## Expressing the Equilibrium Constant of a Gas in Terms of Pressure

#### Learning Objective

• Write the equilibrium expression, KP, in terms of the partial pressures of a gas-phase reaction

#### Key Points

• According to the ideal gas equation, pressure is directly proportional to concentration, assuming volume and temperature are constant.
• Since pressure is directly proportional to concentration, we can write our equilibrium expression for a gas-phase reaction in terms of the partial pressures of each gas. This special equilibrium constant is known as KP.
• KP takes the exact same form as KC. To avoid confusion between the two, do not use brackets ([ ]) when expressing partial pressures.

#### Terms

• equilibriumThe state of a reaction in which the rates of the forward and reverse reactions are the same.
• partial pressureThe pressure that one component of a mixture of gases contributes to the total pressure.

## Equilibrium Constants for Gases

Up to this point, we have been discussing equilibrium constants in terms of concentration. For gas-specific reactions, however, we can also express the equilibrium constant in terms of the partial pressures of the gases involved. Take the general gas-phase reaction:

$aA(g)+bB(g)\rightleftharpoons cC(g)+dD(g)$

Our equilibrium constant in terms of partial pressures, designated KP, is given as:

$K_P=\frac{P^c_CP^d_D}{P^a_AP^b_B}$

Note that this expression is extremely similar to KC, the equilibrium expression written in terms of concentrations. In order to prevent confusion, do not use brackets ([ ]), when writing KP expressions.

## KP and the Ideal Gas Law

The reason we are allowed to write a K expression in terms of partial pressures for gases can be found by looking at the ideal gas law. Recall that the ideal gas law is given by:

$PV=nRT$

Re-writing this expression in terms of P, we have:

$P=\frac{n}{V}RT$

Note that in order for K to be constant, temperature must be constant as well. Therefore, the term RT is a constant in the above expression. As for n/V (moles per unit volume) this is simply a measure of concentration. Pressure is directly proportional to concentration, so we are justified in our use of KP.

Lastly, there is a very important equation that relates KP and KC. It is given as follows:

$K_{p} = K_{c}(RT) ^{ \Delta n}$

In this expression, $\Delta n$ is a measure of the change in number of moles of gas in the reaction. For instance, if a reaction produces three moles of gas, and consumes two moles of gas, then $\Delta n=(3-2)=1$. Liquefied gasInside this tank, propane is compressed into a liquid, which is in equilibrium with its gaseous headspace. The internal pressure of the gaseous propane is a function of temperature.
{"ft_lang_label":"__label__en","ft_lang_prob":0.88093525,"math_prob":0.99283963,"size":2528,"snap":"2021-43-2021-49","text_gpt3_token_len":574,"char_repetition_ratio":0.15570523,"word_repetition_ratio":0.04568528,"special_character_ratio":0.22033228,"punctuation_ratio":0.09832636,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.998762,"pos_list":[0,1,2],"im_url_duplicate_count":[null,3,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-16T20:50:00Z\",\"WARC-Record-ID\":\"<urn:uuid:b1b09e59-fe6f-465b-a9db-c69cd123d92b>\",\"Content-Length\":\"32372\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:5be3da86-9fec-421e-82a6-da48dd17b257>\",\"WARC-Concurrent-To\":\"<urn:uuid:7f20bbb2-085a-45c8-94c4-39c882184f15>\",\"WARC-IP-Address\":\"23.185.0.1\",\"WARC-Target-URI\":\"https://courses.lumenlearning.com/introchem/chapter/expressing-the-equilibrium-constant-of-a-gas-in-terms-of-pressure/\",\"WARC-Payload-Digest\":\"sha1:ZRLLDXZC6C3QBMSH6M322KT2WHHOM7UO\",\"WARC-Block-Digest\":\"sha1:QKBCE7YHWHKRQ7USPGKDDCQCZWL7LR5N\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323585025.23_warc_CC-MAIN-20211016200444-20211016230444-00660.warc.gz\"}"}