $s_{meth}^2$, and the variance due to sampling, $s_{samp}^2$. As we learned earlier, we can improve the sampling variance by collecting more samples of the proper size. Increasing the number of times we analyze each sample improves the method’s variance. If $s_{samp}^2$ is significantly greater than $s_{meth}^2$, we can ignore the method’s contribution to the overall variance and use Equation \ref{7.5} to estimate the number of samples to analyze. Analyzing any sample more than once will not improve the overall variance, because the method’s variance is insignificant.
If $s_{meth}^2$ is significantly greater than $s_{samp}^2$, then we need to collect and analyze only one sample. The number of replicate analyses, nrep, we need to minimize the error due to the method is given by an equation similar to Equation \ref{7.5}.
$n_{rep}=\frac{t^{2} s_{m e t h}^{2}}{e^{2}} \nonumber$
Unfortunately, the simple situations described above often are the exception. For many analyses, both the sampling variance and the method variance are significant, and both multiple samples and replicate analyses of each sample are necessary. The overall error in this case is
$e=t \sqrt{\frac{s_{samp}^{2}}{n_{samp}} + \frac{s_{meth}^{2}}{n_{sam p} n_{rep}}} \label{7.6}$
Equation \ref{7.6} does not have a unique solution as different combinations of nsamp and nrep give the same overall error. How many samples we collect and how many times we analyze each sample is determined by other concerns, such as the cost of collecting and analyzing samples, and the amount of available sample.
Example 7.2.7
An analytical method has a relative sampling variance of 0.40% and a relative method variance of 0.070%. Evaluate the percent relative error ($\alpha = 0.05$) if you collect 5 samples and analyze each twice, and if you collect 2 samples and analyze each 5 times.
Solution
Both sampling strategies require a total of 10 analyses. From Appendix 4 we find that the value of t(0.05, 9) is 2.262. Using Equation \ref{7.6}, the relative error for the first sampling strategy is
$e=2.262 \sqrt{\frac{0.40}{5}+\frac{0.070}{5 \times 2}}=0.67 \% \nonumber$
and that for the second sampling strategy is
$e=2.262 \sqrt{\frac{0.40}{2}+\frac{0.070}{2 \times 5}}=1.0 \% \nonumber$
Because the method variance is smaller than the sampling variance, we obtain a smaller relative error if we collect more samples and analyze each sample fewer times.
Exercise 7.2.3
An analytical method has a relative sampling variance of 0.10% and a relative method variance of 0.20%. The cost of collecting a sample is $20 and the cost of analyzing a sample is$50. Propose a sampling strategy that provides a maximum relative error of ±0.50% ($\alpha = 0.05$) and a maximum cost of $700. Answer If we collect a single sample (cost$20), then we can analyze that sample 13 times (cost $650) and stay within our budget. For this scenario, the percent relative error is $e=t \sqrt{\frac{s_{samp}^{2}}{n_{samp}} + \frac{s_{meth}^{2}}{n_{sam p} n_{rep}}} = 2.179 \sqrt{\frac{0.10}{1}+\frac{0.20}{1 \times 13}}=0.74 \% \nonumber$ where t(0.05, 12) is 2.179. Because this percent relative error is larger than ±0.50%, this is not a suitable sampling strategy. Next, we try two samples (cost$40), analyzing each six times (cost $600). For this scenario, the percent relative error is $e=t \sqrt{\frac{s_{samp}^{2}}{n_{samp}} + \frac{s_{meth}^{2}}{n_{sam p} n_{rep}}} = 2.2035 \sqrt{\frac{0.10}{2}+\frac{0.20}{2 \times 6}}=0.57 \% \nonumber$ where t(0.05, 11) is 2.2035. Because this percent relative error is larger than ±0.50%, this also is not a suitable sampling strategy. Next we try three samples (cost$60), analyzing each four times (cost $600). For this scenario, the percent relative error is $e=t \sqrt{\frac{s_{samp}^{2}}{n_{samp}} + \frac{s_{meth}^{2}}{n_{sam p} n_{rep}}} = 2.2035 \sqrt{\frac{0.10}{3}+\frac{0.20}{3 \times 4}}=0.49 \% \nonumber$ where t(0.05, 11) is 2.2035. Because both the total cost ($660) and the percent relative error meet our requirements, this is a suitable sampling strategy.
There are other suitable sampling strategies that meet both goals. The strategy that requires the least expense is to collect eight samples, analyzing each once for a total cost of $560 and a percent relative error of ±0.46%. Collecting 10 samples and analyzing each one time, gives a percent relative error of ±0.39% at a cost of$700. | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/07%3A_Obtaining_and_Preparing_Samples_for_Analysis/7.02%3A_Designing_a_Sampling_Plan.txt |
Implementing a sampling plan usually involves three steps: physically removing the sample from its target population, preserving the sample, and preparing the sample for analysis. Except for in situ sampling, we analyze a sample after we have removed it from its target population. Because sampling exposes the target population to potential contamination, our sampling device must be inert and clean.
Once we remove a sample from its target population, there is a danger that it will undergo a chemical or physical change before we can complete its analysis. This is a serious problem because the sample’s properties will no longer e representative of the target population. To prevent this problem, we often preserve samples before we transport them to the laboratory for analysis. Even when we analyze a sample in the field, preservation may still be necessary.
The initial sample is called the primary or gross sample, and it may be a single increment drawn from the target population or a composite of several increments. In many cases we cannot analyze the gross sample without first preparing the sample for analyze by reducing the sample’s particle size, by converting the sample into a more readily analyzable form, or by improving its homogeneity.
Although you may never work with the specific samples highlighted in this section, the case studies presented here may help you in envisioning potential problems associated with your samples.
Solutions
There are many good examples of solution samples: commercial solvents; beverages, such as milk or fruit juice; natural waters, including lakes, streams, seawater, and rain; bodily fluids, such as blood and urine; and, suspensions, such as those found in many oral medications. Let’s use the sampling of natural waters and wastewaters as a case study in how to sample a solution.
Sample Collection
The chemical composition of a surface water—such as a stream, river, lake, estuary, or ocean—is influenced by flow rate and depth. Rapidly flowing shallow streams and rivers, and shallow (<5 m) lakes usually are well mixed and show little stratification with depth. To collect a grab sample we submerge a capped bottle below the surface, remove the cap and allow the bottle to fill completely, and replace the cap. Collecting a sample this way avoids the air–water interface, which may be enriched with heavy metals or contaminated with oil [Duce, R. A.; Quinn, J. G. Olney, C. E.; Piotrowicz, S. R.; Ray, S. J.; Wade, T. L. Science 1972, 176, 161–163].
Slowly moving streams and rivers, lakes deeper than five meters, estuaries, and oceans may show substantial stratification with depth. Grab samples from near the surface are collected as described above, and samples at greater depths are collected using a sample bottle lowered to the desired depth (Figure 7.3.1 ).
Wells for sampling groundwater are purged before we collect samples because the chemical composition of water in a well-casing may differ significantly from that of the groundwater. These differences may result from contaminants introduced while drilling the well or by a change in the groundwater’s redox potential following its exposure to atmospheric oxygen. In general, a well is purged by pumping out a volume of water equivalent to several well-casing volumes or by pumping until the water’s temperature, pH, or specific conductance is constant. A municipal water supply, such as a residence or a business, is purged before sampling because the chemical composition of water standing in a pipe may differ significantly from the treated water supply. Samples are collected at faucets after flushing the pipes for 2-3 minutes.
Samples from municipal wastewater treatment plants and industrial discharges often are collected as a 24-hour composite. An automatic sampler periodically removes an individual grab sample, adding it to those collected previously. The volume of each sample and the frequency of sampling may be constant, or may vary in response to changes in flow rate.
Sample containers for collecting natural waters and wastewaters are made from glass or plastic. Kimax and Pyrex brand borosilicate glass have the advantage of being easy to sterilize, easy to clean, and inert to all solutions except those that are strongly alkaline. The disadvantages of glass containers are cost, weight, and the ease of breakage. Plastic containers are made from a variety of polymers, including polyethylene, polypropylene, polycarbonate, polyvinyl chloride, and Teflon. Plastic containers are light-weight, durable, and, except for those manufactured from Teflon, inexpensive. In most cases glass or plastic bottles are used interchangeably, although polyethylene bottles generally are preferred because of their lower cost. Glass containers are always used when collecting samples for the analysis of pesticides, oil and grease, and organics because these species often interact with plastic surfaces. Because glass surfaces easily adsorb metal ions, plastic bottles are preferred when collecting samples for the analysis of trace metals.
In most cases the sample bottle has a wide mouth, which makes it easy to fill and to remove the sample. A narrow-mouth sample bottle is used if exposing the sample to the container’s cap or to the outside environment is a problem. Unless exposure to plastic is a problem, caps for sample bottles are manufactured from polyethylene. When polyethylene must be avoided, the container’s cap includes an inert interior liner of neoprene or Teflon.
Sample Preservation and Preparation
Here our concern is only with the need to prepare the gross sample by converting it into a form suitable for analysis. Some analytical methods require additional sample preparation steps, such as concentrating or diluting the analyte, or adjusting the analyte’s chemical form. We will consider these forms of sample preparation in later chapters that focus on specific analytical methods.
After removing a sample from its target population, its chemical composition may change as a result of chemical, biological, or physical processes. To prevent a change in composition, samples are preserved by controlling the sample’s pH and temperature, by limiting its exposure to light or to the atmosphere, or by adding a chemical preservative. After preserving a sample, it is safely stored for later analysis. The maximum holding time between preservation and analysis depends on the analyte’s stability and the effectiveness of sample preservation. Table 7.3.1 summarizes preservation methods and maximum holding times for several analytes of importance in the analysis of natural waters and wastewaters.
Table 7.3.1 . Preservation Methods and Maximum Holding Times for Selected Analytes in Natural Waters and Wastewaters
analyte preservation method maximum holding time
ammonia
cool to 4oC; add H2SO4 to pH < 2
28 days
chloride none required 28 days
metals: Cr(VI) cool to 4oC 24 hours
metals: Hg
HNO3 to pH < 2
28 days
metals: all others
HNO3 to pH < 2
6 months
nitrate
none required
48 hours
organochlorine pesticides
1 mL of 10 mg/mL HgCl2 or
immediate extraction with a suitable non-aqueous solvent
7 days without extraction;
40 days with extraction
pH
none required
analyze immediately
Other than adding a preservative, solution samples generally do not need additional preparation before analysis. This is the case for samples of natural waters and wastewaters. Solution samples with particularly complex matricies—blood and milk are two common examples—may need addi- tional processing to separate analytes from interferents, a topic covered later in this chapter.
Gases
Typical examples of gaseous samples include automobile exhaust, emissions from industrial smokestacks, atmospheric gases, and compressed gases. Also included in this category are aerosol particulates—the fine solid particles and liquid droplets that form smoke and smog. Let’s use the sampling of urban air as a case study in how to sample a gas.
Sample Collection
One approach for collecting a sample of urban air is to fill a stainless steel canister or a Tedlar/Teflon bag. A pump pulls the air into the container and, after purging, the container is sealed. This method has the advantage of being simple and of collecting a representative sample. Disadvantages include the tendency for some analytes to adsorb to the container’s walls, the presence of analytes at concentrations too low to detect with suitable accuracy and precision, and the presence of reactive analytes, such as ozone and nitrogen oxides, that may react with the container or that may otherwise alter the sample’s chemical composition during storage. When using a stainless steel canister, cryogenic cooling, which changes the sample from a gaseous state to a liquid state, may limit some of these disadvantages.
Most urban air samples are collected by filtration or by using a trap that contains a solid sorbent. Solid sorbents are used for volatile gases (a vapor pressure more than 10–6 atm) and for semi-volatile gases (a vapor pressure between 10–6 atm and 10–12 atm). Filtration is used to collect aerosol particulates. Trapping and filtering allow for sampling larger volumes of gas—an important concern for an analyte with a small concentration—and stabilizes the sample between its collection and its analysis.
In solid sorbent sampling, a pump pulls the urban air through a canister packed with sorbent particles. Typically 2–100 L of air are sampled when collecting a volatile compound and 2–500 m3 when collecting a semi-volatile gas. A variety of inorganic, organic polymer, and carbon sorbents have been used. Inorganic sorbents, such as silica gel, alumina, magnesium aluminum silicate, and molecular sieves, are efficient collectors for polar compounds. Their efficiency at absorbing water, however, limits their capacity for many organic analytes.
1 m3 is equivalent to 103 L.
Organic polymeric sorbents include polymeric resins of 2,4-diphenyl-p-phenylene oxide or styrene-divinylbenzene for volatile compounds, and polyurethane foam for semi-volatile compounds. These materials have a low affinity for water and are efficient for sampling all but the most highly volatile organic compounds and some lower molecular weight alcohols and ketones. Carbon sorbents are superior to organic polymer resins, which makes them useful for highly volatile organic compounds that will not absorb onto polymeric resins, although removing the compounds may be difficult.
Non-volatile compounds normally are present either as solid particulates or are bound to solid particulates. Samples are collected by pulling a large volume of urban air through a filtering unit and collecting the particulates on glass fiber filters.
The short term exposure of humans, animals, and plants to atmospheric pollutants is more severe than that for pollutants in other matrices. Because the composition of atmospheric gases can vary significantly over a time, the continuous monitoring of atmospheric gases such as O3, CO, SO2, NH3, H2O2, and NO2 by in situ sampling is important [Tanner, R. L. in Keith, L. H., ed. Principles of Environmental Sampling, American Chemical Society: Washington, D. C., 1988, 275–286].
Sample Preservation
After collecting a gross sample of urban air, generally there is little need for sample preservation or preparation. The chemical composition of a gas sample usually is stable when it is collected using a solid sorbent, a filter, or by cryogenic cooling. When using a solid sorbent, gaseous compounds are released for analysis by thermal desorption or by extracting with a suitable solvent. If the sorbent is selective for a single analyte, the increase in the sorbent’s mass is used to determine the amount of analyte in the sample.
Solids
Typical examples of solid samples include large particulates, such as those found in ores; smaller particulates, such as soils and sediments; tablets, pellets, and capsules used for dispensing pharmaceutical products and animal feeds; sheet materials, such as polymers and rolled metals; and tissue samples from biological specimens. Solids usually are heterogeneous and we must collect samples carefully if they are to be representative of the target population. Let’s use the sampling of sediments, soils, and ores as a case study in how to sample solids.
Sample Collection
Sediments from the bottom of streams, rivers, lakes, estuaries, and oceans are collected with a bottom grab sampler or with a corer. A bottom grab sampler (Figure 7.3.2 ) is equipped with a pair of jaws that close when they contact the sediment, scooping up sediment in the process. Its principal advantages are ease of use and the ability to collect a large sample. Disadvantages include the tendency to lose finer grain sediment particles as water flows out of the sampler, and the loss of spatial information—both laterally and with depth—due to mixing of the sample.
An alternative method for collecting sediments is the cylindrical coring device shown in Figure 7.3.3 ). The corer is dropped into the sediment, collecting a column of sediment and the water in contact with the sediment. With the possible exception of sediment at the surface, which may experience mixing, samples collected with a corer maintain their vertical profile, which preserves information about how the sediment’s composition changes with depth.
Collecting soil samples at depths of up to 30 cm is accomplished with a scoop or a shovel, although the sampling variance generally is high. A better tool for collecting soil samples near the surface is a soil punch, which is a thin-walled steel tube that retains a core sample after it is pushed into the soil and removed. Soil samples from depths greater than 30 cm are collected by digging a trench and collecting lateral samples with a soil punch. Alternatively, an auger is used to drill a hole to the desired depth and the sample collected with a soil punch.
For particulate materials, particle size often determines the sampling method. Larger particulate solids, such as ores, are sampled using a riffle (Figure 7.3.4 ), which is a trough with an even number of compartments. Because adjoining compartments empty onto opposite sides of the riffle, dumping a gross sample into the riffle divides it in half. By repeatedly passing half of the separated material back through the riffle, a sample of the desired size is collected.
A sample thief (Figure 7.3.5 ) is used for sampling smaller particulate materials, such as powders. A typical sample thief consists of two tubes that are nestled together. Each tube has one or more slots aligned down the length of the sample thief. Before inserting the sample thief into the material being sampled, the slots are closed by rotating the inner tube. When the sample thief is in place, rotating the inner tube opens the slots, which fill with individual samples. The inner tube is then rotated to the closed position and the sample thief withdrawn.
Sample Preservation
Without preservation, a solid sample may undergo a change in composition due to the loss of volatile material, biodegradation, or chemical reactivity (particularly redox reactions). Storing samples at lower temperatures makes them less prone to biodegradation and to the loss of volatile material, but fracturing of solids and phase separations may present problems. To minimize the loss of volatile compounds, the sample container is filled completely, eliminating a headspace where gases collect. Samples that have not been exposed to O2 particularly are susceptible to oxidation reactions. For example, samples of anaerobic sediments must be prevented from coming into contact with air.
Sample Preparation
Unlike gases and liquids, which generally require little sample preparation, a solid sample usually needs some processing before analysis. There are two reasons for this. First, as discussed in Chapter 7.2, the standard deviation for sampling, ssamp, is a function of the number of particles in the sample, not the combined mass of the particles. For a heterogeneous material that consists of large particulates, the gross sample may be too large to analyze. For example, a Ni-bearing ore with an average particle size of 5 mm may require a sample that weighs one ton to obtain a reasonable ssamp. Reducing the sample’s average particle size allows us to collect the same number of particles with a smaller, more manageable mass. Second, many analytical techniques require that the analyte be in solution.
Reducing Particle Size
A reduction in particle size is accomplished by crushing and grinding the gross sample. The resulting particulates are then thoroughly mixed and divided into subsamples of smaller mass. This process seldom occurs in a single step. Instead, subsamples are cycled through the process several times until a final laboratory sample is obtained.
Crushing and grinding uses mechanical force to break larger particles into smaller particles. A variety of tools are used depending on the particle’s size and hardness. Large particles are crushed using jaw crushers that can reduce particles to diameters of a few millimeters. Ball mills, disk mills, and mortars and pestles are used to further reduce particle size.
A significant change in the gross sample’s composition may occur during crushing and grinding. Decreasing particle size increases the available surface area, which increases the risk of losing volatile components. This problem is made worse by the frictional heat that accompanies crushing and grinding. Increasing the surface area also exposes interior portions of the sample to the atmosphere where oxidation may alter the gross sample’s composition. Other problems include contamination from the materials used to crush and grind the sample, and differences in the ease with which particles are reduced in size. For example, softer particles are easier to reduce in size and may be lost as dust before the remaining sample is processed. This is a particular problem if the analyte’s distribution between different types of particles is not uniform.
The gross sample is reduced to a uniform particle size by intermittently passing it through a sieve. Those particles not passing through the sieve receive additional processing until the entire sample is of uniform size. The resulting material is mixed thoroughly to ensure homogeneity and a subsample obtained with a riffle, or by coning and quartering. As shown in Figure 7.3.6 , the gross sample is piled into a cone, flattened, and divided into four quarters. After discarding two diagonally opposed quarters, the remaining material is cycled through the process of coning and quartering until a suitable laboratory sample remains.
Bringing Solid Samples Into Solution
If you are fortunate, your sample will dissolve easily in a suitable solvent, requiring no more effort than gently swirling and heating. Distilled water usually is the solvent of choice for inorganic salts, but organic solvents, such as methanol, chloroform, and toluene, are useful for organic materials.
When a sample is difficult to dissolve, the next step is to try digesting it with an acid or a base. Table 7.3.2 lists several common acids and bases, and summarizes their use. Digestions are carried out in an open container, usually a beaker, using a hot-plate as a source of heat. The main advantage of an open-vessel digestion is cost because it requires no special equipment. Volatile reaction products, however, are lost, which results in a determinate error if they include the analyte.
Table 7.3.2 . Acids and Bases Used for Digesting Samples
solution uses and properties
HCl (37% w/w)
• dissolves metals more easily reduced than H2 (Eo < 0)
• dissolves insoluble carbonate, sulfides, phosphates, fluorides, sulfates, and many oxides
HNO3 (70% w/w)
• strong oxidizing agent
• dissolves most common metals except Al, Au, Pt, and Cr
• decomposes organics and biological samples (wet ashing)
H2SO4 (98% w/w)
• dissolves many metals and alloys
• decomposes organics by oxidation and dehydration
HF (50% w/w)
• dissolves silicates by forming volatile SiF4
HClO4 (70% w/w)
• hot, concentrated solutions are strong oxidizing agents
• dissolves many metals and alloys
• decomposes organics (Caution: reactions with organics often are explosive; use only in a specially equipped hood with a blast shield and after prior decomposition with HNO3)
HCl:HNO3 (3:1 v/v)
• also known as aqua regia
• dissolves Au and Pt
NaOH
• dissolves Al and amphoteric oxides of Sn, Pb, Zn, and Cr
Many digestions now are carried out in a closed container using microwave radiation as the source of energy. Vessels for microwave digestion are manufactured using Teflon (or some other fluoropolymer) or fused silica. Both materials are thermally stable, chemically resistant, transparent to microwave radiation, and capable of withstanding elevated pressures. A typical microwave digestion vessel, as shown in Figure 7.3.7 , consists of an insulated vessel body and a cap with a pressure relief valve. The vessels are placed in a microwave oven (a typical oven can accommodate 6–14 vessels) and microwave energy is controlled by monitoring the temperature or pressure within one of the vessels.
Figure 7.3.7 . Microwave digestion unit: on the left is a view of the unit’s interior showing the carousel that holds the digestion vessels; on the right is a close-up of a Teflon digestion vessel, which is encased in a thermal sleeve. The pressure relief value, which is part of the vessel’s blue cap, contains a membrane that ruptures if the internal pressure becomes too high.
Inorganic samples that resist decomposition by digesting with acids or bases often are brought into solution by fusing with a large excess of an alkali metal salt, called a flux. After mixing the sample and the flux in a crucible, they are heated to a molten state and allowed to cool slowly to room temperature. The resulting melt usually dissolves readily in distilled water or dilute acid. Table 7.3.3 summarizes several common fluxes and their uses. Fusion works when other methods of decomposition do not because of the high temperature and the flux’s high concentration in the molten liquid. Disadvantages include contamination from the flux and the crucible, and the loss of volatile materials.
Table 7.3.3 . Common Fluxes for Decomposing Inorganic Samples
flux melting temperature (oC) crucible typical samples
Na2CO3 85 Pt silicates, oxides, phosphate, sulfides
Li2B4O7 930 Pt, graphite aluminosilicates, carbontes
LiBO2 845 Pt, graphite aluminosilicates, carbonates
NaOH 318 Au, Ag silicates, silicon carbide
KOH 380 Au, Ag silicates, silicon carbide
Na2O2 Ni silicates, chromium steels, Pt alloys
K2S2O7 300 Ni, porcelain oxides
B2O3 577 Pt silicates, oxides
Finally, we can decompose organic materials by dry ashing. In this method the sample is placed in a suitable crucible and heated over a flame or in a furnace. The carbon present in the sample oxidizes to CO2, and hydrogen, sulfur, and nitrogen are volatilized as H2O, SO2, and N2. These gases can be trapped and weighed to determine their concentration in the organic material. Often the goal of dry ashing is to remove the organic material, leaving behind an inorganic residue, or ash, that can be further analyzed. | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/07%3A_Obtaining_and_Preparing_Samples_for_Analysis/7.03%3A_Implementing_the_Sampling_Plan.txt |
When an analytical method is selective for the analyte, analyzing a sample is a relatively simple task. For example, a quantitative analysis for glucose in honey is relatively easy to accomplish if the method is selective for glucose, even in the presence of other reducing sugars, such as fructose. Unfortunately, few analytical methods are selective toward a single species.
In the absence of an interferent, the relationship between the sample’s signal, Ssamp, and the analyte’s concentration, CA, is
$S_{samp}=k_{A} C_{A} \label{7.1}$
where kA is the analyte’s sensitivity.
In Equation \ref{7.1}, and the equations that follow, you can replace the analyte’s concentration, CA, with the moles of analyte, nA, when working with methods, such as gravimetry, that respond to the absolute amount of analyte in a sample. In this case the interferent also is expressed in terms of moles.
If an interferent, is present, then Equation \ref{7.1} becomes
$S_{samp}=k_{A} C_{A}+k_{I} C_{I} \label{7.2}$
where kI and CI are, respectively, the interferent’s sensitivity and concentration. A method’s selectivity for the analyte is determined by the relative difference in its sensitivity toward the analyte and the interferent. If kA is greater than kI, then the method is more selective for the analyte. The method is more selective for the interferent if kI is greater than kA.
Even if a method is more selective for an interferent, we can use it to determine CA if the interferent’s contribution to Ssamp is insignificant. The selectivity coefficient, KA,I, which we introduced in Chapter 3, provides a way to characterize a method’s selectivity.
$K_{A, I}=\frac{k_{I}}{k_{A}} \label{7.3}$
Solving Equation \ref{7.3} for kI, substituting into Equation \ref{7.2}, and simplifying, gives
$S_{samp}=k_{A}\left(C_{A}+K_{A, I} \times C_{I}\right) \label{7.4}$
An interferent, therefore, does not pose a problem as long as the product of its concentration and its selectivity coefficient is significantly smaller than the analyte’s concentration.
$K_{A, I} \times C_{I}<<C_{A} \nonumber$
If we cannot ignore an interferent’s contribution to the signal, then we must begin our analysis by separating the analyte and the interferent.
7.05: General Theory of Separation Effiiciency
The goal of an analytical separation is to remove either the analyte or the interferent from the sample’s matrix. To achieve this separation we must identify at least one significant difference between the analyte’s and the interferent’s chemical or physical properties. A significant difference in properties, however, is not sufficient to effect a separation if the conditions that favor the extraction of interferent from the sample also removes a small amount of analyte.
Two factors limit a separation’s efficiency: failing to recover all the analyte and failing to remove all the interferent. We define the analyte’s recovery, RA, as
$R_{A}=\frac{C_{A}}{\left(C_{A}\right)_{\mathrm{o}}} \label{7.1}$
where CA is the concentration of analyte that remains after the separation, and (CA)o is the analyte’s initial concentration. A recovery of 1.00 means that no analyte is lost during the separation. The interferent’s recovery, RI, is defined in the same manner
$R_{I}=\frac{C_{I}}{\left(C_{I}\right)_{o}} \label{7.2}$
where CI is the concentration of interferent that remains after the separation, and (CI)o is the interferent’s initial concentration. We define the extent of the separation using a separation factor, SI,A [(a) Sandell, E. B. Colorimetric Determination of Trace Metals, Interscience Publishers: New York, 1950, pp. 19–20; (b) Sandell, E. B. Anal. Chem. 1968, 40, 834–835].
$S_{I, A}=\frac{R_{I}}{R_{A}} \label{7.3}$
In general, an SI,A of approximately 10–7 is needed for the quantitative analysis of a trace analyte in the presence of a macro interferent, and 10–3 when the analyte and interferent are present in approximately equal amounts.
The meaning of trace and macro, as well as other terms for describing the concentrations of analytes and interferents, is presented in Chapter 2.
Example 7.5.1
An analytical method for determining Cu in an industrial plating bath gives poor results in the presence of Zn. To evaluate a method for separating the analyte from the interferent, samples with known concentrations of Cu or Zn were prepared and analyzed. When a sample of 128.6 ppm Cu was taken through the separation, the concentration of Cu that remained was 127.2 ppm. Taking a 134.9 ppm solution of Zn through the separation left behind a concentration of 4.3 ppm Zn. Calculate the recoveries for Cu and Zn, and the separation factor.
Solution
Using Equation \ref{7.1} and Equation \ref{7.2}, the recoveries for the analyte and interferent are
$R_{\mathrm{Cu}}=\frac{127.2 \ \mathrm{ppm}}{128.6 \ \mathrm{ppm}}=0.9891 \text { or } 98.91 \% \nonumber$
$R_{\mathrm{zn}}=\frac{4.3 \ \mathrm{ppm}}{134.9 \ \mathrm{ppm}}=0.032 \text { or } 3.2 \% \nonumber$
and the separation factor is
$S_{\mathrm{Zn}, \mathrm{Cu}}=\frac{R_{\mathrm{Zn}}}{R_{\mathrm{Cu}}}=\frac{0.032}{0.9891}=0.032 \nonumber$
Recoveries and separation factors are useful tools for evaluating a separation’s potential effectiveness; they do not, however, give a direct indication of the error that results from failing to remove all the interferent or from failing to completely recover the analyte. The relative error due to the separation, E, is
$E=\frac{S_{s a m p}-S_{s a m p}^*}{S_{samp}} \label{7.4}$
where $S_{samp}^*$ is the sample’s signal for an ideal separation in which we completely recover the analyte.
$S_{samp}^{*}=k_{A}\left(C_{A}\right)_{\mathrm{o}} \label{7.5}$
Substituting equation 7.4.4 and Equation \ref{7.5} into Equation \ref{7.4}, and rearranging
$E=\frac{k_{A}\left(C_{A}+K_{A, l} \times C_{I}\right)-k_{A}\left(C_{A}\right)_{o}}{k_{A}\left(C_{A}\right)_{o}} \nonumber$
$E=\frac{C_{A}+K_{A, I} \times C_{I}-\left(C_{A}\right)_{\circ}}{\left(C_{A}\right)_{\circ}} \nonumber$
$E=\frac{C_{A}}{\left(C_{A}\right)_{\mathrm{o}}}-\frac{\left(C_{A}\right)_{o}}{\left(C_{A}\right)_{o}}+\frac{K_{A, I} \times C_{I}}{\left(C_{A}\right)_{o}} \nonumber$
leaves us with
$E=\left(R_{A}-1\right)+\frac{K_{A, I} \times C_{I}}{\left(C_{A}\right)_{o}} \label{7.6}$
A more useful equation is obtained by solving Equation \ref{7.2} for CI and substituting into Equation \ref{7.6}.
$E=\left(R_{A}-1\right)+\frac{K_{A, I} \times\left(C_{I}\right)_{o}}{\left(C_{A}\right)_{o}} \times R_{I} \label{7.7}$
The first term of Equation \ref{7.7} accounts for the analyte’s incomplete recovery and the second term accounts for a failure to remove all the interferent.
Example 7.5.2
Following the separation outlined in Example 7.5.1 , an analysis is carried out to determine the concentration of Cu in an industrial plating bath. Analysis of standard solutions that contain either Cu or Zn give the following linear calibrations.
$S_{\mathrm{Cu}}=1250 \ \mathrm{ppm}^{-1} \times C_{\mathrm{Cu}} \text { and } S_{\mathrm{Zn}}=2310 \ \mathrm{ppm}^{-1} \times C_{\mathrm{Zn}} \nonumber$
(a) What is the relative error if we analyze a sample without removing the Zn? Assume the initial concentration ratio, Cu:Zn, is 7:1. (b) What is the relative error if we first complete the separation with the recoveries determined in Example 7.5.1 ? (c) What is the maximum acceptable recovery for Zn if the recovery for Cu is 1.00 and if the error due to the separation must be no greater than 0.10%?
Solution
(a) If we complete the analysis without separating Cu and Zn, then RCu and RZn are exactly 1 and Equation \ref{7.7} simplifies to
$E=\frac{K_{\mathrm{Cu}, \mathrm{Zn}} \times\left(C_{\mathrm{Zn}}\right)_{\mathrm{o}}}{\left(C_{\mathrm{Cu}}\right)_{\mathrm{o}}} \nonumber$
Using equation 7.4.3, we find that the selectivity coefficient is
$K_{\mathrm{Cu}, \mathrm{Zn}}=\frac{k_{\mathrm{Zn}}}{k_{\mathrm{Cu}}}=\frac{2310 \ \mathrm{ppm}^{-1}}{1250 \ \mathrm{ppm}^{-1}}=1.85 \nonumber$
Given the initial concentration ratio of 7:1 for Cu and Zn, the relative error without the separation is
$E=\frac{1.85 \times 1}{7}=0.264 \text { or } 26.4 \% \nonumber$
(b) To calculate the relative error we substitute the recoveries from Example 7.5.1 into Equation \ref{7.7}, obtaining
$E=(0.9891-1)+\frac{1.85 \times 1}{7} \times 0.032= -0.0109+0.085=-0.0024 \nonumber$
or –0.24%. Note that the negative determinate error from failing to recover all the analyte is offset partially by the positive determinate error from failing to remove all the interferent.
(c) To determine the maximum recovery for Zn, we make appropriate substitutions into Equation \ref{7.7}
$E=0.0010=(1-1)+\frac{1.85 \times 1}{7} \times R_{\mathrm{Zn}} \nonumber$
and solve for RZn, obtaining a recovery of 0.0038, or 0.38%. Thus, we must remove at least
$100.00 \%-0.38 \%=99.62 \% \nonumber$
of the Zn to obtain an error of 0.10% when RCu is exactly 1. | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/07%3A_Obtaining_and_Preparing_Samples_for_Analysis/7.04%3A_Separating_the_Analyte_From_Interferents.txt |
We can separate an analyte and an interferent if there is a significant difference in at least one of their chemical or physical properties. Table 7.6.1 provides a partial list of separation techniques, organized by the chemical or physical property affecting the separation.
Table 7.6.1 . Classification of Separation Techniques
basis of separation separation technique(s)
size filtration; dialaysis; size-exclusion chromatography
mass or density centrifugation
complex formation masking
change in physical state distillation; sublimation; recrystalization
change in chemical state precipitation; electrodeposition; volatilization
partitioning between phases extraction; chromatography
Separations Based on Size
Size is the simplest physical property we can exploit in a separation. To accomplish the separation we use a porous medium through which only the analyte or the interferent can pass. Examples of size-based separations include filtration, dialysis, and size-exclusion.
In a filtration we separate a particulate interferent from soluble analytes using a filter with a pore size that will retain the interferent. The solution that passes through the filter is called the filtrate, and the material retained by the filter is the retentate. Gravity filtration and suction filtration using filter paper are techniques with which you should already be familiar. A membrane filter is the method of choice for particulates that are too small to be retained by filter paper. Figure 7.6.1 provides information about three types of membrane filters. For applications of gravity filtration and suction filtration in gravimetric methods of analysis, see Chapter 8.
Dialysis is another example of a separation technique in which size is used to separate the analyte and the interferent. A dialysis membrane usually is made using cellulose and fashioned into tubing, bags, or cassettes. Figure 7.6.2 shows an example of a commercially available dialysis cassette. The sample is injected into the dialysis membrane, which is sealed tightly by a gasket, and the unit is placed in a container filled with a solution with a composition different from the sample. If there is a difference in a species’ concentration on the membrane’s two sides, the resulting concentration gradient provides a driving force for its diffusion across the membrane. While small species freely pass through the membrane, larger species are unable to pass. Dialysis frequently is used to purify proteins, hormones, and enzymes. During kidney dialysis, metabolic waste products, such as urea, uric acid, and creatinine, are removed from blood by passing it over a dialysis membrane.
Size-exclusion chromatography is a third example of a separation technique that uses size as a means to effect a separation. In this technique a column is packed with small, approximately 10-μm, porous polymer beads of cross-linked dextrin or polyacrylamide. The pore size of the particles is controlled by the degree of cross-linking, with more cross-linking producing smaller pore sizes. The sample is placed into a stream of solvent that is pumped through the column at a fixed flow rate. Those species too large to enter the pores pass through the column at the same rate as the solvent. Species that enter into the pores take longer to pass through the column, with smaller species requiring more time to pass through the column. Size-exclusion chromatography is widely used in the analysis of polymers, and in biochemistry, where it is used for the separation of proteins. A more detailed treatment of size-exclusion chromatography, which also is called gel permeation chromatography, is in Chapter 12.
Separations Based on Mass or Density
If the analyte and the interferent have different masses or densities, then a separation using centrifugation may be possible. The sample is placed in a centrifuge tube and spun at a high angular velocity, measured in revolutions per minute (rpm). The sample’s constituents experience a centrifugal force that pulls them toward the bottom of the centrifuge tube. Those species that experience the greatest centrifugal force have the fastest sedimentation rate and are the first to reach the bottom of the centrifuge tube. If two species have the same density, their separation is based on a difference in mass, with the heavier species having the greater sedimentation rate. If the species are of equal mass, then the species with the larger density has the greatest sedimentation rate.
Centrifugation is an important separation technique in biochemistry. Table 7.6.2 , for example, lists conditions for separating selected cellular components. We can separate lysosomes from other cellular components by several differential centrifugations, in which we divide the sample into a solid residue and a supernatant solution. After destroying the cells, the solution is centrifuged for 20 minutes at $15000 \times g$ (a centrifugal force that is 15 000 times the earth’s gravitational force), leaving a solid residue of cell membranes and mitochondria. The supernatant, which contains the lysosomes, is isolated by decanting it from the residue and then centrifuged for 30 minutes at $30000 \times g$, leaving a solid residue of lysosomes. Figure 7.6.3 shows a typical centrifuge capable of producing the centrifugal forces needed for biochemical separations.
Table 7.6.2 . Conditions for Separating Selected Cellular Components by Centrifugation
components centrifugal force ($\times g$) time (min)
eukaryotic cells 1000 5
cell membranes; nuclei 4000 10
mitochondria, bacterial cells 15000 20
lysosomes; bacterial membranes 30000 30
ribosomes 100000 180
Source: Adapted from Zubay, G. Biochemistry, 2nd ed. Macmillan: New York, 1988, p.120.
An alternative approach to differential centrifugation is a density gradient centrifugation. To prepare a sucrose density gradient, for example, a solution with a smaller concentration of sucrose—and, thus, of lower density—is gently layered upon a solution with a higher concentration of sucrose. Repeating this process several times, fills the centrifuge tube with a multi-layer density gradient. The sample is placed on top of the density gradient and centrifuged using a force greater than $150000 \times g$. During centrifugation, each of the sample’s components moves through the gradient until it reaches a position where its density matches the surrounding sucrose solution. Each component is isolated as a separate band positioned where its density is equal to that of the local density within the gradient. Figure 7.6.4 provides an example of a typical sucrose density centrifugation for separating plant thylakoid membranes.
Separations Based on Complexation Reactions (Masking)
One widely used technique for preventing an interference is to bind the interferent in a strong, soluble complex that prevents it from interfering in the analyte’s determination. This process is known as masking. As shown in Table 7.6.3 , a wide variety of ions and molecules are useful masking agents, and, as a result, selectivity is usually not a problem.
Technically, masking is not a separation technique because we do not physically separate the analyte and the interferent. We do, however, chemically isolate the interferent from the analyte, resulting in a pseudo-separation.
Table 7.6.3 . Selected Inorganic and Organic Masking Agents for Metal Ions
masking agent elements whose ions are masked
CN
Ag, Au, Cd, Co, Cu, Fe, Hg, Mn, Ni, Pd, Pt, Zn
SCN
Ag, Cd, Co, Cu, Fe, Ni, Pd, Pt, Zn
NH3
Ag, Co, Ni, Cu, Zn
F Al, Co, Cr, Mg, Mn, Sn, Zn
$\text{S}_2\text{O}_3^{2-}$
Au, Ce, Co, Cu, Fe, Hg, Mn, Pb, Pd, Pt, Sb, Sn, Zn
tartrate Al, Ba, Bi, Ca, Ce, Co, Cr, Cu, Fe, Hg, Mn, Pb, Pd, Pt, Sb, Sn, Zn
oxalate
Al, Fe, Mg, Mn
thioglycolic acid
Cu, Fe, Sn
Source: Meites, L. Handbook of Analytical Chemistry, McGraw-Hill: New York, 1963.
Example 7.6.1
Using Table 7.6.3 , suggest a masking agent for the analysis of aluminum in the presence of iron.
Solution
A suitable masking agent must form a complex with the interferent, but not with the analyte. Oxalate, for example, is not a suitable masking agent because it binds both Al and Fe. Thioglycolic acid, on the other hand, is a selective masking agent for Fe in the presence of Al. Other acceptable masking agents are cyanide (CN) thiocyanate (SCN), and thiosulfate ($\text{S}_2\text{O}_3^{2-}$).
Exercise 7.6.1
Using Table 7.6, suggest a masking agent for the analysis of Fe in the presence of Al.
Answer
The fluoride ion, F, is a suitable masking agent as it binds with Al3+ to form the stable $\text{AlF}_6^{3-}$ complex, leaving iron in solution.
As shown in Example 7.6.2 , we can judge a masking agent’s effectiveness by considering the relevant equilibrium constants.
Example 7.6.2
Show that CN is an appropriate masking agent for Ni2+ in a method where nickel’s complexation with EDTA is an interference.
Solution
The relevant reactions and formation constants are
$\mathrm{Ni}^{2+}(a q)+\mathrm{Y}^{4-}(a q)\rightleftharpoons \mathrm{NiY}^{2-}(a q) \quad K_{1}=4.2 \times 10^{18} \nonumber$
$\mathrm{Ni}^{2+}(a q)+4 \mathrm{CN}^{-}(a q)\rightleftharpoons \mathrm{Ni}(\mathrm{CN})_{4}^{2-}(a q) \quad \beta_{4}=1.7 \times 10^{30} \nonumber$
where Y4– is an abbreviation for EDTA. Cyanide is an appropriate masking agent because the formation constant for $\text{Ni(CN)}_4^{2-}$ is greater than that for the Ni–EDTA complex. In fact, the equilibrium constant for the reaction in which EDTA displaces the masking agent
$\mathrm{Ni}(\mathrm{CN})_{4}^{2-}(a q)+\mathrm{Y}^{4-}(a q) \rightleftharpoons \mathrm{NiY}^{2-}(a q)+4 \mathrm{CN}^{-}(a q) \nonumber$
$K=\frac{K_{1}}{\beta_{4}}=\frac{4.2 \times 10^{18}}{1.7 \times 10^{30}}=2.5 \times 10^{-12} \nonumber$
is sufficiently small that $\text{Ni(CN)}_4^{2-}$ is relatively inert in the presence of EDTA.
Exercise 7.6.2
Use the formation constants in Appendix 12 to show that 1,10-phenanthroline is a suitable masking agent for Fe2+ in the presence of Fe3+. Use a ladder diagram to define any limitations on using 1,10-phenanthroline as a masking agent. See Chapter 6 for a review of ladder diagrams.
Answer
The relevant reactions and equilibrium constants are
$\begin{array}{ll}{\mathrm{Fe}^{2+}(a q)+3 \mathrm{phen}(a q)} & {\rightleftharpoons\mathrm{Fe}(\mathrm{phen})_{3}^{2+}(a q) \quad \beta_{3}=5 \times 10^{20}} \ {\mathrm{Fe}^{3+}(a q)+3 \mathrm{phen}(a q)} & {\rightleftharpoons \mathrm{Fe}(\mathrm{phen})_{3}^{3+}(a q) \quad \beta_{3}=6 \times 10^{13}}\end{array} \nonumber$
where phen is an abbreviation for 1,10-phenanthroline. Because $\beta_3$ is larger for the complex with Fe2+ than it is for the complex with Fe3+,1,10-phenanthroline will bind Fe2+ before it binds Fe3+. A ladder diagram for this system (as shown below) suggests that an equilibrium p(phen) between 5.6 and 5.9 will fully complex Fe2+ without any significant formation of the $\text{Fe(phen)}_3^{3+}$ complex. Adding a stoichiometrically equivalent amount of 1,10-phenanthroline to a solution of Fe2+ is sufficient to mask Fe2+ in the presence of Fe3+. A large excess of 1,10-phenanthroline, however, decreases p(phen) and allows for the formation of both metal–ligand complexes.
Separations Based on a Change of State
Because an analyte and its interferent are usually in the same phase, we can achieve a separation if one of them undergoes a change in its physical state or its chemical state.
Changes in Physical State
When the analyte and the interferent are miscible liquids, separation by distillation is possible if their boiling points are significantly different. Figure 7.6.5 shows the progress of a distillation as a plot of temperature versus the composition of mixture’s vapor-phase and liquid-phase. The initial liquid mixture (point A), contains more interferent than analyte. When this solution is brought to its boiling point, the vapor phase in equilibrium with the liquid phase is enriched in analyte (point B). The horizontal line that connects points A and B represents this vaporization equilibrium. Condensing the vapor phase at point B, by lowering the temperature, creates a new liquid phase with a composition identical to that in the vapor phase (point C). The vertical line that connects points B and C represents this condensation equilibrium. The liquid phase at point C has a lower boiling point than the original mixture, and is in equilibrium with the vapor phase at point D. This process of repeated vaporization and condensation gradually separates the analyte and the interferent.
Two experimental set-ups for distillations are shown in Figure 7.6.6 . The simple distillation apparatus shown in Figure 7.6.6 a is useful only for separating a volatile analyte (or interferent) from a non-volatile interferent (or analyte), or for separating an analyte and an interferent whose boiling points differ by more than 150oC. A more efficient separation is achieved using the fractional distillation apparatus in Figure 7.6.6 b. Packing the fractionating column with a high surface area material, such as a steel sponge or glass beads, provides more opportunity for the repeated process of vaporization and condensation necessary to effect a complete separation.
When the sample is a solid, sublimation may provide a useful separation of the analyte and the interferent. The sample is heated at a temperature and pressure below the analyte’s triple point, allowing it to vaporize without passing through a liquid state. Condensing the vapor recovers the purified analyte (Figure 7.6.7 ). A useful analytical example of sublimation is the isolation of amino acids from fossil mollusk shells and deep-sea sediments [Glavin, D. P.; Bada, J. L. Anal. Chem. 1998, 70, 3119–3122].
Recrystallization is another method for purifying a solid. A solvent is chosen in which the analyte’s solubility is significant when the solvent is hot and minimal when the solvent is cold. The interferents must be less soluble in the hot solvent than the analyte or present in much smaller amounts. After heating a portion of the solvent in an Erlenmeyer flask, small amounts of sample are added until undissolved sample is visible. Additional hot solvent is added until the sample redissolves, or until only insoluble impurities remain. This process of adding sample and solvent is repeated until the entire sample is added to the Erlenmeyer flask. Any insoluble impurities are removed by filtering the hot solution. The solution is allowed to cool slowly, which promotes the growth of large, pure crystals, and then cooled in an ice bath to minimize solubility losses. The purified sample is isolated by filtration and rinsed to remove any soluble impurities. Finally, the sample is dried to remove any remaining traces of the solvent. Further purification, if necessary, is accomplished by additional recrystallizations.
Changes in Chemical State
Distillation, sublimation, and recrystallization use a change in physical state to effect a separation. Chemical reactivity also is a useful tool for separating analytes and interferents. For example, we can separate SiO2 from a sample by reacting it with HF to form SiF4. Because SiF4 is volatile, it is easy to remove by evaporation. If we wish to collect the reaction’s volatile product, then a distillation is possible. For example, we can isolate the $\text{NH}_4^+$ in a sample by making the solution basic and converting it to NH3. The ammonia is then removed by distillation. Table 7.6.4 provides additional examples of this approach for isolating inorganic ions.
Table 7.6.4 . Examples of Using a Chemical Reaction and a Distillation to Separate an Inorganic Analyte From Interferents
analyte treatment isolated species
$\text{CO}_3^{2-}$ $\mathrm{CO}_{3}^{2-}(a q)+2 \mathrm{H}_{3} \mathrm{O}^{+}(a q) \rightarrow \mathrm{CO}_{2}(g)+3 \mathrm{H}_{2} \mathrm{O}(l)$ CO2
$\text{NH}_4^+$ $\mathrm{NH}_{4}^{+}(a q)+\mathrm{OH}^{-}(a q) \rightarrow \mathrm{NH}_{3}(a q)+\mathrm{H}_{2} \mathrm{O}(l)$ NH3
$\text{SO}_3^-$ $\mathrm{SO}_{3}^{2-}(a q)+2 \mathrm{H}_{3} \mathrm{O}^{+}(a q) \rightarrow \mathrm{SO}_{2}(g)+3 \mathrm{H}_{2} \mathrm{O}(l)$ SO2
S2– $\text{S}^{2-}(a q)+2 \text{H}_{3} \mathrm{O}^{+}(a q) \rightarrow \text{H}_{2} \text{S}(g)+2 \text{H}_{2} \text{O}(l)$ H2S
Another reaction for separating analytes and interferents is precipitation. Two important examples of using a precipitation reaction in a separation are the pH-dependent solubility of metal oxides and hydroxides, and the pH-dependent solubility of metal sulfides.
Separations based on the pH-dependent solubility of oxides and hydroxides usually use a strong acid, a strong base, or an NH3/NH4Cl buffer to adjust the pH. Most metal oxides and hydroxides are soluble in hot concentrated HNO3, although a few oxides, such as WO3, SiO2, and SnO2 remain insoluble even under these harsh conditions. To determine the amount of Cu in brass, for example, we can avoid an interference from Sn by dissolving the sample with a strong acid and filtering to remove the solid residue of SnO2.
Most metals form a hydroxide precipitate in the presence of concentrated NaOH. Those metals that form amphoteric hydroxides, however, do not precipitate because they react to form higher-order hydroxo-complexes. For example, Zn2+ and Al3+ do not precipitate in concentrated NaOH because they form the soluble complexes $\text{Zn(OH)}_3^-$ and $\text{Al(OH)}_4^-$. The solubility of Al3+ in concentrated NaOH allows us to isolate aluminum from impure samples of bauxite, an ore of Al2O3. After crushing the ore, we place it in a solution of concentrated NaOH, dissolving the Al2O3 and forming $\text{Al(OH)}_4^-$. Other oxides in the ore, such as Fe2O3 and SiO2, remain insoluble. After filtering, we recover the aluminum as a precipitate of Al(OH)3 by neutralizing some of the OH with acid.
The pH of an NH3/NH4Cl buffer (pKa = 9.26) is sufficient to precipitate most metals as the hydroxide. The alkaline earths and alkaline metals, however, do not precipitate at this pH. In addition, metal ions that form soluble complexes with NH3, such as Cu2+, Zn2+, Ni2+, and Co2+ also do not precipitate under these conditions.
The use of S2– as a precipitating reagent is one of the earliest examples of a separation technique. In Fresenius’s 1881 text A System of Instruction in Quantitative Chemical Analysis, sulfide frequently is used to separate metal ions from the remainder of the sample’s matrix [Fresenius. C. R. A System of Instruction in Quantitative Chemical Analysis, John Wiley and Sons: New York, 1881]. Sulfide is a useful reagent for separating metal ions for two reasons: (1) most metal ions, except for the alkaline earths and alkaline metals, form insoluble sulfides; and (2) these metal sulfides show a substantial variation in solubility. Because the concentration of S2– is pH-dependent, we can control which metal ions precipitate by adjusting the pH. For example, in Fresenius’s gravimetric procedure for the determination of Ni in ore samples (see Figure 1.1.1 for a schematic diagram of this procedure), sulfide is used three times to separate Co2+ and Ni2+ from Cu2+ and, to a lesser extent, from Pb2+.
Separations Based on a Partitioning Between Phases
The most important group of separation techniques uses a selective partitioning of the analyte or interferent between two immiscible phases. If we bring a phase that contains the solute, S, into contact with a second phase, the solute will partition itself between the two phases, as shown by the following equilibrium reaction.
$S_{\text { phase } 1} \rightleftharpoons S_{\text { phase } 2} \label{7.1}$
The equilibrium constant for reaction \ref{7.1}
$K_{\mathrm{D}}=\frac{\left[S_{\mathrm{phase} \ 2}\right]}{\left[S_{\mathrm{phase} \ 1}\right]} \nonumber$
is called the distribution constant or the partition coefficient. If KD is sufficiently large, then the solute moves from phase 1 to phase 2. The solute will remain in phase 1 if the partition coefficient is sufficiently small. When we bring a phase that contains two solutes into contact with a second phase, a separation of the solutes is possible if KD is favorable for only one of the solutes. The physical states of the phases are identified when we describe the separation process, with the phase that contains the sample listed first. For example, if the sample is in a liquid phase and the second phase is a solid, then the separation involves liquid–solid partitioning.
Extraction Between Two Phases
We call the process of moving a species from one phase to another phase an extraction. Simple extractions are particularly useful for separations where only one component has a favorable partition coefficient. Several important separation techniques are based on a simple extraction, including liquid–liquid, liquid–solid, solid–liquid, and gas–solid extractions.
Liquid-Liquid Extractions
A liquid–liquid extraction usually is accomplished using a separatory funnel (Figure 7.6.8 ). After placing the two liquids in the separatory funnel, we shake the funnel to increase the surface area between the phases. When the extraction is complete, we allow the liquids to separate. The stopcock at the bottom of the separatory funnel allows us to remove the two phases.
We also can carry out a liquid–liquid extraction without a separatory funnel by adding the extracting solvent to the sample’s container. Pesticides in water, for example, are preserved in the field by extracting them into a small volume of hexane. A liquid–liquid microextraction, in which the extracting phase is a 1-µL drop suspended from a microsyringe (Figure 7.6.9 ), also has been described [Jeannot, M. A.; Cantwell, F. F. Anal. Chem. 1997, 69, 235–239]. Because of its importance, a more thorough discussion of liquid–liquid extractions is in Chapter7.7.
Solid Phase Extractions
In a solid phase extraction of a liquid sample, we pass the sample through a cartridge that contains a solid adsorbent, several examples of which are shown in Figure 7.6.10 . The choice of adsorbent is determined by the species we wish to separate. Table 7.6.5 provides several representative examples of solid adsorbents and their applications.
Table 7.6.5 . Representative Adsorbents for the Solid Phase Extraction of Liquid Samples
absorbent structure properties and uses
silica
• retains low to moderate polarity species from organic matrices
• fat soluble vitamins, steroids
aminopropyl
• retains polar compounds
• carbohydrates, organic acids
cyanopropyl
• retains wide variety of species from aqueous and organic matrices
• pesticides, hydrophobic peptides
diol
• retains wide variety of species from aqueous and organic matrices
• proteins, peptides, fungicides
octadecyl (C–18) —C18H37
• retains hydrophobic species from aqueous matrices
• caffeine, sedatives, polyaromatic hydrocarbons, carbohydrates, pesticides
octyl (C–8) —C8H17
• similar to C-18
As an example, let’s examine a procedure for isolating the sedatives secobarbital and phenobarbital from serum samples using a C-18 solid adsorbent [Alltech Associates Extract-Clean SPE Sample Preparation Guide, Bulletin 83]. Before adding the sample, the solid phase cartridge is rinsed with 6 mL each of methanol and water. Next, a 500-μL sample of serum is pulled through the cartridge, with the sedatives and matrix interferents retained following a liquid–solid extraction (Figure 7.6.11 a). Washing the cartridge with distilled water removes any interferents (Figure 7.6.11 b). Finally, we elute the sedatives using 500 μL of acetone (Figure 7.6.11 c). In comparison to a liquid–liquid extraction, a solid phase extraction has the advantage of being easier, faster, and requires less solvent.
Continuous Extractions
An extraction is possible even if the analyte has an unfavorable partition coefficient, provided that the sample’s other components have significantly smaller partition coefficients. Because the analyte’s partition coefficient is unfavorable, a single extraction will not recover all the analyte. Instead we continuously pass the extracting phase through the sample until we achieve a quantitative extraction.
A continuous extraction of a solid sample is carried out using a Soxhlet extractor (Figure 7.6.12 ). The extracting solvent is placed in the lower reservoir and heated to its boiling point. Solvent in the vapor phase moves upward through the tube on the far right side of the apparatus, reaching the condenser where it condenses back to the liquid state. The solvent then passes through the sample, which is held in a porous cellulose filter thimble, collecting in the upper reservoir. When the solvent in the upper reservoir reaches the return tube’s upper bend, the solvent and extracted analyte are siphoned back to the lower reservoir. Over time the analyte’s concentration in the lower reservoir increases.
Microwave-assisted extractions have replaced Soxhlet extractions in some applications [Renoe, B. W. Am. Lab August 1994, 34–40]. The process is the same as that described earlier for a microwave digestion. After placing the sample and the solvent in a sealed digestion vessel, a microwave oven is used to heat the mixture. Using a sealed digestion vessel allows the extraction to take place at a higher temperature and pressure, reducing the amount of time needed for a quantitative extraction. In a Soxhlet extraction the temperature is limited by the solvent’s boiling point at atmospheric pressure. When acetone is the solvent, for example, a Soxhlet extraction is limited to 56oC, but a microwave extraction can reach 150oC.
Two other continuous extractions deserve mention. Volatile organic compounds (VOCs) can be quantitatively removed from a liquid sample by a liquid–gas extraction. As shown in Figure 7.6.13 , an inert purging gas, such as He, is passed through the sample. The purge gas removes the VOCs, which are swept to a primary trap where they collect on a solid absorbent. When the extraction is complete, the VOCs are removed from the primary trap by rapidly heating the tube while flushing with He. This technique is known as a purge-and-trap. Because the analyte’s recovery may not be reproducible, an internal standard is required for quantitative work.
Continuous extractions also can be accomplished using supercritical fluids [McNally, M. E. Anal. Chem. 1995, 67, 308A–315A]. If we heat a substance above its critical temperature and pressure it forms a supercritical fluid whose properties are between those of a gas and a liquid. A supercritical fluid is a better solvent than a gas, which makes it a better reagent for extractions. In addition, a supercritical fluid’s viscosity is significantly less than that of a liquid, which makes it easier to push it through a particulate sample. One example of a supercritical fluid extraction is the determination of total petroleum hydrocarbons (TPHs) in soils, sediments, and sludges using supercritical CO2 [“TPH Extraction by SFE,” ISCO, Inc. Lincoln, NE, Revised Nov. 1992]. An approximately 3-g sample is placed in a 10-mL stainless steel cartridge and supercritical CO2 at a pressure of 340 atm and a temperature of 80oC is passed through the cartridge for 30 minutes at flow rate of 1–2 mL/min. To collect the TPHs, the effluent from the cartridge is passed through 3 mL of tetrachloroethylene at room temperature. At this temperature the CO2 reverts to the gas phase and is released to the atmosphere.
Chromatographic Separations
In an extraction, the sample originally is in one phase and we extract the analyte or the interferent into a second phase. We also can separate the analyte and interferents by continuously passing one sample-free phase, called the mobile phase, over a second sample-free phase that remains fixed or stationary. The sample is injected into the mobile phase and the sample’s components partition themselves between the mobile phase and the stationary phase. Those components with larger partition coefficients are more likely to move into the stationary phase and take longer time to pass through the system. This is the basis of all chromatographic separations. Chromatography provides both a separation of analytes and interferents, and a means for performing a qualitative or quantitative analysis for the analyte. For this reason a more thorough treatment of chromatography is found in Chapter 12. | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/07%3A_Obtaining_and_Preparing_Samples_for_Analysis/7.06%3A_Classifying_Separation_Techniques.txt |
A liquid–liquid extraction is an important separation technique for environmental, clinical, and industrial laboratories. A standard environmental analytical method illustrates the importance of liquid–liquid extractions. Municipal water departments routinely monitor public water supplies for trihalomethanes (CHCl3, CHBrCl2, CHBr2Cl, and CHBr3) because they are known or suspected carcinogens. Before their analysis by gas chromatography, trihalomethanes are separated from their aqueous matrix using a liquid–liquid extraction with pentane [“The Analysis of Trihalomethanes in Drinking Water by Liquid Extraction,”EPAMethod501.2 (EPA 500-Series, November 1979)].
The Environmental Protection Agency (EPA) also publishes two additional methods for trihalomethanes. Method 501.1 and Method 501.3 use a purge-and-trap to collect the trihalomethanes prior to a gas chromatographic analysis with a halide-specific detector (Method 501.1) or a mass spectrometer as the detector (Method 501.3). You will find more details about gas chromatography, including detectors, in Chapter 12.
In a simple liquid–liquid extraction the solute partitions itself between two immiscible phases. One phase usually is an aqueous solvent and the other phase is an organic solvent, such as the pentane used to extract trihalomethanes from water. Because the phases are immiscible they form two layers, with the denser phase on the bottom. The solute initially is present in one of the two phases; after the extraction it is present in both phases. Extraction efficiency—that is, the percentage of solute that moves from one phase to the other—is determined by the equilibrium constant for the solute’s partitioning between the phases and any other side reactions that involve the solute. Examples of other reactions that affect extraction efficiency include acid–base reactions and complexation reactions.
Partition Coefficients and Distribution Ratios
As we learned earlier in this chapter, a solute’s partitioning between two phases is described by a partition coefficient, KD. If we extract a solute from an aqueous phase into an organic phase
$S_{a q} \rightleftharpoons S_{o r g} \nonumber$
then the partition coefficient is
$K_{\mathrm{D}}=\frac{\left[S_{org}\right]}{\left[S_{a q}\right]} \nonumber$
A large value for KD indicates that extraction of solute into the organic phase is favorable.
To evaluate an extraction’s efficiency we must consider the solute’s total concentration in each phase, which we define as a distribution ratio, D.
$D=\frac{\left[S_{o r g}\right]_{\text { total }}}{\left[S_{a q}\right]_{\text { total }}} \nonumber$
The partition coefficient and the distribution ratio are identical if the solute has only one chemical form in each phase; however, if the solute exists in more than one chemical form in either phase, then KD and D usually have different values. For example, if the solute exists in two forms in the aqueous phase, A and B, only one of which, A, partitions between the two phases, then
$D=\frac{\left[S_{o r g}\right]_{A}}{\left[S_{a q}\right]_{A}+\left[S_{a q}\right]_{B}} \leq K_{\mathrm{D}}=\frac{\left[S_{o r g}\right]_{A}}{\left[S_{a q}\right]_{A}} \nonumber$
This distinction between KD and D is important. The partition coefficient is a thermodynamic equilibrium constant and has a fixed value for the solute’s partitioning between the two phases. The distribution ratio’s value, however, changes with solution conditions if the relative amounts of A and B change. If we know the solute’s equilibrium reactions within each phase and between the two phases, we can derive an algebraic relationship between KD and D.
Liquid-Liquid Extraction With No Secondary Reactions
In a simple liquid–liquid extraction, the only reaction that affects the extraction efficiency is the solute’s partitioning between the two phases (Figure 7.7.1 ).
In this case the distribution ratio and the partition coefficient are equal.
$D=\frac{\left[S_{o r g}\right]_{\text { total }}}{\left[S_{aq}\right]_{\text { total }}} = K_\text{D} = \frac {[S_{org}]} {[S_{aq}]} \label{7.1}$
Let’s assume the solute initially is present in the aqueous phase and that we wish to extract it into the organic phase. A conservation of mass requires that the moles of solute initially present in the aqueous phase equal the combined moles of solute in the aqueous phase and the organic phase after the extraction.
$\left(\operatorname{mol} \ S_{a q}\right)_{0}=\left(\operatorname{mol} \ S_{a q}\right)_{1}+\left(\operatorname{mol} \ S_{org}\right)_{1} \label{7.2}$
where the subscripts indicate the extraction number with 0 representing the system before the extraction and 1 the system following the first extraction. After the extraction, the solute’s concentration in the aqueous phase is
$\left[S_{a q}\right]_{1}=\frac{\left(\operatorname{mol} \ S_{a q}\right)_{1}}{V_{a q}} \label{7.3}$
and its concentration in the organic phase is
$\left[S_{o r g}\right]_{1}=\frac{\left(\operatorname{mol} \ S_{o r g}\right)_{1}}{V_{o r g}} \label{7.4}$
where Vaq and Vorg are the volumes of the aqueous phase and the organic phase. Solving Equation \ref{7.2} for (mol Sorg)1 and substituting into Equation \ref{7.4} leave us with
$\left[S_{o r g}\right]_{1} = \frac{\left(\operatorname{mol} \ S_{a q}\right)_{0}-\left(\operatorname{mol} \ S_{a q}\right)_{1}}{V_{o r g}} \label{7.5}$
Substituting Equation \ref{7.3} and Equation \ref{7.5} into Equation \ref{7.1} gives
$D = \frac {\frac {(\text{mol }S_{aq})_0-(\text{mol }S_{aq})_1} {V_{org}}} {\frac {(\text{mol }S_{aq})_1} {V_{aq}}} = \frac{\left(\operatorname{mol} \ S_{a q}\right)_{0} \times V_{a q}-\left(\operatorname{mol} \ S_{a q}\right)_{1} \times V_{a q}}{\left(\operatorname{mol} \ S_{a q}\right)_{1} \times V_{o r g}} \nonumber$
Rearranging and solving for the fraction of solute that remains in the aqueous phase after one extraction, (qaq)1, gives
$\left(q_{aq}\right)_{1} = \frac{\left(\operatorname{mol} \ S_{aq}\right)_{1}}{\left(\operatorname{mol} \ S_{a q}\right)_{0}} = \frac{V_{aq}}{D V_{o r g}+V_{a q}} \label{7.6}$
The fraction present in the organic phase after one extraction, (qorg)1, is
$\left(q_{o r g}\right)_{1}=\frac{\left(\operatorname{mol} S_{o r g}\right)_{1}}{\left(\operatorname{mol} S_{a q}\right)_{0}}=1-\left(q_{a q}\right)_{1}=\frac{D V_{o r g}}{D V_{o r g}+V_{a q}} \nonumber$
Example 7.7.1 shows how we can use Equation \ref{7.6} to calculate the efficiency of a simple liquid-liquid extraction.
Example 7.7.1
A solute has a KD between water and chloroform of 5.00. Suppose we extract a 50.00-mL sample of a 0.050 M aqueous solution of the solute using 15.00 mL of chloroform. (a) What is the separation’s extraction efficiency? (b) What volume of chloroform do we need if we wish to extract 99.9% of the solute?
Solution
For a simple liquid–liquid extraction the distribution ratio, D, and the partition coefficient, KD, are identical.
(a) The fraction of solute that remains in the aqueous phase after the extraction is given by Equation \ref{7.6}.
$\left(q_{aq}\right)_{1}=\frac{V_{a q}}{D V_{org}+V_{a q}}=\frac{50.00 \ \mathrm{mL}}{(5.00)(15.00 \ \mathrm{mL})+50.00 \ \mathrm{mL}}=0.400 \nonumber$
The fraction of solute in the organic phase is 1–0.400, or 0.600. Extraction efficiency is the percentage of solute that moves into the extracting phase; thus, the extraction efficiency is 60.0%.
(b) To extract 99.9% of the solute (qaq)1 must be 0.001. Solving Equation \ref{7.6} for Vorg, and making appropriate substitutions for (qaq)1 and Vaq gives
$V_{o r g}=\frac{V_{a q}-\left(q_{a q}\right)_{1} V_{a q}}{\left(q_{a q}\right)_{1} D}=\frac{50.00 \ \mathrm{mL}-(0.001)(50.00 \ \mathrm{mL})}{(0.001)(5.00 \ \mathrm{mL})}=999 \ \mathrm{mL} \nonumber$
This is large volume of chloroform. Clearly, a single extraction is not reasonable under these conditions.
In Example 7.7.1 , a single extraction provides an extraction efficiency of only 60%. If we carry out a second extraction, the fraction of solute remaining in the aqueous phase, (qaq)2, is
$\left(q_{a q}\right)_{2}=\frac{\left(\operatorname{mol} \ S_{a q}\right)_{2}}{\left(\operatorname{mol} \ S_{a q}\right)_{1}}=\frac{V_{a q}}{D V_{org}+V_{a q}} \nonumber$
If Vaq and Vorg are the same for both extractions, then the cumulative fraction of solute that remains in the aqueous layer after two extractions, (Qaq)2, is the product of (qaq)1 and (qaq)2, or
$\left(Q_{aq}\right)_{2}=\frac{\left(\operatorname{mol} \ S_{aq}\right)_{2}}{\left(\operatorname{mol} \ S_{aq}\right)_{0}}=\left(q_{a q}\right)_{1} \times\left(q_{a q}\right)_{2}=\left(\frac{V_{a q}}{D V_{o r g}+V_{a q}}\right)^{2} \nonumber$
In general, for a series of n identical extractions, the fraction of analyte that remains in the aqueous phase after the last extraction is
$\left(Q_{a q}\right)_{n}=\left(\frac{V_{a q}}{D V_{o r g}+V_{a q}}\right)^{n} \label{7.7}$
Example 7.7.2
For the extraction described in Example 7.7.1 , determine (a) the extraction efficiency for two identical extractions and for three identical extractions; and (b) the number of extractions required to ensure that we extract 99.9% of the solute.
Solution
(a) The fraction of solute remaining in the aqueous phase after two extractions and three extractions is
$\left(Q_{aq}\right)_{2}=\left(\frac{50.00 \ \mathrm{mL}}{(5.00)(15.00 \ \mathrm{mL})+50.00 \ \mathrm{mL}}\right)^{2}=0.160 \nonumber$
$\left(Q_{a q}\right)_{3}=\left(\frac{50.0 \ \mathrm{mL}}{(5.00)(15.00 \ \mathrm{mL})+50.00 \ \mathrm{mL}}\right)^{3}=0.0640 \nonumber$
The extraction efficiencies are 84.0% for two extractions and 93.6% for three extractions.
(b) To determine the minimum number of extractions for an efficiency of 99.9%, we set (Qaq)n to 0.001 and solve for n using Equation \ref{7.7}.
$0.001=\left(\frac{50.00 \ \mathrm{mL}}{(5.00)(15.00 \ \mathrm{mL})+50.00 \ \mathrm{mL}}\right)^{n}=(0.400)^{n} \nonumber$
Taking the log of both sides and solving for n
\begin{aligned} \log (0.001) &=n \log (0.400) \ n &=7.54 \end{aligned} \nonumber
we find that a minimum of eight extractions is necessary.
The last two examples provide us with an important observation—for any extraction efficiency, we need less solvent if we complete several extractions using smaller portions of solvent instead of one extraction using a larger volume of solvent. For the conditions in Example 7.7.1 and Example 7.7.2 , an extraction efficiency of 99.9% requires one extraction with 9990 mL of chloroform, or 120 mL when using eight 15-mL portions of chloroform. Although extraction efficiency increases dramatically with the first few multiple, the effect diminishes quickly as we increase the number of extractions (Figure 7.7.2 ). In most cases there is little improvement in extraction efficiency after five or six extractions. For the conditions in Example 7.7.2 , we reach an extraction efficiency of 99% after five extractions and need three additional extractions to obtain the extra 0.9% increase in extraction efficiency.
Exercise 7.7.1
To plan a liquid–liquid extraction we need to know the solute’s distribution ratio between the two phases. One approach is to carry out the extraction on a solution that contains a known amount of solute. After the extraction, we isolate the organic phase and allow it to evaporate, leaving behind the solute. In one such experiment, 1.235 g of a solute with a molar mass of 117.3 g/mol is dissolved in 10.00 mL of water. After extracting with 5.00 mL of toluene, 0.889 g of the solute is recovered in the organic phase. (a) What is the solute’s distribution ratio between water and toluene? (b) If we extract 20.00 mL of an aqueous solution that contains the solute using 10.00 mL of toluene, what is the extraction efficiency? (c) How many extractions will we need to recover 99.9% of the solute?
Answer
(a) The solute’s distribution ratio between water and toluene is
$D=\frac{\left[S_{o r g}\right]}{\left[S_{a q}\right]}=\frac{0.889 \ \mathrm{g} \times \frac{1 \ \mathrm{mol}}{117.3 \ \mathrm{g}} \times \frac{1}{0.00500 \ \mathrm{L}}}{(1.235 \ \mathrm{g}-0.889 \ \mathrm{g}) \times \frac{1 \ \mathrm{mol}}{117.3 \ \mathrm{g}} \times \frac{1}{0.01000 \ \mathrm{L}}}=5.14 \nonumber$
(b) The fraction of solute remaining in the aqueous phase after one extraction is
$\left(q_{a q}\right)_{1}=\frac{V_{a q}}{D V_{org}+V_{a q}}=\frac{20.00 \ \mathrm{mL}}{(5.14)(10.00 \ \mathrm{mL})+20.00 \ \mathrm{mL}}=0.280 \nonumber$
The extraction efficiency, therefore, is 72.0%.
(c) To extract 99.9% of the solute requires
$\left(Q_{aq}\right)_{n}=0.001=\left(\frac{20.00 \ \mathrm{mL}}{(5.14)(10.00 \ \mathrm{mL})+20.00 \ \mathrm{mL}}\right)^{n}=(0.280)^{n} \nonumber$
\begin{aligned} \log (0.001) &=n \log (0.280) \ n &=5.4 \end{aligned} \nonumber
a minimum of six extractions.
Liquid-Liquid Extractions Involving Acid-Base Equilibria
As we see in Equation \ref{7.1}, in a simple liquid–liquid extraction the distribution ratio and the partition coefficient are identical. As a result, the distribution ratio does not depend on the composition of the aqueous phase or the organic phase. A change in the pH of the aqueous phase, for example, will not affect the solute’s extraction efficiency when KD and D have the same value.
If the solute participates in one or more additional equilibrium reactions within a phase, then the distribution ratio and the partition coefficient may not be the same. For example, Figure 7.7.3 shows the equilibrium reactions that affect the extraction of the weak acid, HA, by an organic phase in which ionic species are not soluble.
In this case the partition coefficient and the distribution ratio are
$K_{\mathrm{D}}=\frac{\left[\mathrm{HA}_{org}\right]}{\left[\mathrm{HA}_{a q}\right]} \label{7.8}$
$D=\frac{\left[\mathrm{HA}_{org}\right]_{\text { total }}}{\left[\mathrm{HA}_{a q}\right]_{\text { total }}} =\frac{\left[\mathrm{HA}_{org}\right]}{\left[\mathrm{HA}_{a q}\right]+\left[\mathrm{A}_{a q}^{-}\right]} \label{7.9}$
Because the position of an acid–base equilibrium depends on pH, the distribution ratio, D, is pH-dependent. To derive an equation for D that shows this dependence, we begin with the acid dissociation constant for HA.
$K_{\mathrm{a}}=\frac{\left[\mathrm{H}_{3} \mathrm{O}_{\mathrm{aq}}^{+}\right]\left[\mathrm{A}_{\mathrm{aq}}^{-}\right]}{\left[\mathrm{HA}_{\mathrm{aq}}\right]} \label{7.10}$
Solving Equation \ref{7.10} for the concentration of A in the aqueous phase
$\left[\mathrm{A}_{a q}^{-}\right]=\frac{K_{\mathrm{a}} \times\left[\mathrm{HA}_{a q}\right]}{\left[\mathrm{H}_{3} \mathrm{O}_{a q}^{+}\right]} \nonumber$
and substituting into Equation \ref{7.9} gives
$D = \frac {[\text{HA}_{org}]} {[\text{HA}_{aq}] + \frac {K_a \times [\text{HA}_{aq}]}{[\text{H}_3\text{O}_{aq}^+]}} \nonumber$
Factoring [HAaq] from the denominator, replacing [HAorg]/[HAaq] with KD (Equation \ref{7.8}), and simplifying leaves us with the following relationship between the distribution ratio, D, and the pH of the aqueous solution.
$D=\frac{K_{\mathrm{D}}\left[\mathrm{H}_{3} \mathrm{O}_{aq}^{+}\right]}{\left[\mathrm{H}_{3} \mathrm{O}_{aq}^{+}\right]+K_{a}} \label{7.11}$
Example 7.7.3
An acidic solute, HA, has a Ka of $1.00 \times 10^{-5}$ and a KD between water and hexane of 3.00. Calculate the extraction efficiency if we extract a 50.00 mL sample of a 0.025 M aqueous solution of HA, buffered to a pH of 3.00, with 50.00 mL of hexane. Repeat for pH levels of 5.00 and 7.00.
Solution
When the pH is 3.00, [$\text{H}_3\text{O}_{aq}^+$] is $1.0 \times 10^{-3}$ and the distribution ratio is
$D=\frac{(3.00)\left(1.0 \times 10^{-3}\right)}{1.0 \times 10^{-3}+1.00 \times 10^{-5}}=2.97 \nonumber$
The fraction of solute that remains in the aqueous phase is
$\left(Q_{aq}\right)_{1}=\frac{50.00 \ \mathrm{mL}}{(2.97)(50.00 \ \mathrm{mL})+50.00 \ \mathrm{mL}}=0.252 \nonumber$
The extraction efficiency, therefore, is almost 75%. The same calculation at a pH of 5.00 gives the extraction efficiency as 60%. At a pH of 7.00 the extraction efficiency is just 3% .
The extraction efficiency in Example 7.7.3 is greater at more acidic pH levels because HA is the solute’s predominate form in the aqueous phase. At a more basic pH, where A is the solute’s predominate form, the extraction efficiency is smaller. A graph of extraction efficiency versus pH is shown in Figure 7.7.4 . Note that extraction efficiency essentially is independent of pH for pH levels more acidic than the HA’s pKa, and that it is essentially zero for pH levels more basic than HA’s pKa. The greatest change in extraction efficiency occurs at pH levels where both HA and A are predominate species. The ladder diagram for HA along the graph’s x-axis helps illustrate this effect.
Exercise 7.7.2
The liquid–liquid extraction of the weak base B is governed by the following equilibrium reactions:
$\begin{array}{c}{\mathrm{B}(a q) \rightleftharpoons \mathrm{B}(org) \quad K_{D}=5.00} \ {\mathrm{B}(a q)+\mathrm{H}_{2} \mathrm{O}(l)\rightleftharpoons \mathrm{OH}^{-}(a q)+\mathrm{HB}^{+}(a q) \quad K_{b}=1.0 \times 10^{-4}}\end{array} \nonumber$
Derive an equation for the distribution ratio, D, and calculate the extraction efficiency if 25.0 mL of a 0.025 M solution of B, buffered to a pH of 9.00, is extracted with 50.0 mL of the organic solvent.
Answer
Because the weak base exists in two forms, only one of which extracts into the organic phase, the partition coefficient, KD, and the distribution ratio, D, are not identical.
$K_{\mathrm{D}}=\frac{\left[\mathrm{B}_{org}\right]}{\left[\mathrm{B}_{aq}\right]} \nonumber$
$D = \frac {[\text{B}_{org}]} {[\text{B}_{aq}]} = \frac {[\text{B}_{org}]} {[\text{B}_{aq}] + [\text{HB}_{aq}^+]} \nonumber$
Using the Kb expression for the weak base
$K_{\mathrm{b}}=\frac{\left[\mathrm{OH}_{a q}^{-}\right]\left[\mathrm{HB}_{a q}^{+}\right]}{\left[\mathrm{B}_{a q}\right]} \nonumber$
we solve for the concentration of HB+ and substitute back into the equation for D, obtaining
$D = \frac {[\text{B}_{org}]} {[\text{B}_{aq}] + \frac {K_b \times [\text{B}_{aq}]} {[\text{OH}_{aq}^-]}} = \frac {[\text{B}_{org}]} {[\text{B}_{aq}]\left(1+\frac {K_b} {[\text{OH}_{aq}^+]} \right)} =\frac{K_{D}\left[\mathrm{OH}_{a q}^{-}\right]}{\left[\mathrm{OH}_{a q}^{-}\right]+K_{\mathrm{b}}} \nonumber$
At a pH of 9.0, the [OH] is $1 \times 10^{-5}$ M and the distribution ratio has a value of
$D=\frac{K_{D}\left[\mathrm{OH}_{a q}^{-}\right]}{\left[\mathrm{OH}_{aq}^{-}\right]+K_{\mathrm{b}}}=\frac{(5.00)\left(1.0 \times 10^{-5}\right)}{1.0 \times 10^{-5}+1.0 \times 10^{-4}}=0.455 \nonumber$
After one extraction, the fraction of B remaining in the aqueous phase is
$\left(q_{aq}\right)_{1}=\frac{25.00 \ \mathrm{mL}}{(0.455)(50.00 \ \mathrm{mL})+25.00 \ \mathrm{mL}}=0.524 \nonumber$
The extraction efficiency, therefore, is 47.6%. At a pH of 9, most of the weak base is present as HB+, which explains why the overall extraction efficiency is so poor.
Liquid-Liquid Extraction of a Metal-Ligand Complex
One important application of a liquid–liquid extraction is the selective extraction of metal ions using an organic ligand. Unfortunately, many organic ligands are not very soluble in water or undergo hydrolysis or oxidation reactions in aqueous solutions. For these reasons the ligand is added to the organic solvent instead of the aqueous phase. Figure 7.7.5 shows the relevant equilibrium reactions (and equilibrium constants) for the extraction of Mn+ by the ligand HL, including the ligand’s extraction into the aqueous phase (KD,HL), the ligand’s acid dissociation reaction (Ka), the formation of the metal–ligand complex ($\beta_n$), and the complex’s extraction into the organic phase (KD,c).
If the ligand’s concentration is much greater than the metal ion’s concentration, then the distribution ratio is
$D=\frac{\beta_{n} K_{\mathrm{D}, c}\left(K_{a}\right)^{n}\left(C_{\mathrm{HL}}\right)^{n}}{\left(K_{\mathrm{D}, \mathrm{HL}}\right)^{n}\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]^{n}+\beta_{n}\left(K_{\mathrm{a}}\right)^{n}\left(C_{\mathrm{HL}}\right)^{n}} \label{7.12}$
where CHL is the ligand’s initial concentration in the organic phase. As shown in Example 7.7.4 , the extraction efficiency for metal ions shows a marked pH dependency.
Example 7.7.4
A liquid–liquid extraction of the divalent metal ion, M2+, uses the scheme outlined in Figure 7.7.5 . The partition coefficients for the ligand, KD,HL, and for the metal–ligand complex, KD,c, are $1.0 \times 10^4$ and $7.0 \times 10^4$, respectively. The ligand’s acid dissociation constant, Ka, is $5.0 \times 10^{-5}$, and the formation constant for the metal–ligand complex, $\beta_2$, is $2.5 \times 10^{16}$. What is the extraction efficiency if we extract 100.0 mL of a $1.0 \times 10^{-6}$ M aqueous solution of M2+, buffered to a pH of 1.00, with 10.00 mL of an organic solvent that is 0.1 mM in the chelating agent? Repeat the calculation at a pH of 3.00.
Solution
When the pH is 1.00 the distribution ratio is
$D=\frac{\left(2.5 \times 10^{16}\right)\left(7.0 \times 10^{4}\right)\left(5.0 \times 10^{-5}\right)^{2}\left(1.0 \times 10^{-4}\right)^{2}}{\left(1.0 \times 10^{4}\right)^{2}(0.10)^{2}+\left(2.5 \times 10^{16}\right)\left(5.0 \times 10^{-5}\right)^{2}\left(1.0 \times 10^{-4}\right)^{2}} \nonumber$
or a D of 0.0438. The fraction of metal ion that remains in the aqueous phase is
$\left(Q_{aq}\right)_{1}=\frac{100.0 \ \mathrm{mL}}{(0.0438)(10.00 \ \mathrm{mL})+100.0 \ \mathrm{mL}}=0.996 \nonumber$
At a pH of 1.00, we extract only 0.40% of the metal into the organic phase. Changing the pH to 3.00, however, increases the extraction efficiency to 97.8%. Figure 7.7.6 shows how the pH of the aqueous phase affects the extraction efficiency for M2+.
One advantage of using a ligand to extract a metal ion is the high degree of selectivity that it brings to a liquid–liquid extraction. As seen in Figure 7.7.6 , a divalent metal ion’s extraction efficiency increases from approximately 0% to 100% over a range of 2 pH units. Because a ligand’s ability to form a metal–ligand complex varies substantially from metal ion to metal ion, significant selectivity is possible if we carefully control the pH. Table 7.7.1 shows the minimum pH for extracting 99% of a metal ion from an aqueous solution using an equal volume of 4 mM dithizone in CCl4.
Table 7.7.1 . Minimum pH for Extracting 99% of an Aqueous Metal Ion Using 4.0 mM Dithizone in $\text{CCl}_4$ $V_{aq} = V_{org}$
metal ion minimum pH
Hg2+ –8.7
Ag+ –1.7
Cu2+ –0.8
Bi3+ 0.9
Zn2+ 2.3
Cd2+ 3.6
Co2+ 3.6
Pb2+ 4.1
Ni2+ 6.0
Tl+ 8.7
Example 7.7.5
Using Table 7.7.1 , explain how we can separate the metal ions in an aqueous mixture of Cu2+, Cd2+, and Ni2+ by extracting with an equal volume of dithizone in CCl4.
Solution
From Table 7.7.1 , a quantitative separation of Cu2+ from Cd2+ and from Ni2+ is possible if we acidify the aqueous phase to a pH of less than 1. This pH is greater than the minimum pH for extracting Cu2+ and significantly less than the minimum pH for extracting either Cd2+ or Ni2+. After the extraction of Cu2+ is complete, we shift the pH of the aqueous phase to 4.0, which allows us to extract Cd2+ while leaving Ni2+ in the aqueous phase. | textbooks/chem/Analytical_Chemistry/Analytical_Chemistry_2.1_(Harvey)/07%3A_Obtaining_and_Preparing_Samples_for_Analysis/7.07%3A_Liquid-Liquid_Extractions.txt |
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