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2.1
Let $X$ represent the quantity $V^2$ with dimensions $(\tx{length})^6$. Give a reason that $X$ is or is not an extensive property. Give a reason that $X$ is or is not an intensive property.
2.2
Calculate the relative uncertainty (the uncertainty divided by the value) for each of the measurement methods listed in Table 2.2, using the typical values shown. For each of the five physical quantities listed, which measurement method has the smallest relative uncertainty?
Table 2.5 Helium at a fixed temperature
2.3
Table 2.5 lists data obtained from a constant-volume gas thermometer containing samples of varying amounts of helium maintained at a certain fixed temperature $T_2$ in the gas bulb (K. H. Berry, Metrologia, 15, 89–115, 1979). The molar volume $V\m$ of each sample was evaluated from its pressure in the bulb at a reference temperature of $T_1=7.1992\K$, corrected for gas nonideality with the known value of the second virial coefficient at that temperature.
Use these data and Eq. 2.2.2 to evaluate $T_2$ and the second virial coefficient of helium at temperature $T_2$. (You can assume the third and higher virial coefficients are negligible.)
2.4
Discuss the proposition that, to a certain degree of approximation, a living organism is a steady-state system.
2.5
The value of $\Del U$ for the formation of one mole of crystalline potassium iodide from its elements at $25\units{\(\degC$}\) and $1\br$ is $-327.9\units{kJ}$. Calculate $\Del m$ for this process. Comment on the feasibility of measuring this mass change. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/02%3A_Systems_and_Their_Properties/2.07%3A_Chapter_2_Problems.txt |
In science, a law is a statement or mathematical relation that concisely describes reproducible experimental observations. Classical thermodynamics is built on a foundation of three laws, none of which can be derived from principles that are any more fundamental. This chapter discusses theoretical aspects of the first law; gives examples of reversible and irreversible processes and the heat and work that occur in them; and introduces the extensive state function heat capacity.
03: The First Law
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The box below gives two forms of the first law of thermodynamics.
The equation $\dif U=\dq+\dw$ is the differential form of the first law, and $\Del U=q+w$ is the integrated form.
The heat and work appearing in the first law are two different modes of energy transfer. They can be defined in a general way as follows.
• An infinitesimal quantity of energy transferred as heat at a surface element of the boundary is written $\dq$, and a finite quantity is written $q$ (Sec. 2.5). To obtain the total finite heat for a process from $q=\int\!\dq$ (Eq. 2.5.3), we must integrate over the total boundary surface and the entire path of the process.
An infinitesimal quantity of work is $\dw$, and a finite quantity is $w=\int\!\dw$. To obtain $w$ for a process, we integrate all kinds of work over the entire path of the process.
The first-law equation $\Del U=q+w$ sets up a balance sheet for the energy of the system, measured in the local frame, by equating its change during a process to the net quantity of energy transferred by means of heat and work. Note that the equation applies only to a closed system. If the system is open, energy can also be brought across the boundary by the transport of matter.
An important part of the first law is the idea that heat and work are quantitative energy transfers. That is, when a certain quantity of energy enters the system in the form of heat, the same quantity leaves the surroundings. When the surroundings perform work on the system, the increase in the energy of the system is equal in magnitude to the decrease in the energy of the surroundings. The principle of conservation of energy is obeyed: the total energy (the sum of the energies of the system and surroundings) remains constant over time.
Strictly speaking, it is the sum of the energies of the system, the surroundings, and any potential energy shared by both that is constant. The shared potential energy is usually negligible or essentially constant (Sec. G.5).
Heat transfer may occur by conduction, convection, or radiation. (Some thermodynamicists treat radiation as a separate contribution to $\Del U$, in addition to $q$ and $w$.) We can reduce conduction with good thermal insulation at the boundary, we can eliminate conduction and convection with a vacuum gap, and we can minimize radiation with highly reflective surfaces at both sides of the vacuum gap. The only way to completely prevent heat during a process is to arrange conditions in the surroundings so there is no temperature gradient at any part of the boundary. Under these conditions the process is adiabatic, and any energy transfer in a closed system is then solely by means of work.
3.1.1 The concept of thermodynamic work
Appendix G gives a detailed analysis of energy and work based on the behavior of a collection of interacting particles moving according to the principles of classical mechanics. The analysis shows how we should evaluate mechanical thermodynamic work. Suppose the displacement responsible for the work comes from linear motion of a portion of the boundary in the $+x$ or $-x$ direction of the local frame. The differential and integrated forms of the work are then given by $\dw=F\sur_x\dx \qquad w=\int_{x_1}^{x_2}\!\!F\sur_x\dx \tag{3.1.1}$ (These equations are Eq. G.6.11 with a change of notation.) Here $F\sur_x$ is the component in the $+x$ direction of the force exerted by the surroundings on the system at the moving portion of the boundary, and $\dx$ is the infinitesimal displacement of the boundary in the local frame. If the displacement is in the same direction as the force, $\dw$ is positive; if the displacement is in the opposite direction, $\dw$ is negative.
The kind of force represented by $F\sur_x$ is a short-range contact force. Appendix G shows that the force exerted by a conservative time-independent external field, such as a gravitational force, should not be included as part of $F\sur_x$. This is because the work done by this kind of force causes changes of potential and kinetic energies that are equal and opposite in sign, with no net effect on the internal energy (see Sec. 3.6).
Newton’s third law of action and reaction says that a force exerted by the surroundings on the system is opposed by a force of equal magnitude exerted in the opposite direction by the system on the surroundings. Thus the expressions in Eq. 3.1.1 can be replaced by $\dw=-F\sups{sys}_x\dx \qquad w=-\int_{x_1}^{x_2}\!\!F\sups{sys}_x\dx \tag{3.1.2}$ where $F\sups{sys}_x$ is the component in the $+x$ direction of the contact force exerted by the system on the surroundings at the moving portion of the boundary.
An alternative to using the expressions in Eqs. 3.1.1 or 3.1.2 for evaluating $w$ is to imagine that the only effect of the work on the system’s surroundings is a change in the elevation of a weight in the surroundings. The weight must be one that is linked mechanically to the source of the force $F\sur_x$. Then, provided the local frame is a stationary lab frame, the work is equal in magnitude and opposite in sign to the change in the weight’s potential energy: $w = -mg\Del h$ where $m$ is the weight’s mass, $g$ is the acceleration of free fall, and $h$ is the weight’s elevation in the lab frame. This interpretation of work can be helpful for seeing whether work occurs and for deciding on its sign, but of course cannot be used to determine its value if the actual surroundings include no such weight.
The procedure of evaluating $w$ from the change of an external weight’s potential energy requires that this change be the only mechanical effect of the process on the surroundings, a condition that in practice is met only approximately. For example, Joule’s paddle-wheel experiment using two weights linked to the system by strings and pulleys, described latter in Sec. 3.7.2, required corrections for (1) the kinetic energy gained by the weights as they sank, (2) friction in the pulley bearings, and (3) elasticity of the strings (see Prob. 3.10).
In the first-law relation $\Del U=q+w$, the quantities $\Del U$, $q$, and $w$ are all measured in an arbitrary local frame. We can write an analogous relation for measurements in a stationary lab frame: $\Del E\sys=q\lab+w\lab \tag{3.1.3}$ Suppose the chosen local frame is not a lab frame, and we find it more convenient to measure the heat $q\lab$ and the work $w\lab$ in a lab frame than to measure $q$ and $w$ in the local frame. What corrections are needed to find $q$ and $w$ in this case?
If the Cartesian axes of the local frame do not rotate relative to the lab frame, then the heat is the same in both frames: $q=q\lab$ (Sec. G.7).
The expressions for $\dw\lab$ and $w\lab$ are the same as those for $\dw$ and $w$ in Eqs. 3.1.1 and 3.1.2, with $\dx$ interpreted as the displacement in the lab frame. There is an especially simple relation between $w$ and $w\lab$ when the local frame is a center-of-mass frame—one whose origin moves with the system’s center of mass and whose axes do not rotate relative to the lab frame (Eq. G.8.12): $w = w\lab - \onehalf m \Del\!\left(v^2\cm\right) - mg\Del z\cm \tag{3.1.4}$ In this equation $m$ is the mass of the system, $v\cm$ is the velocity of its center of mass in the lab frame, $g$ is the acceleration of free fall, and $z\cm$ is the height of the center of mass above an arbitrary zero of elevation in the lab frame. In typical thermodynamic processes the quantities $v\cm$ and $z\cm$ change to only a negligible extent, if at all, so that usually to a good approximation $w$ is equal to $w\lab$.
When the local frame is a center-of-mass frame, we can combine the relations $\Del U=q+w$ and $q=q\lab$ with Eqs. 3.1.3 and 3.1.4 to obtain $\Del E\sys = \Del E\subs{k} + \Del E\subs{p} + \Del U \tag{3.1.5}$ where $E\subs{k} = \onehalf mv^2\cm$ and $E\subs{p} = mgz\cm$ are the kinetic and potential energies of the system as a whole in the lab frame.
A more general relation for $w$ can be written for any local frame that has no rotational motion and whose origin has negligible acceleration in the lab frame (Eq. G.7.3): $w = w\lab - mg \Del z\subs{loc} \tag{3.1.6}$ Here $z\subs{loc}$ is the elevation in the lab frame of the origin of the local frame. $\Del z\subs{loc}$ is usually small or zero, so again $w$ is approximately equal to $w\lab$. The only kinds of processes for which we may need to use Eq. 3.1.4 or 3.1.6 to calculate a non-negligible difference between $w$ and $w\lab$ are those in which massive parts of the system undergo substantial changes in elevation in the lab frame.
Simple relations such as these between $q$ and $q\lab$, and between $w$ and $w\lab$, do not exist if the local frame has rotational motion relative to a lab frame.
Hereafter in this e-book, thermodynamic work $w$ will be called simply work. For all practical purposes you can assume the local frames for most of the processes to be described are stationary lab frames. The discussion above shows that the values of heat and work measured in these frames are usually the same, or practically the same, as if they were measured in a local frame moving with the system’s center of mass. A notable exception is the local frame needed to treat the thermodynamic properties of a liquid solution in a centrifuge cell. In this case the local frame is fixed in the spinning rotor of the centrifuge and has rotational motion. This special case will be discussed in Sec. 9.8.2.
3.1.2 Work coefficients and work coordinates
If a process has only one kind of work, it can be expressed in the form $\dw = Y\dif X \qquad \tx{or} \qquad w = \int_{X_1}^{X_2} \! Y\dif X \tag{3.1.7}$ where $Y$ is a generalized force called a work coefficient and $X$ is a generalized displacement called a work coordinate. The work coefficient and work coordinate are conjugate variables. They are not necessarily actual forces and displacements. For example, we shall see in Sec. 3.4.2 that reversible expansion work is given by $\dw=-p\dif V$; in this case, the work coefficient is $-p$ and the work coordinate is $V$.
A process may have more than one kind of work, each with its own work coefficient and conjugate work coordinate. In this case the work can be expressed as a sum over the different kinds labeled by the index $i$: $\dw = \sum_i Y_i \dif X_i \qquad \tx{or} \qquad w = \sum_i \int_{X_{i,1}}^{X_{i,2}} \! Y_i \dif X_i \tag{3.1.8}$
3.1.3 Heat and work as path functions
Consider the apparatus shown in Fig. 3.1. The system consists of the water together with the immersed parts: stirring paddles attached to a shaft (a paddle wheel) and an electrical resistor attached to wires. In equilibrium states of this system, the temperature and pressure are uniform and the paddle wheel is stationary. The system is open to the atmosphere, so the pressure is constrained to be constant. We may describe the equilibrium states of this system by a single independent variable, the temperature $T$. (The angular position of the shaft is irrelevant to the state and is not a state function for equilibrium states of this system.)
Here are three experiments with different processes. Each process has the same initial state defined by $T_1 = 300.0\K$, and each has the same final state.
• Although the paths in the three experiments are entirely different, the overall change of state is the same. In fact, a person who observes only the initial and final states and has no knowledge of the intermediate states or the changes in the surroundings will be ignorant of the path. Did the paddle wheel turn? Did an electric current pass through the resistor? How much energy was transferred by work and how much by heat? The observer cannot tell from the change of state, because heat and work are not state functions. The change of state depends on the sum of heat and work. This sum is the change in the state function $U\!$, as expressed by the integrated form of the first law, $\Del U = q + w$.
It follows from this discussion that neither heat nor work are quantities possessed by the system. A system at a given instant does not have or contain a particular quantity of heat or a particular quantity of work. Instead, heat and work depend on the path of a process occurring over a period of time. They are path functions.
3.1.4 Heat and heating
In everyday speech the noun heat is often used somewhat differently. Here are three statements with similar meanings that could be misleading:
“Heat is transferred from a laboratory hot plate to a beaker of water.”
“Heat flows from a warmer body to a cooler body.”
“To remove heat from a hot body, place it in cold water.”
Statements such as these may give the false impression that heat is like a substance that retains its identity as it moves from one body to another. Actually heat, like work, does not exist as an entity once a process is completed. Nevertheless, the wording of statements such as these is embedded in our everyday language, and no harm is done if we interpret them correctly. This e-book, for conciseness, often refers to “heat transfer” and “heat flow,” instead of using the technically more correct phrase “energy transfer by means of heat.”
Another common problem is failure to distinguish between thermodynamic “heat” and the process of “heating.” To heat a system is to cause its temperature to increase. A heated system is one that has become warmer. This process of heating does not necessarily involve thermodynamic heat; it can also be carried out with work as illustrated by experiments 1 and 2 of the preceding section.
The notion of heat as an indestructible substance was the essence of the caloric theory. This theory was finally disproved by the cannon-boring experiments of Benjamin Thompson (Count Rumford) in the late eighteenth century, and in a more quantitative way by the measurement of the mechanical equivalent of heat by James Joule in the 1840s (see Sec. 3.7.2).
3.1.5 Heat capacity
The heat capacity of a closed system is defined as the ratio of an infinitesimal quantity of heat transferred across the boundary under specified conditions and the resulting infinitesimal temperature change: \begin{gather} \s{ \tx{heat capacity} \defn \frac{\dq}{\dif T} } \tag{3.1.9} \cond{(closed system)} \end{gather} Since $q$ is a path function, the value of the heat capacity depends on the specified conditions, usually either constant volume or constant pressure. $C_V$ is the heat capacity at constant volume and $C_p$ is the heat capacity at constant pressure. These are extensive state functions that will be discussed more fully in Sec. 5.6.
3.1.6 Thermal energy
It is sometimes useful to use the concept of thermal energy. It can be defined as the kinetic energy of random translational motions of atoms and molecules relative to the local frame, plus the vibrational and rotational energies of molecules. The thermal energy of a body or phase depends on its temperature, and increases when the temperature increases. The thermal energy of a system is a contribution to the internal energy.
It is important to understand that a change of the system’s thermal energy during a process is not necessarily the same as energy transferred across the system boundary as heat. The two quantities are equal only if the system is closed and there is no work, volume change, phase change, or chemical reaction. This is illustrated by the three experiments described in Sec. 3.1.3: the thermal energy change is the same in each experiment, but only in experiment 3 is the work negligible and the thermal energy change equal to the heat. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/03%3A_The_First_Law/3.01%3A_Heat_Work_and_the_First_Law.txt |
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A spontaneous process is a process that can actually occur in a finite time period under the existing conditions. Any change over time in the state of a system that we observe experimentally is a spontaneous process.
A spontaneous process is sometimes called a natural process, feasible process, possible process, allowed process, or real process.
3.2.1 Reversible processes
A reversible process is an important concept in thermodynamics. This concept is needed for the chain of reasoning that will allow us to define entropy changes in the next chapter, and will then lead on to the establishment of criteria for spontaneity and for various kinds of equilibria.
Before reversible processes can be discussed, it is necessary to explain the meaning of the reverse of a process. If a particular process takes the system from an initial state A through a continuous sequence of intermediate states to a final state B, then the reverse of this process is a change over time from state B to state A with the same intermediate states occurring in the reverse time sequence. To visualize the reverse of any process, imagine making a movie film of the events of the process. Each frame of the film is a “snapshot” picture of the state at one instant. If you run the film backward through a movie projector, you see the reverse process: the values of system properties such as $p$ and $V$ appear to change in reverse chronological order, and each velocity changes sign.
The concept of a reversible process is not easy to describe or to grasp. Perhaps the most confusing aspect is that a reversible process is not a process that ever actually occurs, but is only approached as a hypothetical limit. During a reversible process the system passes through a continuous sequence of equilibrium states. These states are ones that can be approached, as closely as desired, by the states of a spontaneous process carried out sufficiently slowly. As the spontaneous process is carried out more and more slowly, it approaches the reversible limit. Thus, a reversible process is an idealized process with a sequence of equilibrium states that are those of a spontaneous process in the limit of infinite slowness.
This e-book has many equations expressing relations among heat, work, and state functions during various kinds of reversible processes. What is the use of an equation for a process that can never actually occur? The point is that the equation can describe a spontaneous process to a high degree of accuracy, if the process is carried out slowly enough for the intermediate states to depart only slightly from exact equilibrium states. For example, for many important spontaneous processes we will assume the temperature and pressure are uniform throughout the system, which strictly speaking is an approximation.
A reversible process of a closed system, as used in this e-book, has all of the following characteristics:
• We must imagine the reversible process to proceed at a finite rate, otherwise there would be no change of state over time. The precise rate of the change is not important. Imagine a gas whose volume, temperature, and pressure are changing at some finite rate while the temperature and pressure magically stay perfectly uniform throughout the system. This is an entirely imaginary process, because there is no temperature or pressure gradient—no physical “driving force”—that would make the change tend to occur in a particular direction. This imaginary process is a reversible process—one whose states of uniform temperature and pressure are approached by the states of a real process as the real process takes place more and more slowly.
It is a good idea, whenever you see the word “reversible,” to think “in the reversible limit.” Thus a reversible process is a process in the reversible limit, reversible work is work in the reversible limit, and so on.
3.2.2 Irreversible processes
An irreversible process is a spontaneous process whose reverse is neither spontaneous nor reversible. That is, the reverse of an irreversible process can never actually occur and is impossible. If a movie is made of a spontaneous process, and the time sequence of the events depicted by the film when it is run backward could not occur in reality, the spontaneous process is irreversible.
A good example of a spontaneous, irreversible process is experiment 1 in Section 3.1.3, in which the sinking of an external weight immersed in water causes a paddle wheel to rotate and the temperature of the water to increase. During this experiment mechanical energy is dissipated into thermal energy. Suppose you insert a thermometer in the water and make a movie film of the experiment. Then when you run the film backward in a projector, you will see the paddle wheel rotating in the direction that raises the weight, and the water becoming cooler according to the thermometer. Clearly, this reverse process is impossible in the real physical world, and the process occurring during the experiment is irreversible. It is not difficult to understand why it is irreversible when we consider events on the microscopic level: it is extremely unlikely that the H$_2$O molecules next to the paddles would happen to move simultaneously over a period of time in the concerted motion needed to raise the weight.
3.2.3 Purely mechanical processes
There is a class of spontaneous processes that are also spontaneous in reverse; that is, spontaneous but not irreversible. These are purely mechanical processes involving the motion of perfectly-elastic macroscopic bodies without friction, temperature gradients, viscous flow, or other irreversible changes.
A simple example of a purely mechanical process and its reverse is shown in Fig. 3.2. The ball can move spontaneously in either direction. Another example is a flywheel with frictionless bearings rotating in a vacuum.
A purely mechanical process proceeding at a finite rate is not reversible, for its states are not equilibrium states. Such a process is an idealization, of a different kind than a reversible process, and is of little interest in chemistry. Later chapters of this e-book will ignore such processes and will treat the terms spontaneous and irreversible as synonyms. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/03%3A_The_First_Law/3.02%3A_Spontaneous_Reversible_and_Irreversible_Processes.txt |
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This section describes irreversible and reversible heat transfer. Keep in mind that when this e-book refers to heat transfer or heat flow, energy is being transferred across the boundary on account of a temperature gradient at the boundary. The transfer is always in the direction of decreasing temperature.
We may sometimes wish to treat the temperature as if it is discontinuous at the boundary, with different values on either side. The transfer of energy is then from the warmer side to the cooler side. The temperature is not actually discontinuous; instead there is a thin zone with a temperature gradient.
3.3.1 Heating and cooling
As an illustration of irreversible heat transfer, consider a system that is a solid metal sphere. This spherical body is immersed in a well-stirred water bath whose temperature we can control. The bath and the metal sphere are initially equilibrated at temperature $T_1=300.0\K$, and we wish to raise the temperature of the sphere by one kelvin to a final uniform temperature $T_2=301.0\K$.
One way to do this is to rapidly increase the external bath temperature to $301.0\K$ and keep it at that temperature. The temperature difference across the surface of the immersed sphere then causes a spontaneous flow of heat through the system boundary into the sphere. It takes time for all parts of the sphere to reach the higher temperature, so a temporary internal temperature gradient is established. Thermal energy flows spontaneously from the higher temperature at the boundary to the lower temperature in the interior. Eventually the temperature in the sphere becomes uniform and equal to the bath temperature of $301.0\K$.
Figure 3.3(a) graphically depicts temperatures within the sphere at different times during the heating process. Note the temperature gradient in the intermediate states. Because of the gradient, these states cannot be characterized by a single value of the temperature. If we were to suddenly isolate the system (the sphere) with a thermally-insulated jacket while it is in one of these states, the state would change as the temperature gradient rapidly disappears. Thus, the intermediate states of the spontaneous heating process are not equilibrium states, and the rapid heating process is not reversible.
To make the intermediate states more nearly uniform in temperature, with smaller temperature gradients, we can raise the temperature of the bath at a slower rate. The sequence of states approached in the limit of infinite slowness is indicated in Fig. 3.3(b). In each intermediate state of this limiting sequence, the temperature is perfectly uniform throughout the sphere and is equal to the external bath temperature. That is, each state has thermal equilibrium both internally and with respect to the surroundings. A single temperature now suffices to define the state at each instant. Each state is an equilibrium state because it would have no tendency to change if we isolated the system with thermal insulation. This limiting sequence of states is a reversible heating process.
The reverse of the reversible heating process is a reversible cooling process in which the temperature is again uniform in each state. The sequence of states of this reverse process is the limit of the spontaneous cooling process depicted in Fig. 3.3(c) as we decrease the bath temperature more and more slowly.
In any real heating process occurring at a finite rate, the sphere’s temperature could not be perfectly uniform in intermediate states. If we raise the bath temperature very slowly, however, the temperature in all parts of the sphere will be very close to that of the bath. At any point in this very slow heating process, it would then take only a small decrease in the bath temperature to start a cooling process; that is, the practically-reversible heating process would be reversed.
The important thing to note about the temperature gradients shown in Fig. 3.3(c) for the spontaneous cooling process is that none resemble the gradients in Fig. 3.3(a) for the spontaneous heating process—the gradients are in opposite directions. It is physically impossible for the sequence of states of either process to occur in the reverse chronological order, for that would have thermal energy flowing in the wrong direction along the temperature gradient. These considerations show that a spontaneous heat transfer is irreversible. Only in the reversible limits do the heating and cooling processes have the same intermediate states; these states have no temperature gradients.
Although the spontaneous heating and cooling processes are irreversible, the energy transferred into the system during heating can be fully recovered as energy transferred back to the surroundings during cooling, provided there is no irreversible work. This recoverability of irreversible heat is in distinct contrast to the behavior of irreversible work.
3.3.2 Spontaneous phase transitions
Consider a different kind of system, one consisting of the liquid and solid phases of a pure substance. At a given pressure, this kind of system can be in transfer equilibrium at only one temperature: for example, water and ice at $1.01\br$ and $273.15\K$. Suppose the system is initially at this pressure and temperature. Heat transfer into the system will then cause a phase transition from solid to liquid (Sec. 2.2.2). We can carry out the heat transfer by placing the system in thermal contact with an external water bath at a higher temperature than the equilibrium temperature, which will cause a temperature gradient in the system and the melting of an amount of solid proportional to the quantity of energy transferred.
The closer the external temperature is to the equilibrium temperature, the smaller are the temperature gradients and the closer are the states of the system to equilibrium states. In the limit as the temperature difference approaches zero, the system passes through a sequence of equilibrium states in which the temperature is uniform and constant, energy is transferred into the system by heat, and the substance is transformed from solid to liquid. This idealized process is an equilibrium phase transition, and it is a reversible process. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/03%3A_The_First_Law/3.03%3A_Heat_Transfer.txt |
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Figure 3.9 depicts a spherical body, such as a glass marble, immersed in a liquid or gas in the presence of an external gravitational field. The vessel is stationary on a lab bench, and the local reference frame for work is a stationary lab frame. The variable $z$ is the body’s elevation above the bottom of the vessel. All displacements are parallel to the vertical $z$ axis. From Eq. 3.1.1, the work is given by $\dw=F\sur_z\dif z$ where $F\sur_z$ is the upward component of the net contact force exerted by the surroundings on the system at the moving portion of the boundary. There is also a downward gravitational force on the body, but as explained in Sec. 3.1.1, this force does not contribute to $F\sur_z$.
Consider first the simple process in Fig. 3.9(a) in which the body falls freely through the fluid. This process is clearly spontaneous. Here are two choices for the definition of the system:
• The buoyant force is a consequence of the pressure gradient that exists in the fluid in a gravitational field (see Sec. 8.1.4). We ignore this gradient when we treat the fluid as a uniform phase.
Next, consider the arrangement in Fig. 3.9(b) in which the body is suspended by a thin string. The string is in the surroundings and provides a means for the surroundings to exert an upward contact force on the system. As before, there are two appropriate choices for the system:
• When we compare Eqs. 3.6.3 and 3.6.5, we see that the work when the system is the body is greater by the quantity $\left( F\subs{buoy}+F\fric \right)\dif z$ than the work when the system is the combination of body and fluid, just as in the case of the freely-falling body. The difference in the quantity of work is due to the different choices of the system boundary where contact forces are exerted by the surroundings. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/03%3A_The_First_Law/3.06%3A_Work_in_a_Gravitational_Field.txt |
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Shaft work refers to energy transferred across the boundary by a rotating shaft.
The complete apparatus is depicted in Fig. 3.13. In use, two lead weights sank and caused the paddle wheel to rotate. Joule evaluated the stirring work done on the system (the vessel, its contents, and the lid) from the change of the vertical position $h$ of the weights. To a first approximation, this work is the negative of the change of the weights’ potential energy: $w = -mg\Del h$ where $m$ is the combined mass of the two weights. Joule made corrections for the kinetic energy gained by the weights, the friction in the connecting strings and pulley bearings, the elasticity of the strings, and the heat gain from the air surrounding the system.
A typical experiment performed by Joule is described in Prob. 3.10. His results for the mechanical equivalent of heat, based on 40 such experiments at average temperatures in the range $13{\thinspace}\degC$–$16{\thinspace}\degC$ and expressed as the work needed to increase the temperature of one gram of water by one kelvin, was $4.165\units{J}$. This value is close to the modern value of $4.1855\units{J}$ for the “$15{\thinspace}\degC$ calorie,” the energy needed to raise the temperature of one gram of water from $14.5{\thinspace}\degC$ to $15.5{\thinspace}\degC$.
The thermochemical calorie (cal), often used as an energy unit in the older literature, is defined as $4.184\units{J}$. Thus $1\units{kcal} = 4.184\units{kJ}$. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/03%3A_The_First_Law/3.07%3A_Shaft_Work.txt |
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Consider an irreversible adiabatic process of a closed system in which a work coordinate $X$ changes at a finite rate along the path, starting and ending with equilibrium states. For a given initial state and a given change $\Del X$, the work is found to be less positive or more negative the more slowly is the rate of change of $X$. The work is least positive or most negative in the reversible limit—that is, the least work needs to be done on the system, or the most work can be done by the system on the surroundings. This minimal work principle has already been illustrated in the sections of this chapter describing expansion work, work in a gravitational field, and electrical work with a galvanic cell.
Let the work during an irreversible adiabatic process be $w\irr$, and the reversible adiabatic work for the same initial state and the same value of $\Del X$ be $w\rev$. $w\irr$ is algebraically greater than $w\rev$, and we can treat the difference $w\irr{-}w\rev$ as excess work $w\subs{ex}$ that is positive for an irreversible process and zero for a reversible process.
Conceptually, we can attribute the excess work of an irreversible adiabatic process to internal friction that dissipates other forms of energy into thermal energy within the system. Internal friction occurs only during a process with work that is irreversible . Internal friction is not involved when, for example, a temperature gradient causes heat to flow spontaneously across the system boundary, or an irreversible chemical reaction takes place spontaneously in a homogeneous phase. Nor is internal friction necessarily involved when positive work increases the thermal energy: during an infinitely slow adiabatic compression of a gas, the temperature and thermal energy increase but internal friction is absent—the process is reversible.
During a process with irreversible work, energy dissipation can be either partial or complete. Dissipative work, such as the stirring work and electrical heating described in previous sections, is irreversible work with complete energy dissipation. The final equilibrium state of an adiabatic process with dissipative work can also be reached by a path in which positive heat replaces the dissipative work. This is a special case of the minimal work principle.
Figure 3.18 Cylinder and piston with internal sliding friction. The dashed rectangle indicates the system boundary. P—piston; R—internal rod attached to the piston; B—lubricated bushing fixed inside the cylinder. A fixed amount of an ideal gas fills the remaining space inside the cylinder.
As a model for work with partial energy dissipation, consider the gas-filled cylinder-and-piston device depicted in Fig. 3.18. This device has an obvious source of internal friction in the form of a rod sliding through a bushing. The contact between the rod and bushing is assumed to be lubricated to allow the piston to move at velocities infinitesimally close to zero. The system consists of the contents of the cylinder to the left of the piston, including the gas, the rod, and the bushing; the piston and cylinder wall are in the surroundings.
From Eq. 3.1.2, the energy transferred as work across the boundary of this system is $w = -\int_{x_1}^{x_2}\!\! F\sups{sys}\dx \tag{3.9.1}$ where $x$ is the piston position and $F\sups{sys}$ is the component in the direction of increasing $x$ of the force exerted by the system on the surroundings at the moving boundary.
For convenience, we let $V$ be the volume of the gas rather than that of the entire system. The relation between changes of $V$ and $x$ is $\dif V=\As\dx$ where $\As$ is the cross-section area of the cylinder. With $V$ replacing $x$ as the work coordinate, Eq. 3.9.1 becomes $w=-\int_{V_1}^{V_2}\!\!\left(F\sups{sys}/\As\right)\dif V \tag{3.9.2}$ Equation 3.9.2 shows that a plot of $F\sups{sys}/\As$ as a function of $V$ is an indicator diagram (Sec. 3.5.4), and that the work is equal to the negative of the area under the curve of this plot.
We can write the force $F\sups{sys}$ as the sum of two contributions: $F\sups{sys} = p\As + F\fric\in \tag{3.9.3}$ (This equation assumes that the gas pressure is uniform, and that a term for the acceleration of the rod is negligible.) Here $p$ is the gas pressure, and $F\fric\in$ is the force on the rod due to internal friction with sign opposite to that of the piston velocity $\dx/\dt$. Substitution of this expression for $F\sups{sys}$ in Eq. 3.9.2 gives $w=-\int_{V_1}^{V_2}\!\!\!p\dif V - \int_{V_1}^{V_2}\!\!(F\fric\in/\As)\dif V \tag{3.9.4}$ The first term on the right is the work of expanding or compressing the gas. The second term is the frictional work: $w\fric = -\int(F\fric\in/\As)\dif V$. The frictional work is positive or zero, and represents the energy dissipated within the system by internal sliding friction.
Consider the situation when the piston moves very slowly in one direction or the other. In the limit of infinite slowness $F\fric\in$ and $w\fric$ vanish, and the process is reversible with expansion work given by $w=-\int\!\!p\dif V$.
The situation is different when the piston moves at an appreciable finite rate. The frictional work $w\fric$ is then positive. As a result, the irreversible work of expansion is less negative than the reversible work for the same volume increase, and the irreversible work of compression is more positive than the reversible work for the same volume decrease. These effects of piston velocity on the work are consistent with the minimal work principle.
The piston velocity, besides affecting the frictional force on the rod, has an effect on the force exerted by the gas on the piston as described in Sec. 3.4.1. At large finite velocities, this latter effect tends to further decrease $F\sups{sys}$ during expansion and increase it during compression, and so is an additional contribution to internal friction. If turbulent flow is present within the system, that would also be a contribution.
Figure 3.19 Indicator diagrams for the system of Fig. 3.18.
Solid curves: $F\sups{sys}/\As$ for irreversible adiabatic volume changes at finite rates in the directions indicated by the arrows.
Dashed curves: $F\sups{sys}/\As=p$ along a reversible adiabat.
Open circles: initial and final equilibrium states.
(a) Adiabatic expansion.
(b) Adiabatic compression.
Figure 3.19 shows indicator diagrams for adiabatic expansion and compression with internal friction. The solid curves are for irreversible processes at finite rates, and the dashed curves are for reversible processes with the same initial states as the irreversible processes. The areas under the curves confirm that the work for expansion is less negative along the irreversible path than along the reversible path, and that for compression the work is more positive along the irreversible path than along the reversible path.
Because of these differences in work, the final states of the irreversible processes have greater internal energies and higher temperatures and pressures than the final states of the reversible processes with the same volume change, as can be seen from the positions on the indicator diagrams of the points for the final equilibrium states. The overall change of state during the irreversible expansion or compression is the same for a path in which the reversible adiabatic volume change is followed by positive heat at constant volume. Since $\Del U$ must be the same for both paths, the heat has the same value as the excess work $w\subs{ex}=w\irr{-}w\rev$. The excess work and frictional work are not equal, because the thermal energy released by frictional work increases the gas pressure, making $w\subs{ex}$ less than $w\fric$ for expansion and greater than $w\fric$ for compression. There seems to be no general method by which the energy dissipated by internal friction can be evaluated, and it would be even more difficult for an irreversible process with both work and heat.
Figure 3.20 Adiabatic expansion work with internal friction for a fixed magnitude of $\Del V$, as a function of the average rate of volume change. The open circles indicate the reversible limits.
Figure 3.20 shows the effect of the rate of change of the volume on the adiabatic work for a fixed magnitude of the volume change. Note that the work of expansion and the work of compression have opposite signs, and that it is only in the reversible limit that they have the same magnitude. The figure resembles Fig. 3.17 for electrical work of a galvanic cell with the horizontal axis reversed, and is typical of irreversible work with partial energy dissipation. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/03%3A_The_First_Law/3.09%3A_Irreversible_Work_and_Internal_Friction.txt |
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This section summarizes some general characteristics of processes in closed systems. Some of these statements will be needed to develop aspects of the second law in Chap. 4.
• Table 3.1 lists general formulas for various kinds of work, including those that were described in detail in Secs. 3.4–3.8. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/03%3A_The_First_Law/3.10%3A_Reversible_and_Irreversible_Processes-_Generalities.txt |
An underlined problem number or problem-part letter indicates that the numerical answer appears in Appendix I.
3.1 Assume you have a metal spring that obeys Hooke's law: $F=c\left(l-l_{0}\right)$, where $F$ is the force exerted on the spring of length $l, l_{0}$ is the length of the unstressed spring, and $c$ is the spring constant. Find an expression for the work done on the spring when you reversibly compress it from length $l_{0}$ to a shorter length $l^{\prime}$.
3.2 The apparatus shown in Fig. $3.22$ consists of fixed amounts of water and air and an incompressible solid glass sphere (a marble), all enclosed in a rigid vessel resting on a lab bench. Assume the marble has an adiabatic outer layer so that its temperature cannot change, and that the walls of the vessel are also adiabatic.
Initially the marble is suspended above the water. When released, it falls through the air into the water and comes to rest at the bottom of the vessel, causing the water and air (but not the marble) to become slightly warmer. The process is complete when the system returns to an equilibrium state. The system energy change during this process depends on the frame of reference and on how the system is defined. $\Delta E_{\text {sys }}$ is the energy change in a lab frame, and $\Delta U$ is the energy change in a specified local frame.
For each of the following definitions of the system, give the sign (positive, negative, or zero) of both $\Delta E_{\text {sys }}$ and $\Delta U$, and state your reasoning. Take the local frame for each system to be a center-of-mass frame.
(a) The system is the marble.
(b) The system is the combination of water and air.
(c) The system is the combination of water, air, and marble.
3.3 Figure $3.23$ shows the initial state of an apparatus consisting of an ideal gas in a bulb, a stopcock, a porous plug, and a cylinder containing a frictionless piston. The walls are diathermal, and the surroundings are at a constant temperature of $300.0 \mathrm{~K}$ and a constant pressure of $1.00 \mathrm{bar}$.
When the stopcock is opened, the gas diffuses slowly through the porous plug, and the piston moves slowly to the right. The process ends when the pressures are equalized and the piston stops moving. The system is the gas. Assume that during the process the temperature throughout the system differs only infinitesimally from $300.0 \mathrm{~K}$ and the pressure on both sides of the piston differs only infinitesimally from $1.00$ bar.
(a) Which of these terms correctly describes the process: isothermal, isobaric, isochoric, reversible, irreversible?
(b) Calculate $q$ and $w$.
3.4 Consider a horizontal cylinder-and-piston device similar to the one shown in Fig. $3.5$ on page 72. The piston has mass $m$. The cylinder wall is diathermal and is in thermal contact with a heat reservoir of temperature $T_{\text {ext }}$. The system is an amount $n$ of an ideal gas confined in the cylinder by the piston.
The initial state of the system is an equilibrium state described by $p_{1}$ and $T=T_{\text {ext }}$. There is a constant external pressure $p_{\text {ext }}$, equal to twice $p_{1}$, that supplies a constant external force on the piston. When the piston is released, it begins to move to the left to compress the gas. Make the idealized assumptions that (1) the piston moves with negligible friction; and (2) the gas remains practically uniform (because the piston is massive and its motion is slow) and has a practically constant temperature $T=T_{\text {ext }}$ (because temperature equilibration is rapid).
(a) Describe the resulting process.
(b) Describe how you could calculate $w$ and $q$ during the period needed for the piston velocity to become zero again.
(c) Calculate $w$ and $q$ during this period for $0.500 \mathrm{~mol}$ gas at $300 \mathrm{~K}$.
3.5 This problem is designed to test the assertion on page 60 that for typical thermodynamic processes in which the elevation of the center of mass changes, it is usually a good approximation to set $w$ equal to $w_{\text {lab. }}$. The cylinder shown in Fig. $3.24$ on the preceding page has a vertical orientation, so the elevation of the center of mass of the gas confined by the piston changes as the piston slides up or down. The system is the gas. Assume the gas is nitrogen $\left(M=28.0 \mathrm{~g} \mathrm{~mol}^{1}\right)$ at $300 \mathrm{~K}$, and initially the vertical length $l$ of the gas column is one meter. Treat the nitrogen as an ideal gas, use a center-of-mass local frame, and take the center of mass to be at the midpoint of the gas column. Find the difference between the values of $w$ and $w_{\text {lab }}$, expressed as a percentage of $w$, when the gas is expanded reversibly and isothermally to twice its initial volume.
3.6 Figure $3.25$ shows an ideal gas confined by a frictionless piston in a vertical cylinder. The system is the gas, and the boundary is adiabatic. The downward force on the piston can be varied by changing the weight on top of it.
(a) Show that when the system is in an equilibrium state, the gas pressure is given by $p=$ $m g h / V$ where $m$ is the combined mass of the piston and weight, $g$ is the acceleration of free fall, and $h$ is the elevation of the piston shown in the figure.
(b) Initially the combined mass of the piston and weight is $m_{1}$, the piston is at height $h_{1}$, and the system is in an equilibrium state with conditions $p_{1}$ and $V_{1}$. The initial temperature is $T_{1}=p_{1} V_{1} / n R$. Suppose that an additional weight is suddenly placed on the piston, so that $m$ increases from $m_{1}$ to $m_{2}$, causing the piston to sink and the gas to be compressed adiabatically and spontaneously. Pressure gradients in the gas, a form of friction, eventually cause the piston to come to rest at a final position $h_{2}$. Find the final volume, $V_{2}$, as a function of $p_{1}, p_{2}, V_{1}$, and $C_{V} .$ (Assume that the heat capacity of the gas, $C_{V}$, is independent of temperature.) Hint: The potential energy of the surroundings changes by $m_{2} g \Delta h$; since the kinetic energy of the piston and weights is zero at the beginning and end of the process, and the boundary is adiabatic, the internal energy of the gas must change by $-m_{2} g \Delta h=-m_{2} g \Delta V / A_{\mathrm{s}}=-p_{2} \Delta V$.
(c) It might seem that by making the weight placed on the piston sufficiently large, $V_{2}$ could be made as close to zero as desired. Actually, however, this is not the case. Find expressions for $V_{2}$ and $T_{2}$ in the limit as $m_{2}$ approaches infinity, and evaluate $V_{2} / V_{1}$ in this limit if the heat capacity is $C_{V}=(3 / 2) n R$ (the value for an ideal monatomic gas at room temperature).
3.7 The solid curve in Fig. $3.7$ shows the path of a reversible adiabatic expansion or compression of a fixed amount of an ideal gas. Information about the gas is given in the figure caption. For compression along this path, starting at $V=0.3000 \mathrm{dm}^{3}$ and $T=300.0 \mathrm{~K}$ and ending at $V=0.1000 \mathrm{dm}^{3}$, find the final temperature to $0.1 \mathrm{~K}$ and the work.
3.8 Figure $3.26$ shows the initial state of an apparatus containing an ideal gas. When the stopcock is opened, gas passes into the evacuated vessel. The system is the gas. Find $q, w$, and $\Delta U$ under the following conditions.
(a) The vessels have adiabatic walls.
(b) The vessels have diathermal walls in thermal contact with a water bath maintained at $300 . \mathrm{K}$, and the final temperature in both vessels is $T=300 . \mathrm{K}$.
3.9 Consider a reversible process in which the shaft of system A in Fig. $3.11$ makes one revolution in the direction of increasing $\vartheta$. Show that the gravitational work of the weight is the same as the shaft work given by $w=m g r \Delta \vartheta$.
Table 3.2 Data for Problem 3.10. The values are from Joule's 1850 paper $^{a}$ and have been converted to SI units.
${ }^{a}$ Ref. [91], p. 67, experiment $5 .$
${ }^{b}$ Calculated from the masses and specific heat capacities of the materials.
3.10 This problem guides you through a calculation of the mechanical equivalent of heat using data from one of James Joule's experiments with a paddle wheel apparatus (see Sec. 3.7.2). The experimental data are collected in Table 3.2.
In each of his experiments, Joule allowed the weights of the apparatus to sink to the floor twenty times from a height of about $1.6 \mathrm{~m}$, using a crank to raise the weights before each descent (see Fig. $3.14$ on page 89). The paddle wheel was engaged to the weights through the roller and strings only while the weights descended. Each descent took about 26 seconds, and the entire experiment lasted 35 minutes. Joule measured the water temperature with a sensitive mercury-in-glass thermometer at both the start and finish of the experiment.
For the purposes of the calculations, define the system to be the combination of the vessel, its contents (including the paddle wheel and water), and its lid. All energies are measured in a lab frame. Ignore the small quantity of expansion work occurring in the experiment. It helps conceptually to think of the cellar room in which Joule set up his apparatus as being effectively isolated from the rest of the universe; then the only surroundings you need to consider for the calculations are the part of the room outside the system.
(a) Calculate the change of the gravitational potential energy $E_{\mathrm{p}}$ of the lead weights during each of the descents. For the acceleration of free fall at Manchester, England (where Joule carried out the experiment) use the value $g=9.813 \mathrm{~m} \mathrm{~s}^{-2}$. This energy change represents a decrease in the energy of the surroundings, and would be equal in magnitude and opposite in sign to the stirring work done on the system if there were no other changes in the surroundings.
(b) Calculate the kinetic energy $E_{\mathrm{k}}$ of the descending weights just before they reached the floor. This represents an increase in the energy of the surroundings. (This energy was dissipated into thermal energy in the surroundings when the weights came to rest on the floor.)
(c) Joule found that during each descent of the weights, friction in the strings and pulleys decreased the quantity of work performed on the system by $2.87 \mathrm{~J}$. This quantity represents an increase in the thermal energy of the surroundings. Joule also considered the slight stretching of the strings while the weights were suspended from them: when the weights came to rest on the floor, the tension was relieved and the potential energy of the strings changed by $-1.15 \mathrm{~J}$. Find the total change in the energy of the surroundings during the entire experiment from all the effects described to this point. Keep in mind that the weights descended 20 times during the experiment.
(d) Data in Table $3.2$ show that change of the temperature of the system during the experiment was
$\Delta T=(289.148-288.829) \mathrm{K}=+0.319 \mathrm{~K}$
The paddle wheel vessel had no thermal insulation, and the air temperature was slighter warmer, so during the experiment there was a transfer of some heat into the system. From a correction procedure described by Joule, the temperature change that would have occurred if the vessel had been insulated is estimated to be $+0.317 \mathrm{~K}$.
Use this information together with your results from part (c) to evaluate the work needed to increase the temperature of one gram of water by one kelvin. This is the "mechanical equivalent of heat" at the average temperature of the system during the experiment. (As mentioned on p. 87 , Joule obtained the value $4.165 \mathrm{~J}$ based on all 40 of his experiments.)
3.11 Refer to the apparatus depicted in Fig. $3.1$ on page 61. Suppose the mass of the external weight is $m=1.50 \mathrm{~kg}$, the resistance of the electrical resistor is $R_{\mathrm{el}}=5.50 \mathrm{k} \Omega$, and the acceleration of free fall is $g=9.81 \mathrm{~m} \mathrm{~s}^{-2}$. For how long a period of time will the external cell need to operate, providing an electric potential difference $|\Delta \phi|=1.30 \mathrm{~V}$, to cause the same change in the state of the system as the change when the weight sinks $20.0 \mathrm{~cm}$ without electrical work? Assume both processes occur adiabatically. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/03%3A_The_First_Law/3.11%3A_Chapter_3_Problems.txt |
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The second law of thermodynamics concerns entropy and the spontaneity of processes. This chapter discusses theoretical aspects and practical applications.
We have seen that the first law allows us to set up a balance sheet for energy changes during a process, but says nothing about why some processes occur spontaneously and others are impossible. The laws of physics explain some spontaneous changes. For instance, unbalanced forces on a body cause acceleration, and a temperature gradient at a diathermal boundary causes heat transfer. But how can we predict whether a phase change, a transfer of solute from one solution phase to another, or a chemical reaction will occur spontaneously under the existing conditions? The second law provides the principle we need to answer these and other questions—a general criterion for spontaneity in a closed system.
04: The Second Law
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Any conceivable process is either spontaneous, reversible, or impossible. These three possibilities were discussed in Sec. 3.2 and are summarized below.
• A spontaneous process is a real process that can actually take place in a finite time period.
• A reversible process is an imaginary, idealized process in which the system passes through a continuous sequence of equilibrium states. This sequence of states can be approached by a spontaneous process in the limit of infinite slowness, and so also can the reverse sequence of states.
• An impossible process is a change that cannot occur under the existing conditions, even in a limiting sense. It is also known as an unnatural or disallowed process. Sometimes it is useful to describe a hypothetical impossible process that we can imagine but that does not occur in reality. The second law of thermodynamics will presently be introduced with two such impossible processes.
The spontaneous processes relevant to chemistry are irreversible. An irreversible process is a spontaneous process whose reverse is an impossible process.
There is also the special category, of little interest to chemists, of purely mechanical processes. A purely mechanical process is a spontaneous process whose reverse is also spontaneous.
It is true that reversible processes and purely mechanical processes are idealized processes that cannot occur in practice, but a spontaneous process can be practically reversible if carried out sufficiently slowly, or practically purely mechanical if friction and temperature gradients are negligible. In that sense, they are not impossible processes. This e-book will reserve the term “impossible” for a process that cannot be approached by any spontaneous process, no matter how slowly or how carefully it is carried out. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/04%3A_The_Second_Law/4.01%3A_Types_of_Processes.txt |
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A description of the mathematical statement of the second law is given in the box below.
The box includes three distinct parts. First, there is the assertion that a property called entropy, $S$, is an extensive state function.
Second, there is an equation for calculating the entropy change of a closed system during a reversible change of state: $\dif S$ is equal to $\dq/T\bd$. During a reversible process, the temperature usually has the same value $T$ throughout the system, in which case we can simply write $\dif S = \dq/T$. The equation $\dif S = \dq/T\bd$ allows for the possibility that in an equilibrium state the system has phases of different temperatures separated by internal adiabatic partitions.
Third, there is a criterion for spontaneity: $\dif S$ is greater than $\dq/T\bd$ during an irreversible change of state. The temperature $T\bd$ is a thermodynamic temperature, which will be defined in Sec. 4.3.4.
Each of the three parts is an essential component of the second law, but is somewhat abstract. What fundamental principle, based on experimental observation, may we take as the starting point to obtain them? Two principles are available, one associated with Clausius and the other with Kelvin and Planck. Both principles are equivalent statements of the second law. Each asserts that a certain kind of process is impossible, in agreement with common experience.
Next consider the impossible process shown in Fig. 4.2(a). A Joule paddle wheel rotates in a container of water as a weight rises. As the weight gains potential energy, the water loses thermal energy and its temperature decreases. Energy is conserved, so there is no violation of the first law. This process is just the reverse of the Joule paddle-wheel experiment (Sec. 3.7.2) and its impossibility has already been discussed.
We might again attempt to use some sort of device operating in a cycle to accomplish the same overall process, as in Fig. 4.2(b). A closed system that operates in a cycle and does net work on the surroundings is called a heat engine. The heat engine shown in Fig. 4.2(b) is a special one. During one cycle, a quantity of energy is transferred by heat from a heat reservoir to the engine, and the engine performs an equal quantity of work on a weight, causing it to rise. At the end of the cycle, the engine has returned to its initial state. This would be a very desirable engine, because it could convert thermal energy into an equal quantity of useful mechanical work with no other effect on the surroundings. (This hypothetical process is called “perpetual motion of the second kind.”) The engine could power a ship; it would use the ocean as a heat reservoir and require no fuel. Unfortunately, it is impossible to construct such a heat engine!
The principle was expressed by William Thomson (Lord Kelvin) in 1852 as follows: “It is impossible by means of inanimate material agency to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of the surrounding objects.” Max Planck in 1922 gave this statement: “It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and the cooling of a heat-reservoir.” For the purposes of this chapter, the principle can be reworded as follows.
• Both the Clausius statement and the Kelvin–Planck statement assert that certain processes, although they do not violate the first law, are nevertheless impossible.
These processes would not be impossible if we could control the trajectories of large numbers of individual particles. Newton’s laws of motion are invariant to time reversal. Suppose we could measure the position and velocity of each molecule of a macroscopic system in the final state of an irreversible process. Then, if we could somehow arrange at one instant to place each molecule in the same position with its velocity reversed, and if the molecules behaved classically, they would retrace their trajectories in reverse and we would observe the reverse “impossible” process.
Carnot engines and Carnot cycles are admittedly outside the normal experience of chemists, and using them to derive the mathematical statement of the second law may seem arcane. G. N. Lewis and M. Randall, in their classic 1923 book Thermodynamics and the Free Energy of Chemical Substances, complained of the presentation of “cyclical processes limping about eccentric and not quite completed cycles.” There seems, however, to be no way to carry out a rigorous general derivation without invoking thermodynamic cycles. You may avoid the details by skipping Secs. 4.3–4.5. (Incidently, the cycles described in these sections are complete!) | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/04%3A_The_Second_Law/4.02%3A_Statements_of_the_Second_Law.txt |
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4.3.1 Carnot engines and Carnot cycles
Could the efficiency of the Carnot engine be different from the efficiency the heat pump would have when run in reverse as a Carnot engine? If so, either the supersystem is an impossible Clausius device as shown in Fig. 4.7(b), or the supersystem operated in reverse (with the engine and heat pump switching roles) is an impossible Clausius device as shown in Fig. 4.7(d). We conclude that all Carnot engines operating between the same two temperatures have the same efficiency.
This is a good place to pause and think about the meaning of this statement in light of the fact that the steps of a Carnot engine, being reversible changes, cannot take place in a real system (Sec. 3.2). How can an engine operate that is not real? The statement is an example of a common kind of thermodynamic shorthand. To express the same idea more accurately, one could say that all heat engines (real systems) operating between the same two temperatures have the same limiting efficiency, where the limit is the reversible limit approached as the steps of the cycle are carried out more and more slowly. You should interpret any statement involving a reversible process in a similar fashion: a reversible process is an idealized limiting process that can be approached but never quite reached by a real system.
Thus, the efficiency of a Carnot engine must depend only on the values of $T\subs{c}$ and $T\subs{h}$ and not on the properties of the working substance. Since the efficiency is given by $\epsilon = 1+q\subs{c}/q\subs{h}$, the ratio $q\subs{c}/q\subs{h}$ must be a unique function of $T\subs{c}$ and $T\subs{h}$ only. To find this function for temperatures on the ideal-gas temperature scale, it is simplest to choose as the working substance an ideal gas.
An ideal gas has the equation of state $pV=nRT$. Its internal energy change in a closed system is given by $\dif U = C_V\dif T$ (Eq. 3.5.3), where $C_V$ (a function only of $T$) is the heat capacity at constant volume. Reversible expansion work is given by $\dw = -p\dif V$, which for an ideal gas becomes $\dw = -(nRT/V)\dif V$. Substituting these expressions for $\dif U$ and $\dw$ in the first law, $\dif U = \dq + \dw$, and solving for $\dq$, we obtain \begin{gather} \s{ \dq = C_V \dif T + \frac{nRT}{V} \dif V } \tag{4.3.4} \cond{(ideal gas, reversible} \nextcond{expansion work only)} \end{gather} Dividing both sides by $T$ gives \begin{gather} \s{ \frac{\dq}{T} = \frac{C_V \dif T}{T} + nR\frac{\dif V}{V}} \tag{4.3.5} \cond{(ideal gas, reversible} \nextcond{expansion work only)} \end{gather} In the two adiabatic steps of the Carnot cycle, $\dq$ is zero. We obtain a relation among the volumes of the four labeled states shown in Fig. 4.3 by integrating Eq. 4.3.5 over these steps and setting the integrals equal to zero: $\tx{Path B$\ra$C:} \qquad \int\! \frac {\dq}{T} = \int_{T\subs{h}}^{T\subs{c}}\frac{C_V \dif T}{T} + nR\ln \frac{V\subs{C}}{V\subs{B}} = 0 \tag{4.3.6}$
$\tx{Path D$\ra$A:} \qquad \int\! \frac {\dq}{T} = \int_{T\subs{c}}^{T\subs{h}}\frac{C_V \dif T}{T} + nR\ln \frac{V\subs{A}}{V\subs{D}} = 0 \tag{4.3.7}$ } Adding these two equations (the integrals shown with limits cancel) gives the relation $nR\ln \frac{V\subs{A} V\subs{C}}{V\subs{B} V\subs{D}}=0 \tag{4.3.8}$ which we can rearrange to \begin{gather} \s{ \ln (V\subs{B} / V\subs{A}) = -\ln (V\subs{D} / V\subs{C}) } \tag{4.3.9} \cond{(ideal gas, Carnot cycle)} \end{gather} We obtain expressions for the heat in the two isothermal steps by integrating Eq. 4.3.4 with $\dif T$ set equal to 0. $\tx{Path A$\ra$B}: \qquad q\subs{h} = nRT\subs{h}\ln (V\subs{B} / V\subs{A}) \tag{4.3.10}$ $\tx{Path C$\ra$D}: \qquad q\subs{c} = nRT\subs{c}\ln (V\subs{D} / V\subs{C}) \tag{4.3.11}$ The ratio of $q\subs{c}$ and $q\subs{h}$ obtained from these expressions is $\frac{q\subs{c}}{q\subs{h}} = \frac{T\subs{c}}{T\subs{h}} \times \frac{\ln (V\subs{D} / V\subs{C})}{\ln (V\subs{B} / V\subs{A})} \tag{4.3.12}$ By means of Eq. 4.3.9, this ratio becomes \begin{gather} \s{ \frac{q\subs{c}}{q\subs{h}}=-\frac{T\subs{c}}{T\subs{h}} } \tag{4.3.13} \cond{(Carnot cycle)} \end{gather} Accordingly, the unique function of $T\subs{c}$ and $T\subs{h}$ we seek that is equal to $q\subs{c}/q\subs{h}$ is the ratio $-T\subs{c}/T\subs{h}$. The efficiency, from Eq. 4.3.3, is then given by \begin{gather} \s{ \epsilon = 1 - \frac{T\subs{c}}{T\subs{h}} } \tag{4.3.14} \cond{(Carnot engine)} \end{gather} In Eqs. 4.3.13 and 4.3.14, $T\subs{c}$ and $T\subs{h}$ are temperatures on the ideal-gas scale. As we have seen, these equations must be valid for any working substance; it is not necessary to specify as a condition of validity that the system is an ideal gas.
The ratio $T\subs{c}/T\subs{h}$ is positive but less than one, so the efficiency is less than one as deduced earlier. This conclusion is an illustration of the Kelvin–Planck statement of the second law: A heat engine cannot have an efficiency of unity—that is, it cannot in one cycle convert all of the energy transferred by heat from a single heat reservoir into work. The example shown in Fig. 4.5, with $\epsilon = 1/4$, must have $T\subs{c}/T\subs{h} = 3/4$ (e.g., $T\subs{c} = 300\K$ and $T\subs{h} = 400\K$).
Keep in mind that a Carnot engine operates reversibly between two heat reservoirs. The expression of Eq. 4.3.14 gives the efficiency of this kind of idealized heat engine only. If any part of the cycle is carried out irreversibly, dissipation of mechanical energy will cause the efficiency to be lower than the theoretical value given by Eq. 4.3.14.
4.3.4 Thermodynamic temperature
The negative ratio $q\subs{c}/q\subs{h}$ for a Carnot cycle depends only on the temperatures of the two heat reservoirs. Kelvin (1848) proposed that this ratio be used to establish an “absolute” temperature scale. The physical quantity now called thermodynamic temperature is defined by the relation \begin{gather} \s{ \frac{T\subs{c}}{T\subs{h}}=-\frac{q\subs{c}}{q\subs{h}} } \tag{4.3.15} \cond{(Carnot cycle)} \end{gather} That is, the ratio of the thermodynamic temperatures of two heat reservoirs is equal, by definition, to the ratio of the absolute quantities of heat transferred in the isothermal steps of a Carnot cycle operating between these two temperatures. In principle, a measurement of $q\subs{c}/q\subs{h}$ during a Carnot cycle, combined with a defined value of the thermodynamic temperature of one of the heat reservoirs, can establish the thermodynamic temperature of the other heat reservoir. This defined value is provided by the triple point of H$_2$O; its thermodynamic temperature is defined as exactly $273.16$ kelvins.
Just as measurements with a gas thermometer in the limit of zero pressure establish the ideal-gas temperature scale (Sec. 2.3.5), the behavior of a heat engine in the reversible limit establishes the thermodynamic temperature scale. Note, however, that a reversible Carnot engine used as a “thermometer” to measure thermodynamic temperature is only a theoretical concept and not a practical instrument, since a completely-reversible process cannot occur in practice.
It is now possible to justify the statement in Sec. 2.3.5 that the ideal-gas temperature scale is proportional to the thermodynamic temperature scale. Both Eq. 4.3.13 and Eq. 4.3.15 equate the ratio $T\subs{c}/T\subs{h}$ to $-q\subs{c}/q\subs{h}$; but whereas $T\subs{c}$ and $T\subs{h}$ refer in Eq. 4.3.13 to the ideal-gas temperatures of the heat reservoirs, in Eq. 4.3.15 they refer to the thermodynamic temperatures. This means that the ratio of the ideal-gas temperatures of two bodies is equal to the ratio of the thermodynamic temperatures of the same bodies, and therefore the two scales are proportional to one another. The proportionality factor is arbitrary, but must be unity if the same unit (e.g., kelvins) is used in both scales. Thus, as stated in Sec. 2.3.5, the two scales expressed in kelvins are identical. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/04%3A_The_Second_Law/4.03%3A_Concepts_Developed_with_Carnot_Engines.txt |
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4.4.1 The existence of the entropy function
This section derives the existence and properties of the state function called entropy.
Consider an arbitrary cyclic process of a closed system. To avoid confusion, this system will be the “experimental system” and the process will be the “experimental process” or “experimental cycle.” There are no restrictions on the contents of the experimental system—it may have any degree of complexity whatsoever. The experimental process may involve more than one kind of work, phase changes and reactions may occur, there may be temperature and pressure gradients, constraints and external fields may be present, and so on. All parts of the process must be either irreversible or reversible, but not impossible.
Figure 4.8 Experimental system, Carnot engine (represented by a small square box), and heat reservoir. The dashed lines indicate the boundary of the supersystem.
(a) Reversible heat transfer between heat reservoir and Carnot engine.
(b) Heat transfer between Carnot engine and experimental system.
The infinitesimal quantities $\dq'$ and $\dq$ are positive for transfer in the directions indicated by the arrows.
We imagine that the experimental cycle is carried out in a special way that allows us to apply the Kelvin–Planck statement of the second law. The heat transferred across the boundary of the experimental system in each infinitesimal path element of the cycle is exchanged with a hypothetical Carnot engine. The combination of the experimental system and the Carnot engine is a closed supersystem (see Fig. 4.8). In the surroundings of the supersystem is a heat reservoir of arbitrary constant temperature $T\subs{res}$. By allowing the supersystem to exchange heat with only this single heat reservoir, we will be able to apply the Kelvin–Planck statement to a cycle of the supersystem.
This procedure is similar to ones described by A. B. Pippard (Elements of Classical Thermodynamics for Advanced Students of Physics, Cambridge University Press, Cambridge, 1966, Chap. 4); C. J. Adkins (Equilibrium Thermodynamics, 3rd edition, Cambridge University Press, Cambridge, 1983, Chap. 5); and Peter T. Landsberg (Thermodynamics and Statistical Mechanics, Dover Publications, Inc., New York, 1990, p. 53).
We assume that we are able to control changes of the work coordinates of the experimental system from the surroundings of the supersystem. We are also able to control the Carnot engine from these surroundings, for example by moving the piston of a cylinder-and-piston device containing the working substance. Thus the energy transferred by work across the boundary of the experimental system, and the work required to operate the Carnot engine, is exchanged with the surroundings of the supersystem.
During each stage of the experimental process with nonzero heat, we allow the Carnot engine to undergo many infinitesimal Carnot cycles with infinitesimal quantities of heat and work. In one of the isothermal steps of each Carnot cycle, the Carnot engine is in thermal contact with the heat reservoir, as depicted in Fig. 4.8(a). In this step the Carnot engine has the same temperature as the heat reservoir, and reversibly exchanges heat $\dq'$ with it. The sign convention is that $\dq'$ is positive if heat is transferred in the direction of the arrow, from the heat reservoir to the Carnot engine.
In the other isothermal step of the Carnot cycle, the Carnot engine is in thermal contact with the experimental system at a portion of the system’s boundary. as depicted in Fig. 4.8(b). The Carnot engine now has the same temperature, $T\bd$, as the experimental system at this part of the boundary, and exchanges heat with it. The heat $\dq$ is positive if the transfer is into the experimental system.
The relation between temperatures and heats in the isothermal steps of a Carnot cycle is given by Eq. 4.3.15. From this relation we obtain, for one infinitesimal Carnot cycle, the relation $T\bd/T\subs{res}=\dq/\dq'$, or $\dq'=T\subs{res}\frac{\dq}{T\bd} \tag{4.4.1}$
After many infinitesimal Carnot cycles, the experimental cycle is complete, the experimental system has returned to its initial state, and the Carnot engine has returned to its initial state in thermal contact with the heat reservoir. Integration of Eq. 4.4.1 around the experimental cycle gives the net heat entering the supersystem during the process: \begin{gather} q'=T\subs{res} \tag{4.4.2} \oint\!\frac{\dq}{T\bd} \end{gather} The integration here is over each path element of the experimental process and over each surface element of the boundary of the experimental system.
Keep in mind that the value of the cyclic integral $\oint\dq/T\bd$ depends only on the path of the experimental cycle, that this process can be reversible or irreversible, and that $T\subs{res}$ is a positive constant.
In this experimental cycle, could the net heat $q'$ transferred to the supersystem be positive? If so, the net work would be negative (to make the internal energy change zero) and the supersystem would have converted heat from a single heat reservoir completely into work, a process the Kelvin–Planck statement of the second law says is impossible. Therefore it is impossible for $q'$ to be positive, and from Eq. 4.4.2 we obtain the relation \begin{gather} \s{\oint\!\frac{\dq}{T\bd}\leq 0} \tag{4.4.3} \cond{(cyclic process of a closed system)} \end{gather} This relation is known as the Clausius inequality. It is valid only if the integration is taken around a cyclic path in a direction with nothing but reversible and irreversible changes—the path must not include an impossible change, such as the reverse of an irreversible change. The Clausius inequality says that if a cyclic path meets this specification, it is impossible for the cyclic integral $\oint(\dq/T\bd)$ to be positive.
If the entire experimental cycle is adiabatic (which is only possible if the process is reversible), the Carnot engine is not needed and Eq. 4.4.3 can be replaced by $\oint(\dq/T\bd)=0$.
Next let us investigate a reversible nonadiabatic process of the closed experimental system. Starting with a particular equilibrium state A, we carry out a reversible process in which there is a net flow of heat into the system, and in which $\dq$ is either positive or zero in each path element. The final state of this process is equilibrium state B. If each infinitesimal quantity of heat $\dq$ is positive or zero during the process, then the integral $\int_{\tx{A}}^{\tx{B}}(\dq/T\bd)$ must be positive. In this case the Clausius inequality tells us that if the system completes a cycle by returning from state B back to state A by a different path, the integral $\int_{\tx{B}}^{\tx{A}}(\dq/T\bd)$ for this second path must be negative. Therefore the change B$\ra$A cannot be carried out by any adiabatic process.
Any reversible process can be carried out in reverse. Thus, by reversing the reversible nonadiabatic process, it is possible to change the state from B to A by a reversible process with a net flow of heat out of the system and with $\dq$ either negative or zero in each element of the reverse path. In contrast, the absence of an adiabatic path from B to A means that it is impossible to carry out the change A$\ra$B by a reversible adiabatic process.
The general rule, then, is that whenever equilibrium state A of a closed system can be changed to equilibrium state B by a reversible process with finite “one-way” heat (i.e., the flow of heat is either entirely into the system or else entirely out of it), it is impossible for the system to change from either of these states to the other by a reversible adiabatic process.
A simple example will relate this rule to experience. We can increase the temperature of a liquid by allowing heat to flow reversibly into the liquid. It is impossible to duplicate this change of state by a reversible process without heat—that is, by using some kind of reversible work. The reason is that reversible work involves the change of a work coordinate that brings the system to a different final state. There is nothing in the rule that says we can’t increase the temperature irreversibly without heat, as we can for instance with stirring work.
States A and B can be arbitrarily close. We conclude that every equilibrium state of a closed system has other equilibrium states infinitesimally close to it that are inaccessible by a reversible adiabatic process. This is Carathéodory’s principle of adiabatic inaccessibility. (Constantin Carathéodory in 1909 combined this principle with a mathematical theorem$—$Carathéodory’s theorem$—$to deduce the existence of the entropy function. The derivation outlined here avoids the complexities of that mathematical treatment and leads to the same results.)
Next let us consider the reversible adiabatic processes that are possible. To carry out a reversible adiabatic process, starting at an initial equilibrium state, we use an adiabatic boundary and slowly vary one or more of the work coordinates. A certain final temperature will result. It is helpful in visualizing this process to think of an $N$-dimensional space in which each axis represents one of the $N$ independent variables needed to describe an equilibrium state. A point in this space represents an equilibrium state, and the path of a reversible process can be represented as a curve in this space.
A suitable set of independent variables for equilibrium states of a closed system of uniform temperature consists of the temperature $T$ and each of the work coordinates (Sec. 3.10). We can vary the work coordinates independently while keeping the boundary adiabatic, so the paths for possible reversible adiabatic processes can connect any arbitrary combinations of work coordinate values.
There is, however, the additional dimension of temperature in the $N$-dimensional space. Do the paths for possible reversible adiabatic processes, starting from a common initial point, lie in a volume in the $N$-dimensional space? Or do they fall on a surface described by $T$ as a function of the work coordinates? If the paths lie in a volume, then every point in a volume element surrounding the initial point must be accessible from the initial point by a reversible adiabatic path. This accessibility is precisely what Carathéodory’s principle of adiabatic inaccessibility denies. Therefore, the paths for all possible reversible adiabatic processes with a common initial state must lie on a unique surface. This is an $(N-1)$-dimensional hypersurface in the $N$-dimensional space, or a curve if $N$ is $2$. One of these surfaces or curves will be referred to as a reversible adiabatic surface.
Now consider the initial and final states of a reversible process with one-way heat (i.e., each nonzero infinitesimal quantity of heat $\dq$ has the same sign). Since we have seen that it is impossible for there to be a reversible adiabatic path between these states, the points for these states must lie on different reversible adiabatic surfaces that do not intersect anywhere in the $N$-dimensional space. Consequently, there is an infinite number of nonintersecting reversible adiabatic surfaces filling the $N$-dimensional space. (To visualize this for $N=3$, think of a flexed stack of paper sheets; each sheet represents a different reversible adiabatic surface in three-dimensional space.) A reversible, nonadiabatic process with one-way heat is represented by a path beginning at a point on one reversible adiabatic surface and ending at a point on a different surface. If $q$ is positive, the final surface lies on one side of the initial surface, and if $q$ is negative, the final surface is on the opposite side.
4.4.2 Using reversible processes to define the entropy
The existence of reversible adiabatic surfaces is the justification for defining a new state function $S$, the entropy. $S$ is specified to have the same value everywhere on one of these surfaces, and a different, unique value on each different surface. In other words, the reversible adiabatic surfaces are surfaces of constant entropy in the $N$-dimensional space. The fact that the surfaces fill this space without intersecting ensures that $S$ is a state function for equilibrium states, because any point in this space represents an equilibrium state and also lies on a single reversible adiabatic surface with a definite value of $S$.
Figure 4.9 A family of reversible adiabatic curves (two-dimensional reversible adiabatic surfaces) for an ideal gas with $V$ and $T$ as independent variables. A reversible adiabatic process moves the state of the system along a curve, whereas a reversible process with positive heat moves the state from one curve to another above and to the right. The curves are calculated for $n = 1\mol$ and $\CVm = (3/2)R$. Adjacent curves differ in entropy by $1\units{J K\(^{-1}$}\).
We know the entropy function must exist, because the reversible adiabatic surfaces exist. For instance, Fig. 4.9 shows a family of these surfaces for a closed system of a pure substance in a single phase. In this system, $N$ is equal to 2, and the surfaces are two-dimensional curves. Each curve is a contour of constant $S$. At this stage in the derivation, our assignment of values of $S$ to the different curves is entirely arbitrary.
How can we assign a unique value of $S$ to each reversible adiabatic surface? We can order the values by letting a reversible process with positive one-way heat, which moves the point for the state to a new surface, correspond to an increase in the value of $S$. Negative one-way heat will then correspond to decreasing $S$. We can assign an arbitrary value to the entropy on one particular reversible adiabatic surface. (The third law of thermodynamics is used for this purpose—see Sec. 6.1.) Then all that is needed to assign a value of $S$ to each equilibrium state is a formula for evaluating the difference in the entropies of any two surfaces.
Figure 4.10 Reversible paths in $V$–$T$ space. The thin curves are reversible adiabatic surfaces.
(a) Two paths connecting the same pair of reversible adiabatic surfaces.
(b) A cyclic path.
Consider a reversible process with positive one-way heat that changes the system from state A to state B. The path for this process must move the system from a reversible adiabatic surface of a certain entropy to a different surface of greater entropy. An example is the path A$\ra$B in Fig. 4.10(a). (The adiabatic surfaces in this figure are actually two-dimensional curves.) As before, we combine the experimental system with a Carnot engine to form a supersystem that exchanges heat with a single heat reservoir of constant temperature $T\subs{res}$. The net heat entering the supersystem, found by integrating Eq. 4.4.1, is $q' = T\subs{res} \int_{\tx{A}}^{\tx{B}} \frac{\dq}{T\bd} \tag{4.4.4}$ and it is positive.
Suppose the same experimental system undergoes a second reversible process, not necessarily with one-way heat, along a different path connecting the same pair of reversible adiabatic surfaces. This could be path C$\ra$D in Fig. 4.10(a). The net heat entering the supersystem during this second process is $q''$: $q'' = T\subs{res} \int_{\tx{C}}^{\tx{D}} \frac{\dq}{T\bd} \tag{4.4.5}$ We can then devise a cycle of the supersystem in which the experimental system undergoes the reversible path A$\ra$B$\ra$D$\ra$C$\ra$A, as shown in Fig. 4.10(b). Step A$\ra$B is the first process described above, step D$\ra$C is the reverse of the second process described above, and steps B$\ra$D and C$\ra$A are reversible and adiabatic. The net heat entering the supersystem in the cycle is $q' - q''$. In the reverse cycle the net heat is $q''-q'$. In both of these cycles the heat is exchanged with a single heat reservoir; therefore, according to the Kelvin–Planck statement, neither cycle can have positive net heat. Therefore $q'$ and $q''$ must be equal, and Eqs. 4.4.4 and 4.4.5 then show the integral $\int(\dq/T\bd)$ has the same value when evaluated along either of the reversible paths from the lower to the higher entropy surface.
Note that since the second path (C$\ra$D) does not necessarily have one-way heat, it can take the experimental system through any sequence of intermediate entropy values, provided it starts at the lower entropy surface and ends at the higher. Furthermore, since the path is reversible, it can be carried out in reverse resulting in reversal of the signs of $\Del S$ and $\int(\dq/T\bd)$.
It should now be apparent that a satisfactory formula for defining the entropy change of a reversible process in a closed system is \begin{gather} \s{ \Del S = \int\!\frac{\dq}{T\bd} } \tag{4.4.6} \cond{(reversible process,} \nextcond{closed system)} \end{gather} This formula satisfies the necessary requirements: it makes the value of $\Del S$ positive if the process has positive one-way heat, negative if the process has negative one-way heat, and zero if the process is adiabatic. It gives the same value of $\Del S$ for any reversible change between the same two reversible adiabatic surfaces, and it makes the sum of the $\Del S$ values of several consecutive reversible processes equal to $\Del S$ for the overall process.
In Eq. 4.4.6, $\Del S$ is the entropy change when the system changes from one arbitrary equilibrium state to another. If the change is an infinitesimal path element of a reversible process, the equation becomes \begin{gather} \s{ \dif S = \frac{\dq}{T\bd} } \tag{4.4.7} \cond{(reversible process,} \nextcond{closed system)} \end{gather} It is common to see this equation written in the form $\dif S = \dq\rev/T$, where $\dq\rev$ denotes an infinitesimal quantity of heat in a reversible process.
In Eq. 4.4.7, the quantity $1/T\bd$ is called an integrating factor for $\dq$, a factor that makes the product $(1/T\bd)\dq$ be the infinitesimal change of a state function. The quantity $c/T\bd$, where $c$ is any nonzero constant, would also be a satisfactory integrating factor; so the definition of entropy, using $c{=}1$, is actually one of an infinite number of possible choices for assigning values to the reversible adiabatic surfaces.
4.4.3 Some properties of the entropy
It is not difficult to show that the entropy of a closed system in an equilibrium state is an extensive property. Suppose a system of uniform temperature $T$ is divided into two closed subsystems A and B. When a reversible infinitesimal change occurs, the entropy changes of the subsystems are $\dif S\subs{A} = \dq\subs{A}/T$ and $\dif S\subs{B} = \dq\subs{B}/T$ and of the system $\dif S = \dq/T$. But $\dq$ is the sum of $\dq\subs{A}$ and $\dq\subs{B}$, which gives $\dif S = \dif S\subs{A} + \dif S\subs{B}$. Thus, the entropy changes are additive, so that entropy must be extensive: $S=S\subs{A}+S\subs{B}$. (The argument is not quite complete, because we have not shown that when each subsystem has an entropy of zero, so does the entire system. The zero of entropy will be discussed in Sec. 6.1.)
How can we evaluate the entropy of a particular equilibrium state of the system? We must assign an arbitrary value to one state and then evaluate the entropy change along a reversible path from this state to the state of interest using $\Del S=\int(\dq/T\bd)$.
We may need to evaluate the entropy of a nonequilibrium state. To do this, we imagine imposing hypothetical internal constraints that change the nonequilibrium state to a constrained equilibrium state with the same internal structure. Some examples of such internal constraints were given in Sec. 2.4.4, and include rigid adiabatic partitions between phases of different temperature and pressure, semipermeable membranes to prevent transfer of certain species between adjacent phases, and inhibitors to prevent chemical reactions.
We assume that we can, in principle, impose or remove such constraints reversibly without heat, so there is no entropy change. If the nonequilibrium state includes macroscopic internal motion, the imposition of internal constraints involves negative reversible work to bring moving regions of the system to rest. This concept amounts to defining the entropy of a state with macroscopic internal motion to be the same as the entropy of a state with the same internal structure but without the motion, i.e., the same state frozen in time. By this definition, $\Del S$ for a purely mechanical process (Sec. 3.2.3) is zero.
If the system is nonuniform over its extent, the internal constraints will partition it into practically-uniform regions whose entropy is additive. The entropy of the nonequilibrium state is then found from $\Del S=\int(\dq/T\bd)$ using a reversible path that changes the system from an equilibrium state of known entropy to the constrained equilibrium state with the same entropy as the state of interest. This procedure allows every possible state (at least conceptually) to have a definite value of $S$. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/04%3A_The_Second_Law/4.04%3A__Derivation_of_the_Mathematical_Statement_of_the_Second_Law.txt |
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We know that during a reversible process of a closed system, each infinitesimal entropy change $\dif S$ is equal to $\dq/T\bd$ and the finite change $\Del S$ is equal to the integral $\int(\dq/T\bd)$—but what can we say about $\dif S$ and $\Del S$ for an irreversible process?
The derivation of this section will show that for an infinitesimal irreversible change of a closed system, $\dif S$ is greater than $\dq/T\bd$, and for an entire process $\Del S$ is greater than $\int(\dq/T\bd)$. That is, the equalities that apply to a reversible process are replaced, for an irreversible process, by inequalities.
The derivation begins with irreversible processes that are adiabatic, and is then extended to irreversible processes in general.
4.5.1 Irreversible adiabatic processes
Consider an arbitrary irreversible adiabatic process of a closed system starting with a particular initial state A. The final state B depends on the path of this process. We wish to investigate the sign of the entropy change $\Del S\subs{A\(\ra$B}\). Our reasoning will depend on whether or not there is work during the process.
If there is work along any infinitesimal path element of the irreversible adiabatic process ($\dw \ne 0$), we know from experience that this work would be different if the work coordinate or coordinates were changing at a different rate, because energy dissipation from internal friction would then be different. In the limit of infinite slowness, an adiabatic process with initial state A and the same change of work coordinates would become reversible, and the net work and final internal energy would differ from those of the irreversible process. Because the final state of the reversible adiabatic process is different from B, there is no reversible adiabatic path with work between states A and B.
All states of a reversible process, including the initial and final states, must be equilibrium states. There is therefore a conceptual difficulty in considering reversible paths between two states if either of these states are nonequilibrium states. In such a case we will assume that the state has been replaced by a constrained equilibrium state of the same entropy as described in Sec. 4.4.3.
If, on the other hand, there is no work along any infinitesimal path element of the irreversible adiabatic process ($\dw{=}0$), the process is taking place at constant internal energy $U$ in an isolated system. A reversible limit cannot be reached without heat or work (Sec. 3.2.1). Thus any reversible adiabatic change from state A would require work, causing a change of $U$ and preventing the system from reaching state B by any reversible adiabatic path.
So regardless of whether or not an irreversible adiabatic process A$\ra$B involves work, there is no reversible adiabatic path between A and B. The only reversible paths between these states must be nonadiabatic. It follows that the entropy change $\Del S\subs{A\(\ra$B}\), given by the value of $\dq/T\bd$ integrated over a reversible path from A to B, cannot be zero.
Next we ask whether $\Del S\subs{A\(\ra$B}\) could be negative. In each infinitesimal path element of the irreversible adiabatic process A$\ra$B, $\dq$ is zero and the integral $\int_{\tx{A}}^{\tx{B}}(\dq/T\bd)$ along the path of this process is zero. Suppose the system completes a cycle by returning along a different, reversible path from state B back to state A. The Clausius inequality (Eq. 4.4.3) tells us that in this case the integral $\int_{\tx{B}}^{\tx{A}}(\dq/T\bd)$ along the reversible path cannot be positive. But this integral for the reversible path is equal to $-\Del S\subs{A\(\ra$B}\), so $\Del S\subs{A\(\ra$B}\) cannot be negative.
We conclude that because the entropy change of the irreversible adiabatic process A$\ra$B cannot be zero, and it cannot be negative, it must be positive.
In this derivation, the initial state A is arbitrary and the final state B is reached by an irreversible adiabatic process. If the two states are only infinitesimally different, then the change is infinitesimal. Thus for an infinitesimal change that is irreversible and adiabatic, $\dif S$ must be positive.
4.5.2 Irreversible processes in general
Figure 4.11 Supersystem including the experimental system, a Carnot engine (square box), and a heat reservoir. The dashed rectangle indicates the boundary of the supersystem.
To treat an irreversible process of a closed system that is nonadiabatic, we proceed as follows. As in Sec. 4.4.1, we use a Carnot engine for heat transfer across the boundary of the experimental system. We move the boundary of the supersystem of Fig. 4.8 so that the supersystem now includes the experimental system, the Carnot engine, and a heat reservoir of constant temperature $T\subs{res}$, as depicted in Fig. 4.11. During an irreversible change of the experimental system, the Carnot engine undergoes many infinitesimal cycles. During each cycle, the Carnot engine exchanges heat $\dq'$ at temperature $T\subs{res}$ with the heat reservoir and heat $\dq$ at temperature $T\bd$ with the experimental system, as indicated in the figure. We use the sign convention that $\dq'$ is positive if heat is transferred to the Carnot engine, and $\dq$ is positive if heat is transferred to the experimental system, in the directions of the arrows in the figure.
The supersystem exchanges work, but not heat, with its surroundings. During one infinitesimal cycle of the Carnot engine, the net entropy change of the Carnot engine is zero, the entropy change of the experimental system is $\dif S$, the heat transferred between the Carnot engine and the experimental system is $\dq$, and the heat transferred between the heat reservoir and the Carnot engine is given by $\dq'=T\subs{res}\dq/T\bd$ (Eq. 4.4.1). The heat transfer between the heat reservoir and Carnot engine is reversible, so the entropy change of the heat reservoir is $\dif S\subs{res} = -\frac{\dq'}{T\subs{res}} = -\frac{\dq}{T\bd} \tag{4.5.1}$ The entropy change of the supersystem is the sum of the entropy changes of its parts: $\dif S\subs{ss} = \dif S + \dif S\subs{res} = \dif S - \frac{\dq}{T\bd} \tag{4.5.2}$ The process within the supersystem is adiabatic and includes an irreversible change within the experimental system, so according to the conclusions of Sec. 4.5.1, $\dif S\subs{ss}$ is positive. Equation 4.5.2 then shows that $\dif S$, the infinitesimal entropy change during the irreversible change of the experimental system, must be greater than $\dq/T\bd$: \begin{gather} \s{ \dif S > \frac{\dq}{T\bd} } \tag{4.5.3} \cond{(irreversible change, closed system)} \end{gather} This relation includes the case of an irreversible adiabatic change, because it shows that if $\dq$ is zero, $\dif S$ is greater than zero.
By integrating both sides of Eq. 4.5.3 between the initial and final states of the irreversible process, we obtain a relation for the finite entropy change corresponding to many infinitesimal cycles of the Carnot engine: \begin{gather} \s{ \Del S > \int\!\frac{\dq}{T\bd} } \tag{4.5.4} \cond{(irreversible process, closed system)} \end{gather} | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/04%3A_The_Second_Law/4.05%3A_Irreversible_Processes.txt |
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The lengthy derivation in Secs. 4.3–4.5 is based on the Kelvin–Planck statement describing the impossibility of converting completely into work the energy transferred into the system by heat from a single heat reservoir. The derivation has now given us all parts of the mathematical statement of the second law shown in the box in Sec. 4.2. The mathematical statement includes an equality, $\dif S=\dq/T\bd$, that applies to an infinitesimal reversible change, and an inequality, $\dif S > \dq/T\bd$, that applies to an infinitesimal irreversible change. It is convenient to combine the equality and inequality in a single relation that is a general mathematical statement of the second law: \begin{gather} \s{\dif S \geq \frac{\dq}{T\bd}} \tag{4.6.1} \cond{(${}\subs{ rev}\sups{ irrev}$, closed system)} \end{gather} The inequality refers to an irreversible change and the equality to a reversible change, as indicated by the notation ${}\subs{ rev}\sups{ irrev}$ in the conditions of validity. The integrated form of this relation is \begin{gather} \s{\Del S \geq \int\!\frac{\dq}{T\bd}} \tag{4.6.2} \cond{(${}\subs{ rev}\sups{ irrev}$, closed system)} \end{gather}
During a reversible process, the states are equilibrium states and the temperature is usually uniform throughout the system. The only exception is if the system happens to have internal adiabatic partitions that allow phases of different temperatures in an equilibrium state. When the process is reversible and the temperature is uniform, we can replace $\dif S=\dq/T\bd$ by $\dif S=\dq/T$.
The rest of Sec. 4.6 will apply Eqs. 4.6.1 and 4.6.2 to various reversible and irreversible processes.
4.6.1 Reversible heating
The definition of the heat capacity $C$ of a closed system is given by Eq. 3.1.9: $C \defn \dq/\dif T$. For reversible heating or cooling of a homogeneous phase, $\dq$ is equal to $T\dif S$ and we can write $\Del S = \int_{T_1}^{T_2}\!\frac{C}{T}\dif T \tag{4.6.3}$ where $C$ should be replaced by $C_V$ if the volume is constant, or by $C_p$ if the pressure is constant (Sec. 3.1.5). If the heat capacity has a constant value over the temperature range from $T_1$ to $T_2$, the equation becomes $\Del S = C\ln\frac{T_2}{T_1} \tag{4.6.4}$ Heating increases the entropy, and cooling decreases it.
4.6.2 Reversible expansion of an ideal gas
When the volume of an ideal gas, or of any other fluid, is changed reversibly and adiabatically, there is of course no entropy change.
When the volume of an ideal gas is changed reversibly and isothermally, there is expansion work given by $w=-nRT\ln(V_2/V_1)$ (Eq. 3.5.1). Since the internal energy of an ideal gas is constant at constant temperature, there must be heat of equal magnitude and opposite sign: $q=nRT\ln(V_2/V_1)$. The entropy change is therefore \begin{gather} \s{\Del S = nR\ln\frac{V_2}{V_1}} \tag{4.6.5} \cond{(reversible isothermal volume} \nextcond{change of an ideal gas)} \end{gather} Isothermal expansion increases the entropy, and isothermal compression decreases it.
Since the change of a state function depends only on the initial and final states, Eq. 4.6.5 gives a valid expression for $\Del S$ of an ideal gas under the less stringent condition $T_2=T_1$; it is not necessary for the intermediate states to be equilibrium states of the same temperature.
4.6.3 Spontaneous changes in an isolated system
An isolated system is one that exchanges no matter or energy with its surroundings. Any change of state of an isolated system that actually occurs is spontaneous, and arises solely from conditions within the system, uninfluenced by changes in the surroundings—the process occurs by itself, of its own accord. The initial state and the intermediate states of the process must be nonequilibrium states, because by definition an equilibrium state would not change over time in the isolated system.
Unless the spontaneous change is purely mechanical, it is irreversible. According to the second law, during an infinitesimal change that is irreversible and adiabatic, the entropy increases. For the isolated system, we can therefore write \begin{gather} \s{\dif S > 0} \tag{4.6.6} \cond{(irreversible change, isolated system)} \end{gather} In later chapters, the inequality of Eq. 4.6.6 will turn out to be one of the most useful for deriving conditions for spontaneity and equilibrium in chemical systems: The entropy of an isolated system continuously increases during a spontaneous, irreversible process until it reaches a maximum value at equilibrium.
If we treat the universe as an isolated system (although cosmology provides no assurance that this is a valid concept), we can say that as spontaneous changes occur in the universe, its entropy continuously increases. Clausius summarized the first and second laws in a famous statement: Die Energie der Welt ist constant; die Entropie der Welt strebt einem Maximum zu (the energy of the universe is constant; the entropy of the universe strives toward a maximum).
4.6.4 Internal heat flow in an isolated system
Suppose the system is a solid body whose temperature initially is nonuniform. Provided there are no internal adiabatic partitions, the initial state is a nonequilibrium state lacking internal thermal equilibrium. If the system is surrounded by thermal insulation, and volume changes are negligible, this is an isolated system. There will be a spontaneous, irreversible internal redistribution of thermal energy that eventually brings the system to a final equilibrium state of uniform temperature.
In order to be able to specify internal temperatures at any instant, we treat the system as an assembly of phases, each having a uniform temperature that can vary with time. To describe a region that has a continuous temperature gradient, we approximate the region with a very large number of very small phases or parcels, each having a temperature infinitesimally different from its neighbors.
We use Greek letters to label the phases. The temperature of phase $\pha$ at any given instant is $T\aph$. We can treat each phase as a subsystem with a boundary across which there can be energy transfer in the form of heat. Let $\dq_{\pha\phb}$ represent an infinitesimal quantity of heat transferred during an infinitesimal interval of time to phase $\pha$ from phase $\phb$. The heat transfer, if any, is to the cooler from the warmer phase. If phases $\pha$ and $\phb$ are in thermal contact and $T\aph$ is less than $T\bph$, then $\dq_{\pha\phb}$ is positive; if the phases are in thermal contact and $T\aph$ is greater than $T\bph$, $\dq_{\pha\phb}$ is negative; and if neither of these conditions is satisfied, $\dq_{\pha\phb}$ is zero.
To evaluate the entropy change, we need a reversible path from the initial to the final state. The net quantity of heat transferred to phase $\pha$ during an infinitesimal time interval is $\dq\aph = \sum_{\phb\neq\pha}\dq_{\pha\phb}$. The entropy change of phase $\pha$ is the same as it would be for the reversible transfer of this heat from a heat reservoir of temperature $T\aph$: $\dif S\aph=\dq\aph/T\aph$. The entropy change of the entire system along the reversible path is found by summing over all phases: $\begin{split} \dif S & = \sum_\pha\dif S\aph = \sum_\pha \frac{\dq\aph}{T\aph} = \sum_\pha\sum_{\phb\neq\pha}\frac{\dq_{\pha\phb}}{T\aph}\ & = \sum_\pha\sum_{\phb > \pha} \left(\frac{\dq_{\pha\phb}}{T\aph} +\frac{\dq_{\phb\pha}}{T\bph}\right) \end{split} \tag{4.6.7}$ There is also the condition of quantitative energy transfer, $\dq_{\phb\pha}=-\dq_{\pha\phb}$, which we use to rewrite Eq. 4.6.7 in the form $\dif S = \sum_\pha\sum_{\phb > \pha} \left(\frac{1}{T\aph}-\frac{1}{T\bph} \right)\dq_{\pha\phb} \tag{4.6.8}$
Consider an individual term of the sum on the right side of Eq. 4.6.8 that has a nonzero value of $\dq_{\pha\phb}$ due to finite heat transfer between phases $\pha$ and $\phb$. If $T\aph$ is less than $T\bph$, then both $\dq_{\pha\phb}$ and $(1/T\aph-1/T\bph)$ are positive. If, on the other hand, $T\aph$ is greater than $T\bph$, both $\dq_{\pha\phb}$ and $(1/T\aph-1/T\bph)$ are negative. Thus each term of the sum is either zero or positive, and as long as phases of different temperature are present, $\dif S$ is positive.
This derivation shows that during a spontaneous thermal equilibration process in an isolated system, starting with any initial distribution of the internal temperatures, the entropy continuously increases until the system reaches a state of thermal equilibrium with a single uniform temperature throughout. The result agrees with Eq. 4.6.6. Harvey S. Leff (Am. J. Phys., 45, 252–254, 1977) obtains the same result by a more complicated derivation.
4.6.5 Free expansion of a gas
Consider the free expansion of a gas shown in Fig. 3.8. The system is the gas. Assume that the vessel walls are rigid and adiabatic, so that the system is isolated. When the stopcock between the two vessels is opened, the gas expands irreversibly into the vacuum without heat or work and at constant internal energy. To carry out the same change of state reversibly, we confine the gas at its initial volume and temperature in a cylinder-and-piston device and use the piston to expand the gas adiabatically with negative work. Positive heat is then needed to return the internal energy reversibly to its initial value. Because the reversible path has positive heat, the entropy change is positive.
This is an example of an irreversible process in an isolated system for which a reversible path between the initial and final states has both heat and work.
4.6.6 Adiabatic process with work
In general (Sec. 3.10), an adiabatic process with a given initial equilibrium state and a given change of a work coordinate has the least positive or most negative work in the reversible limit. Consider an irreversible adiabatic process with work $w\subs{irr}$. The same change of state can be accomplished reversibly by the following two steps: (1) a reversible adiabatic change of the work coordinate with work $w\rev$, followed by (2) reversible transfer of heat $q\rev$ with no further change of the work coordinate. Since $w\rev$ is algebraically less than $w\subs{irr}$, $q\rev$ must be positive in order to make $\Del U$ the same in the irreversible and reversible paths. The positive heat increases the entropy along the reversible path, and consequently the irreversible adiabatic process has a positive entropy change. This conclusion agrees with the second-law inequality of Eq. 4.6.1. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/04%3A_The_Second_Law/4.06%3A_Applications.txt |
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Some of the important terms and definitions discussed in this chapter are as follows.
• The derivation of the mathematical statement of the second law shows that during a reversible process of a closed system, the infinitesimal quantity $\dq/T\bd$ equals the infinitesimal change of a state function called the entropy, $S$. Here $\dq$ is heat transferred at the boundary where the temperature is $T\bd$.
In each infinitesimal path element of a process of a closed system, $\dif S$ is equal to $\dq/T\bd$ if the process is reversible, and is greater than $\dq/T\bd$ if the process is irreversible, as summarized by the relation $\dif S \geq \dq/T\bd$.
The second law establishes no general relation between entropy changes and heat in an open system, or for an impossible process. The entropy of an open system may increase or decrease depending on whether matter enters or leaves. It is possible to imagine different impossible processes in which $\dif S$ is less than, equal to, and greater than $\dq/T\bd$. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/04%3A_The_Second_Law/4.07%3A_Summary.txt |
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Because entropy is such an important state function, it is natural to seek a description of its meaning on the microscopic level.
Entropy is sometimes said to be a measure of “disorder.” According to this idea, the entropy increases whenever a closed system becomes more disordered on a microscopic scale. This description of entropy as a measure of disorder is highly misleading. It does not explain why entropy is increased by reversible heating at constant volume or pressure, or why it increases during the reversible isothermal expansion of an ideal gas. Nor does it seem to agree with the freezing of a supercooled liquid or the formation of crystalline solute in a supersaturated solution; these processes can take place spontaneously in an isolated system, yet are accompanied by an apparent decrease of disorder.
Thus we should not interpret entropy as a measure of disorder. We must look elsewhere for a satisfactory microscopic interpretation of entropy.
A rigorous interpretation is provided by the discipline of statistical mechanics, which derives a precise expression for entropy based on the behavior of macroscopic amounts of microscopic particles. Suppose we focus our attention on a particular macroscopic equilibrium state. Over a period of time, while the system is in this equilibrium state, the system at each instant is in a microstate, or stationary quantum state, with a definite energy. The microstate is one that is accessible to the system—that is, one whose wave function is compatible with the system’s volume and with any other conditions and constraints imposed on the system. The system, while in the equilibrium state, continually jumps from one accessible microstate to another, and the macroscopic state functions described by classical thermodynamics are time averages of these microstates.
The fundamental assumption of statistical mechanics is that accessible microstates of equal energy are equally probable, so that the system while in an equilibrium state spends an equal fraction of its time in each such microstate. The statistical entropy of the equilibrium state then turns out to be given by the equation $S\subs{stat} = k \ln W + C \label{4.8.1}$ where $k$ is the Boltzmann constant $k=R/N\subs{A}$, $W$ is the number of accessible microstates, and $C$ is a constant.
In the case of an equilibrium state of a perfectly-isolated system of constant internal energy $U$, the accessible microstates are the ones that are compatible with the constraints and whose energies all have the same value, equal to the value of $U$.
It is more realistic to treat an equilibrium state with the assumption the system is in thermal equilibrium with an external constant-temperature heat reservoir. The internal energy then fluctuates over time with extremely small deviations from the average value $U$, and the accessible microstates are the ones with energies close to this average value. In the language of statistical mechanics, the results for an isolated system are derived with a microcanonical ensemble, and for a system of constant temperature with a canonical ensemble.
A change $\Del S\subs{stat}$ of the statistical entropy function given by Eq. 4.8.1 is the same as the change $\Del S$ of the macroscopic second-law entropy, because the derivation of Eq. 4.8.1 is based on the macroscopic relation $\dif S\subs{stat}=\dq/T=(\dif U-\dw)/T$ with $\dif U$ and $\dw$ given by statistical theory. If the integration constant $C$ is set equal to zero, $S\subs{stat}$ becomes the third-law entropy $S$ to be described in Chap. 6.
Equation 4.8.1 shows that a reversible process in which entropy increases is accompanied by an increase in the number of accessible microstates of equal, or nearly equal, internal energies. This interpretation of entropy increase has been described as the spreading and sharing of energy (Harvey S. Leff, Am. J. Phys., 64, 1261–1271, 1996) and as the dispersal of energy (Frank L. Lambert, J. Chem. Educ., 79, 1241–1246, 2002). It has even been proposed that entropy should be thought of as a “spreading function” with its symbol $S$ suggesting spreading (Frank L. Lambert and Harvey S. Leff, J. Chem. Educ., 86, 94–98, 2009). The symbol $S$ for entropy seems originally to have been an arbitrary choice by Clausius.
4.09: Chapter 4 Problems
An underlined problem number or problem-part letter indicates that the numerical answer appears in Appendix I.
4.1 Explain why an electric refrigerator, which transfers energy by means of heat from the cold food storage compartment to the warmer air in the room, is not an impossible "Clausius device."
4.2 A system consisting of a fixed amount of an ideal gas is maintained in thermal equilibrium with a heat reservoir at temperature $T$. The system is subjected to the following isothermal cycle:
1. The gas, initially in an equilibrium state with volume $V_{0}$, is allowed to expand into a vacuum and reach a new equilibrium state of volume $V^{\prime}$.
2. The gas is reversibly compressed from $V^{\prime}$ to $V_{0}$.
For this cycle, find expressions or values for $w, \oint \mathrm{d} q / T$, and $\oint \mathrm{d} S .$
4.3 In an irreversible isothermal process of a closed system:
(a) Is it possible for $\Delta S$ to be negative?
(b) Is it possible for $\Delta S$ to be less than $q / T$ ?
4.4 Suppose you have two blocks of copper, each of heat capacity $C_{V}=200.0 \mathrm{JK}^{-1}$. Initially one block has a uniform temperature of $300.00 \mathrm{~K}$ and the other $310.00 \mathrm{~K}$. Calculate the entropy change that occurs when you place the two blocks in thermal contact with one another and surround them with perfect thermal insulation. Is the sign of $\Delta S$ consistent with the second law? (Assume the process occurs at constant volume.)
4.5 Refer to the apparatus shown in Figs. $3.23$ on page 101 and $3.26$ on page 103 and described in Probs. $3.3$ and 3.8. For both systems, evaluate $\Delta S$ for the process that results from opening the stopcock. Also evaluate $\int \mathrm{d} q / T_{\text {ext }}$ for both processes (for the apparatus in Fig. 3.26, assume the vessels have adiabatic walls). Are your results consistent with the mathematical statement of the second law?
Figure 4.13
4.6 Figure $4.13$ shows the walls of a rigid thermally-insulated box (cross hatching). The system is the contents of this box. In the box is a paddle wheel immersed in a container of water, connected by a cord and pulley to a weight of mass $m$. The weight rests on a stop located a distance $h$ above the bottom of the box. Assume the heat capacity of the system, $C_{V}$, is independent of temperature. Initially the system is in an equilibrium state at temperature $T_{1}$.
When the stop is removed, the weight irreversibly sinks to the bottom of the box, causing the paddle wheel to rotate in the water. Eventually the system reaches a final equilibrium state with thermal equilibrium. Describe a reversible process with the same entropy change as this irreversible process, and derive a formula for $\Delta S$ in terms of $m, h, C_{V}$, and $T_{1}$. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/04%3A_The_Second_Law/4.08%3A_The_Statistical_Interpretation_of_Entropy.txt |
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This chapter begins with a discussion of mathematical properties of the total differential of a dependent variable. Three extensive state functions with dimensions of energy are introduced: enthalpy, Helmholtz energy, and Gibbs energy. These functions, together with internal energy, are called thermodynamic potentials. (The term thermodynamic potential should not be confused with the chemical potential, $\mu$, to be introduced in Sec. 5.2.) Some formal mathematical manipulations of the four thermodynamic potentials are described that lead to expressions for heat capacities, surface work, and criteria for spontaneity in closed systems.
05: Thermodynamic Potentials
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Recall from Sec. 2.4.1 that the state of the system at each instant is defined by a certain minimum number of state functions, the independent variables. State functions not treated as independent variables are dependent variables. Infinitesimal changes in any of the independent variables will, in general, cause an infinitesimal change in each dependent variable.
A dependent variable is a function of the independent variables. The total differential of a dependent variable is an expression for the infinitesimal change of the variable in terms of the infinitesimal changes of the independent variables. As explained in Sec. F.2 of Appendix F, the expression can be written as a sum of terms, one for each independent variable. Each term is the product of a partial derivative with respect to one of the independent variables and the infinitesimal change of that independent variable. For example, if the system has two independent variables, and we take these to be $T$ and $V$, the expression for the total differential of the pressure is $\difp = \Pd{p}{T}{V}\dif T + \Pd{p}{V}{T}\dif V \tag{5.1.1}$ Thus, in the case of a fixed amount of an ideal gas with pressure given by $p=nRT/V$, the total differential of the pressure can be written $\difp = \frac{nR}{V}\dif T - \frac{nRT}{V^2}\dif V \tag{5.1.2}$ | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/05%3A_Thermodynamic_Potentials/5.01%3A_Total_Differential_of_a_Dependent_Variable.txt |
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For a closed system undergoing processes in which the only kind of work is expansion work, the first law becomes $\dif U=\dq+\dw=\dq-p\bd\dif V$. Since it will often be useful to make a distinction between expansion work and other kinds of work, this e-book will sometimes write the first law in the form \begin{gather} \s{ \dif U = \dq - p\bd \dif V + \dw' } \tag{5.2.1} \cond{(closed system)} \end{gather} where $\dw'$ is nonexpansion work—that is, any thermodynamic work that is not expansion work.
Consider a closed system of one chemical component (e.g., a pure substance) in a single homogeneous phase. The only kind of work is expansion work, with $V$ as the work variable. This kind of system has two independent variables (Sec. 2.4.3). During a reversible process in this system, the heat is $\dq=T\dif S$, the work is $\dw=-p\dif V$, and an infinitesimal internal energy change is given by \begin{gather} \s{ \dif U = T \dif S - p \dif V } \tag{5.2.2} \cond{(closed system, $C{=}1$,} \nextcond{$P{=}1$, $\dw'{=}0$)} \end{gather} In the conditions of validity shown next to this equation, $C{=}1$ means there is one component ($C$ is the number of components) and $P{=}1$ means there is one phase ($P$ is the number of phases).
The appearance of the intensive variables $T$ and $p$ in Eq. 5.2.2 implies, of course, that the temperature and pressure are uniform throughout the system during the process. If they were not uniform, the phase would not be homogeneous and there would be more than two independent variables. The temperature and pressure are strictly uniform only if the process is reversible; it is not necessary to include “reversible” as one of the conditions of validity.
A real process approaches a reversible process in the limit of infinite slowness. For all practical purposes, therefore, we may apply Eq. 5.2.2 to a process obeying the conditions of validity and taking place so slowly that the temperature and pressure remain essentially uniform—that is, for a process in which the system stays very close to thermal and mechanical equilibrium.
Because the system under consideration has two independent variables, Eq. 5.2.2 is an expression for the total differential of $U$ with $S$ and $V$ as the independent variables. In general, an expression for the differential $\dif X$ of a state function $X$ is a total differential if
1. Note that the work coordinate of any kind of dissipative work—work without a reversible limit—cannot appear in the expression for a total differential, because it is not a state function (Sec. 3.10).
As explained in Appendix F, we may identify the coefficient of each term in an expression for the total differential of a state function as a partial derivative of the function. We identify the coefficients on the right side of Eq. 5.2.2 as follows: $T=\Pd{U}{S}{V} \qquad -p=\Pd{U}{V}{S} \tag{5.2.3}$
The quantity given by the third partial derivative, $\pd{U}{n}{S,V}$, is represented by the symbol $\mu$ (mu). This quantity is an intensive state function called the chemical potential.
With these substitutions, Eq. 5.2.4 becomes \begin{gather} \s{\dif U = T \dif S - p \dif V + \mu \dif n} \tag{5.2.5} \cond{(pure substance,} \nextcond{$P{=}1$, $\dw'{=}0$)} \end{gather} and this is a valid expression for the total differential of $U$ under the given conditions.
If a system contains a mixture of $s$ different substances in a single phase, and the system is open so that the amount of each substance can vary independently, there are $2+s$ independent variables and the total differential of $U$ can be written \begin{gather} \s{\dif U = T \dif S - p \dif V + \sum_{i=1}^s \mu_i \dif n_i} \tag{5.2.6} \cond{(open system,} \nextcond{$P{=}1$, $\dw'{=}0$)} \end{gather} The coefficient $\mu_i$ is the chemical potential of substance $i$. We identify it as the partial derivative $\pd{U}{n_i}{S,V,n_{j\ne i}}$.
The term $-p\dif V$ on the right side of Eq. 5.2.6 is the reversible work. However, the term $T\dif S$ does not equal the reversible heat as it would if the system were closed. This is because the entropy change $\dif S$ is partly due to the entropy of the matter transferred across the boundary. It follows that the remaining term, $\sum_i\mu_i\dif n_i$ (sometimes called the “chemical work”), should not be interpreted as the energy brought into the system by the transfer of matter.
Suppose that in addition to expansion work, other kinds of reversible work are possible. Each work coordinate adds an additional independent variable. Thus, for a closed system of one component in one phase, with reversible nonexpansion work given by $\dw'=Y\dif X$, the total differential of $U$ becomes \begin{gather} \s{\dif U = T\dif S - p\dif V + Y\dif X} \tag{5.2.7} \cond{(closed system,} \nextcond{$C{=}1$, $P{=}1$)} \end{gather} | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/05%3A_Thermodynamic_Potentials/5.02%3A_Total_Differential_of_the_Internal_Energy.txt |
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For the moment we shall confine our attention to closed systems with one component in one phase. The total differential of the internal energy in such a system is given by Eq. 5.2.2: $\dif U = T \dif S - p \dif V$. The independent variables in this equation, $S$ and $V$, are called the natural variables of $U$.
In the laboratory, entropy and volume may not be the most convenient variables to measure and control. Entropy is especially inconvenient, as its value cannot be measured directly. The way to change the independent variables is to make Legendre transforms, as explained in Sec. F.4 in Appendix F.
A Legendre transform of a dependent variable is made by subtracting one or more products of conjugate variables. In the total differential $\dif U = T \dif S - p \dif V$, $T$ and $S$ are conjugates (that is, they comprise a conjugate pair), and $-p$ and $V$ are conjugates. Thus the products that can be subtracted from $U$ are either $TS$ or $-pV$, or both. Three Legendre transforms of the internal energy are possible, defined as follows: $\textbf{Enthalpy} \qquad H \defn U+pV \tag{5.3.1}$ $\textbf{Helmholtz energy} \qquad A \defn U-TS \tag{5.3.2}$ $\textbf{Gibbs energy} \qquad G \defn U-TS+pV = H-TS \tag{5.3.3}$ These definitions are used whether or not the system has only two independent variables.
The enthalpy, Helmholtz energy, and Gibbs energy are important functions used extensively in thermodynamics. They are state functions (because the quantities used to define them are state functions) and are extensive (because $U$, $S$, and $V$ are extensive). If temperature or pressure are not uniform in the system, we can apply the definitions to constituent phases, or to subsystems small enough to be essentially uniform, and sum over the phases or subsystems.
Alternative names for the Helmholtz energy are Helmholtz function, Helmholtz free energy, and work function. Alternative names for the Gibbs energy are Gibbs function and Gibbs free energy. Both the Helmholtz energy and Gibbs energy have been called simply free energy, and the symbol $F$ has been used for both. The nomenclature in this e-book follows the recommendations of the IUPAC Green Book (E. Richard Cohen et al, Quantities, Units and Symbols in Physical Chemistry, 3rd edition, RSC Publishing, Cambridge, 2007).
Expressions for infinitesimal changes of $H$, $A$, and $G$ are obtained by applying the rules of differentiation to their defining equations: $\dif H = \dif U + p \dif V + V \difp \tag{5.3.4}$ $\dif A = \dif U - T \dif S - S \dif T \tag{5.3.5}$ $\dif G = \dif U - T\dif S - S\dif T + p\dif V + V\difp \tag{5.3.6}$ These expressions for $\dif H$, $\dif A$, and $\dif G$ are general expressions for any system or phase with uniform $T$ and $p$. They are not total differentials of $H$, $A$, and $G$, as the variables in the differentials in each expression are not independent.
A useful property of the enthalpy in a closed system can be found by replacing $\dif U$ in Eq. 5.3.4 by the first law expression $\dq-p\dif V+\dw'$, to obtain $\dif H = \dq + V\difp + \dw'$. Thus, in a process at constant pressure ($\difp = 0$) with expansion work only ($\dw'{=}0$), we have \begin{gather} \s{ \dif H = \dq } \tag{5.3.7} \cond{(closed system, constant $p$,} \nextcond{$\dw'{=}0$)} \end{gather} The enthalpy change under these conditions is equal to the heat. The integrated form of this relation is $\int\!\dif H=\int\!\dq$, or \begin{gather} \s{\Del H = q} \tag{5.3.8} \cond{(closed system, constant $p$,} \nextcond{$w'{=}0$)} \end{gather}
Equation 5.3.7 is analogous to the following relation involving the internal energy, obtained from the first law: \begin{gather} \s{ \dif U = \dq} \tag{5.3.9} \cond{(closed system, constant $V$,} \nextcond{$\dw'{=}0$)} \end{gather} That is, in a process at constant volume with expansion work only, the internal energy change is equal to the heat. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/05%3A_Thermodynamic_Potentials/5.03%3A_Enthalpy_Helmholtz_Energy_and_Gibbs_Energy.txt |
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$\newcommand{\CVm}{C_{V,\text{m}}} % molar heat capacity at const.V$
$\newcommand{\Cpm}{C_{p,\text{m}}} % molar heat capacity at const.p$
$\newcommand{\kT}{\kappa_T} % isothermal compressibility$
$\newcommand{\A}{_{\text{A}}} % subscript A for solvent or state A$
$\newcommand{\B}{_{\text{B}}} % subscript B for solute or state B$
$\newcommand{\bd}{_{\text{b}}} % subscript b for boundary or boiling point$
$\newcommand{\C}{_{\text{C}}} % subscript C$
$\newcommand{\f}{_{\text{f}}} % subscript f for freezing point$
$\newcommand{\mA}{_{\text{m},\text{A}}} % subscript m,A (m=molar)$
$\newcommand{\mB}{_{\text{m},\text{B}}} % subscript m,B (m=molar)$
$\newcommand{\mi}{_{\text{m},i}} % subscript m,i (m=molar)$
$\newcommand{\fA}{_{\text{f},\text{A}}} % subscript f,A (for fr. pt.)$
$\newcommand{\fB}{_{\text{f},\text{B}}} % subscript f,B (for fr. pt.)$
$\newcommand{\xbB}{_{x,\text{B}}} % x basis, B$
$\newcommand{\xbC}{_{x,\text{C}}} % x basis, C$
$\newcommand{\cbB}{_{c,\text{B}}} % c basis, B$
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$\newcommand{\kHi}{k_{\text{H},i}} % Henry's law constant, x basis, i$
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$\renewcommand{\in}{\sups{int}} % internal$
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$\newcommand{\cm}{\subs{cm}} % center of mass$
$\newcommand{\rev}{\subs{rev}} % reversible$
$\newcommand{\irr}{\subs{irr}} % irreversible$
$\newcommand{\fric}{\subs{fric}} % friction$
$\newcommand{\diss}{\subs{diss}} % dissipation$
$\newcommand{\el}{\subs{el}} % electrical$
$\newcommand{\cell}{\subs{cell}} % cell$
$\newcommand{\As}{A\subs{s}} % surface area$
$\newcommand{\E}{^\mathsf{E}} % excess quantity (superscript)$
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$\newcommand{\pha}{\alpha} % phase alpha$
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$\newcommand{\dBar}{\mathop{}\!\mathrm{d}\hspace-.3em\raise1.05ex{\Rule{.8ex}{.125ex}{0ex}}} % inexact differential$
$\newcommand{\dq}{\dBar q} % heat differential$
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$\newcommand{\G}{\varGamma} % activity coefficient of a reference state (pressure factor)$
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$\newcommand{\Eeq}{E\subs{cell, eq}} % equilibrium cell potential$
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In order to find expressions for the total differentials of $H$, $A$, and $G$ in a closed system with one component in one phase, we must replace $\dif U$ in Eqs. 5.3.4–5.3.6 with $\dif U = T\dif S-p\dif V \tag{5.4.1}$ to obtain $\dif H = T \dif S + V \difp \tag{5.4.2}$ $\dif A = -S \dif T - p \dif V \tag{5.4.3}$ $\dif G = -S \dif T + V \difp \tag{5.4.4}$ Equations 5.4.1–5.4.4 are sometimes called the Gibbs equations. They are expressions for the total differentials of the thermodynamic potentials $U$, $H$, $A$, and $G$ in closed systems of one component in one phase with expansion work only. Each equation shows how the dependent variable on the left side varies as a function of changes in two independent variables (the natural variables of the dependent variable) on the right side.
By identifying the coefficients on the right side of Eqs. 5.4.1–5.4.4, we obtain the following relations (which again are valid for a closed system of one component in one phase with expansion work only):
from Eq. 5.4.1: $\Pd{U}{S}{V} = T \tag{5.4.5}$ $\Pd{U}{V}{S} = -p \tag{5.4.6}$
from Eq. 5.4.2: $\Pd{H}{S}{p} = T \tag{5.4.7}$ $\Pd{H}{p}{S} = V \tag{5.4.8}$
from Eq. 5.4.3: $\Pd{A}{T}{V} = -S \tag{5.4.9}$ $\Pd{A}{V}{T} = -p \tag{5.4.10}$
from Eq. 5.4.4: $\Pd{G}{T}{p} = -S \tag{5.4.11}$ $\Pd{G}{p}{T} = V \tag{5.4.12}$
This e-book now uses for the first time an extremely useful mathematical tool called the reciprocity relation of a total differential (Sec. F.2). Suppose the independent variables are $x$ and $y$ and the total differential of a dependent state function $f$ is given by $\df = a\dx + b\dif y \tag{5.4.13}$ where $a$ and $b$ are functions of $x$ and $y$. Then the reciprocity relation is $\Pd{a}{y}{x} = \Pd{b}{x}{y} \tag{5.4.14}$
The reciprocity relations obtained from the Gibbs equations (Eqs. 5.4.1–5.4.4) are called Maxwell relations (again valid for a closed system with $C{=}1$, $P{=}1$, and $\dw'{=}0$):
from Eq. 5.4.1: $\Pd{T}{V}{S} = -\Pd{p}{S}{V} \tag{5.4.15}$
from Eq. 5.4.2: $\Pd{T}{p}{S} = \Pd{V}{S}{p} \tag{5.4.16}$
from Eq. 5.4.3: $\Pd{S}{V}{T} = \Pd{p}{T}{V} \tag{5.4.17}$
from Eq. 5.4.4: $-\Pd{S}{p}{T} = \Pd{V}{T}{p} \tag{5.4.18}$ | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/05%3A_Thermodynamic_Potentials/5.04%3A_Closed_Systems.txt |
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An open system of one substance in one phase, with expansion work only, has three independent variables. The total differential of $U$ is given by Eq. 5.2.5: $\dif U = T \dif S - p \dif V + \mu \dif n \tag{5.5.1}$ In this open system the natural variables of $U$ are $S$, $V$, and $n$. Substituting this expression for $\dif U$ into the expressions for $\dif H$, $\dif A$, and $\dif G$ given by Eqs. 5.3.4–5.3.6, we obtain the following total differentials: $\dif H = T \dif S + V \difp + \mu \dif n \tag{5.5.2}$ $\dif A = -S \dif T - p \dif V + \mu \dif n \tag{5.5.3}$ $\dif G = -S \dif T + V \difp + \mu \dif n \tag{5.5.4}$ Note that these are the same as the four Gibbs equations (Eqs. 5.4.1–5.4.4) with the addition of a term $\mu \dif n$ to allow for a change in the amount of substance.
Identification of the coefficient of the last term on the right side of each of these equations shows that the chemical potential can be equated to four different partial derivatives: $\mu = \Pd{U}{n}{S,V} = \Pd{H}{n}{S,p} = \Pd{A}{n}{T,V} = \Pd{G}{n}{T,p} \tag{5.5.5}$ All four of these partial derivatives must have the same value for a given state of the system; the value, of course, depends on what that state is.
The last partial derivative on the right side of Eq. 5.5.5, $\pd{G}{n}{T,p}$, is especially interesting because it is the rate at which the Gibbs energy increases with the amount of substance added to a system whose intensive properties remain constant. Thus, $\mu$ is revealed to be equal to $G\m$, the molar Gibbs energy of the substance.
Suppose the system contains several substances or species in a single phase (a mixture) whose amounts can be varied independently. We again assume the only work is expansion work. Then, making use of Eq. 5.2.6, we find the total differentials of the thermodynamic potentials are given by
$\dif U = T \dif S - p \dif V + \sum_i\mu_i\dif n_i \tag{5.5.6}$ $\dif H = T \dif S + V \difp + \sum_i\mu_i\dif n_i \tag{5.5.7}$ $\dif A = -S \dif T - p \dif V + \sum_i\mu_i\dif n_i \tag{5.5.8}$ $\dif G = -S \dif T + V \difp + \sum_i\mu_i\dif n_i \tag{5.5.9}$ The independent variables on the right side of each of these equations are the natural variables of the corresponding thermodynamic potential. Section F.4 shows that all of the information contained in an algebraic expression for a state function is preserved in a Legendre transform of the function. What this means for the thermodynamic potentials is that an expression for any one of them, as a function of its natural variables, can be converted to an expression for each of the other thermodynamic potentials as a function of its natural variables.
Willard Gibbs, after whom the Gibbs energy is named, called Eqs. 5.5.6–5.5.9 the fundamental equations of thermodynamics, because from any single one of them not only the other thermodynamic potentials but also all thermal, mechanical, and chemical properties of the system can be deduced (J. Willard Gibbs, in Henry Andrews Bumstead and Ralph Gibbs Van Name, editors, The Scientific Papers of J. Willard Gibbs, Vol. I, Ox Bow Press, Woodbridge, Connecticut, 1993, p. 86). Problem 5.4 illustrates this useful application of the total differential of a thermodynamic potential.
In Eqs. 5.5.6–5.5.9, the coefficient $\mu_i$ is the chemical potential of species $i$. The equations show that $\mu_i$ can be equated to four different partial derivatives, similar to the equalities shown in Eq. 5.5.5 for a pure substance: $\mu_i = \Pd{U}{n_i}{S,V,n_{j \ne i}} = \Pd{H}{n_i}{S,p,n_{j \ne i}} = \Pd{A}{n_i}{T,V,n_{j \ne i}} = \Pd{G}{n_i}{T,p,n_{j \ne i}} \tag{5.5.10}$ The partial derivative $\pd{G}{n_i}{T,P,n_{j \ne i}}$ is called the partial molar Gibbs energy of species $i$, another name for the chemical potential as will be discussed in Sec. 9.2.6. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/05%3A_Thermodynamic_Potentials/5.05%3A_Open_Systems.txt |
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$\newcommand{\kT}{\kappa_T} % isothermal compressibility$
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$\newcommand{\C}{_{\text{C}}} % subscript C$
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$\newcommand{\cm}{\subs{cm}} % center of mass$
$\newcommand{\rev}{\subs{rev}} % reversible$
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$\newcommand{\diss}{\subs{diss}} % dissipation$
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$\newcommand{\dBar}{\mathop{}\!\mathrm{d}\hspace-.3em\raise1.05ex{\Rule{.8ex}{.125ex}{0ex}}} % inexact differential$
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As explained in Sec. 3.1.5, the heat capacity of a closed system is defined as the ratio of an infinitesimal quantity of heat transferred across the boundary under specified conditions and the resulting infinitesimal temperature change: $\tx{heat capacity} \defn \dq/\dif T$. The heat capacities of isochoric (constant volume) and isobaric (constant pressure) processes are of particular interest.
The heat capacity at constant volume, $C_V$, is the ratio $\dq/\dif T$ for a process in a closed constant-volume system with no nonexpansion work—that is, no work at all. The first law shows that under these conditions the internal energy change equals the heat: $\dif U=\dq$ (Eq. 5.3.9). We can replace $\dq$ by $\dif U$ and write $C_V$ as a partial derivative: \begin{gather} \s{ C_V = \Pd{U}{T}{V} } \tag{5.6.1} \cond{(closed system)} \end{gather}
If the closed system has more than two independent variables, additional conditions are needed to define $C_V$ unambiguously. For instance, if the system is a gas mixture in which reaction can occur, we might specify that the system remains in reaction equilibrium as $T$ changes at constant $V$.
Equation 5.6.1 does not require the condition $\dw'{=}0$, because all quantities appearing in the equation are state functions whose relations to one another are fixed by the nature of the system and not by the path. Thus, if heat transfer into the system at constant $V$ causes $U$ to increase at a certain rate with respect to $T$, and this rate is defined as $C_V$, the performance of electrical work on the system at constant $V$ will cause the same rate of increase of $U$ with respect to $T$ and can equally well be used to evaluate $C_V$.
Note that $C_V$ is a state function whose value depends on the state of the system—that is, on $T$, $V$, and any additional independent variables. $C_V$ is an extensive property: the combination of two identical phases has twice the value of $C_V$ that one of the phases has by itself.
For a phase containing a pure substance, the molar heat capacity at constant volume is defined by $\CVm \defn C_V/n$. $\CVm$ is an intensive property.
If the system is an ideal gas, its internal energy depends only on $T$, regardless of whether $V$ is constant, and Eq. 5.6.1 can be simplified to \begin{gather} \s{ C_V = \frac{\dif U}{\dif T}} \tag{5.6.2} \cond{(closed system, ideal gas)} \end{gather} Thus the internal energy change of an ideal gas is given by $\dif U=C_V\dif T$, as mentioned earlier in Sec. 3.5.3.
The heat capacity at constant pressure, $C_p$, is the ratio $\dq/\dif T$ for a process in a closed system with a constant, uniform pressure and with expansion work only. Under these conditions, the heat $\dq$ is equal to the enthalpy change $\dif H$ (Eq. 5.3.7), and we obtain a relation analogous to Eq. 5.6.1: \begin{gather} \s{ C_p = \Pd{H}{T}{\!p} } \tag{5.6.3} \cond{(closed system)} \end{gather} $C_p$ is an extensive state function. For a phase containing a pure substance, the molar heat capacity at constant pressure is $\Cpm = C_p/n$, an intensive property.
Since the enthalpy of a fixed amount of an ideal gas depends only on $T$ (Prob. 5.1), we can write a relation analogous to Eq. 5.6.2: \begin{gather} \s{ C_p = \frac{\dif H}{\dif T}} \tag{5.6.4} \cond{(closed system, ideal gas)} \end{gather} | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/05%3A_Thermodynamic_Potentials/5.06%3A_Expressions_for_Heat_Capacity.txt |
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Sometimes we need more than the usual two independent variables to describe an equilibrium state of a closed system of one substance in one phase. This is the case when, in addition to expansion work, another kind of work is possible. The total differential of $U$ is then given by $\dif U = T\dif S - p\dif V + Y\dif X$ (Eq. 5.2.7), where $Y\dif X$ represents the nonexpansion work $\dw'$.
A good example of this situation is surface work in a system in which surface area is relevant to the description of the state.
A liquid–gas interface behaves somewhat like a stretched membrane. The upper and lower surfaces of the liquid film in the device depicted in Fig. 5.1 exert a force $F$ on the sliding rod, tending to pull it in the direction that reduces the surface area. We can measure the force by determining the opposing force $F\subs{ext}$ needed to prevent the rod from moving. This force is found to be proportional to the length of the rod and independent of the rod position $x$. The force also depends on the temperature and pressure.
The surface tension or interfacial tension, $\g$, is the force exerted by an interfacial surface per unit length. The film shown in Fig. 5.1 has two surfaces, so we have $\g = F/2l$ where $l$ is the rod length.
To increase the surface area of the film by a practically-reversible process, we slowly pull the rod to the right in the $+x$ direction. The system is the liquid. The $x$ component of the force exerted by the system on the surroundings at the moving boundary, $F_x\sups{sys}$, is equal to $-F$ ($F$ is positive and $F_x\sups{sys}$ is negative). The displacement of the rod results in surface work given by Eq. 3.1.2: $\dw' = -F_x\sups{sys}\dx = 2\g l\dx$. The increase in surface area, $\dif A\subs{s}$, is $2l\dx$, so the surface work is $\dw' = \g \dif A\subs{s}$ where $\g$ is the work coefficient and $A\subs{s}$ is the work coordinate. Equation 5.2.7 becomes $\dif U=T\dif S -p\dif V+\g\dif A\subs{s} \tag{5.7.1}$ Substitution into Eq. 5.3.6 gives $\dif G = -S \dif T + V \difp + \g \dif A\subs{s} \tag{5.7.2}$ which is the total differential of $G$ with $T$, $p$, and $A\subs{s}$ as the independent variables. Identifying the coefficient of the last term on the right side as a partial derivative, we find the following expression for the surface tension: $\g = \Pd{G}{A\subs{s}}{T,p} \tag{5.7.3}$ That is, the surface tension is not only a force per unit length, but also a Gibbs energy per unit area.
From Eq. 5.7.2, we obtain the reciprocity relation $\Pd{\g}{T}{p,A\subs{s}} = -\Pd{S}{A\subs{s}}{T,p} \tag{5.7.4}$ It is valid to replace the partial derivative on the left side by $\pd{\g}{T}{p}$ because $\g$ is independent of $A\subs{s}$. Thus, the variation of surface tension with temperature tells us how the entropy of the liquid varies with surface area. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/05%3A_Thermodynamic_Potentials/5.07%3A_Surface_Work.txt |
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$\newcommand{\aph}{^{\alpha}} % alpha phase superscript$
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In this section we combine the first and second laws in order to derive some general relations for changes during a reversible or irreversible process of a closed system. The temperature and pressure will be assumed to be practically uniform during the process, even if the process is irreversible. For example, the volume might be changing at a finite rate but very slowly, or there might be a spontaneous homogeneous reaction in a mixture of uniform temperature and pressure.
The second law states that $\dif S$ is equal to $\dq/T$ if the process is reversible, and is greater than $\dq/T$ if the process is irreversible: \begin{gather} \s{\dif S \ge \dq/T} \tag{5.8.1} \cond{(${}\subs{ rev}\sups{ irrev}$, closed system)} \end{gather} or \begin{gather} \s{ \dq \le T \dif S } \tag{5.8.2} \cond{(${}\subs{ rev}\sups{ irrev}$, closed system)} \end{gather} The inequalities in these relations refer to an irreversible process and the equalities to a reversible process, as indicated by the notation ${}\subs{ rev}\sups{ irrev}$.
When we substitute $\dq$ from Eq. 5.8.2 into the first law in the form $\dif U=\dq-p\dif V+\dw'$, where $\dw'$ is nonexpansion work, we obtain the relation \begin{gather} \s{ \dif U \le T \dif S - p \dif V + \dw' } \tag{5.8.3} \cond{(${}\subs{ rev}\sups{ irrev}$, closed system)} \end{gather} We substitute this relation for $\dif U$ into the differentials of enthalpy, Helmholtz energy, and Gibbs energy given by Eqs. 5.3.4–5.3.6 to obtain three more relations:
\begin{gather} \s{ \dif H \le T \dif S + V \difp + \dw' } \tag{5.8.4} \cond{(${}\subs{ rev}\sups{ irrev}$, closed system)} \end{gather} \begin{gather} \s{ \dif A \le -S \dif T - p \dif V + \dw' } \tag{5.8.5} \cond{(${}\subs{ rev}\sups{ irrev}$, closed system)} \end{gather} \begin{gather} \s{ \dif G \le -S \dif T + V \difp + \dw' } \tag{5.8.6} \cond{(${}\subs{ rev}\sups{ irrev}$, closed system)} \end{gather}
The last two of these relations provide valuable criteria for spontaneity under common laboratory conditions. Equation 5.8.5 shows that during a spontaneous irreversible change at constant temperature and volume, $\dif A$ is less than $\dw'$. If the only work is expansion work (i.e., $\dw'$ is zero), the Helmholtz energy decreases during a spontaneous process at constant $T$ and $V$ and has its minimum value when the system reaches an equilibrium state.
Equation 5.8.6 is especially useful. From it, we can conclude the following:
• Ben-Amotz and Honig (J. Chem. Phys., 118, 5932–5936, 2003; J. Chem. Educ., 83, 132–137, 2006) developed a “rectification” procedure that simplifies the mathematical manipulation of inequalities. Following this procedure, we can write $\dif S = \dq/T + \dBar\theta \tag{5.8.7}$ where $\dBar\theta$ is an excess entropy function that is positive for an irreversible change and zero for a reversible change ($\dBar\theta \geq 0$). Solving for $\dq$ gives the expression $\dq=T\dif S-T\dBar\theta$ that, when substituted in the first law expression $\dif U=\dq-p\dif V+\dw'$, produces $\dif U = T\dif S-p\dif V+\dw'-T\dBar\theta \tag{5.8.8}$ The equality of this equation is equivalent to the combined equality and inequality of Eq. 5.8.3. Then by substitution of this expression for $\dif U$ into Eqs. 5.3.4–5.3.6, we obtain equalities equivalent to Eqs. 5.8.4–5.8.6, for example $\dif G = -S\dif T + V\difp + \dw' - T\dBar\theta \tag{5.8.9}$ Equation 5.8.9 tells us that during a process at constant $T$ and $p$, with expansion work only ($\dw'{=}0$), $\dif G$ has the same sign as $-T\dBar\theta$: negative for an irreversible change and zero for a reversible change. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/05%3A_Thermodynamic_Potentials/5.08%3A_Criteria_for_Spontaneity.txt |
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$\newcommand{\cond}[1]{\$-2.5pt]{}\tag*{#1}}$ $\newcommand{\nextcond}[1]{\[-5pt]{}\tag*{#1}}$ $\newcommand{\R}{8.3145\units{J\,K\per\,mol\per}} % gas constant value$ $\newcommand{\Rsix}{8.31447\units{J\,K\per\,mol\per}} % gas constant value - 6 sig figs$ $\newcommand{\jn}{\hspace3pt\lower.3ex{\Rule{.6pt}{2ex}{0ex}}\hspace3pt}$ $\newcommand{\ljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}} \hspace3pt}$ $\newcommand{\lljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace1.4pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace3pt}$ An underlined problem number or problem-part letter indicates that the numerical answer appears in Appendix I. 5.1 Show that the enthalpy of a fixed amount of an ideal gas depends only on the temperature. 5.2 From concepts in this chapter, show that the heat capacities $C_V$ and $C_p$ of a fixed amount of an ideal gas are functions only of $T$. 5.3 During the reversible expansion of a fixed amount of an ideal gas, each increment of heat is given by the expression $\dq=C_V \dif T + (nRT/V)\dif V$ (Eq. 4.3.4). (a) A necessary and sufficient condition for this expression to be an exact differential is that the reciprocity relation must be satisfied for the independent variables $T$ and $V$ (see Appendix F). Apply this test to show that the expression is not an exact differential, and that heat therefore is not a state function. (b) By the same method, show that the entropy increment during the reversible expansion, given by the expression $\dif S=\dq/T$, is an exact differential, so that entropy is a state function. 5.4 This problem illustrates how an expression for one of the thermodynamic potentials as a function of its natural variables contains the information needed to obtain expressions for the other thermodynamic potentials and many other state functions. From statistical mechanical theory, a simple model for a hypothetical “hard-sphere” liquid (spherical molecules of finite size without attractive intermolecular forces) gives the following expression for the Helmholtz energy with its natural variables $T$, $V$, and $n$ as the independent variables: \[ A = -nRT\ln\left[cT^{3/2}\left(\frac{V}{n}-b\right)\right] - nRT + na$ Here $a$, $b$, and $c$ are constants. Derive expressions for the following state functions of this hypothetical liquid as functions of $T$, $V$, and $n$.
(a) The entropy, $S$
(b) The pressure, $p$
(c) The chemical potential, $\mu$
(d) The internal energy, $U$
(e) The enthalpy, $H$
(f) The Gibbs energy, $G$
(g) The heat capacity at constant volume, $C_V$
(h) The heat capacity at constant pressure, $C_p$ (hint: use the expression for $p$ to solve for $V$ as a function of $T$, $p$, and $n$; then use $H=U+pV$)
5.6
Use the data in Table 5.1 to evaluate $\pd{S}{A\subs{s}}{T,p}$ at $25\units{\(\degC$}\), which is the rate at which the entropy changes with the area of the air–water interface at this temperature.
5.7
When an ordinary rubber band is hung from a clamp and stretched with constant downward force $F$ by a weight attached to the bottom end, gentle heating is observed to cause the rubber band to contract in length. To keep the length $l$ of the rubber band constant during heating, $F$ must be increased. The stretching work is given by $\dw'=F\dif l$. From this information, find the sign of the partial derivative $\pd{T}{l}{S,p}$; then predict whether stretching of the rubber band will cause a heating or a cooling effect.
(Hint: make a Legendre transform of $U$ whose total differential has the independent variables needed for the partial derivative, and write a reciprocity relation.)
You can check your prediction experimentally by touching a rubber band to the side of your face before and after you rapidly stretch it. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/05%3A_Thermodynamic_Potentials/5.09%3A_Chapter_5_Problems.txt |
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The third law of thermodynamics concerns the entropy of perfectly-ordered crystals at zero kelvins.
When a chemical reaction or phase transition is studied at low temperatures, and all substances are pure crystals presumed to be perfectly ordered, the entropy change is found to approach zero as the temperature approaches zero kelvins: \begin{gather} \s{ \lim_{T\!\ra 0} \Del S=0 } \tag{6.0.1} \cond{(pure, perfectly-ordered crystals)} \end{gather} Equation 6.0.1 is the mathematical statement of the Nernst heat theorem or third law of thermodynamics. It is true in general only if each reactant and product is a pure crystal with identical unit cells arranged in perfect spatial order.
Nernst preferred to avoid the use of the entropy function and to use in its place the partial derivative $-\pd{A}{T}{V}$ (Eq. 5.4.9). The original 1906 version of his heat theorem was in the form $\lim_{T\!\ra 0} \pd{\Del A}{T}{V}{=}0$ (William H. Cropper, J. Chem. Educ., 64, 3–8, 1987).
06: The Third Law and Cryogenics
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There is no theoretical relation between the entropies of different chemical elements. We can arbitrarily choose the entropy of every pure crystalline element to be zero at zero kelvins. Then the experimental observation expressed by Eq. 6.0.1 requires that the entropy of every pure crystalline compound also be zero at zero kelvins, in order that the entropy change for the formation of a compound from its elements will be zero at this temperature.
A classic statement of the third law principle appears in the 1923 book Thermodynamics and the Free Energy of Chemical Substances by G. N. Lewis and M. Randall (McGraw-Hill, New York, p. 448):
“If the entropy of each element in some crystalline state be taken as zero at the absolute zero of temperature: every substance has a finite positive entropy, but at the absolute zero of temperature the entropy may become zero, and does so become in the case of perfect crystalline substances.”
According to this principle, every substance (element or compound) in a pure, perfectly-ordered crystal at $0\K$, at any pressure, has a molar entropy of zero: \begin{gather} \s{ S\m \tx{($0$ K)}=0 } \tag{6.1.1} \cond{(pure, perfectly-ordered crystal)} \end{gather} This convention establishes a scale of absolute entropies at temperatures above zero kelvins called third-law entropies, as explained in Sec. 6.2.
The entropy becomes independent of pressure as $T$ approaches zero kelvins. This behavior can be deduced from the relation $\pd{S}{p}{T} = -\alpha V$ (Table 7.1) combined with the experimental observation that the cubic expansion coefficient $\alpha$ approaches zero as $T$ approaches zero kelvins. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/06%3A_The_Third_Law_and_Cryogenics/6.01%3A_The_Zero_of_Entropy.txt |
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With the convention that the entropy of a pure, perfectly-ordered crystalline solid at zero kelvins is zero, we can establish the third-law value of the molar entropy of a pure substance at any temperature and pressure. Absolute values of $S\m$ are what are usually tabulated for calculational use.
6.2.1 Third-law molar entropies
Suppose we wish to evaluate the entropy of an amount $n$ of a pure substance at a certain temperature $T'$ and a certain pressure. The same substance, in a perfectly-ordered crystal at zero kelvins and the same pressure, has an entropy of zero. The entropy at the temperature and pressure of interest, then, is the entropy change $\Del S = \int_{0}^{T'}\!\dq / T$ of a reversible heating process at constant pressure that converts the perfectly-ordered crystal at zero kelvins to the state of interest.
Consider a reversible isobaric heating process of a pure substance while it exists in a single phase. The definition of heat capacity as $\dq/\dif T$ (Eq. 3.1.9) allows us to substitute $C_p\dif T$ for $\dq$, where $C_p$ is the heat capacity of the phase at constant pressure.
If the substance in the state of interest is a liquid or gas, or a crystal of a different form than the perfectly-ordered crystal present at zero kelvins, the heating process will include one or more equilibrium phase transitions under conditions where two phases are in equilibrium at the same temperature and pressure (Sec. 2.2.2). For example, a reversible heating process at a pressure above the triple point that transforms the crystal at $0\K$ to a gas may involve transitions from one crystal form to another, and also melting and vaporization transitions.
Each such reversible phase transition requires positive heat $q\subs{trs}$. Because the pressure is constant, the heat is equal to the enthalpy change (Eq. 5.3.8). The ratio $q\subs{trs}/n$ is called the molar heat or molar enthalpy of the transition, $\Delsub{trs}H$ (see Sec. 8.3.1). Because the phase transition is reversible, the entropy change during the transition is given by $\Delsub{trs}S=q\subs{trs}/nT\subs{trs}$ where $T\subs{trs}$ is the transition temperature.
With these considerations, we can write the following expression for the entropy change of the entire heating process: $\Del S = \int_{0}^{T'}\! \frac{C_p}{T}\dif T + \sum\frac{n\Delsub{trs}H}{T\subs{trs}} \tag{6.2.1}$ The resulting operational equation for the calculation of the molar entropy of the substance at the temperature and pressure of interest is \begin{gather} \s{ S\m(T') = \frac{\Del S}{n} = \int_{0}^{T'} \frac{\Cpm}{T}\dif T + \sum\frac{\Delsub{trs}H}{T\subs{trs}} } \tag{6.2.2} \cond{(pure substance,} \nextcond{constant $p$)} \end{gather} where $\Cpm=C_p/n$ is the molar heat capacity at constant pressure. The summation is over each equilibrium phase transition occurring during the heating process.
Since $\Cpm$ is positive at all temperatures above zero kelvins, and $\Delsub{trs}H$ is positive for all transitions occurring during a reversible heating process, the molar entropy of a substance is positive at all temperatures above zero kelvins.
The heat capacity and transition enthalpy data required to evaluate $S\m(T')$ using Eq. 6.2.2 come from calorimetry. The calorimeter can be cooled to about $10\K$ with liquid hydrogen, but it is difficult to make measurements below this temperature. Statistical mechanical theory may be used to approximate the part of the integral in Eq. 6.2.2 between zero kelvins and the lowest temperature at which a value of $\Cpm$ can be measured. The appropriate formula for nonmagnetic nonmetals comes from the Debye theory for the lattice vibration of a monatomic crystal. This theory predicts that at low temperatures (from $0\K$ to about $30\K$), the molar heat capacity at constant volume is proportional to $T^3$: $\CVm = aT^3$, where $a$ is a constant. For a solid, the molar heat capacities at constant volume and at constant pressure are practically equal. Thus for the integral on the right side of Eq. 6.2.2 we can, to a good approximation, write $\int_{0}^{T'}\frac{\Cpm}{T}\dif T = a\int_{0}^{T''} \!\! T^2\dif T + \int_{T''}^{T'}\frac{\Cpm}{T}\dif T \tag{6.2.3}$ where $T''$ is the lowest temperature at which $\Cpm$ is measured. The first term on the right side of Eq. 6.2.3 is $a\int_{0}^{T''} \!\! T^2\dif T = \left.(aT^3/3)\right|_0^{T''} = a(T'')^3/3 \tag{6.2.4}$ But $a(T'')^3$ is the value of $\Cpm$ at $T''$, so Eq. 6.2.2 becomes \begin{gather} \s{ S\m(T') = \frac{\Cpm (T'')}{3} + \int_{T''}^{T'}\frac{\Cpm}{T}\dif T + \sum\frac{\Delsub{trs}H}{T\subs{trs}} } \tag{6.2.5} \cond{(pure substance,} \nextcond{constant $p$)} \end{gather}
In the case of a metal, statistical mechanical theory predicts an electronic contribution to the molar heat capacity, proportional to $T$ at low temperature, that should be added to the Debye $T^3$ term: $\Cpm = aT^3 + bT$. The error in using Eq. 6.2.5, which ignores the electronic term, is usually negligible if the heat capacity measurements are made down to about $10\K$.
We may evaluate the integral on the right side of Eq. 6.2.5 by numerical integration. We need the area under the curve of $\Cpm/T$ plotted as a function of $T$ between some low temperature, $T''$, and the temperature $T'$ at which the molar entropy is to be evaluated. Since the integral may be written in the form $\int_{T''}^{T'}\frac{\Cpm}{T}\dif T = \int_{T=T''}^{T=T'}\Cpm\dif\ln(T/\tx{K}) \tag{6.2.6}$ we may also evaluate the integral from the area under a curve of $\Cpm$ plotted as a function of $\ln(T/K)$.
Ideally, the molar entropy values obtained by the calorimetric (third-law) method for a gas should agree closely with the values calculated from spectroscopic data. Table 6.1 shows that for some substances this agreement is not present. The table lists values of $S\m\st$ for ideal gases at $298.15\K$ evaluated by both the calorimetric and spectroscopic methods. The quantity $S\subs{m,0}$ in the last column is the difference between the two $S\m\st$ values, and is called the molar residual entropy.
In the case of HCl, the experimental value of the residual entropy is comparable to its uncertainty, indicating good agreement between the calorimetric and spectroscopic methods. This agreement is typical of most substances, particularly those like HCl whose molecules are polar and asymmetric with a large energetic advantage of forming perfectly-ordered crystals.
The other substances listed in Table 6.1 have residual entropies that are greater than zero within the uncertainty of the data. What is the meaning of this discrepancy between the calorimetric and spectroscopic results? We can assume that the true values of $S\m\st$ at $298.15\K$ are the spectroscopic values, because their calculation assumes the solid has only one microstate at $0\K$, with an entropy of zero, and takes into account all of the possible accessible microstates of the ideal gas. The calorimetric values, on the other hand, are based on Eq. 6.2.2 which assumes the solid becomes a perfectly-ordered crystal as the temperature approaches $0\K$.
The calorimetric values in Table 6.1 were calculated as follows. Measurements of heat capacities and heats of transition were used in Eq. 6.2.2 to find the third-law value of $S\m$ for the vapor at the boiling point of the substance at $p=1\units{atm}$. This calculated value for the gas was corrected to that for the ideal gas at $p=1\br$ and adjusted to $T=298.15\K$ with spectroscopic data.
The conventional explanation of a nonzero residual entropy is the presence of random rotational orientations of molecules in the solid at the lowest temperature at which the heat capacity can be measured, so that the crystals are not perfectly ordered. The random structure is established as the crystals form from the liquid, and becomes frozen into the crystals as the temperature is lowered below the freezing point. This tends to happen with almost-symmetric molecules with small dipole moments which in the crystal can have random rotational orientations of practically equal energy. In the case of solid H$_2$O it is the arrangement of intermolecular hydrogen bonds that is random. Crystal imperfections such as dislocations can also contribute to the residual entropy. If such crystal imperfection is present at the lowest experimental temperature, the calorimetric value of $S\m\st$ for the gas at $298.15\K$ is the molar entropy increase for the change at $1\br$ from the imperfectly-ordered solid at $0\K$ to the ideal gas at $298.15\K$, and the residual entropy $S\subs{m,0}$ is the molar entropy of this imperfectly-ordered solid. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/06%3A_The_Third_Law_and_Cryogenics/6.02%3A_Molar_Entropies.txt |
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The field of cryogenics involves the production of very low temperatures, and the study of the behavior of matter at these temperatures. These low temperatures are needed to evaluate third-law entropies using calorimetric measurements. There are some additional interesting thermodynamic applications.
6.3.1 Joule–Thomson expansion
A gas can be cooled by expanding it adiabatically with a piston (Sec. 3.5.3), and a liquid can be cooled by pumping on its vapor to cause evaporation (vaporization). An evaporation procedure with a refrigerant fluid is what produces the cooling in an ordinary kitchen refrigerator.
For further cooling of a fluid, a common procedure is to use a continuous throttling process in which the fluid is forced to flow through a porous plug, valve, or other constriction that causes an abrupt drop in pressure. A slow continuous adiabatic throttling of a gas is called the Joule–Thomson experiment, or Joule–Kelvin experiment, after the two scientists who collaborated between 1852 and 1862 to design and analyze this procedure. (William Thomson later became Lord Kelvin.)
Figure 6.3 illustrates the principle of the technique. The solid curve shows the temperature dependence of the entropy of a paramagnetic solid in the absence of an applied magnetic field, and the dashed curve is for the solid in a constant, finite magnetic field. The temperature range shown is from $0\K$ to approximately $1\K$. At $0\K$, the magnetic dipoles are perfectly ordered. The increase of $S$ shown by the solid curve between $0\K$ and $1\K$ is due almost entirely to increasing disorder in the orientations of the magnetic dipoles as heat enters the system.
Path A represents the process that occurs when the paramagnetic solid, surrounded by gaseous helium in thermal contact with liquid helium that has been cooled to about $1\K$, is slowly moved into a strong magnetic field. The process is isothermal magnetization, which partially orients the magnetic dipoles and reduces the entropy. During this process there is heat transfer to the liquid helium, which partially boils away. In path B, the thermal contact between the solid and the liquid helium has been broken by pumping away the gas surrounding the solid, and the sample is slowly moved away from the magnetic field. This step is a reversible adiabatic demagnetization. Because the process is reversible and adiabatic, the entropy change is zero, which brings the state of the solid to a lower temperature as shown.
The sign of $\pd{T}{B}{S,p}$ is of interest because it tells us the sign of the temperature change during a reversible adiabatic demagnetization (path B of Fig. 6.3). To change the independent variables in Eq. 6.3.4 to $S$, $p$, and $B$, we define the Legendre transform $H'\defn U+pV-Bm\subs{mag} \tag{6.3.5}$ ($H'$ is sometimes called the magnetic enthalpy.) From Eqs. 6.3.4 and 6.3.5 we obtain the total differential $\dif H' = T\dif S+V\difp-m\subs{mag}\dif B \tag{6.3.6}$ From it we find the reciprocity relation $\Pd{T}{B}{S,p} = -\Pd{m\subs{mag}}{S}{p,B} \tag{6.3.7}$
According to Curie’s law of magnetization, the magnetic dipole moment $m\subs{mag}$ of a paramagnetic phase at constant magnetic flux density $B$ is proportional to $1/T$. This law applies when $B$ is small, but even if $B$ is not small $m\subs{mag}$ decreases with increasing $T$. To increase the temperature of a phase at constant $B$, we allow heat to enter the system, and $S$ then increases. Thus, $\pd{m\subs{mag}}{S}{p,B}$ is negative and, according to Eq. 6.3.7, $\pd{T}{B}{S,p}$ must be positive. Adiabatic demagnetization is a constant-entropy process in which $B$ decreases, and therefore the temperature also decreases.
We can find the sign of the entropy change during the isothermal magnetization process shown as path A in Fig. 6.3. In order to use $T$, $p$, and $B$ as the independent variables, we define the Legendre transform $G'\defn H'-TS$. Its total differential is $\dif G' = -S\dif T + V\difp -m\subs{mag}\dif B \tag{6.3.8}$ From this total differential, we obtain the reciprocity relation $\Pd{S}{B}{T,p} = \Pd{m\subs{mag}}{T}{p,B} \tag{6.3.9}$ Since $m\subs{mag}$ at constant $B$ decreases with increasing $T$, as explained above, we see that the entropy change during isothermal magnetization is negative.
By repeatedly carrying out a procedure of isothermal magnetization and adiabatic demagnetization, starting each stage at the temperature produced by the previous stage, it has been possible to attain a temperature as low as $0.0015\K$. The temperature can be reduced still further, down to 16 microkelvins, by using adiabatic nuclear demagnetization. However, as is evident from the figure, if in accordance with the third law both of the entropy curves come together at the absolute zero of the kelvin scale, then it is not possible to attain a temperature of zero kelvins in a finite number of stages of adiabatic demagnetization. This conclusion is called the principle of the unattainability of absolute zero. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/06%3A_The_Third_Law_and_Cryogenics/6.03%3A_Cryogenics.txt |
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An underlined problem number or problem-part letter indicates that the numerical answer appears in Appendix I.
6.1
Calculate the molar entropy of carbon disulfide at $25.00\units{\(\degC$}\) and $1\br$ from the heat capacity data for the solid in Table 6.2 and the following data for $p=1\br$. At the melting point, $161.11\K$, the molar enthalpy of fusion is $\Delsub{fus}H = 4.39\timesten{3}\units{J mol\(^{-1}$}\). The molar heat capacity of the liquid in the range 161–300 K is described by $\Cpm = a + bT$, where the constants have the values $a = 74.6\units{J K\(^{-1}$ mol$^{-1}$}\) and $b = 0.0034\units{J K\(^{-2}$ mol$^{-1}$}\). | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/06%3A_The_Third_Law_and_Cryogenics/6.04%3A_Chapter_6_Problem.txt |
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This chapter applies concepts introduced in earlier chapters to the simplest kind of system, one consisting of a pure substance or a single component in a single phase. The system has three independent variables if it is open, and two if it is closed. Relations among various properties of a single phase are derived, including temperature, pressure, and volume. The important concepts of standard states and chemical potential are introduced.
07: Pure Substances in Single Phases
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Two volume properties of a closed system are defined as follows: $\textbf{cubic expansion coefficient} \quad \alpha \defn \frac{1}{V}\Pd{V}{T}{p} \tag{7.1.1}$ $\textbf{isothermal compressibility} \quad \kT \defn -\frac{1}{V}\Pd{V}{p}{T} \tag{7.1.2}$
The cubic expansion coefficient is also called the coefficient of thermal expansion and the expansivity coefficient. Other symbols for the isothermal compressibility are $\beta$ and $\g_T$.
These definitions show that $\alpha$ is the fractional volume increase per unit temperature increase at constant pressure, and $\kT$ is the fractional volume decrease per unit pressure increase at constant temperature. Both quantities are intensive properties. Most substances have positive values of $\alpha$, and all substances have positive values of $\kT$, because a pressure increase at constant temperature requires a volume decrease.
The cubic expansion coefficient is not always positive. $\alpha$ is negative for liquid water below its temperature of maximum density, $3.98\units{\(\degC$}\). The crystalline ceramics zirconium tungstate (ZrW$_2$O$_8$) and hafnium tungstate (HfW$_2$O$_8$) have the remarkable behavior of contracting uniformly and continuously in all three dimensions when they are heated from $0.3\K$ to about $1050\K$; $\alpha$ is negative throughout this very wide temperature range (T. A. Mary et al, Science, 272, 90–92, 1996). The intermetallic compound YbGaGe has been found to have a value of $\alpha$ that is practically zero in the range $100$–$300\K$ (James R. Salvador et al, Nature, 425, 702–705, 2003).
If an amount $n$ of a substance is in a single phase, we can divide the numerator and denominator of the right sides of Eqs. 7.1.1 and 7.1.2 by $n$ to obtain the alternative expressions \begin{gather} \s{ \alpha = \frac{1}{V\m}\Pd{V\m}{T}{\!p} } \tag{7.1.3} \cond{(pure substance, $P{=}1$)} \end{gather} \begin{gather} \s{ \kT = -\frac{1}{V\m}\Pd{V\m}{p}{T} } \tag{7.1.4} \cond{(pure substance, $P{=}1$)} \end{gather} where $V\m$ is the molar volume. $P$ in the conditions of validity is the number of phases. Note that only intensive properties appear in Eqs. 7.1.3 and 7.1.4; the amount of the substance is irrelevant. Figures 7.1 and 7.2 show the temperature variation of $\alpha$ and $\kT$ for several substances.
If we choose $T$ and $p$ as the independent variables of the closed system, the total differential of $V$ is given by $\dif V = \Pd{V}{T}{\!p}\dif T + \Pd{V}{p}{T}\difp \tag{7.1.5}$ With the substitutions $\pd{V}{T}{p} = \alpha V$ (from Eq. 7.1.1) and $\pd{V}{p}{T} = -\kT V$ (from Eq. 7.1.2), the expression for the total differential of $V$ becomes \begin{gather} \s{ \dif V = \alpha V \dif T - \kT V \difp } \tag{7.1.6} \cond{(closed system,} \nextcond{$C{=}1$, $P{=}1$)} \end{gather} To find how $p$ varies with $T$ in a closed system kept at constant volume, we set $\dif V$ equal to zero in Eq. 7.1.6: $0 = \alpha V\dif T - \kappa _T V\difp$, or $\difp/\dif T = \alpha /\kappa _T$. Since $\difp/\dif T$ under the condition of constant volume is the partial derivative $\pd{p}{T}{V}$, we have the general relation \begin{gather} \s{ \Pd{p}{T}{V} = \frac{\alpha}{\kT} } \tag{7.1.7} \cond{(closed system,} \nextcond{$C{=}1$, $P{=}1$)} \end{gather} | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/07%3A_Pure_Substances_in_Single_Phases/7.01%3A_Volume_Properties.txt |
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The partial derivative $\pd{U}{V}{T}$ applied to a fluid phase in a closed system is called the internal pressure. (Note that $U$ and $pV$ have dimensions of energy; therefore, $U/V$ has dimensions of pressure.)
To relate the internal pressure to other properties, we divide Eq. 5.2.2 by $\dif V$: $\dif U/\dif V = T(\dif S/\dif V) - p$. Then we impose a condition of constant $T$: $\pd{U}{V}{T} = T\pd{S}{V}{T}-p$. When we make a substitution for $\pd{S}{V}{T}$ from the Maxwell relation of Eq. 5.4.17, we obtain \begin{gather} \s{ \Pd{U}{V}{T} = T \Pd{p}{T}{V} - p } \tag{7.2.1} \cond{(closed system,} \nextcond{fluid phase, $C{=}1$)} \end{gather} This equation is sometimes called the “thermodynamic equation of state” of the fluid.
For an ideal-gas phase, we can write $p=nRT/V$ and then $\Pd{p}{T}{V} = \frac{nR}{V} = \frac{p}{T} \tag{7.2.2}$ Making this substitution in Eq. 7.2.1 gives us \begin{gather} \s{ \Pd{U}{V}{T} = 0 } \tag{7.2.3} \cond{(closed system of an ideal gas)} \end{gather} showing that the internal pressure of an ideal gas is zero.
In Sec. 3.5.1, an ideal gas was defined as a gas (1) that obeys the ideal gas equation, and (2) for which $U$ in a closed system depends only on $T$. Equation 7.2.3, derived from the first part of this definition, expresses the second part. It thus appears that the second part of the definition is redundant, and that we could define an ideal gas simply as a gas obeying the ideal gas equation. This argument is valid only if we assume the ideal-gas temperature is the same as the thermodynamic temperature (Secs. 2.3.5 and 4.3.4) since this assumption is required to derive Eq. 7.2.3. Without this assumption, we can’t define an ideal gas solely by $pV = nRT$, where $T$ is the ideal gas temperature.
Here is a simplified interpretation of the significance of the internal pressure. When the volume of a fluid increases, the average distance between molecules increases and the potential energy due to intermolecular forces changes. If attractive forces dominate, as they usually do unless the fluid is highly compressed, expansion causes the potential energy to increase. The internal energy is the sum of the potential energy and thermal energy. The internal pressure, $\pd{U}{V}{T}$, is the rate at which the internal energy changes with volume at constant temperature. At constant temperature, the thermal energy is constant so that the internal pressure is the rate at which just the potential energy changes with volume. Thus, the internal pressure is a measure of the strength of the intermolecular forces and is positive if attractive forces dominate. (These attractive intermolecular forces are the cohesive forces that can allow a negative pressure to exist in a liquid; see Sec. 2.3.4.) In an ideal gas, intermolecular forces are absent and therefore the internal pressure of an ideal gas is zero.
With the substitution $\pd{p}{T}{V} = \alpha/\kT$ (Eq. 7.1.7), Eq. 7.2.1 becomes \begin{gather} \s{ \Pd{U}{V}{T} = \frac{\alpha T}{\kT} - p } \tag{7.2.4} \cond{(closed system,} \nextcond{fluid phase, $C{=}1$)} \end{gather} The internal pressure of a liquid at $p = 1\br$ is typically much larger than $1\br$ (see Prob. 7.6). Equation 7.2.4 shows that, in this situation, the internal pressure is approximately equal to $\alpha T/\kT$. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/07%3A_Pure_Substances_in_Single_Phases/7.02%3A_Internal_Pressure.txt |
For convenience in derivations to follow, expressions from Chap. 5 are repeated here that apply to processes in a closed system in the absence of nonexpansion work (i.e., đ $w^{\prime}=0$ ). For a process at constant volume we have $^{3}$
$\mathrm{d} U=\mathrm{d} q \quad C_{V}=\left(\frac{\partial U}{\partial T}\right)_{V}$
and for a process at constant pressure we have ${ }^{4}$
$\mathrm{d} H=\mathrm{d} q \quad C_{p}=\left(\frac{\partial H}{\partial T}\right)_{p}$
A closed system of one component in a single phase has only two independent variables. In such a system, the partial derivatives above are complete and unambiguous definitions of $C_{V}$ and $C_{p}$ because they are expressed with two independent variables- $T$ and $V$ for $C_{V}$, and $T$ and $p$ for $C_{p}$. As mentioned on page 146, additional conditions would have to be specified to define $C_{V}$ for a more complicated system; the same is true for $C_{p}$.
For a closed system of an ideal gas we have 5
$C_{V}=\frac{\mathrm{d} U}{\mathrm{~d} T} \quad C_{p}=\frac{\mathrm{d} H}{\mathrm{~d} T}$
7.3.1 The relation between $C_{V, \mathrm{~m}}$ and $C_{p, \mathrm{~m}}$
The value of $C_{p, \mathrm{~m}}$ for a substance is greater than $C_{V, \mathrm{~m}}$. The derivation is simple in the case of a fixed amount of an ideal gas. Using substitutions from Eq. 7.3.3, we write
$C_{p}-C_{V}=\frac{\mathrm{d} H}{\mathrm{~d} T}-\frac{\mathrm{d} U}{\mathrm{~d} T}=\frac{\mathrm{d}(H-U)}{\mathrm{d} T}=\frac{\mathrm{d}(p V)}{\mathrm{d} T}=n R$
Division by $n$ to obtain molar quantities and rearrangement then gives
$C_{p, \mathrm{~m}}=C_{V, \mathrm{~m}}+R$
For any phase in general, we proceed as follows. First we write
$C_{p}=\left(\frac{\partial H}{\partial T}\right)_{p}=\left[\frac{\partial(U+p V)}{\partial T}\right]_{p}=\left(\frac{\partial U}{\partial T}\right)_{p}+p\left(\frac{\partial V}{\partial T}\right)_{p}$
Then we write the total differential of $U$ with $T$ and $V$ as independent variables and identify one of the coefficients as $C_{V}$ :
$\mathrm{d} U=\left(\frac{\partial U}{\partial T}\right)_{V} \mathrm{~d} T+\left(\frac{\partial U}{\partial V}\right)_{T} \mathrm{~d} V=C_{V} \mathrm{~d} T+\left(\frac{\partial U}{\partial V}\right)_{T} \mathrm{~d} V$
When we divide both sides of the preceding equation by $\mathrm{d} T$ and impose a condition of constant $p$, we obtain
$\left(\frac{\partial U}{\partial T}\right)_{p}=C_{V}+\left(\frac{\partial U}{\partial V}\right)_{T}\left(\frac{\partial V}{\partial T}\right)_{p}$
Substitution of this expression for $(\partial U / \partial T)_{p}$ in the equation for $C_{p}$ yields
$C_{p}=C_{V}+\left[\left(\frac{\partial U}{\partial V}\right)_{T}+p\right]\left(\frac{\partial V}{\partial T}\right)_{p}$
Finally we set the partial derivative $(\partial U / \partial V)_{T}$ (the internal pressure) equal to $\left(\alpha T / \kappa_{T}\right)-p$ (Eq. 7.2.4) and $(\partial V / \partial T)_{p}$ equal to $\alpha V$ to obtain
$C_{p}=C_{V}+\frac{\alpha^{2} T V}{\kappa_{T}}$
and divide by $n$ to obtain molar quantities:
$C_{p, \mathrm{~m}}=C_{V, \mathrm{~m}}+\frac{\alpha^{2} T V_{\mathrm{m}}}{\kappa_{T}}$
Since the quantity $\alpha^{2} T V_{\mathrm{m}} / \kappa_{T}$ must be positive, $C_{p, \mathrm{~m}}$ is greater than $C_{V, \mathrm{~m}}$.
7.3.2 The measurement of heat capacities
The most accurate method of evaluating the heat capacity of a phase is by measuring the temperature change resulting from heating with electrical work. The procedure in general is called calorimetry, and the apparatus containing the phase of interest and the electric heater is a calorimeter. The principles of three commonly-used types of calorimeters with electrical heating are described below.
Adiabatic calorimeters
An adiabatic calorimeter is designed to have negligible heat flow to or from its surroundings. The calorimeter contains the phase of interest, kept at either constant volume or constant pressure, and also an electric heater and a temperature-measuring device such as a platinum resistance thermometer, thermistor, or quartz crystal oscillator. The contents may be stirred to ensure temperature uniformity.
To minimize conduction and convection, the calorimeter usually is surrounded by a jacket separated by an air gap or an evacuated space. The outer surface of the calorimeter and inner surface of the jacket may be polished to minimize radiation emission from these surfaces. These measures, however, are not sufficient to ensure a completely adiabatic boundary, because energy can be transferred by heat along the mounting hardware and through the electrical leads. Therefore, the temperature of the jacket, or of an outer metal shield, is adjusted throughout the course of the experiment so as to be as close as possible to the varying temperature of the calorimeter. This goal is most easily achieved when the temperature change is slow.
To make a heat capacity measurement, a constant electric current is passed through the heater circuit for a known period of time. The system is the calorimeter and its contents. The electrical work $w_{\text {el }}$ performed on the system by the heater circuit is calculated from the integrated form of Eq. $3.8 .5$ on page 91: $w_{\mathrm{el}}=I^{2} R_{\mathrm{el}} \Delta t$, where $I$ is the electric current, $R_{\mathrm{el}}$ is the electric resistance, and $\Delta t$ is the time interval. We assume the boundary is adiabatic and write the first law in the form
$\mathrm{d} U=-p \mathrm{~d} V+\mathrm{d} w_{\mathrm{el}}+\mathrm{d} w_{\mathrm{cont}}$
where $-p \mathrm{~d} V$ is expansion work and $w_{\text {cont }}$ is any continuous mechanical work from stirring (the subscript "cont" stands for continuous). If electrical work is done on the system by a
thermometer using an external electrical circuit, such as a platinum resistance thermometer, this work is included in $w_{\text {cont }}$.
Consider first an adiabatic calorimeter in which the heating process is carried out at constant volume. There is no expansion work, and Eq. $7.3 .12$ becomes
$\mathrm{d} U=\mathrm{d} w_{\mathrm{el}}+\mathrm{d} w_{\mathrm{cont}}$ (constant $V$ )
An example of a measured heating curve (temperature $T$ as a function of time $t$ ) is shown in Fig. 7.3. We select two points on the heating curve, indicated in the figure by open circles. Time $t_{1}$ is at or shortly before the instant the heater circuit is closed and electrical heating begins, and time $t_{2}$ is after the heater circuit has been opened and the slope of the curve has become essentially constant.
In the time periods before $t_{1}$ and after $t_{2}$, the temperature may exhibit a slow rate of increase due to the continuous work $w_{\text {cont }}$ from stirring and temperature measurement. If this work is performed at a constant rate throughout the course of the experiment, the slope is constant and the same in both time periods as shown in the figure.
The relation between the slope and the rate of work is given by a quantity called the energy equivalent, $\epsilon$. The energy equivalent is the heat capacity of the calorimeter under the conditions of an experiment. The heat capacity of a constant-volume calorimeter is given by $\epsilon=(\partial U / \partial T)_{V}$ (Eq. 5.6.1). Thus, at times before $t_{1}$ or after $t_{2}$, when đ $w_{\text {el }}$ is zero and $\mathrm{d} U$ equals $w_{\text {cont }}$, the slope $r$ of the heating curve is given by
$r=\frac{\mathrm{d} T}{\mathrm{~d} t}=\frac{\mathrm{d} T}{\mathrm{~d} U} \frac{\mathrm{d} U}{\mathrm{~d} t}=\frac{1}{\epsilon} \frac{\mathrm{d} w_{\text {cont }}}{\mathrm{d} t}$
The rate of the continuous work is therefore $\mathrm{d} w_{\text {cont }} / \mathrm{d} t=\epsilon r$. This rate is constant throughout the experiment. In the time interval from $t_{1}$ to $t_{2}$, the total quantity of continuous work is $w_{\text {cont }}=\epsilon r\left(t_{2}-t_{1}\right)$, where $r$ is the slope of the heating curve measured outside this time interval.
To find the energy equivalent, we integrate Eq. $7.3 .13$ between the two points on the curve:
$\Delta U=w_{\mathrm{el}}+w_{\mathrm{cont}}=w_{\mathrm{el}}+\epsilon r\left(t_{2}-t_{1}\right)$ (constant $V$ )
Then the average heat capacity between temperatures $T_{1}$ and $T_{2}$ is
$\epsilon=\frac{\Delta U}{T_{2}-T_{1}}=\frac{w_{\mathrm{el}}+\epsilon r\left(t_{2}-t_{1}\right)}{T_{2}-T_{1}}$
Solving for $\epsilon$, we obtain
$\epsilon=\frac{w_{\mathrm{el}}}{T_{2}-T_{1}-r\left(t_{2}-t_{1}\right)}$
The value of the denominator on the right side is indicated by the vertical line in Fig. 7.3. It is the temperature change that would have been observed if the same quantity of electrical work had been performed without the continuous work.
Next, consider the heating process in a calorimeter at constant pressure. In this case the enthalpy change is given by $\mathrm{d} H=\mathrm{d} U+p \mathrm{~d} V$ which, with substitution from Eq. 7.3.12, becomes
$\mathrm{d} H=\mathrm{d} w_{\mathrm{el}}+\mathrm{d} w_{\mathrm{cont}}$
(constant $p$ )
We follow the same procedure as for the constant-volume calorimeter, using Eq. $7.3 .18$ in place of Eq. $7.3 .13$ and equating the energy equivalent $\epsilon$ to $(\partial H / \partial T)_{p}$, the heat capacity of the calorimeter at constant pressure (Eq. 5.6.3). We obtain the relation
$\Delta H=w_{\mathrm{el}}+w_{\mathrm{cont}}=w_{\mathrm{el}}+\epsilon r\left(t_{2}-t_{1}\right)$
(constant $p$ )
in place of Eq. $7.3 .15$ and end up again with the expression of Eq. $7.3 .17$ for $\epsilon$.
The value of $\epsilon$ calculated from Eq. $7.3 .17$ is an average value for the temperature interval from $T_{1}$ to $T_{2}$, and we can identify this value with the heat capacity at the temperature of the midpoint of the interval. By taking the difference of values of $\epsilon$ measured with and without the phase of interest present in the calorimeter, we obtain $C_{V}$ or $C_{p}$ for the phase alone.
It may seem paradoxical that we can use an adiabatic process, one without heat, to evaluate a quantity defined by heat (heat capacity $=\mathrm{d} q / \mathrm{d} T$ ). The explanation is that energy transferred into the adiabatic calorimeter as electrical work, and dissipated completely to thermal energy, substitutes for the heat that would be needed for the same change of state without electrical work.
Isothermal-jacket calorimeters
A second common type of calorimeter is similar in construction to an adiabatic calorimeter, except that the surrounding jacket is maintained at constant temperature. It is sometimes called an isoperibol calorimeter. A correction is made for heat transfer resulting from the difference in temperature across the gap separating the jacket from the outer surface of the calorimeter. It is important in making this correction that the outer surface have a uniform temperature without "hot spots."
Assume the outer surface of the calorimeter has a uniform temperature $T$ that varies with time, the jacket temperature has a constant value $T_{\text {ext }}$, and convection has been eliminated by evacuating the gap. Then heat transfer is by conduction and radiation, and its rate
is given by Newton's law of cooling
$\frac{\mathrm{d} q}{\mathrm{~d} t}=-k\left(T-T_{\mathrm{ext}}\right)$
where $k$ is a constant (the thermal conductance). Heat flows from a warmer to a cooler body, so đ $q / \mathrm{d} t$ is positive if $T$ is less than $T_{\text {ext }}$ and negative if $T$ is greater than $T_{\text {ext }}$.
The possible kinds of work are the same as for the adiabatic calorimeter: expansion work $-p \mathrm{~d} V$, intermittent work $w_{\mathrm{el}}$ done by the heater circuit, and continuous work $w_{\text {cont }}$. By combining the first law and Eq. 7.3.20, we obtain the following relation for the rate at which the internal energy changes:
$\frac{\mathrm{d} U}{\mathrm{~d} t}=\frac{\mathrm{d} q}{\mathrm{~d} t}+\frac{\mathrm{d} w}{\mathrm{~d} t}=-k\left(T-T_{\mathrm{ext}}\right)-p \frac{\mathrm{d} V}{\mathrm{~d} t}+\frac{\mathrm{d} w_{\mathrm{el}}}{\mathrm{d} t}+\frac{\mathrm{d} w_{\text {cont }}}{\mathrm{d} t}$
For heating at constant volume $(\mathrm{d} V / \mathrm{d} t=0)$, this relation becomes
$\frac{\mathrm{d} U}{\mathrm{~d} t}=-k\left(T-T_{\mathrm{ext}}\right)+\frac{\mathrm{d} w_{\mathrm{el}}}{\mathrm{d} t}+\frac{\mathrm{d} w_{\mathrm{cont}}}{\mathrm{d} t}$(constant $V$ )
An example of a heating curve is shown in Fig. 7.4. In contrast to the curve of Fig. $7.3$, the slopes are different before and after the heating interval due to changed rate of heat flow. Times $t_{1}$ and $t_{2}$ are before and after the heater circuit is closed. In any time interval before time $t_{1}$ or after time $t_{2}$, the system behaves as if it is approaching a steady state of constant temperature $T_{\infty}$ (called the convergence temperature), which it would eventually reach if the experiment were continued without closing the heater circuit. $T_{\infty}$ is greater than $T_{\text {ext }}$ because of the energy transferred to the system by stirring and electrical temperature measurement. By setting $\mathrm{d} U / \mathrm{d} t$ and $\mathrm{d} w_{\mathrm{el}} / \mathrm{d} t$ equal to zero and $T$ equal to $T_{\infty}$ in Eq. 7.3.22, we obtain đ $w_{\text {cont }} / \mathrm{d} t=k\left(T_{\infty}-T_{\text {ext }}\right) .$ We assume d $w_{\text {cont }} / \mathrm{d} t$ is constant. Substituting this expression into Eq. $7.3 .22$ gives us a general expression for the rate at which $U$ changes in terms of the unknown quantities $k$ and $T_{\infty}$ :
$\frac{\mathrm{d} U}{\mathrm{~d} t}=-k\left(T-T_{\infty}\right)+\frac{\mathrm{d} w_{\mathrm{el}}}{\mathrm{d} t}$
(constant $V$ )
This relation is valid throughout the experiment, not only while the heater circuit is closed. If we multiply by $\mathrm{d} t$ and integrate from $t_{1}$ to $t_{2}$, we obtain the internal energy change in the time interval from $t_{1}$ to $t_{2}$ :
$\Delta U=-k \int_{t_{1}}^{t_{2}}\left(T-T_{\infty}\right) \mathrm{d} t+w_{\mathrm{el}}$
(constant $V$ )
All the intermittent work $w_{\mathrm{el}}$ is performed in this time interval.
The derivation of Eq. $7.3 .24$ is a general one. The equation can be applied also to a isothermal-jacket calorimeter in which a reaction is occurring. Section $11.5 .2$ will mention the use of this equation for an internal energy correction of a reaction calorimeter with an isothermal jacket.
The average value of the energy equivalent in the temperature range $T_{1}$ to $T_{2}$ is
$\epsilon=\frac{\Delta U}{T_{2}-T_{1}}=\frac{-\epsilon(k / \epsilon) \int_{t_{1}}^{t_{2}}\left(T-T_{\infty}\right) \mathrm{d} t+w_{\mathrm{el}}}{T_{2}-T_{1}}$
Solving for $\epsilon$, we obtain
$\epsilon=\frac{w_{\mathrm{el}}}{\left(T_{2}-T_{1}\right)+(k / \epsilon) \int_{t_{1}}^{t_{2}}\left(T-T_{\infty}\right) \mathrm{d} t}$
The value of $w_{\mathrm{el}}$ is known from $w_{\mathrm{el}}=I^{2} R_{\mathrm{el}} \Delta t$, where $\Delta t$ is the time interval during which the heater circuit is closed. The integral can be evaluated numerically once $T_{\infty}$ is known.
For heating at constant pressure, $\mathrm{d} H$ is equal to $\mathrm{d} U+p \mathrm{~d} V$, and we can write
$\frac{\mathrm{d} H}{\mathrm{~d} t}=\frac{\mathrm{d} U}{\mathrm{~d} t}+p \frac{\mathrm{d} V}{\mathrm{~d} t}=-k\left(T-T_{\mathrm{ext}}\right)+\frac{\mathrm{d} w_{\mathrm{el}}}{\mathrm{d} t}+\frac{\mathrm{d} w_{\mathrm{cont}}}{\mathrm{d} t}$
(constant $p$ )
which is analogous to Eq. 7.3.22. By the procedure described above for the case of constant $V$, we obtain
$\Delta H=-k \int_{t_{1}}^{t_{2}}\left(T-T_{\infty}\right) \mathrm{d} t+w_{\mathrm{el}}$
(constant $p$ )
At constant $p$, the energy equivalent is equal to $C_{p}=\Delta H /\left(T_{2}-T_{1}\right)$, and the final expression for $\epsilon$ is the same as that given by Eq. 7.3.26.
To obtain values of $k / \epsilon$ and $T_{\infty}$ for use in Eq. 7.3.26, we need the slopes of the heating curve in time intervals (rating periods) just before $t_{1}$ and just after $t_{2}$. Consider the case of constant volume. In these intervals, $\mathrm{d} w_{\mathrm{el}} / \mathrm{d} t$ is zero and $\mathrm{d} U / \mathrm{d} t$ equals $-k\left(T-T_{\infty}\right)$ (from Eq. 7.3.23). The heat capacity at constant volume is $C_{V}=\mathrm{d} U / \mathrm{d} T$. The slope $r$ in general is then given by
$r=\frac{\mathrm{d} T}{\mathrm{~d} t}=\frac{\mathrm{d} T}{\mathrm{~d} U} \frac{\mathrm{d} U}{\mathrm{~d} t}=-(k / \epsilon)\left(T-T_{\infty}\right)$
Applying this relation to the points at times $t_{1}$ and $t_{2}$, we have the following simultaneous equations in the unknowns $k / \epsilon$ and $T_{\infty}$ :
$r_{1}=-(k / \epsilon)\left(T_{1}-T_{\infty}\right) \quad r_{2}=-(k / \epsilon)\left(T_{2}-T_{\infty}\right)$
The solutions are
$(k / \epsilon)=\frac{r_{1}-r_{2}}{T_{2}-T_{1}} \quad T_{\infty}=\frac{r_{1} T_{2}-r_{2} T_{1}}{r_{1}-r_{2}}$
Finally, $k$ is given by
$k=(k / \epsilon) \epsilon=\left(\frac{r_{1}-r_{2}}{T_{2}-T_{1}}\right) \epsilon$
When the pressure is constant, this procedure yields the same relations for $k / \epsilon, T_{\infty}$, and $k .$
Continuous-flow calorimeters
A flow calorimeter is a third type of calorimeter used to measure the heat capacity of a fluid phase. The gas or liquid flows through a tube at a known constant rate past an electrical heater of known constant power input. After a steady state has been achieved in the tube, the temperature increase $\Del T$ at the heater is measured.
If $\dw\el/\dt$ is the rate at which electrical work is performed (the electric power) and $\dif m/\dt$ is the mass flow rate, then in time interval $\Del t$ a quantity $w=(\dw\el/\dt)\Del t$ of work is performed on an amount $n=(\dif m/\dt)\Del t/M$ of the fluid (where $M$ is the molar mass). If heat flow is negligible, the molar heat capacity of the substance is given by $\Cpm = \frac{w}{n\Del T} = \frac{M(\dw\el/\dt)}{\Del T(\dif m/\dt)} \tag{7.3.33}$ To correct for the effects of heat flow, $\Del T$ is usually measured over a range of flow rates and the results extrapolated to infinite flow rate.
7.3.3 Typical values
Figure 7.5 Temperature dependence of molar heat capacity at constant pressure ($p=1\br$) of H$_2$O, N$_2$, and C(graphite).
Figure 7.5 shows the temperature dependence of $\Cpm$ for several substances. The discontinuities seen at certain temperatures occur at equilibrium phase transitions. At these temperatures the heat capacity is in effect infinite, since the phase transition of a pure substance involves finite heat with zero temperature change. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/07%3A_Pure_Substances_in_Single_Phases/7.03%3A_Thermal_Properties.txt |
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Consider the process of changing the temperature of a phase at constant volume.
Keeping the volume exactly constant while increasing the temperature is not as simple as it may sound. Most solids expand when heated, unless we arrange to increase the external pressure at the same time. If we use solid walls to contain a fluid phase, the container volume will change with temperature. For practical purposes, these volume changes are usually negligible.
The rate of change of internal energy with $T$ at constant $V$ is the heat capacity at constant volume: $C_V=\pd{U}{T}{V}$ (Eq. 7.3.1). Accordingly, an infinitesimal change of $U$ is given by \begin{gather} \s{ \dif U = C_V \dif T } \tag{7.4.1} \cond{(closed system,} \nextcond{$C{=}1$, $P{=}1$, constant $V$)} \end{gather} and the finite change of $U$ between temperatures $T_1$ and $T_2$ is \begin{gather} \s{ \Del U = \int_{T_1}^{T_2}\!C_V \dif T } \tag{7.4.2} \cond{(closed system,} \nextcond{$C{=}1$, $P{=}1$, constant $V$)} \end{gather}
Three comments, relevant to these and other equations in this chapter, are in order:
1. If, at a fixed volume and over the temperature range $T_1$ to $T_2$, the value of $C_V$ is essentially constant (i.e., independent of $T$), Eq. 7.4.2 becomes \begin{gather} \s{ \Del U = C_V (T_2-T_1) } \tag{7.4.5} \cond{(closed system, $C{=}1$,} \nextcond{$P{=}1$, constant $V$ and $C_V$)} \end{gather}
An infinitesimal entropy change during a reversible process in a closed system is given according to the second law by $\dif S = \dq/T$. At constant volume, $\dq$ is equal to $\dif U$ which in turn equals $C_V\dif T$. Therefore, the entropy change is \begin{gather} \s{ \dif S = \frac{C_V}{T} \dif T } \tag{7.4.6} \cond{(closed system,} \nextcond{$C{=}1$, $P{=}1$, constant $V$)} \end{gather} Integration yields the finite change \begin{gather} \s{ \Del S = \int_{T_1}^{T_2} \frac{C_V}{T} \dif T } \tag{7.4.7} \cond{(closed system,} \nextcond{$C{=}1$, $P{=}1$, constant $V$)} \end{gather} If $C_V$ is treated as constant, Eq. 7.4.7 becomes \begin{gather} \s{ \Del S = C_V \ln \frac{T_2}{T_1} } \tag{7.4.8} \cond{(closed system, $C{=}1$,} \nextcond{$P{=}1$, constant $V$ and $C_V$)} \end{gather} (More general versions of the two preceding equations have already been given in Sec. 4.6.1.)
We may derive relations for a temperature change at constant pressure by the same methods. From $C_p = \pd{H}{T}{p}$ (Eq. 7.3.2), we obtain \begin{gather} \s{ \Del H = \int_{T_1}^{T_2} C_p \dif T } \tag{7.4.9} \cond{(closed system,} \nextcond{$C{=}1$, $P{=}1$, constant $p$)} \end{gather} If $C_p$ is treated as constant, Eq. 7.4.9 becomes \begin{gather} \s{ \Del H = C_p (T_2-T_1) } \tag{7.4.10} \cond{(closed system, $C{=}1$,} \nextcond{$P{=}1$, constant $p$ and $C_p$)} \end{gather} From $\dif S = \dq/T$ and Eq. 7.3.2 we obtain for the entropy change at constant pressure \begin{gather} \s{ \dif S = \frac{C_p}{T} \dif T } \tag{7.4.11} \cond{(closed system,} \nextcond{$C{=}1$, $P{=}1$, constant $p$)} \end{gather} Integration gives \begin{gather} \s{ \Del S = \int_{T_1}^{T_2} \frac{C_p}{T} \dif T } \tag{7.4.12} \cond{(closed system,} \nextcond{$C{=}1$, $P{=}1$, constant $p$)} \end{gather} or, with $C_p$ treated as constant, \begin{gather} \s{ \Del S = C_p \ln \frac{T_2}{T_1} } \tag{7.4.13} \cond{(closed system, $C{=}1$,} \nextcond{$P{=}1$, constant $p$ and $C_p$)} \end{gather} $C_p$ is positive, so heating a phase at constant pressure causes $H$ and $S$ to increase.
The Gibbs energy changes according to $\pd{G}{T}{p}=-S$ (Eq. 5.4.11), so heating at constant pressure causes $G$ to decrease. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/07%3A_Pure_Substances_in_Single_Phases/7.04%3A_Heating_at_Constant_Volume_or_Pressure.txt |
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7.5.1 Tables of partial derivatives
The tables in this section collect useful expressions for partial derivatives of the eight state functions $T$, $p$, $V$, $U$, $H$, $A$, $G$, and $S$ in a closed, single-phase system. Each derivative is taken with respect to one of the three easily-controlled variables $T$, $p$, or $V$ while another of these variables is held constant. We have already seen some of these expressions, and the derivations of the others are indicated below.
We can use these partial derivatives (1) for writing an expression for the total differential of any of the eight quantities, and (2) for expressing the finite change in one of these quantities as an integral under conditions of constant $T$, $p$, or $V$. For instance, given the expressions $\Pd{S}{T}{\!p} = \frac{C_p}{T} \qquad \tx{and} \qquad \Pd{S}{p}{T} = -\alpha V \tag{7.5.1}$ we may write the total differential of $S$, taking $T$ and $p$ as the independent variables, as $\dif S = \frac{C_p}{T}\dif T - \alpha V \difp \tag{7.5.2}$ Furthermore, the first expression is equivalent to the differential form $\dif S = \frac{C_p}{T} \dif T \tag{7.5.3}$ provided $p$ is constant; we can integrate this equation to obtain the finite change $\Del S$ under isobaric conditions as shown in Eq. 7.4.12.
Both general expressions and expressions valid for an ideal gas are given in Tables 7.1, 7.2, and 7.3.
We may derive the general expressions as follows. We are considering differentiation with respect only to $T$, $p$, and $V$. Expressions for $\pd{V}{T}{p}$, $\pd{V}{p}{T}$, and $\pd{p}{T}{V}$ come from Eqs. 7.1.1, 7.1.2, and 7.1.7 and are shown as functions of $\alpha$ and $\kT$. The reciprocal of each of these three expressions provides the expression for another partial derivative from the general relation $\pd{y}{x}{z}=\frac{1}{\pd{x}{y}{z}} \tag{7.5.4}$ This procedure gives us expressions for the six partial derivatives of $T$, $p$, and $V$.
The remaining expressions are for partial derivatives of $U$, $H$, $A$, $G$, and $S$. We obtain the expression for $\pd{U}{T}{V}$ from Eq. 7.3.1, for $\pd{U}{V}{T}$ from Eq. 7.2.4, for $\pd{H}{T}{p}$ from Eq. 7.3.2, for $\pd{A}{T}{V}$ from Eq. 5.4.9, for $\pd{A}{V}{T}$ from Eq. 5.4.10, for $\pd{G}{p}{T}$ from Eq. 5.4.12, for $\pd{G}{T}{p}$ from Eq. 5.4.11, for $\pd{S}{T}{V}$ from Eq. 7.4.6, for $\pd{S}{T}{p}$ from Eq. 7.4.11, and for $\pd{S}{p}{T}$ from Eq. 5.4.18.
We can transform each of these partial derivatives, and others derived in later steps, to two other partial derivatives with the same variable held constant and the variable of differentiation changed. The transformation involves multiplying by an appropriate partial derivative of $T$, $p$, or $V$. For instance, from the partial derivative $\pd{U}{V}{T}=(\alpha T/\kT)-p$, we obtain $\Pd{U}{p}{T} =\Pd{U}{V}{T}\Pd{V}{p}{T} = \left( \frac{\alpha T}{\kT}-p \right) \left( -\kT V \right) = \left(-\alpha T + \kT p\right) V \tag{7.5.5}$ The remaining partial derivatives can be found by differentiating $U=H-pV$, $H=U+pV$, $A=U-TS$, and $G=H-TS$ and making appropriate substitutions. Whenever a partial derivative appears in a derived expression, it is replaced with an expression derived in an earlier step. The expressions derived by these steps constitute the full set shown in Tables 7.1, 7.2, and 7.3.
Bridgman devised a simple method to obtain expressions for these and many other partial derivatives from a relatively small set of formulas (Phys. Rev., 3, 273–281, 1914; The Thermodynamics of Electrical Phenomena in Metals and a Condensed Collection of Thermodynamic Formulas, Dover, New York, 1961, p. 199–241).
7.5.2 The Joule–Thomson coefficient
The Joule–Thomson coefficient of a gas was defined in Eq. 6.3.3 by $\mu\subs{JT}=\pd{T}{p}{H}$. It can be evaluated with measurements of $T$ and $p$ during adiabatic throttling processes as described in Sec. 6.3.1.
To relate $\mu\subs{JT}$ to other properties of the gas, we write the total differential of the enthalpy of a closed, single-phase system in the form $\dif H=\Pd{H}{T}{\!p}\dif T + \Pd{H}{p}{T}\difp \tag{7.5.6}$ and divide both sides by $\difp$: $\frac{\dif H}{\difp}=\Pd{H}{T}{\!p}\frac{\dif T}{\difp} + \Pd{H}{p}{T} \tag{7.5.7}$ Next we impose a condition of constant $H$; the ratio $\dif T/\difp$ becomes a partial derivative: $0=\Pd{H}{T}{\!p}\Pd{T}{p}{H} + \Pd{H}{p}{T} \tag{7.5.8}$ Rearrangement gives $\Pd{T}{p}{H} = -\frac{ \pd{H}{p}{T} }{ \pd{H}{T}{p} } \tag{7.5.9}$ The left side of this equation is the Joule–Thomson coefficient. An expression for the partial derivative $\pd{H}{p}{T}$ is given in Table 7.1, and the partial derivative $\pd{H}{T}{p}$ is the heat capacity at constant pressure (Eq. 5.6.3). These substitutions give us the desired relation $\mu\subs{JT} = \frac{(\alpha T-1)V}{C_p} = \frac{(\alpha T-1)V\m}{\Cpm} \tag{7.5.10}$ | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/07%3A_Pure_Substances_in_Single_Phases/7.05%3A_Partial_Derivatives_with_Respect_to_%28T%29_%28p%29_and_%28V%29.txt |
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In various applications, we will need expressions for the effect of changing the pressure at constant temperature on the internal energy, enthalpy, entropy, and Gibbs energy of a phase. We obtain the expressions by integrating expressions found in Table 7.1. For example, $\Del U$ is given by $\int\pd{U}{p}{T}\difp$. The results are listed in the second column of Table 7.4.
7.6.1 Ideal gases
Simplifications result when the phase is an ideal gas. In this case, we can make the substitutions $V = nRT/p$, $\alpha=1/T$, and $\kT=1/p$, resulting in the expressions in the third column of Table 7.4.
The expressions in the third column of Table 7.4 may be summarized by the statement that, when an ideal gas expands isothermally, the internal energy and enthalpy stay constant, the entropy increases, and the Helmholtz energy and Gibbs energy decrease.
7.6.2 Condensed phases
Solids, and liquids under conditions of temperature and pressure not close to the critical point, are much less compressible than gases. Typically the isothermal compressibility, $\kT$, of a liquid or solid at room temperature and atmospheric pressure is no greater than $1\timesten{-4}\units{bar\(^{-1}$}\) (see Fig. 7.2), whereas an ideal gas under these conditions has $\kT = 1/p =1\units{bar\(^{-1}$}\). Consequently, it is frequently valid to treat $V$ for a liquid or solid as essentially constant during a pressure change at constant temperature. Because $\kT$ is small, the product $\kT\!p$ for a liquid or solid is usually much smaller than the product $\alpha T$. Furthermore, $\kT$ for liquids and solids does not change rapidly with $p$ as it does for gases, and neither does $\alpha$.
With the approximations that $V$, $\alpha$, and $\kT$ are constant during an isothermal pressure change, and that $\kT\!p$ is negligible compared with $\alpha T$, we obtain the expressions in the last column of Table 7.4. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/07%3A_Pure_Substances_in_Single_Phases/7.06%3A_Isothermal_Pressure_Changes.txt |
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It is often useful to refer to a reference pressure, the standard pressure, denoted $p\st$. The standard pressure has an arbitrary but constant value in any given application. Until 1982, chemists used a standard pressure of $1\units{atm}$ ($1.01325\timesten{5}\Pa$). The IUPAC now recommends the value $p\st = 1\br$ (exactly $10^5\Pa$). This e-book uses the latter value unless stated otherwise. (Note that there is no defined standard temperature.)
A superscript degree symbol ($\circ$) denotes a standard quantity or standard-state conditions. An alternative symbol for this purpose, used extensively outside the U.S., is a superscript Plimsoll mark ($\unicode{x29B5}$). (The Plimsoll mark is named after the British merchant Samuel Plimsoll, at whose instigation Parliament passed an act in 1875 requiring the symbol to be placed on the hulls of cargo ships to indicate the maximum depth for safe loading.)
A standard state of a pure substance is a particular reference state appropriate for the kind of phase and is described by intensive variables. This e-book follows the recommendations of the IUPAC Green Book (E. Richard Cohen et al, Quantities, Units and Symbols in Physical Chemistry, 3rd edition, RSC Publishing, Cambridge, 2007, p. 61–62) for various standard states.
• Section 9.7 will introduce additional standard states for constituents of mixtures. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/07%3A_Pure_Substances_in_Single_Phases/7.07%3A_Standard_States_of_Pure_Substances.txt |
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The chemical potential, $\mu$, of a pure substance has as one of its definitions (Sec. 5.5) \begin{gather} \s{ \mu \defn G\m = \frac{G}{n} } \tag{7.8.1} \cond{(pure substance)} \end{gather} That is, $\mu$ is equal to the molar Gibbs energy of the substance at a given temperature and pressure. (Section 9.2.6 will introduce a more general definition of chemical potential that applies also to a constituent of a mixture.) The chemical potential is an intensive state function.
The total differential of the Gibbs energy of a fixed amount of a pure substance in a single phase, with $T$ and $p$ as independent variables, is $\dif G = -S\dif T + V\difp$ (Eq. 5.4.4). Dividing both sides of this equation by $n$ gives the total differential of the chemical potential with these same independent variables: \begin{gather} \s{ \dif \mu = -S\m \dif T + V\m \difp } \tag{7.8.2} \cond{(pure substance, $P{=}1$)} \end{gather} (Since all quantities in this equation are intensive, it is not necessary to specify a closed system; the amount of the substance in the system is irrelevant.)
We identify the coefficients of the terms on the right side of Eq. 7.8.2 as the partial derivatives \begin{gather} \s{ \Pd{\mu}{T}{\!p} = -S\m } \tag{7.8.3} \cond{(pure substance, $P{=}1$)} \end{gather} and \begin{gather} \s{ \Pd{\mu}{p}{T} = V\m } \tag{7.8.4} \cond{(pure substance, $P{=}1$)} \end{gather} Since $V\m$ is positive, Eq. 7.8.4 shows that the chemical potential increases with increasing pressure in an isothermal process.
The standard chemical potential, $\mu\st$, of a pure substance in a given phase and at a given temperature is the chemical potential of the substance when it is in the standard state of the phase at this temperature and the standard pressure $p\st$.
There is no way we can evaluate the absolute value of $\mu$ at a given temperature and pressure, or of $\mu\st$ at the same temperature—at least not to any useful degree of precision. The values of $\mu$ and $\mu\st$ include the molar internal energy whose absolute value can only be calculated from the Einstein relation; see Sec. 2.6.2. We can however measure or calculate the difference $\mu - \mu\st$. The general procedure is to integrate $\dif\mu=V\m\difp$ (Eq. 7.8.2 with $\dif T$ set equal to zero) from the standard state at pressure $p\st$ to the experimental state at pressure $p'$: \begin{gather} \s{ \mu(p') - \mu\st = \int_{p\st}^{p'} V\m \difp} \tag{7.8.5} \cond{(constant $T$)} \end{gather}
7.8.1 Gases
For the standard chemical potential of a gas, this e-book will usually use the notation $\mu\st\gas$ to emphasize the choice of a gas standard state.
An ideal gas is in its standard state at a given temperature when its pressure is the standard pressure. We find the relation of the chemical potential of an ideal gas to its pressure and its standard chemical potential at the same temperature by setting $V\m$ equal to $RT/p$ in Eq. 7.8.5: $\mu(p') - \mu\st = \int_{p\st}^{p'} (RT/p) \difp = RT\ln(p'/p\st)$. The general relation for $\mu$ as a function of $p$, then, is \begin{gather} \s{ \mu = \mu\st\gas + RT\ln\frac{p}{p\st} } \tag{7.8.6} \cond{(pure ideal gas, constant $T$)} \end{gather} This function is shown as the dashed curve in Fig. 7.6.
If a gas is not an ideal gas, its standard state is a hypothetical state. The fugacity, $\fug$, of a real gas (a gas that is not necessarily an ideal gas) is defined by an equation with the same form as Eq. 7.8.6: \begin{gather} \s{ \mu = \mu\st\gas + RT \ln \frac{\fug}{p\st}} \tag{7.8.7} \cond{(pure gas)} \end{gather} or \begin{gather} \s{\fug \defn p\st \exp\left[ \frac{\mu-\mu\st\gas }{RT} \right]} \tag{7.8.8} \cond{(pure gas)} \end{gather} Note that fugacity has the dimensions of pressure. Fugacity is a kind of effective pressure. Specifically, it is the pressure that the hypothetical ideal gas (the gas with intermolecular forces “turned off” ) would need to have in order for its chemical potential at the given temperature to be the same as the chemical potential of the real gas (see point C in Fig. 7.6). If the gas is an ideal gas, its fugacity is equal to its pressure.
To evaluate the fugacity of a real gas at a given $T$ and $p$, we must relate the chemical potential to the pressure–volume behavior. Let $\mu'$ be the chemical potential and $\fug'$ be the fugacity at the pressure $p'$ of interest; let $\mu''$ be the chemical potential and $\fug''$ be the fugacity of the same gas at some low pressure $p''$ (all at the same temperature). Then we use Eq. 7.8.5 to write $\mu'-\mu\st\gas = RT\ln(\fug'/p\st)$ and $\mu''-\mu\st\gas =RT\ln(\fug''/p\st)$, from which we obtain $\mu' - \mu'' = RT\ln\frac{\fug'}{\fug''} \tag{7.8.9}$ By integrating $\dif\mu = V\m\difp$ from pressure $p''$ to pressure $p'$, we obtain $\mu' - \mu'' = \int_{\mu''}^{\mu'}\dif \mu = \int_{p''}^{p'}V\m\difp \tag{7.8.10}$ Equating the two expressions for $\mu' - \mu''$ and dividing by $RT$ gives $\ln\frac{\fug'}{\fug''} = \int_{p''}^{p'}\frac{V\m}{RT} \difp \tag{7.8.11}$
In principle, we could use the integral on the right side of Eq. 7.8.11 to evaluate $\fug'$ by choosing the lower integration limit $p''$ to be such a low pressure that the gas behaves as an ideal gas and replacing $\fug''$ by $p''$. However, because the integrand $V\m/RT$ becomes very large at low pressure, the integral is difficult to evaluate. We avoid this difficulty by subtracting from the preceding equation the identity $\ln\frac{p'}{p''} = \int_{p''}^{p'}\frac{\difp}{p} \tag{7.8.12}$ which is simply the result of integrating the function $1/p$ from $p''$ to $p'$. The result is $\ln\frac{\fug' p''}{\fug'' p'} = \int_{p''}^{p'} \left( \frac{V\m}{RT} - \frac{1}{p} \right) \difp \tag{7.8.13}$ Now we take the limit of both sides of Eq. 7.8.13 as $p''$ approaches zero. In this limit, the gas at pressure $p''$ approaches ideal-gas behavior, $\fug''$ approaches $p''$, and the ratio $\fug' p''/\fug'' p'$ approaches $\fug'/p'$: $\ln\frac{\fug'}{p'} = \int_{0}^{p'} \left( \frac{V\m}{RT} - \frac{1}{p} \right) \difp \tag{7.8.14}$ The integrand $(V\m/RT-1/p)$ of this integral approaches zero at low pressure, making it feasible to evaluate the integral from experimental data.
The fugacity coefficient $\phi$ of a gas is defined by \begin{gather} \s{\phi\defn \frac{\fug}{p} \quad \tx{or} \quad \fug = \phi p } \tag{7.8.15} \cond{(pure gas)} \end{gather} The fugacity coefficient at pressure $p'$ is then given by Eq. 7.8.14: \begin{gather} \s{ \ln \phi(p') = \int_{0}^{p'} \left( \frac{V\m}{RT} - \frac{1}{p} \right) \difp } \tag{7.8.16} \cond{(pure gas, constant $T$)} \end{gather}
The isothermal behavior of real gases at low to moderate pressures (up to at least $1\br$) is usually adequately described by a two-term equation of state of the form given in Eq. 2.2.8: $V\m \approx \frac{RT}{p} + B \tag{7.8.17}$ Here $B$ is the second virial coefficient, a function of $T$. With this equation of state, Eq. 7.8.16 becomes $\ln\phi \approx \frac{Bp}{RT} \tag{7.8.18}$
For a real gas at temperature $T$ and pressure $p$, Eq. 7.8.16 or 7.8.18 allows us to evaluate the fugacity coefficient from an experimental equation of state or a second virial coefficient. We can then find the fugacity from $\fug=\phi p$.
As we will see in Sec. 9.7, the dimensionless ratio $\phi = \fug /p$ is an example of an activity coefficient and the dimensionless ratio $\fug /p\st$ is an example of an activity.
7.8.2 Liquids and solids
The dependence of the chemical potential on pressure at constant temperature is given by Eq. 7.8.5. With an approximation of zero compressibility, this becomes \begin{gather} \s{ \mu \approx \mu\st + V\m (p - p\st) } \tag{7.8.19} \cond{(pure liquid or solid,} \nextcond{constant $T$)} \end{gather} | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/07%3A_Pure_Substances_in_Single_Phases/7.08%3A_Chemical_Potential_and_Fugacity.txt |
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A standard molar quantity of a substance is the molar quantity in the standard state at the temperature of interest. We have seen (Sec. 7.7) that the standard state of a pure liquid or solid is a real state, so any standard molar quantity of a pure liquid or solid is simply the molar quantity evaluated at the standard pressure and the temperature of interest.
The standard state of a gas, however, is a hypothetical state in which the gas behaves ideally at the standard pressure without influence of intermolecular forces. The properties of the gas in this standard state are those of an ideal gas. We would like to be able to relate molar properties of the real gas at a given temperature and pressure to the molar properties in the standard state at the same temperature.
We begin by using Eq. 7.8.7 to write an expression for the chemical potential of the real gas at pressure $p'$: $\begin{split} \mu(p') & = \mu\st\gas + RT \ln \frac{\fug(p')}{p\st} \ & = \mu\st\gas + RT \ln \frac{p'}{p\st} + RT \ln \frac{\fug(p')}{p'} \end{split} \tag{7.9.1}$ We then substitute from Eq. 7.8.14 to obtain a relation between the chemical potential, the standard chemical potential, and measurable properties, all at the same temperature: \begin{gather} \s{ \mu(p') = \mu\st\gas + RT\ln\frac{p'}{p\st} + \int_0^{p'}\!\! \left( V\m - \frac{RT}{p} \right)\difp } \tag{7.9.2} \cond{(pure gas)} \end{gather} Note that this expression for $\mu$ is not what we would obtain by simply integrating $\dif\mu=V\m \difp$ from $p\st$ to $p'$, because the real gas is not necessarily in its standard state of ideal-gas behavior at a pressure of $1\br$.
Recall that the chemical potential $\mu$ of a pure substance is also its molar Gibbs energy $G\m=G/n$. The standard chemical potential $\mu\st\gas$ of the gas is the standard molar Gibbs energy, $G\m\st\gas$. Therefore Eq. 7.9.2 can be rewritten in the form $G\m(p') = G\m\st\gas + RT\ln\frac{p'}{p\st} + \int_0^{p'}\!\! \left( V\m - \frac{RT}{p} \right)\difp \tag{7.9.3}$ The middle column of Table 7.5 contains an expression for $G\m(p')-G\m\st\gas$ taken from this equation. This expression contains all the information needed to find a relation between any other molar property and its standard molar value in terms of measurable properties. The way this can be done is as follows.
The relation between the chemical potential of a pure substance and its molar entropy is given by Eq. 7.8.3: $S\m=-\Pd{\mu}{T}{\!p} \tag{7.9.4}$ The standard molar entropy of the gas is found from Eq. 7.9.4 by changing $\mu$ to $\mu\st\gas$: $S\m\st\gas = -\Pd{\mu\st\gas }{T}{\!p} \tag{7.9.5}$ By substituting the expression for $\mu$ given by Eq. 7.9.2 into Eq. 7.9.4 and comparing the result with Eq. 7.9.5, we obtain $S\m(p') = S\m\st\gas - R\ln\frac{p'}{p\st} - \int_{0}^{p'}\left[ \Pd{V\m}{T}{\!p}- \frac{R}{p} \right] \difp \tag{7.9.6}$ The expression for $S\m-S\m\st\gas$ in the middle column of Table 7.5 comes from this equation. The equation, together with a value of $S\m$ for a real gas obtained by the calorimetric method described in Sec. 6.2.1, can be used to evaluate $S\m\st\gas$.
Now we can use the expressions for $G\m$ and $S\m$ to find expressions for molar quantities such as $H\m$ and $\Cpm$ relative to the respective standard molar quantities. The general procedure for a molar quantity $X\m$ is to write an expression for $X\m$ as a function of $G\m$ and $S\m$ and an analogous expression for $X\m\st\gas$ as a function of $G\m\st\gas$ and $S\m\st\gas$. Substitutions for $G\m$ and $S\m$ from Eqs. 7.9.3 and 7.9.6 are then made in the expression for $X\m$, and the difference $X\m-X\m\st\gas$ taken.
For example, the expression for $U\m-U\m\st\gas$ in the middle column Table 7.5 was derived as follows. The equation defining the Gibbs energy, $G=U-TS+pV$, was divided by the amount $n$ and rearranged to $U\m = G\m + TS\m - pV\m \tag{7.9.7}$ The standard-state version of this relation is $U\m\st\gas =G\m\st\gas +TS\m\st\gas -p\st V\m\st\gas \tag{7.9.8}$ where from the ideal gas law $p\st V\m\st\gas$ can be replaced by $RT$. Substitutions from Eqs. 7.9.3 and 7.9.6 were made in Eq. 7.9.7 and the expression for $U\m\st\gas$ in Eq. 7.9.8 was subtracted, resulting in the expression in the table.
For a real gas at low to moderate pressures, we can approximate $V\m$ by $(RT/p)+B$ where $B$ is the second virial coefficient (Eq. 7.8.17). Equation 7.9.2 then becomes $\mu \approx \mu\st\gas + RT\ln\frac{p}{p\st} + Bp \tag{7.9.9}$ The expressions in the last column of Table 7.5 use this equation of state. We can see what the expressions look like if the gas is ideal simply by setting $B$ equal to zero. They show that when the pressure of an ideal gas increases at constant temperature, $G\m$ and $A\m$ increase, $S\m$ decreases, and $U\m$, $H\m$, and $\Cpm$ are unaffected. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/07%3A_Pure_Substances_in_Single_Phases/7.09%3A_Standard_Molar_Quantities_of_a_Gas.txt |
An underlined problem number or problem-part letter indicates that the numerical answer appears in Appendix I.
7.1 Derive the following relations from the definitions of $\alpha, \kappa_{T}$, and $\rho$ :
$\alpha=-\frac{1}{\rho}\left(\frac{\partial \rho}{\partial T}\right)_{p} \quad \kappa_{T}=\frac{1}{\rho}\left(\frac{\partial \rho}{\partial p}\right)_{T}$
7.2 Use equations in this chapter to derive the following expressions for an ideal gas:
$\alpha=1 / T \quad \kappa_{T}=1 / p$
7.3 For a gas with the simple equation of state
$V_{\mathrm{m}}=\frac{R T}{p}+B$
(Eq. 2.2.8), where $B$ is the second virial coefficient (a function of $T$ ), find expressions for $\alpha$, $\kappa_{T}$, and $\left(\partial U_{\mathrm{m}} / \partial V\right)_{T}$ in terms of $\mathrm{d} B / \mathrm{d} T$ and other state functions.
7.4 Show that when the virial equation $p V_{\mathrm{m}}=R T\left(1+B_{p} p+C_{p} p^{2}+\cdots\right)$ (Eq. 2.2.3) adequately represents the equation of state of a real gas, the Joule-Thomson coefficient is given by
$\mu_{\mathrm{JT}}=\frac{R T^{2}\left[\mathrm{~d} B_{p} / \mathrm{d} T+\left(\mathrm{d} C_{p} / \mathrm{d} T\right) p+\cdots\right]}{C_{p, \mathrm{~m}}}$
Note that the limiting value at low pressure, $R T^{2}\left(\mathrm{~d} B_{p} / \mathrm{d} T\right) / C_{p, \mathrm{~m}}$, is not necessarily equal to zero even though the equation of state approaches that of an ideal gas in this limit.
7.5 The quantity $(\partial T / \partial V)_{U}$ is called the Joule coefficient. James Joule attempted to evaluate this quantity by measuring the temperature change accompanying the expansion of air into a vacuum - the "Joule experiment." Write an expression for the total differential of $U$ with $T$ and $V$ as independent variables, and by a procedure similar to that used in Sec. 7.5.2 show that the Joule coefficient is equal to
$\frac{p-\alpha T / \kappa_{T}}{C_{V}}$
7.6 $p-V-T$ data for several organic liquids were measured by Gibson and Loeffler. ${ }^{11}$ The following formulas describe the results for aniline.
Molar volume as a function of temperature at $p=1$ bar $(298-358 \mathrm{~K})$ :
$V_{\mathrm{m}}=a+b T+c T^{2}+d T^{3}$
where the parameters have the values
$\begin{array}{ll} a=69.287 \mathrm{~cm}^{3} \mathrm{~mol}^{-1} & c=-1.0443 \times 10^{-4} \mathrm{~cm}^{3} \mathrm{~K}^{-2} \mathrm{~mol}^{-1} \ b=0.08852 \mathrm{~cm}^{3} \mathrm{~K}^{-1} \mathrm{~mol}^{-1} & d=1.940 \times 10^{-7} \mathrm{~cm}^{3} \mathrm{~K}^{-3} \mathrm{~mol}^{-1} \end{array}$
Molar volume as a function of pressure at $T=298.15 \mathrm{~K}$ (1-1000 bar):
$V_{\mathrm{m}}=e-f \ln (g+p / \text { bar })$
where the parameter values are
$e=156.812 \mathrm{~cm}^{3} \mathrm{~mol}^{-1} \quad f=8.5834 \mathrm{~cm}^{3} \mathrm{~mol}^{-1} \quad g=2006.6$
(a) Use these formulas to evaluate $\alpha, \kappa_{T},(\partial p / \partial T)_{V}$, and $(\partial U / \partial V)_{T}$ (the internal pressure) for aniline at $T=298.15 \mathrm{~K}$ and $p=1.000$ bar.
(b) Estimate the pressure increase if the temperature of a fixed amount of aniline is increased by $0.10 \mathrm{~K}$ at constant volume.
7.7 (a) From the total differential of $H$ with $T$ and $p$ as independent variables, derive the relation $\left(\partial C_{p, \mathrm{~m}} / \partial p\right)_{T}=-T\left(\partial^{2} V_{\mathrm{m}} / \partial T^{2}\right)_{p}$
(b) Evaluate $\left(\partial C_{p, \mathrm{~m}} / \partial p\right)_{T}$ for liquid aniline at $300.0 \mathrm{~K}$ and 1 bar using data in Prob. $7.6 .$
7.8 (a) From the total differential of $V$ with $T$ and $p$ as independent variables, derive the relation $(\partial \alpha / \partial p)_{T}=-\left(\partial \kappa_{T} / \partial T\right)_{p}$.
(b) Use this relation to estimate the value of $\alpha$ for benzene at $25^{\circ} \mathrm{C}$ and 500 bar, given that the value of $\alpha$ is $1.2 \times 10^{-3} \mathrm{~K}^{-1}$ at $25^{\circ} \mathrm{C}$ and 1 bar. (Use information from Fig. $7.2$ on page 168.)
7.9 Certain equations of state supposed to be applicable to nonpolar liquids and gases are of the form $p=T f\left(V_{\mathrm{m}}\right)-a / V_{\mathrm{m}}^{2}$, where $f\left(V_{\mathrm{m}}\right)$ is a function of the molar volume only and $a$ is a constant.
(a) Show that the van der Waals equation of state $\left(p+a / V_{\mathrm{m}}^{2}\right)\left(V_{\mathrm{m}}-b\right)=R T$ (where $a$ and $b$ are constants) is of this form.
(b) Show that any fluid with an equation of state of this form has an internal pressure equal to $a / V_{\mathrm{m}}^{2}$.
7.10 Suppose that the molar heat capacity at constant pressure of a substance has a temperature dependence given by $C_{p, \mathrm{~m}}=a+b T+c T^{2}$, where $a, b$, and $c$ are constants. Consider the heating of an amount $n$ of the substance from $T_{1}$ to $T_{2}$ at constant pressure. Find expressions for $\Delta H$ and $\Delta S$ for this process in terms of $a, b, c, n, T_{1}$, and $T_{2}$.
7.11 At $p=1 \mathrm{~atm}$, the molar heat capacity at constant pressure of aluminum is given by
$C_{p, \mathrm{~m}}=a+b T$
where the constants have the values
$a=20.67 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1} \quad b=0.01238 \mathrm{JK}^{-2} \mathrm{~mol}^{-1}$
Calculate the quantity of electrical work needed to heat $2.000 \mathrm{~mol}$ of aluminum from $300.00 \mathrm{~K}$ to $400.00 \mathrm{~K}$ at $1 \mathrm{~atm}$ in an adiabatic enclosure.
7.12 The temperature dependence of the standard molar heat capacity of gaseous carbon dioxide in the temperature range $298 \mathrm{~K}-2000 \mathrm{~K}$ is given by
$C_{p, \mathrm{~m}}^{\circ}=a+b T+\frac{c}{T^{2}}$
where the constants have the values
$a=44.2 \mathrm{JK}^{-1} \mathrm{~mol}^{-1} \quad b=8.8 \times 10^{-3} \mathrm{~J} \mathrm{~K}^{-2} \mathrm{~mol}^{-1} \quad c=-8.6 \times 10^{5} \mathrm{~J} \mathrm{~K} \mathrm{~mol}^{-1}$
Calculate the enthalpy and entropy changes when one mole of $\mathrm{CO}_{2}$ is heated at 1 bar from $300.00 \mathrm{~K}$ to $800.00 \mathrm{~K}$. You can assume that at this pressure $C_{p, \mathrm{~m}}$ is practically equal to $C_{p, \mathrm{~m}}^{\circ}$.
7.13 This problem concerns gaseous carbon dioxide. At $400 \mathrm{~K}$, the relation between $p$ and $V_{\mathrm{m}}$ at pressures up to at least 100 bar is given to good accuracy by a virial equation of state truncated
at the second virial coefficient, $B$. In the temperature range $300 \mathrm{~K}-800 \mathrm{~K}$ the dependence of $B$ on temperature is given by
$B=a^{\prime}+b^{\prime} T+c^{\prime} T^{2}+d^{\prime} T^{3}$
where the constants have the values
\begin{aligned} &a^{\prime}=-521 \mathrm{~cm}^{3} \mathrm{~mol}^{-1} \ &b^{\prime}=2.08 \mathrm{~cm}^{3} \mathrm{~K}^{-1} \mathrm{~mol}^{-1} \ &c^{\prime}=-2.89 \times 10^{-3} \mathrm{~cm}^{3} \mathrm{~K}^{-2} \mathrm{~mol}^{-1} \ &d^{\prime}=1.397 \times 10^{-6} \mathrm{~cm}^{3} \mathrm{~K}^{-3} \mathrm{~mol}^{-1} \end{aligned}
(a) From information in Prob. 7.12, calculate the standard molar heat capacity at constant pressure, $C_{p, \mathrm{~m}}^{\circ}$, at $T=400.0 \mathrm{~K}$.
(b) Estimate the value of $C_{p, \mathrm{~m}}$ under the conditions $T=400.0 \mathrm{~K}$ and $p=100.0$ bar.
7.14 A chemist, needing to determine the specific heat capacity of a certain liquid but not having an electrically heated calorimeter at her disposal, used the following simple procedure known as drop calorimetry. She placed $500.0 \mathrm{~g}$ of the liquid in a thermally insulated container equipped with a lid and a thermometer. After recording the initial temperature of the liquid, $24.80^{\circ} \mathrm{C}$, she removed a $60.17$-g block of aluminum metal from a boiling water bath at $100.00^{\circ} \mathrm{C}$ and quickly immersed it in the liquid in the container. After the contents of the container had become thermally equilibrated, she recorded a final temperature of $27.92^{\circ} \mathrm{C}$. She calculated the specific heat capacity $C_{p} / m$ of the liquid from these data, making use of the molar mass of aluminum $\left(M=26.9815 \mathrm{~g} \mathrm{~mol}^{-1}\right)$ and the formula for the molar heat capacity of aluminum given in Prob. $7.11 .$
(a) From these data, find the specific heat capacity of the liquid under the assumption that its value does not vary with temperature. Hint: Treat the temperature equilibration process as adiabatic and isobaric $(\Delta H=0)$, and equate $\Delta H$ to the sum of the enthalpy changes in the two phases.
(b) Show that the value obtained in part (a) is actually an average value of $C_{p} / m$ over the temperature range between the initial and final temperatures of the liquid given by
$\frac{\int_{T_{1}}^{T_{2}}\left(C_{p} / m\right) \mathrm{d} T}{T_{2}-T_{1}}$
7.15 Suppose a gas has the virial equation of state $p V_{\mathrm{m}}=R T\left(1+B_{p} p+C_{p} p^{2}\right)$, where $B_{p}$ and $C_{p}$ depend only on $T$, and higher powers of $p$ can be ignored.
(a) Derive an expression for the fugacity coefficient, $\phi$, of this gas as a function of $p$.
(b) For $\mathrm{CO}_{2}(\mathrm{~g})$ at $0.00^{\circ} \mathrm{C}$, the virial coefficients have the values $B_{p}=-6.67 \times 10^{-3}$ bar $^{-1}$ and $C_{p}=-3.4 \times 10^{-5} \mathrm{bar}^{-2}$. Evaluate the fugacity $f$ at $0.00^{\circ} \mathrm{C}$ and $p=20.0$ bar.
7.16 Table $7.6$ on the next page lists values of the molar volume of gaseous $\mathrm{H}_{2} \mathrm{O}$ at $400.00^{\circ} \mathrm{C}$ and 12 pressures.
(a) Evaluate the fugacity coefficient and fugacity of $\mathrm{H}_{2} \mathrm{O}(\mathrm{g})$ at $400.00^{\circ} \mathrm{C}$ and 200 bar.
(b) Show that the second virial coefficient $B$ in the virial equation of state, $p V_{\mathrm{m}}=R T(1+$ $\left.B / V_{\mathrm{m}}+C / V_{\mathrm{m}}^{2}+\cdots\right)$, is given by
$B=R T \lim _{p \rightarrow 0}\left(\frac{V_{\mathrm{m}}}{R T}-\frac{1}{p}\right)$
where the limit is taken at constant $T$. Then evaluate $B$ for $\mathrm{H}_{2} \mathrm{O}(\mathrm{g})$ at $400.00^{\circ} \mathrm{C}$.
Table 7.6 Molar volume of $\mathrm{H}_{2} \mathrm{O}(\mathrm{g})$ at $400.00^{\circ} \mathrm{C}^{a}$
$\begin{tabular}{cccc} \hline$p / 10^{5} \mathrm{~Pa}$ & $V_{\mathrm{m}} / 10^{-3} \mathrm{~m}^{3} \mathrm{~mol}^{-1}$ & $p / 10^{5} \mathrm{~Pa}$ & $V_{\mathrm{m}} / 10^{-3} \mathrm{~m}^{3} \mathrm{~mol}^{-1}$ \ \hline 1 & $55.896$ & 100 & $0.47575$ \ 10 & $5.5231$ & 120 & $0.37976$ \ 20 & $2.7237$ & 140 & $0.31020$ \ 40 & $1.3224$ & 160 & $0.25699$ \ 60 & $0.85374$ & 180 & $0.21447$ \ 80 & $0.61817$ & 200 & $0.17918$ \ \hline \end{tabular}$
${ }^{a}$ based on data in Ref. [75] | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/07%3A_Pure_Substances_in_Single_Phases/7.10%3A_Chapter_7_Problems.txt |
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A system of two or more phases of a single substance, in the absence of internal constraints, is in an equilibrium state when each phase has the same temperature, the same pressure, and the same chemical potential. This chapter describes the derivation and consequences of this simple principle, the general appearance of phase diagrams of single-substance systems, and quantitative aspects of the equilibrium phase transitions of these systems.
08: Phase Transitions and Equilibria of Pure Substances
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8.1.1 Equilibrium conditions
If the state of an isolated system is an equilibrium state, this state does not change over time (Sec. 2.4.4). We expect an isolated system that is not in an equilibrium state to undergo a spontaneous, irreversible process and eventually to reach an equilibrium state. Just how rapidly this process occurs is a matter of kinetics, not thermodynamics. During this irreversible adiabatic process, the entropy increases until it reaches a maximum in the equilibrium state.
A general procedure will now be introduced for finding conditions for equilibrium with given constraints. The procedure is applied to phase equilibria of single-substance, multiphase systems in the next section, to transfer equilibria in multicomponent, multiphase systems in Sec. 9.2.7, and to reaction equilibria in Sec. 11.7.3.
The procedure has five steps:
1. In this section we consider a system of a single substance in two or more uniform phases with distinctly different intensive properties. For instance, one phase might be a liquid and another a gas. We assume the phases are not separated by internal partitions, so that there is no constraint preventing the transfer of matter and energy among the phases. (A tall column of gas in a gravitational field is a different kind of system in which intensive properties of an equilibrium state vary continuously with elevation; this case will be discussed in Sec. 8.1.4.)
Phase $\pha'$ will be the reference phase. Since internal energy is extensive, we can write $U=U\aphp+\sum_{\pha\ne\pha'}U\aph$ and $\dif U = \dif U\aphp + \sum_{\pha\ne\pha'}\dif U\aph$. We assume any changes are slow enough to allow each phase to be practically uniform at all times. Treating each phase as an open subsystem with expansion work only, we use the relation $\dif U = T \dif S - p\dif V + \mu\dif n$ (Eq. 5.2.5) to replace each $\dif U\aph$ term: $\dif U = (T\aphp\dif S\aphp - p\aphp\dif V\aphp + \mu\aphp\dif n\aphp) + \sum_{\pha\ne\pha'}(T\aph\dif S\aph - p\aph\dif V\aph + \mu\aph\dif n\aph) \tag{8.1.1}$ This is an expression for the total differential of $U$ when there are no constraints.
In an isolated system, an equilibrium state cannot change spontaneously to a different state. Once the isolated system has reached an equilibrium state, an imagined finite change of any of the independent variables consistent with the constraints (a so-called virtual displacement) corresponds to an impossible process with an entropy decrease. Thus, the equilibrium state has the maximum entropy that is possible for the isolated system. In order for $S$ to be a maximum, $\dif S$ must be zero for an infinitesimal change of any of the independent variables of the isolated system.
This requirement is satisfied in the case of the multiphase system only if the coefficient of each term in the sums on the right side of Eq. 8.1.6 is zero. Therefore, in an equilibrium state the temperature of each phase is equal to the temperature $T\aphp$ of the reference phase, the pressure of each phase is equal to $p\aphp$, and the chemical potential in each phase is equal to $\mu\aphp$. That is, at equilibrium the temperature, pressure, and chemical potential are uniform throughout the system. These are, respectively, the conditions described in Sec. 2.4.4 of thermal equilibrium, mechanical equilibrium, and transfer equilibrium. These conditions must hold in order for a multiphase system of a pure substance without internal partitions to be in an equilibrium state, regardless of the process by which the system attains that state.
8.1.3 Simple derivation of equilibrium conditions
Here is a simpler, less formal derivation of the three equilibrium conditions in a multiphase system of a single substance.
It is intuitively obvious that, unless there are special constraints (such as internal partitions), an equilibrium state must have thermal and mechanical equilibrium. A temperature difference between two phases would cause a spontaneous transfer of heat from the warmer to the cooler phase; a pressure difference would cause spontaneous flow of matter.
When some of the substance is transferred from one phase to another under conditions of constant $T$ and $p$, the intensive properties of each phase remains the same including the chemical potential. The chemical potential of a pure phase is the Gibbs energy per amount of substance in the phase. We know that in a closed system of constant $T$ and $p$ with expansion work only, the total Gibbs energy decreases during a spontaneous process and is constant during a reversible process (Eq. 5.8.6). The Gibbs energy will decrease only if there is a transfer of substance from a phase of higher chemical potential to a phase of lower chemical potential, and this will be a spontaneous change. No spontaneous transfer is possible if both phases have the same chemical potential, so this is a condition for an equilibrium state.
8.1.4 Tall column of gas in a gravitational field
The earth’s gravitational field is an example of an external force field that acts on a system placed in it. Usually we ignore its effects on the state of the system. If, however, the system’s vertical extent is considerable we must take the presence of the field into account to explain, for example, why gas pressure varies with elevation in an equilibrium state.
A tall column of gas whose intensive properties are a function of elevation may be treated as an infinite number of uniform phases, each of infinitesimal vertical height. We can approximate this system with a vertical stack of many slab-shaped gas phases, each thin enough to be practically uniform in its intensive properties, as depicted in Fig. 8.1. The system can be isolated from the surroundings by confining the gas in a rigid adiabatic container. In order to be able to associate each of the thin slab-shaped phases with a definite constant elevation, we specify that the volume of each phase is constant so that in the rigid container the vertical thickness of a phase cannot change.
We can use the phase of lowest elevation as the reference phase $\pha'$, as indicated in the figure. We repeat the derivation of Sec. 8.1.2 with one change: for each phase $\pha$ the volume change $\dif V\aph$ is set equal to zero. Then the second sum on the right side of Eq. 8.1.6, with terms proportional to $\dif V\aph$, drops out and we are left with $\dif S = \sum_{\pha\ne\pha'}\frac{T\aphp-T\aph}{T\aphp}\dif S\aph + \sum_{\pha\ne\pha'}\frac{\mu\aphp-\mu\aph}{T\aphp}\dif n\aph \tag{8.1.7}$ In the equilibrium state of the isolated system, $\dif S$ is equal to zero for an infinitesimal change of any of the independent variables. In this state, therefore, the coefficient of each term in the sums on the right side of Eq. 8.1.7 must be zero. We conclude that in an equilibrium state of a tall column of a pure gas, the temperature and chemical potential are uniform throughout. The equation, however, gives us no information about pressure.
We will use this result to derive an expression for the dependence of the fugacity $\fug$ on elevation in an equilibrium state. We pick an arbitrary position such as the earth’s surface for a reference elevation at which $h$ is zero, and define the standard chemical potential $\mu\st\gas$ as the chemical potential of the gas under standard state conditions at this reference elevation. At $h{=}0$, the chemical potential and fugacity are related by Eq. 7.8.7 which we write in the following form, indicating the elevation in parentheses: $\mu(0) = \mu\st\gas + RT\ln\frac{\fug(0)}{p\st} \tag{8.1.8}$
Imagine a small sample of gas of mass $m$ that is initially at elevation $h{=}0$. The vertical extent of this sample should be small enough for the variation of the gravitational force field within the sample to be negligible. The gravitational work needed to raise the gas to an arbitrary elevation $h$ is $w'=mgh$ (Sec. 3.6). We assume this process is carried out reversibly at constant volume and without heat, so that there is no change in $T$, $p$, $V$, $S$, or $\fug$. The internal energy $U$ of the gas must increase by $mgh=nMgh$, where $M$ is the molar mass. Then, because the Gibbs energy $G$ depends on $U$ according to $G=U-TS+pV$, $G$ must also increase by $nMgh$.
The chemical potential $\mu$ is the molar Gibbs energy $G/n$. During the elevation process, $f$ remains the same and $\mu$ increases by $Mgh$: \begin{gather} \s{ \mu(h)=\mu(0)+Mgh } \tag{8.1.9} \cond{($\fug(h){=}\fug(0)$ )} \end{gather} From Eqs. 8.1.8 and 8.1.9, we can deduce the following general relation between chemical potential, fugacity, and elevation: \begin{gather} \s{ \mu(h) = \mu\st\gas + RT\ln\frac{\fug(h)}{p\st} + Mgh } \tag{8.1.10} \cond{(pure gas in} \nextcond{gravitational field)} \end{gather} Compare this relation with the equation that defines the fugacity when the effect of a gravitational field is negligible: $\mu=\mu\st\gas+RT\ln(\fug/p\st)$ (Eq. 7.8.7). The additional term $Mgh$ is needed when the vertical extent of the gas is considerable.
Some thermodynamicists call the expression on the right side of Eq. 8.1.10 the “total chemical potential” or “gravitochemical potential” and reserve the term “chemical potential” for the function $\mu\st\gas+RT\ln(\fug/p\st)$. With these definitions, in an equilibrium state the “total chemical potential” is the same at all elevations and the “chemical potential” decreases with increasing elevation.
This e-book instead defines the chemical potential $\mu$ of a pure substance at any elevation as the molar Gibbs energy at that elevation, as recommended in a 2001 IUPAC technical report (Robert A. Alberty, Pure Appl. Chem., 73, 1349–1380, 2001). When the chemical potential is defined in this way, it has the same value at all elevations in an equilibrium state.
We know that in the equilibrium state of the gas column, the chemical potential $\mu(h)$ has the same value at each elevation $h$. Equation 8.1.10 shows that in order for this to be possible, the fugacity must decrease with increasing elevation. By equating expressions from Eq. 8.1.10 for $\mu(h)$ at an arbitrary elevation $h$, and for $\mu(0)$ at the reference elevation, we obtain $\mu\st\gas + RT \ln \frac{\fug(h)}{p\st} + Mgh = \mu\st\gas + RT \ln \frac{\fug(0)}{p\st} \tag{8.1.11}$ Solving for $\fug(h)$ gives \begin{gather} \s {\fug(h) = \fug(0) e^{-Mgh/RT} } \tag{8.1.12} \cond{(pure gas at equilibrium} \nextcond{in gravitational field)} \end{gather} If we treat the gas as ideal, so that the fugacity equals the pressure, this equation becomes \begin{gather} \s{ p(h) = p(0)e^{-Mgh/RT} } \tag{8.1.13} \cond{(pure ideal gas at equilibrium} \nextcond{in gravitational field)} \end{gather} Equation 8.1.13 is the barometric formula for a pure ideal gas. It shows that in the equilibrium state of a tall column of an ideal gas, the pressure decreases exponentially with increasing elevation.
This derivation of the barometric formula has introduced a method that will be used in Sec. 9.8.1 for dealing with mixtures in a gravitational field. There is, however, a shorter derivation based on Newton’s second law and not involving the chemical potential. Consider one of the thin slab-shaped phases of Fig. 8.1. Let the density of the phase be $\rho$, the area of each horizontal face be $\As$, and the thickness of the slab be $\delta h$. The mass of the phase is then $m=\rho \As\delta h$. The pressure difference between the top and bottom of the phase is $\delta p$. Three vertical forces act on the phase: an upward force $p \As$ at its lower face, a downward force $-(p+\delta p)\As$ at its upper face, and a downward gravitational force $-mg=-\rho \As g\delta h$. If the phase is at rest, the net vertical force is zero: $p \As-(p+\delta p)\As-\rho \As g\delta h = 0$, or $\delta p = -\rho g\delta h$. In the limit as the number of phases becomes infinite and $\delta h$ and $\delta p$ become infinitesimal, this becomes \begin{gather} \s{\difp = -\rho g\dif h} \tag{8.1.14} \cond{(fluid at equilibrium} \nextcond{in gravitational field)} \end{gather} Equation 8.1.14 is a general relation between changes in elevation and hydrostatic pressure in any fluid. To apply it to an ideal gas, we replace the density by $\rho=nM/V=M/V\m=Mp/RT$ and rearrange to $\difp/p=-(gM/RT)\dif h$. Treating $g$ and $T$ as constants, we integrate from $h{=}0$ to an arbitrary elevation $h$ and obtain the same result as Eq. 8.1.13.
8.1.5 The pressure in a liquid droplet
The equilibrium shape of a small liquid droplet surrounded by vapor of the same substance, when the effects of gravity and other external forces are negligible, is spherical. This is the result of the surface tension of the liquid–gas interface which acts to minimize the ratio of surface to volume. The interface acts somewhat like the stretched membrane of an inflated balloon, resulting in a greater pressure inside the droplet than the pressure of the vapor in equilibrium with it.
We can derive the pressure difference by considering a closed system containing a spherical liquid droplet and surrounding vapor. We treat both phases as open subsystems. An infinitesimal change $\dif U$ of the internal energy is the sum of contributions from the liquid and gas phases and from the surface work $\g\dif\As$, where $\g$ is the surface tension of the liquid–gas interface and $\As$ is the surface area of the droplet (Sec. 5.7): $\dif U = \dif U\sups{l} + \dif U\sups{g} + \g\dif\As = T\sups{l} \dif S\sups{l} - p\sups{l} \dif V\sups{l} + \mu\sups{l} \dif n\sups{l} + T\sups{g}\dif S\sups{g} - p\sups{g}\dif V\sups{g} + \mu\sups{g}\dif n\sups{g} + \g\dif\As \tag{8.1.15}$ Note that Eq. 8.1.15 is not an expression for the total differential of $U$, because $V\sups{l}$ and $\As$ are not independent variables. A derivation by a procedure similar to the one used in Sec. 8.1.2 shows that at equilibrium the liquid and gas have equal temperatures and equal chemical potentials, and the pressure in the droplet is greater than the gas pressure by an amount that depends on $r$: $p\sups{l} = p\sups{g} + \frac{2\g}{r} \tag{8.1.16}$ Equation 8.1.16 is the Laplace equation. The pressure difference is significant if $r$ is small, and decreases as $r$ increases. The limit $r\ra\infty$ represents the flat surface of bulk liquid with $p\sups{l}$ equal to $p\sups{g}$.
The derivation of Eq. 8.1.16 is left as an exercise (Prob. 8.1). The Laplace equation is valid also for a liquid droplet in which the liquid and the surrounding gas may both be mixtures (Prob. 9.3).
The Laplace equation can also be applied to the pressure in a gas bubble surrounded by liquid. In this case the liquid and gas phases switch roles, and the equation becomes $p\sups{g} = p\sups{l} + 2\g/r$.
8.1.6 The number of independent variables
From this point on in this e-book, unless stated otherwise, the discussions of multiphase systems will implicitly assume the existence of thermal, mechanical, and transfer equilibrium. Equations will not explicitly show these equilibria as a condition of validity.
In the rest of this chapter, we shall assume the state of each phase can be described by the usual variables: temperature, pressure, and amount. That is, variables such as elevation in a gravitational field, interface surface area, and extent of stretching of a solid, are not relevant.
How many of the usual variables of an open multiphase one-substance equilibrium system are independent? To find out, we go through the following argument. In the absence of any kind of equilibrium, we could treat phase $\pha$ as having the three independent variables $T\aph$, $p\aph$, and $n\aph$, and likewise for every other phase. A system of $P$ phases without thermal, mechanical, or transfer equilibrium would then have $3P$ independent variables.
We must decide how to count the number of phases. It is usually of no thermodynamic significance whether a phase, with particular values of its intensive properties, is contiguous. For instance, splitting a crystal into several pieces is not usually considered to change the number of phases or the state of the system, provided the increased surface area makes no significant contribution to properties such as internal energy. Thus, the number of phases $P$ refers to the number of different kinds of phases.
Each independent relation resulting from equilibrium imposes a restriction on the system and reduces the number of independent variables by one. A two-phase system with thermal equilibrium has the single relation $T\bph = T\aph$. For a three-phase system, there are two such relations that are independent, for instance $T\bph = T\aph$ and $T\gph = T\aph$. (The additional relation $T\gph = T\bph$ is not independent since we may deduce it from the other two.) In general, thermal equilibrium gives $P-1$ independent relations among temperatures.
By the same reasoning, mechanical equilibrium involves $P-1$ independent relations among pressures, and transfer equilibrium involves $P-1$ independent relations among chemical potentials.
The total number of independent relations for equilibrium is $3(P-1)$, which we subtract from $3P$ (the number of independent variables in the absence of equilibrium) to obtain the number of independent variables in the equilibrium system: $3P - 3(P-1) = 3$. Thus, an open single-substance system with any number of phases has at equilibrium three independent variables. For example, in equilibrium states of a two-phase system we may vary $T$, $n\aph$, and $n\bph$ independently, in which case $p$ is a dependent variable; for a given value of $T$, the value of $p$ is the one that allows both phases to have the same chemical potential.
8.1.7 The Gibbs phase rule for a pure substance
The complete description of the state of a system must include the value of an extensive variable of each phase (e.g., the volume, mass, or amount) in order to specify how much of the phase is present. For an equilibrium system of $P$ phases with a total of $3$ independent variables, we may choose the remaining $3 - P$ variables to be intensive. The number of these intensive independent variables is called the number of degrees of freedom or variance, $F$, of the system: \begin{gather} \s{ F = 3 - P } \tag{8.1.17} \cond{(pure substance)} \end{gather}
The application of the phase rule to multicomponent systems will be taken up in Sec. 13.1. Equation 8.1.17 is a special case, for $C = 1$, of the more general Gibbs phase rule $F = C - P + 2$.
We may interpret the variance $F$ in either of two ways:
• A system with two degrees of freedom is called bivariant, one with one degree of freedom is univariant, and one with no degrees of freedom is invariant. For a system of a pure substance, these three cases correspond to one, two, and three phases respectively. For instance, a system of liquid and gaseous H$_2$O (and no other substances) is univariant ($F = 3 - P = 3 - 2 = 1$); we are able to independently vary only one intensive property, such as $T$, while the liquid and gas remain in equilibrium. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/08%3A_Phase_Transitions_and_Equilibria_of_Pure_Substances/8.01%3A_Phase_Equilibria.txt |
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A phase diagram is a two-dimensional map showing which phase or phases are able to exist in an equilibrium state under given conditions. This chapter describes pressure–volume and pressure–temperature phase diagrams for a single substance, and Chap. 13 will describe numerous types of phase diagrams for multicomponent systems.
8.2.1 Features of phase diagrams
Two-dimensional phase diagrams for a single-substance system can be generated as projections of a three-dimensional surface in a coordinate system with Cartesian axes $p$, $V/n$, and $T$. A point on the three-dimensional surface corresponds to a physically-realizable combination of values, for an equilibrium state of the system containing a total amount $n$ of the substance, of the variables $p$, $V/n$, and $T$.
The concepts needed to interpret single-substance phase diagrams will be illustrated with carbon dioxide.
Figure 8.2 Relations among $p$, $V/n$, and $T$ for carbon dioxide (based on data in NIST Chemistry WebBook and in S. Angus, B. Armstrong, and K. M. de Reuck, International Thermodynamic Tables of the Fluid State, Vol. 3, Carbon Dioxide, Pergamon Press, Oxford, 1976). Areas are labeled with the stable phase or phases (scf stands for supercritical fluid). The open circle indicates the critical point.
(a) Three-dimensional $p$–$(V/n)$–$T$ surface. The dashed curve is the critical isotherm at $T=304.21\K$, and the dotted curve is a portion of the critical isobar at $p=73.8\br$.
(b) Pressure–volume phase diagram (projection of the surface onto the $p$–$(V/n)$ plane).
(c) Pressure–temperature phase diagram (projection of the surface onto the $p$–$T$ plane).
Figure 8.3 Three-dimensional $p$–$(V/n)$–$T$ surface for CO$_2$, magnified along the $V/n$ axis compared to Fig. 8.2. The open circle is the critical point, the dashed curve is the critical isotherm, and the dotted curve is a portion of the critical isobar.
Three-dimensional surfaces for carbon dioxide are shown at two different scales in Fig. 8.2 and in Fig. 8.3. In these figures, some areas of the surface are labeled with a single physical state: solid, liquid, gas, or supercritical fluid. A point in one of these areas corresponds to an equilibrium state of the system containing a single phase of the labeled physical state. The shape of the surface in this one-phase area gives the equation of state of the phase (i.e., the dependence of one of the variables on the other two). A point in an area labeled with two physical states corresponds to two coexisting phases. The triple line is the locus of points for all possible equilibrium systems of three coexisting phases, which in this case are solid, liquid, and gas. A point on the triple line can also correspond to just one or two phases.
The two-dimensional projections shown in Figs. 8.2(b) and 8.2(c) are pressure–volume and pressure–temperature phase diagrams. Because all phases of a multiphase equilibrium system have the same temperature and pressure (assuming there are no constraints such as internal adiabatic partitions), the projection of each two-phase area onto the pressure–temperature diagram is a curve, called a coexistence curve or phase boundary, and the projection of the triple line is a point, called a triple point.
How may we use a phase diagram? The two axes represent values of two independent variables, such as $p$ and $V/n$ or $p$ and $T$. For given values of these variables, we place a point on the diagram at the intersection of the corresponding coordinates; this is the system point. Then depending on whether the system point falls in an area or on a coexistence curve, the diagram tells us the number and kinds of phases that can be present in the equilibrium system.
If the system point falls within an area labeled with the physical state of a single phase, only that one kind of phase can be present in the equilibrium system. A system containing a pure substance in a single phase is bivariant ($F = 3 - 1 = 2$), so we may vary two intensive properties independently. That is, the system point may move independently along two coordinates ($p$ and $V/n$, or $p$ and $T$) and still remain in the one-phase area of the phase diagram. When $V$ and $n$ refer to a single phase, the variable $V/n$ is the molar volume $V\m$ in the phase.
If the system point falls in an area of the pressure–volume phase diagram labeled with symbols for two phases, these two phases coexist in equilibrium. The phases have the same pressure and different molar volumes. To find the molar volumes of the individual phases, we draw a horizontal line of constant pressure, called a tie line, through the system point and extending from one edge of the area to the other. The horizontal position of each end of the tie line, where it terminates at the boundary with a one-phase area, gives the molar volume in that phase in the two-phase system. For an example of a tie line, see Fig. 8.9.
The triple line on the pressure–volume diagram represents the range of values of $V/n$ in which three phases (solid, liquid, and gas) can coexist at equilibrium.
Helium is the only substance lacking a solid–liquid–gas triple line. When a system containing the coexisting liquid and gas of ${}^4$He is cooled to $2.17\K$, a triple point is reached in which the third phase is a liquid called He-II, which has the unique property of superfluidity. It is only at high pressures ($10\br$ or greater) that solid helium can exist.
A three-phase one-component system is invariant ($F = 3 - 3 = 0$); there is only one temperature (the triple-point temperature $T\subs{tp}$) and one pressure (the triple-point pressure $p\subs{tp}$) at which the three phases can coexist. The values of $T\subs{tp}$ and $p\subs{tp}$ are unique to each substance, and are shown by the position of the triple point on the pressure–temperature phase diagram. The molar volumes in the three coexisting phases are given by the values of $V/n$ at the three points on the pressure–volume diagram where the triple line touches a one-phase area. These points are at the two ends and an intermediate position of the triple line. If the system point is at either end of the triple line, only the one phase of corresponding molar volume at temperature $T\subs{tp}$ and pressure $p\subs{tp}$ can be present. When the system point is on the triple line anywhere between the two ends, either two or three phases can be present. If the system point is at the position on the triple line corresponding to the phase of intermediate molar volume, there might be only that one phase present.
Figure 8.4 High-pressure pressure–temperature phase diagram of H$_2$O (based on data in D. Eisenberg and W. Kauzmann, The Structure and Properties of Water, Oxford University Press, New York, 1969, Table 3.5, and Carl W. F. T. Pistorius et al, J. Chem. Phys., 38, 600–602, 1963). The roman numerals designate seven forms of ice.
At high pressures, a substance may have additional triple points for two solid phases and the liquid, or for three solid phases. This is illustrated by the pressure–temperature phase diagram of H$_2$O in Fig. 8.4, which extends to pressures up to $30\units{kbar}$. (On this scale, the liquid–gas coexistence curve lies too close to the horizontal axis to be visible.) The diagram shows seven different solid phases of H$_2$O differing in crystal structure and designated ice I, ice II, and so on. Ice I is the ordinary form of ice, stable below $2\br$. On the diagram are four triple points for two solids and the liquid and three triple points for three solids. Each triple point is invariant. Note how H$_2$O can exist as solid ice VI or ice VII above its standard melting point of $273\K$ if the pressure is high enough (“hot ice” ).
8.2.2 Two-phase equilibrium
A system containing two phases of a pure substance in equilibrium is univariant. Both phases have the same values of $T$ and of $p$, but these values are not independent because of the requirement that the phases have equal chemical potentials. We may vary only one intensive variable of a pure substance (such as $T$ or $p$) independently while two phases coexist in equilibrium.
Figure 8.5 An isoteniscope. The liquid to be investigated is placed in the vessel and U-tube, as indicated by shading, and maintained at a fixed temperature in the bath. The pressure in the side tube is reduced until the liquid boils gently and its vapor sweeps out the air. The pressure is adjusted until the liquid level is the same in both limbs of the U-tube; the vapor pressure of the liquid is then equal to the pressure in the side tube, which can be measured with a manometer.
At a given temperature, the pressure at which solid and gas or liquid and gas are in equilibrium is called the vapor pressure or saturation vapor pressure of the solid or liquid. The vapor pressure of a solid is sometimes called the sublimation pressure. We may measure the vapor pressure of a liquid at a fixed temperature with a simple device called an isoteniscope (Fig. 8.5).
In a system of more than one substance, vapor pressure can refer to the partial pressure of a substance in a gas mixture equilibrated with a solid or liquid of that substance. The effect of total pressure on vapor pressure will be discussed in Sec. 12.8.1. This e-book refers to the saturation vapor pressure of a liquid when it is necessary to indicate that it is the pure liquid and pure gas phases that are in equilibrium at the same pressure.
At a given pressure, the melting point or freezing point is the temperature at which solid and liquid are in equilibrium, the boiling point or saturation temperature is the temperature at which liquid and gas are in equilibrium, and the sublimation temperature or sublimation point is the temperature at which solid and gas are in equilibrium.
Figure 8.6 Pressure–temperature phase diagram of H$_2$O. (Based on data in NIST Chemistry WebBook.)
The relation between temperature and pressure in a system with two phases in equilibrium is shown by the coexistence curve separating the two one-phase areas on the pressure–temperature diagram (see Fig. 8.6). Consider the liquid–gas curve. If we think of $T$ as the independent variable, the curve is a vapor-pressure curve showing how the vapor pressure of the liquid varies with temperature. If, however, $p$ is the independent variable, then the curve is a boiling-point curve showing the dependence of the boiling point on pressure.
The normal melting point or boiling point refers to a pressure of one atmosphere, and the standard melting point or boiling point refers to the standard pressure. Thus, the normal boiling point of water ($99.97\units{\(\degC$}\)) is the boiling point at $1\units{atm}$; this temperature is also known as the steam point. The standard boiling point of water ($99.61\units{\(\degC$}\)) is the boiling point at the slightly lower pressure of $1\br$.
Coexistence curves will be discussed further in Sec. 8.4.
8.2.3 The critical point
Every substance has a certain temperature, the critical temperature, above which only one fluid phase can exist at any volume and pressure (Sec. 2.2.3). The critical point is the point on a phase diagram corresponding to liquid–gas coexistence at the critical temperature, and the critical pressure is the pressure at this point.
Figure 8.7 Glass bulb filled with CO$_2$ at a value of $V/n$ close to the critical value, viewed at four different temperatures. The three balls have densities less than, approximately equal to, and greater than the critical density. (Photos by permission of the photographer; they appeared in Jan V. Sengers and Anneke Levelt Sengers, Chem. Eng. News, June 10, 104–118, 1968.)
(a) Supercritical fluid at a temperature above the critical temperature.
(b) Intense opalescence just above the critical temperature.
(c) Meniscus formation slightly below the critical temperature; liquid and gas of nearly the same density.
(d) Temperature well below the critical temperature; liquid and gas of greatly different densities.
To observe the critical point of a substance experimentally, we can evacuate a glass vessel, introduce an amount of the substance such that $V/n$ is approximately equal to the molar volume at the critical point, seal the vessel, and raise the temperature above the critical temperature. The vessel now contains a single fluid phase. When the substance is slowly cooled to a temperature slightly above the critical temperature, it exhibits a cloudy appearance, a phenomenon called critical opalescence (Fig. 8.7). The opalescence is the scattering of light caused by large local density fluctuations. At the critical temperature, a meniscus forms between liquid and gas phases of practically the same density. With further cooling, the density of the liquid increases and the density of the gas decreases.
At temperatures above the critical temperature and pressures above the critical pressure, the one existing fluid phase is called a supercritical fluid. Thus, a supercritical fluid of a pure substance is a fluid that does not undergo a phase transition to a different fluid phase when we change the pressure at constant temperature or change the temperature at constant pressure.
If, however, we increase $p$ at constant $T$, the supercritical fluid will change to a solid. In the phase diagram of H$_2$O, the coexistence curve for ice VII and liquid shown in Fig. 8.4 extends to a higher temperature than the critical temperature of $647\K$. Thus, supercritical water can be converted to ice VII by isothermal compression.
A fluid in the supercritical region can have a density comparable to that of the liquid, and can be more compressible than the liquid. Under supercritical conditions, a substance is often an excellent solvent for solids and liquids. By varying the pressure or temperature, the solvating power can be changed; by reducing the pressure isothermally, the substance can be easily removed as a gas from dissolved solutes. These properties make supercritical fluids useful for chromatography and solvent extraction.
Figure 8.8 Densities of coexisting gas and liquid phases close to the critical point as functions of temperature for (a) CO$_2$ (based on data in A. Michels, B. Blaisse, and C. Michels, Proc. R. Soc. London, Ser. A, 160, 358–375, 1937); (b) SF$_6$ (data of M. W. Pestak et al, Phys. Rev. B, 36, 599–614, 1987, Table VII). Experimental gas densities are shown by open squares and experimental liquid densities by open triangles. The mean density at each experimental temperature is shown by an open circle. The open diamond is at the critical temperature and critical density.
The critical temperature of a substance can be measured quite accurately by observing the appearance or disappearance of a liquid–gas meniscus, and the critical pressure can be measured at this temperature with a high-pressure manometer. To evaluate the density at the critical point, it is best to extrapolate the mean density of the coexisting liquid and gas phases, $(\rho\sups{l} +\rho\sups{g})/2$, to the critical temperature as illustrated in Fig. 8.8. The observation that the mean density closely approximates a linear function of temperature, as shown in the figure, is known as the law of rectilinear diameters, or the law of Cailletet and Matthias. This law is an approximation, as can be seen by the small deviation of the mean density of SF$_6$ from a linear relation very close to the critical point in Fig. 8.8(b). This failure of the law of rectilinear diameters is predicted by recent theoretical treatments (Jingtao Wang and Mikhail A. Anisimov, Phys. Rev. E, 75, 051107, 2007; Hassan Behnejad, Jan V. Sengers, and Mikhail A. Anisimov, in A. R. H. Goodwin, J. V. Sengers, and C. J. Peters, editors, Applied Thermodynamics of Fluids, pages 321–367, Royal Society of Chemistry, Cambridge, 2010).
8.2.4 The lever rule
Consider a single-substance system whose system point is in a two-phase area of a pressure–volume phase diagram. How can we determine the amounts in the two phases?
Figure 8.9 Tie line (dashed) at constant $T$ and $p$ in the liquid–gas area of a pressure–volume phase diagram. Points A and B are at the ends of the tie line, and point S is a system point on the tie line. $L\sups{l}$ and $L\sups{g}$ are the lengths AS and SB, respectively.
As an example, let the system contain a fixed amount $n$ of a pure substance divided into liquid and gas phases, at a temperature and pressure at which these phases can coexist in equilibrium. When heat is transferred into the system at this $T$ and $p$, some of the liquid vaporizes by a liquid–gas phase transition and $V$ increases; withdrawal of heat at this $T$ and $p$ causes gas to condense and $V$ to decrease. The molar volumes and other intensive properties of the individual liquid and gas phases remain constant during these changes at constant $T$ and $p$. On the pressure–volume phase diagram of Fig. 8.9, the volume changes correspond to movement of the system point to the right or left along the tie line AB.
When enough heat is transferred into the system to vaporize all of the liquid at the given $T$ and $p$, the system point moves to point B at the right end of the tie line. $V/n$ at this point must be the same as the molar volume of the gas, $V\m\sups{g}$. We can see this because the system point could have moved from within the one-phase gas area to this position on the boundary without undergoing a phase transition.
When, on the other hand, enough heat is transferred out of the system to condense all of the gas, the system point moves to point A at the left end of the tie line. $V/n$ at this point is the molar volume of the liquid, $V\m\sups{l}$.
When the system point is at position S on the tie line, both liquid and gas are present. Their amounts must be such that the total volume is the sum of the volumes of the individual phases, and the total amount is the sum of the amounts in the two phases: $V = V\sups{l}+V\sups{g} = n\sups{l}V\m\sups{l}+n\sups{g}V\m\sups{g} \tag{8.2.1}$ $n = n\sups{l}+n\sups{g} \tag{8.2.2}$ The value of $V/n$ at the system point is then given by the equation $\frac{V}{n} = \frac{n\sups{l}V\m\sups{l}+n\sups{g}V\m\sups{g}}{n\sups{l}+n\sups{g}} \tag{8.2.3}$ which can be rearranged to $n\sups{l}\left(V\m\sups{l}-\frac{V}{n}\right) =n\sups{g}\left(\frac{V}{n}-V\m\sups{g}\right) \tag{8.2.4}$ The quantities $V\m\sups{l}-V/n$ and $V/n-V\m\sups{g}$ are the lengths $L\sups{l}$ and $L\sups{g}$, respectively, defined in the figure and measured in units of $V/n$. This gives us the lever rule for liquid–gas equilibrium: \begin{gather} \s{ n\sups{l} L\sups{l} = n\sups{g} L\sups{g} \quad \tx{or} \quad \frac{n\sups{g}}{n\sups{l} } = \frac{L\sups{l} }{L\sups{g}} } \tag{8.2.5} \cond{(coexisting liquid and gas} \nextcond{phases of a pure substance)} \end{gather} (The relation is called the lever rule by analogy to a stationary mechanical lever, each end of which has the same value of the product of applied force and distance from the fulcrum.)
In Fig. 8.9 the system point S is positioned on the tie line two thirds of the way from the left end, making length $L\sups{l}$ twice as long as $L\sups{g}$. The lever rule then gives the ratio of amounts: $n\sups{g}/n\sups{l} = L\sups{l}/L\sups{g} =2$. One-third of the total amount is liquid and two-thirds is gas.
We cannot apply the lever rule to a point on the triple line, because we need more than the value of $V/n$ to determine the relative amounts present in three phases.
We can derive a more general form of the lever rule that will be needed in Chap. 13 for phase diagrams of multicomponent systems. This general form can be applied to any two-phase area of a two-dimensional phase diagram in which a tie-line construction is valid, with the position of the system point along the tie line given by the variable $F \defn \frac{a}{b} \tag{8.2.6}$ where $a$ and $b$ are extensive state functions. (In the pressure–volume phase diagram of Fig. 8.9, these functions are $a=V$ and $b=n$ and the system point position is given by $F=V/n$.) We repeat the steps of the derivation above, labeling the two phases by superscripts $\pha$ and $\phb$ instead of $l$ and $g$. The relation corresponding to Eq. 8.2.4 is $b\aph(F\aph-F)=b\bph(F-F\bph) \tag{8.2.7}$ If $L\aph$ and $L\bph$ are lengths measured along the tie line from the system point to the ends of the tie line at single phases $\pha$ and $\phb$, respectively, Eq. 8.2.7 is equivalent to the general lever rule $b\aph L\aph = b\bph L\bph \qquad \tx{or} \qquad \frac{b\bph}{b\aph} = \frac{L\aph}{L\bph} \tag{8.2.8}$
8.2.5 Volume properties
Figure 8.10 Isotherms for the fluid phases of H$_2$O (based on data in NIST Chemistry WebBook). The open circle indicates the critical point, the dashed curve is the critical isotherm at $373.95\units{\(\degC$}\), and the dotted curve encloses the two-phase area of the pressure–volume phase diagram. The triple line lies too close to the bottom of the diagram to be visible on this scale.
Figure 8.10 is a pressure–volume phase diagram for H$_2$O. On the diagram are drawn isotherms (curves of constant $T$). These isotherms define the shape of the three-dimensional $p$–$(V/n)$–$T$ surface. The area containing the horizontal isotherm segments is the two-phase area for coexisting liquid and gas phases. The boundary of this area is defined by the dotted curve drawn through the ends of the horizontal segments. The one-phase liquid area lies to the left of this curve, the one-phase gas area lies to the right, and the critical point lies at the top.
The diagram contains the information needed to evaluate the molar volume at any temperature and pressure in the one-phase region and the derivatives of the molar volume with respect to temperature and pressure. At a system point in the one-phase region, the slope of the isotherm passing through the point is the partial derivative $\pd{p}{V\m}{T}$. Since the isothermal compressibility is given by $\kT = -(1/V\m)\pd{V\m}{p}{T}$, we have $\kT = -\frac{1}{V\m \times \tx{slope of isotherm}} \tag{8.2.9}$ We see from Fig. 8.10 that the slopes of the isotherms are large and negative in the liquid region, smaller and negative in the gas and supercritical fluid regions, and approach zero at the critical point. Accordingly, the isothermal compressibility of the gas and the supercritical fluid is much greater than that of the liquid, approaching infinity at the critical point. The critical opalescence seen in Fig. 8.7 is caused by local density fluctuations, which are large when $\kT$ is large.
Figure 8.11 Isobars for the fluid phases of H$_2$O (based on data in NIST Chemistry WebBook). The open circle indicates the critical point, the dashed curve is the critical isobar at $220.64\br$, and the dotted curve encloses the two-phase area of the temperature–volume phase diagram.
Solid curves: a, $p=200\br$; b, $p=210\br$; c, $p=230\br$; d, $p=240\br$.
Figure 8.11 shows isobars for H$_2$O instead of isotherms. At a system point in the one-phase region, the slope of the isobar passing through the point is the partial derivative $\pd{T}{V\m}{p}$. The cubic expansion coefficient $\alpha$ is equal to $(1/V\m)\pd{V\m}{T}{p}$, so we have $\alpha = \frac{1}{V\m \times \tx{slope of isobar}} \tag{8.2.10}$ The figure shows that the slopes of the isobars are large and positive in the liquid region, smaller and negative in the gas and supercritical fluid regions, and approach zero at the critical point. Thus the gas and the supercritical fluid have much larger cubic expansion coefficients than the liquid. The value of $\alpha$ approaches infinity at the critical point, meaning that in the critical region the density distribution is greatly affected by temperature gradients. This may account for the low position of the middle ball in Fig. 8.7(b). | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/08%3A_Phase_Transitions_and_Equilibria_of_Pure_Substances/8.02%3A_Phase_Diagrams_of_Pure_Substances.txt |
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Recall (Sec. 2.2.2) that an equilibrium phase transition of a pure substance is a process in which some or all of the substance is transferred from one coexisting phase to another at constant temperature and pressure.
8.3.1 Molar transition quantities
The quantity $\Delsub{vap}H$ is the molar enthalpy change for the reversible process in which liquid changes to gas at a temperature and pressure at which the two phases coexist at equilibrium. This quantity is called the molar enthalpy of vaporization. (Because $\Delsub{vap}H$ is an enthalpy change per amount of vaporization, it would be more accurate to call it the “molar enthalpy change of vaporization.”) Since the pressure is constant during the process, $\Delsub{vap}H$ is equal to the heat per amount of vaporization (Eq. 5.3.8). Hence, $\Delsub{vap}H$ is also called the molar heat of vaporization.
The IUPAC Green Book (E. Richard Cohen et al, Quantities, Units and Symbols in Physical Chemistry, 3rd edition, RSC Publishing, Cambridge, 2007, p. 58) recommends that $\Delsub{p}$ be interpreted as an operator symbol: $\Delsub{p} \defn \partial/\partial\xi\subs{p}$, where “p” is the abbreviation for a process at constant $T$ and $p$ (in this case “vap”) and $\xi\subs{p}$ is its advancement. Thus $\Delsub{vap}H$ is the same as $\pd{H}{\xi\subs{vap}}{T,p}$ where $\xi\subs{vap}$ is the amount of liquid changed to gas.
Here is a list of symbols for the molar enthalpy changes of various equilibrium phase transitions:
\begin{array}{ll} \Delsub{vap}H & \tx{molar enthalpy of vaporization (liquid$\ra$gas)} \cr \Delsub{sub}H & \tx{molar enthalpy of sublimation (solid$\ra$gas)} \cr \Delsub{fus}H & \tx{molar enthalpy of fusion (solid$\ra$liquid)} \cr \Delsub{trs}H & \tx{molar enthalpy of a transition between any two phases in general} \end{array}
Molar enthalpies of vaporization, sublimation, and fusion are positive. The reverse processes of condensation (gas$\ra$liquid), condensation or deposition (gas$\ra$solid), and freezing (liquid$\ra$solid) have negative enthalpy changes.
The subscripts in the list above are also used for other molar transition quantities. Thus, there is the molar entropy of vaporization $\Delsub{vap}S$, the molar internal energy of sublimation $\Delsub{sub}U$, and so on.
A molar transition quantity of a pure substance is the change of an extensive property divided by the amount transferred between the phases. For example, when an amount $n$ in a liquid phase is allowed to vaporize to gas at constant $T$ and $p$, the enthalpy change is $\Del H = nH\m\sups{g} - nH\m\sups{l}$ and the molar enthalpy of vaporization is \begin{gather} \s{ \Delsub{vap}H = \frac{\Del H}{n} = H\m\sups{g} - H\m\sups{l} } \tag{8.3.1} \cond{(pure substance)} \end{gather} In other words, $\Delsub{vap}H$ is the enthalpy change per amount vaporized and is also the difference between the molar enthalpies of the two phases.
A molar property of a phase, being intensive, usually depends on two independent intensive variables such as $T$ and $p$. Despite the fact that $\Delsub{vap}H$ is the difference of the two molar properties $H\m\sups{g}$ and $H\m\sups{l}$, its value depends on only one intensive variable, because the two phases are in transfer equilibrium and the system is univariant. Thus, we may treat $\Delsub{vap}H$ as a function of $T$ only. The same is true of any other molar transition quantity.
The molar Gibbs energy of an equilibrium phase transition, $\Delsub{trs}G$, is a special case. For the phase transition $\pha\ra\phb$, we may write an equation analogous to Eq. 8.3.1 and equate the molar Gibbs energy in each phase to a chemical potential (see Eq. 7.8.1): \begin{gather} \s{ \Delsub{trs}G = G\m\bph - G\m\aph = \mu\bph - \mu\aph } \tag{8.3.2} \cond{(pure substance)} \end{gather} But the transition is between two phases at equilibrium, requiring both phases to have the same chemical potential: $\mu\bph - \mu\aph = 0$. Therefore, the molar Gibbs energy of any equilibrium phase transition is zero: \begin{gather} \s{ \Delsub{trs}G = 0 } \tag{8.3.3} \cond{(pure substance)} \end{gather}
Since the Gibbs energy is defined by $G = H - TS$, in phase $\pha$ we have $G\m\aph = G\aph/n\aph = H\m\aph - TS\m\aph$. Similarly, in phase $\phb$ we have $G\m\bph = H\m\bph - TS\m\bph$. When we substitute these expressions in $\Delsub{trs}G = G\m\bph - G\m\aph$ (Eq. 8.3.2) and set $T$ equal to the transition temperature $T\subs{trs}$, we obtain $\begin{split} \Delsub{trs}G = (H\m\bph - H\m\aph) - T\subs{trs}(S\m\bph - S\m\aph) \ = \Delsub{trs}H - T\subs{trs}\Delsub{trs}S \end{split} \tag{8.3.4}$ Then, by setting $\Delsub{trs}G$ equal to zero, we find the molar entropy and molar enthalpy of the equilibrium phase transition are related by \begin{gather} \s{ \Delsub{trs}S = \frac{\Delsub{trs}H}{T\subs{trs}} } \tag{8.3.5} \cond{(pure substance)} \end{gather} where $\Delsub{trs}S$ and $\Delsub{trs}H$ are evaluated at the transition temperature $T\subs{trs}$.
We may obtain Eq. 8.3.5 directly from the second law. With the phases in equilibrium, the transition process is reversible. The second law gives $\Del S = q/T\subs{trs} = \Del H/T\subs{trs}$. Dividing by the amount transferred between the phases gives Eq. 8.3.5.
8.3.2 Calorimetric measurement of transition enthalpies
The most precise measurement of the molar enthalpy of an equilibrium phase transition uses electrical work. A known quantity of electrical work is performed on a system containing coexisting phases, in a constant-pressure adiabatic calorimeter, and the resulting amount of substance transferred between the phases is measured. The first law shows that the electrical work $I^{2}R\el\Del t$ equals the heat that would be needed to cause the same change of state. This heat, at constant $p$, is the enthalpy change of the process.
The method is similar to that used to measure the heat capacity of a phase at constant pressure (Sec. 7.3.2), except that now the temperature remains constant and there is no need to make a correction for the heat capacity of the calorimeter.
8.3.3 Standard molar transition quantities
The standard molar enthalpy of vaporization, $\Delsub{vap}H\st$, is the enthalpy change when pure liquid in its standard state at a specified temperature changes to gas in its standard state at the same temperature, divided by the amount changed.
Note that the initial state of this process is a real one (the pure liquid at pressure $p\st$), but the final state (the gas behaving ideally at pressure $p\st$) is hypothetical. The liquid and gas are not necessarily in equilibrium with one another at pressure $p\st$ and the temperature of interest, and we cannot evaluate $\Delsub{vap}H\st$ from a calorimetric measurement with electrical work without further corrections. The same difficulty applies to the evaluation of $\Delsub{sub}H\st$. In contrast, $\Delsub{vap}H$ and $\Delsub{sub}H$ (without the $\st$ symbol), as well as $\Delsub{fus}H\st$, all refer to reversible transitions between two real phases coexisting in equilibrium.
Let $X$ represent one of the thermodynamic potentials or the entropy of a phase. The standard molar transition quantities $\Delsub{vap}X\st=X\m\st\gas-X\m\s\liquid$ and $\Delsub{sub}X\st=X\m\st\gas-X\m\s\solid$ are functions only of $T$. To evaluate $\Delsub{vap}X\st$ or $\Delsub{sub}X\st$ at a given temperature, we must calculate the change of $X\m$ for a path that connects the standard state of the liquid or solid with that of the gas. The simplest choice of path is one of constant temperature $T$ with the following steps:
1. The sum of $\Del X\m$ for these three steps is the desired quantity $\Delsub{vap}X\st$ or $\Delsub{sub}X\st$. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/08%3A_Phase_Transitions_and_Equilibria_of_Pure_Substances/8.03%3A_Phase_Transitions.txt |
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$\newcommand{\bd}{_{\text{b}}} % subscript b for boundary or boiling point$
$\newcommand{\C}{_{\text{C}}} % subscript C$
$\newcommand{\f}{_{\text{f}}} % subscript f for freezing point$
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$\newcommand{\cbB}{_{c,\text{B}}} % c basis, B$
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$\newcommand{\bph}{^{\beta}} % beta phase superscript$
$\newcommand{\gph}{^{\gamma}} % gamma phase superscript$
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$\newcommand{\bphp}{^{\beta'}} % beta prime phase superscript$
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A coexistence curve on a pressure–temperature phase diagram shows the conditions under which two phases can coexist in equilibrium, as explained in Sec. 8.2.2.
8.4.1 Chemical potential surfaces
From the relation $\pd{\mu}{T}{p}=-S\m$, we see that at constant $p$ the slope of $\mu$ versus $T$ is negative since molar entropy is always positive. Furthermore, the magnitude of the slope increases on going from solid to liquid and from liquid to gas, because the molar entropies of sublimation and vaporization are positive. This difference in slope is illustrated by the curves for H$_2$O in Fig. 8.13(a). The triple-point pressure of H$_2$O is $0.0062\br$. At a pressure of $0.03\br$, greater than the triple-point pressure, the curves for solid and liquid intersect at a melting point (point A) and the curves for liquid and gas intersect at a boiling point (point B).
From $\pd{\mu}{p}{T}=V\m$, we see that a pressure reduction at constant temperature lowers the chemical potential of a phase. The result of a pressure reduction from $0.03\br$ to $0.003\br$ (below the triple-point pressure of H$_2$O) is a downward shift of each of the curves of Fig. 8.13(a) by a distance proportional to the molar volume of the phase. The shifts of the solid and liquid curves are too small to see ($\Del \mu$ is only $-0.002\units{kJ mol\(^{-1}$}\)). Because the gas has a large molar volume, the gas curve shifts substantially to a position where it intersects with the solid curve at a sublimation point (point C). At $0.003\br$, or any other pressure below the triple-point pressure, only a solid–gas equilibrium is possible for H$_2$O. The liquid phase is not stable at any pressure below the triple-point pressure, as shown by the pressure–temperature phase diagram of H$_2$O in Fig. 8.13(b).
8.4.2 The Clapeyron equation
If we start with two coexisting phases, $\pha$ and $\phb$, of a pure substance and change the temperature of both phases equally without changing the pressure, the phases will no longer be in equilibrium, because their chemical potentials change unequally. In order for the phases to remain in equilibrium during the temperature change $\dif T$ of both phases, there must be a certain simultaneous change $\difp$ in the pressure of both phases. The changes $\dif T$ and $\difp$ must be such that the chemical potentials of both phases change equally so as to remain equal to one another: $\dif\mu\aph = \dif\mu\bph$.
The infinitesimal change of $\mu$ in a phase is given by $\dif \mu = -S\m\dif T + V\m\difp$ (Eq. 7.8.2). Thus, the two phases remain in equilibrium if $\dif T$ and $\difp$ satisfy the relation $-S\m\aph\dif T + V\m\aph\difp = -S\m\bph\dif T + V\m\bph\difp \tag{8.4.2}$ which we rearrange to $\frac{\difp}{\dif T} = \frac{S\m\bph-S\m\aph}{V\m\bph-V\m\aph} \tag{8.4.3}$ or \begin{gather} \s{ \frac{\difp}{\dif T} = \frac{\Delsub{trs}S}{\Delsub{trs}V} } \tag{8.4.4} \cond{(pure substance)} \end{gather} Equation 8.4.4 is one form of the Clapeyron equation, which contains no approximations. We find an alternative form by substituting $\Delsub{trs}S = \Delsub{trs}H/T\subs{trs}$ (Eq. 8.3.5): \begin{gather} \s{ \frac{\difp}{\dif T} = \frac{\Delsub{trs}H}{T\Delsub{trs}V} } \tag{8.4.5} \cond{(pure substance)} \end{gather}
Equations 8.4.4 and 8.4.5 give the slope of the coexistence curve, $\difp/\dif T$, as a function of quantities that can be measured. For the sublimation and vaporization processes, both $\Delsub{trs}H$ and $\Delsub{trs}V$ are positive. Therefore, according to Eq. 8.4.5, the solid–gas and liquid–gas coexistence curves have positive slopes. For the fusion process, however, $\Delsub{fus}H$ is positive, but $\Delsub{fus}V$ may be positive or negative depending on the substance, so that the slope of the solid–liquid coexistence curve may be either positive or negative. The absolute value of $\Delsub{fus}V$ is small, causing the solid–liquid coexistence curve to be relatively steep; see Fig. 8.13(b) for an example.
Most substances expand on melting, making the slope of the solid–liquid coexistence curve positive. This is true of carbon dioxide, although in Fig. 8.2(c) the curve is so steep that it is difficult to see the slope is positive. Exceptions at ordinary pressures, substances that contract on melting, are H$_2$O, rubidium nitrate, and the elements antimony, bismuth, and gallium.
The phase diagram for H$_2$O in Fig. 8.4 clearly shows that the coexistence curve for ice I and liquid has a negative slope due to ordinary ice being less dense than liquid water. The high-pressure forms of ice are more dense than the liquid, causing the slopes of the other solid–liquid coexistence curves to be positive. The ice VII–ice VIII coexistence curve is vertical, because these two forms of ice have identical crystal structures, except for the orientations of the H$_2$O molecule; therefore, within experimental uncertainty, the two forms have equal molar volumes.
We may rearrange Eq. 8.4.5 to give the variation of $p$ with $T$ along the coexistence curve: $\difp = \frac{\Delsub{trs}H}{\Delsub{trs}V} \cdot \frac{\dif T}{T} \tag{8.4.6}$ Consider the transition from solid to liquid (fusion). Because of the fact that the cubic expansion coefficient and isothermal compressibility of a condensed phase are relatively small, $\Delsub{fus}V$ is approximately constant for small changes of $T$ and $p$. If $\Delsub{fus}H$ is also practically constant, integration of Eq. 8.4.6 yields the relation $p_2-p_1 \approx \frac{\Delsub{fus}H}{\Delsub{fus}V} \ln \frac{T_2}{T_1} \tag{8.4.7}$ or \begin{gather} \s{ T_2 \approx T_1 \exp\left[ \frac{\Delsub{fus}V(p_2-p_1)}{\Delsub{fus}H}\right] } \tag{8.4.8} \cond{(pure substance)} \end{gather} from which we may estimate the dependence of the melting point on pressure.
8.4.3 The Clausius–Clapeyron equation
When the gas phase of a substance coexists in equilibrium with the liquid or solid phase, and provided $T$ and $p$ are not close to the critical point, the molar volume of the gas is much greater than that of the condensed phase. Thus, we may write for the processes of vaporization and sublimation $\Delsub{vap}V = V\m\sups{g}-V\m\sups{l} \approx V\m\sups{g} \qquad \Delsub{sub}V = V\m\sups{g}-V\m\sups{s} \approx V\m\sups{g} \tag{8.4.9}$ The further approximation that the gas behaves as an ideal gas, $V\m\sups{g} \approx RT/p$, then changes Eq. 8.4.5 to \begin{gather} \s{ \frac{\difp}{\dif T} \approx \frac{p\Delsub{trs}H}{RT^2} } \tag{8.4.10} \cond{(pure substance,} \nextcond{vaporization or sublimation)} \end{gather}
Equation 8.4.10 is the Clausius–Clapeyron equation. It gives an approximate expression for the slope of a liquid–gas or solid–gas coexistence curve. The expression is not valid for coexisting solid and liquid phases, or for coexisting liquid and gas phases close to the critical point.
At the temperature and pressure of the triple point, it is possible to carry out all three equilibrium phase transitions of fusion, vaporization, and sublimation. When fusion is followed by vaporization, the net change is sublimation. Therefore, the molar transition enthalpies at the triple point are related by $\Delsub{fus}H + \Delsub{vap}H = \Delsub{sub}H \tag{8.4.11}$ Since all three of these transition enthalpies are positive, it follows that $\Delsub{sub}H$ is greater than $\Delsub{vap}H$ at the triple point. Therefore, according to Eq. 8.4.10, the slope of the solid–gas coexistence curve at the triple point is slightly greater than the slope of the liquid–gas coexistence curve.
We divide both sides of Eq. 8.4.10 by $p\st$ and rearrange to the form $\frac{\dif(p/p\st)}{p/p\st} \approx \frac{\Delsub{trs}H}{R}\cdot\frac{\dif T}{T^2} \tag{8.4.12}$ Then, using the mathematical identities $\dif(p/p\st)/(p/p\st) = \dif\ln(p/p\st)$ and $\dif T/T^2 = -\dif(1/T)$, we can write Eq. 8.4.12 in three alternative forms:
\begin{gather} \s{ \frac{\dif\ln (p/p\st)}{\dif T} \approx \frac{\Delsub{trs}H}{RT^2} } \tag{8.4.13} \cond{(pure substance,} \nextcond{vaporization or sublimation)} \end{gather} \begin{gather} \s{ \dif\ln (p/p\st) \approx -\frac{\Delsub{trs}H}{R}\dif (1/T) } \tag{8.4.14} \cond{(pure substance,} \nextcond{vaporization or sublimation)} \end{gather} \begin{gather} \s{ \frac{\dif\ln (p/p\st)}{\dif (1/T)} \approx -\frac{\Delsub{trs}H}{R} } \tag{8.4.15} \cond{(pure substance,} \nextcond{vaporization or sublimation)} \end{gather}
Equation 8.4.15 shows that the curve of a plot of $\ln(p/p\st)$ versus $1/T$ (where $p$ is the vapor pressure of a pure liquid or solid) has a slope at each temperature equal, usually to a high degree of accuracy, to $-\Delsub{vap}H/R$ or $-\Delsub{sub}H/R$ at that temperature. This kind of plot provides an alternative to calorimetry for evaluating molar enthalpies of vaporization and sublimation.
If we use the recommended standard pressure of $1\br$, the ratio $p/p\st$ appearing in these equations becomes $p/\tx{bar}$. That is, $p/p\st$ is simply the numerical value of $p$ when $p$ is expressed in bars. For the purpose of using Eq. 8.4.15 to evaluate $\Delsub{trs}H$, we can replace $p\st$ by any convenient value. Thus, the curves of plots of $\ln(p/\tx{bar})$ versus $1/T$, $\ln(p/\tx{Pa})$ versus $1/T$, and $\ln(p/\tx{Torr})$ versus $1/T$ using the same temperature and pressure data all have the same slope (but different intercepts) and yield the same value of $\Delsub{trs}H$.
If we assume $\Delsub{vap}H$ or $\Delsub{sub}H$ is essentially constant in a temperature range, we may integrate Eq. 8.4.14 from an initial to a final state along the coexistence curve to obtain \begin{gather} \s{ \ln\frac{p_2}{p_1} \approx -\frac{\Delsub{trs}H}{R} \left( \frac{1}{T_2}-\frac{1}{T_1} \right) } \tag{8.4.16} \cond{(pure substance,} \nextcond{vaporization or sublimation)} \end{gather} Equation 8.4.16 allows us to estimate any one of the quantities $p_1$, $p_2$, $T_1$, $T_2$, or $\Delsub{trs}H$, given values of the other four. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/08%3A_Phase_Transitions_and_Equilibria_of_Pure_Substances/8.04%3A_Coexistence_Curves.txt |
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$\newcommand{\cond}[1]{\$-2.5pt]{}\tag*{#1}}$ $\newcommand{\nextcond}[1]{\[-5pt]{}\tag*{#1}}$ $\newcommand{\R}{8.3145\units{J\,K\per\,mol\per}} % gas constant value$ $\newcommand{\Rsix}{8.31447\units{J\,K\per\,mol\per}} % gas constant value - 6 sig figs$ $\newcommand{\jn}{\hspace3pt\lower.3ex{\Rule{.6pt}{2ex}{0ex}}\hspace3pt}$ $\newcommand{\ljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}} \hspace3pt}$ $\newcommand{\lljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace1.4pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace3pt}$ An underlined problem number or problem-part letter indicates that the numerical answer appears in Appendix I. 8.1 Consider the system described in Sec. 8.1.5 containing a spherical liquid droplet of radius $r$ surrounded by pure vapor. Starting with Eq. 8.1.15, find an expression for the total differential of $U$. Then impose conditions of isolation and show that the equilibrium conditions are $T\sups{g} = T\sups{l}$, $\mu\sups{g} = \mu\sups{l}$, and $p\sups{l} = p\sups{g} + 2\g /r$, where $\g$ is the surface tension. 8.2 This problem concerns diethyl ether at $T=298.15\K$. At this temperature, the standard molar entropy of the gas calculated from spectroscopic data is $S\m\st\gas = 342.2\units{J K\(^{-1}$ mol$^{-1}$}\). The saturation vapor pressure of the liquid at this temperature is $0.6691\br$, and the molar enthalpy of vaporization is $\Delsub{vap}H = 27.10\units{kJ mol\(^{-1}$}\). The second virial coefficient of the gas at this temperature has the value $B=-1.227\timesten{-3}\units{m\(^3$ mol$^{-1}$}\), and its variation with temperature is given by $\dif B/\dif T = 1.50\timesten{-5}\units{m\(^3$ K$^{-1}$ mol$^{-1}$}\). (a) Use these data to calculate the standard molar entropy of liquid diethyl ether at $298.15\K$. A small pressure change has a negligible effect on the molar entropy of a liquid, so that it is a good approximation to equate $S\m\st\liquid$ to $S\m\liquid$ at the saturation vapor pressure. (b) Calculate the standard molar entropy of vaporization and the standard molar enthalpy of vaporization of diethyl ether at $298.15\K$. It is a good approximation to equate $H\m\st\liquid$ to $H\m\liquid$ at the saturation vapor pressure. 8.3 Explain why the chemical potential surfaces shown in Fig. 8.12 are concave downward; that is, why $\pd{\mu}{T}{p}$ becomes more negative with increasing $T$ and $\pd{\mu}{p}{T}$ becomes less positive with increasing $p$. 8.4 Potassium has a standard boiling point of $773\units{\(\degC$}\) and a molar enthalpy of vaporization $\Delsub{vap}H = 84.9\units{kJ mol\(^{-1}$}\). Estimate the saturation vapor pressure of liquid potassium at $400.\units{\(\degC$}\). 8.5 Naphthalene has a melting point of $78.2\units{\(\degC$}\) at $1\br$ and $81.7\units{\(\degC$}\) at $100\br$. The molar volume change on melting is $\Delsub{fus}V=0.019\units{cm\(^{3}$ mol$^{-1}$}\). Calculate the molar enthalpy of fusion to two significant figures. 8.6 The dependence of the vapor pressure of a liquid on temperature, over a limited temperature range, is often represented by the Antoine equation, $\log_{10}(p/\tx{Torr})=A-B/(t+C)$, where $t$ is the Celsius temperature and $A$, $B$, and $C$ are constants determined by experiment. A variation of this equation, using a natural logarithm and the thermodynamic temperature, is \[ \ln(p/\tx{bar}) = a - \frac{b}{T+c}$ The vapor pressure of liquid benzene at temperatures close to $298\K$ is adequately represented by the preceding equation with the following values of the constants: $a = 9.25092 \qquad b = 2771.233\K \qquad c = -53.262\K$
(a) Find the standard boiling point of benzene.
(b) Use the Clausius–Clapeyron equation to evaluate the molar enthalpy of vaporization of benzene at $298.15\K$.
8.7
At a pressure of one atmosphere, water and steam are in equilibrium at $99.97\units{\(\degC$}\) (the normal boiling point of water). At this pressure and temperature, the water density is $0.958\units{g cm\(^{-3}$}\), the steam density is $5.98\timesten{-4}\units{g cm\(^{-3}$}\), and the molar enthalpy of vaporization is $40.66\units{kJ mol\(^{-1}$}\).
(a) Use the Clapeyron equation to calculate the slope $\difp/\dif T$ of the liquid–gas coexistence curve at this point.
(b) Repeat the calculation using the Clausius–Clapeyron equation.
(c) Use your results to estimate the standard boiling point of water. (Note: The experimental value is $99.61\units{\(\degC$}\).)
8.8
At the standard pressure of $1\br$, liquid and gaseous H$_2$O coexist in equilibrium at $372.76\K$, the standard boiling point of water.
(a) Do you expect the standard molar enthalpy of vaporization to have the same value as the molar enthalpy of vaporization at this temperature? Explain.
(b) The molar enthalpy of vaporization at $372.76\K$ has the value $\Delsub{vap}H=40.67\units{kJ mol\(^{-1}$}\). Estimate the value of $\Delsub{vap}H\st$ at this temperature with the help of Table 7.5 and the following data for the second virial coefficient of gaseous H$_2$O at $372.76\K$: $B=-4.60\timesten{-4}\units{m$^3$ mol$^{-1}$} \qquad \dif B/\dif T=3.4\timesten{-6}\units{m$^3$ K$^{-1}$ mol$^{-1}$}$
(c) Would you expect the values of $\Delsub{fus}H$ and $\Delsub{fus}H\st$ to be equal at the standard freezing point of water? Explain.
8.9
The standard boiling point of H$_2$O is $99.61\units{\(\degC$}\). The molar enthalpy of vaporization at this temperature is $\Delsub{vap}H=40.67\units{kJ mol\(^{-1}$}\). The molar heat capacity of the liquid at temperatures close to this value is given by \begin{equation*} \Cpm=a+b(t-c) \end{equation*} where $t$ is the Celsius temperature and the constants have the values $a=75.94\units{J K$^{-1}$ mol$^{-1}$} \qquad b = 0.022\units{J K$^{-2}$ mol$^{-1}$} \qquad c = 99.61\units{$\degC$}$ Suppose $100.00\mol$ of liquid H$_2$O is placed in a container maintained at a constant pressure of $1\br$, and is carefully heated to a temperature $5.00\units{\(\degC$}\) above the standard boiling point, resulting in an unstable phase of superheated water. If the container is enclosed with an adiabatic boundary and the system subsequently changes spontaneously to an equilibrium state, what amount of water will vaporize? (Hint: The temperature will drop to the standard boiling point, and the enthalpy change will be zero.) | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/08%3A_Phase_Transitions_and_Equilibria_of_Pure_Substances/8.05%3A_Chapter_8_Problems.txt |
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A homogeneous mixture is a phase containing more than one substance. This chapter discusses composition variables and partial molar quantities of mixtures in which no chemical reaction is occurring. The ideal mixture is defined. Chemical potentials, activity coefficients, and activities of individual substances in both ideal and nonideal mixtures are discussed.
Except for the use of fugacities to determine activity coefficients in condensed phases, a discussion of phase equilibria involving mixtures will be postponed to Chap. 13.
09: Mixtures
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A composition variable is an intensive property that indicates the relative amount of a particular species or substance in a phase.
9.1.1 Species and substances
We sometimes need to make a distinction between a species and a substance. A species is any entity of definite elemental composition and charge and can be described by a chemical formula, such as H$_2$O, H$_3$O$^+$, NaCl, or Na$^+$. A substance is a species that can be prepared in a pure state (e.g., N$_2$ and NaCl). Since we cannot prepare a macroscopic amount of a single kind of ion by itself, a charged species such as H$_3$O$^+$ or Na$^+$ is not a substance. Chap. 10 will discuss the special features of mixtures containing charged species.
9.1.2 Mixtures in general
The mole fraction of species $i$ is defined by \begin{gather} \s{ x_i \defn \frac{n_i}{\sum_j n_j} \qquad \tx{or} \qquad y_i \defn \frac{n_i}{\sum_j n_j} } \tag{9.1.1} \cond{($P{=}1$)} \end{gather} where $n_i$ is the amount of species $i$ and the sum is taken over all species in the mixture. The symbol $x_i$ is used for a mixture in general, and $y_i$ is used when the mixture is a gas.
The mass fraction, or weight fraction, of species $i$ is defined by \begin{gather} \s{ w_i \defn \frac{m(i)}{m} = \frac{n_i M_i}{\sum_j n_j M_j} } \tag{9.1.2} \cond{($P{=}1$)} \end{gather} where $m(i)$ is the mass of species $i$ and $m$ is the total mass.
The concentration, or molarity, of species $i$ in a mixture is defined by \begin{gather} \s{ c_i \defn \frac{n_i}{V} } \tag{9.1.3} \cond{($P{=}1$)} \end{gather} The symbol M is often used to stand for units of mol L$^{-1}$, or mol dm$^{-3}$. Thus, a concentration of $0.5\units{M}$ is $0.5$ moles per liter, or $0.5$ molar.
Concentration is sometimes called “amount concentration” or “molar concentration” to avoid confusion with number concentration (the number of particles per unit volume). An alternative notation for $c\A$ is [A].
A binary mixture is a mixture of two substances.
9.1.3 Solutions
A solution, strictly speaking, is a mixture in which one substance, the solvent, is treated in a special way. Each of the other species comprising the mixture is then a solute. The solvent is denoted by A and the solute species by B, C, and so on. (Some chemists denote the solvent by subscript $1$ and use $2$, $3$, and so on for solutes.) Although in principle a solution can be a gas mixture, in this section we will consider only liquid and solid solutions.
We can prepare a solution of varying composition by gradually mixing one or more solutes with the solvent so as to continuously increase the solute mole fractions. During this mixing process, the physical state (liquid or solid) of the solution remains the same as that of the pure solvent. When the sum of the solute mole fractions is small compared to $x\A$ (i.e., $x\A$ is close to unity), the solution is called dilute. As the solute mole fractions increase, we say the solution becomes more concentrated.
Mole fraction, mass fraction, and concentration can be used as composition variables for both solvent and solute, just as they are for mixtures in general. A fourth composition variable, molality, is often used for a solute. The molality of solute species B is defined by \begin{gather} \s{ m\B \defn \frac{n\B}{m(\tx{A})} } \tag{9.1.4} \cond{(solution)} \end{gather} where $m(\tx{A})=n\A M\A$ is the mass of solvent. The symbol m is sometimes used to stand for units of mol kg$^{-1}$, although this should be discouraged because m is also the symbol for meter. For example, a solute molality of $0.6\units{m}$ is $0.6$ moles of solute per kilogram of solvent, or $0.6$ molal.
9.1.4 Binary solutions
We may write simplified equations for a binary solution of two substances, solvent A and solute B. Equations 9.1.1–9.1.4 become
\begin{gather} \s{ x\B = \frac{n\B}{n\A + n\B} } \tag{9.1.5} \cond{(binary solution)} \end{gather} \begin{gather} \s{ w\B = \frac{n\B M\B}{n\A M\A + n\B M\B} } \tag{9.1.6} \cond{(binary solution)} \end{gather} \begin{gather} \s{ c\B = \frac{n\B}{V} = \frac{n\B \rho}{n\A M\A + n\B M\B} } \tag{9.1.7} \cond{(binary solution)} \end{gather} \begin{gather} \s{ m\B = \frac{n\B}{n\A M\A} } \tag{9.1.8} \cond{(binary solution)} \end{gather} The right sides of Eqs. 9.1.5–9.1.8 express the solute composition variables in terms of the amounts and molar masses of the solvent and solute and the density $\rho$ of the solution.
To be able to relate the values of these composition variables to one another, we solve each equation for $n\B$ and divide by $n\A$ to obtain an expression for the mole ratio $n\B/n\A$: \begin{gather} \s{\tx{from Eq. 9.1.5}} \tag{9.1.9} \qquad \s{\frac{n\B}{n\A} = \frac{x\B}{1-x\B}} \cond{(binary solution)} \end{gather} \begin{gather} \s{\tx{from Eq. 9.1.6}} \tag{9.1.10} \qquad \s{\frac{n\B}{n\A} = \frac{M\A w\B}{M\B(1-w\B)}} \cond{(binary solution)} \end{gather} \begin{gather} \s{\tx{from Eq. 9.1.7}} \tag{9.1.11} \qquad \s{\frac{n\B}{n\A} = \frac{M\A c\B}{\rho - M\B c\B}} \cond{(binary solution)} \end{gather} \begin{gather} \s{\tx{from Eq. 9.1.8}} \tag{9.1.12} \qquad \s{\frac{n\B}{n\A} = M\A m\B} \cond{(binary solution)} \end{gather} These expressions for $n\B/n\A$ allow us to find one composition variable as a function of another. For example, to find molality as a function of concentration, we equate the expressions for $n\B/n\A$ on the right sides of Eqs. 9.1.11 and 9.1.12 and solve for $m\B$ to obtain $m\B = \frac{c\B}{\rho - M\B c\B} \tag{9.1.13}$
A binary solution becomes more dilute as any of the solute composition variables becomes smaller. In the limit of infinite dilution, the expressions for $n\B/n\A$ become: \begin{gather} \s{\begin{split} \frac{n\B}{n\A} & = x\B \cr & = \frac{M\A}{M\B}w\B \cr & = \frac{M\A}{\rho\A^*}c\B = V\mA^* c\B \cr & = \s{ M\A m\B } \end{split}} \tag{9.1.14} \cond{(binary solution at} \nextcond{infinite dilution)} \end{gather}
where a superscript asterisk (${}^*$) denotes a pure phase. We see that, in the limit of infinite dilution, the composition variables $x\B$, $w\B$, $c\B$, and $m\B$ are proportional to one another. These expressions are also valid for solute B in a multisolute solution in which each solute is very dilute; that is, in the limit $x\A\ra 1$.
The rule of thumb that the molarity and molality values of a dilute aqueous solution are approximately equal is explained by the relation $M\A c\B/\rho\A^*=M\A m\B$ (from Eq. 9.1.14), or $c\B/\rho\A^* = m\B$, and the fact that the density $\rho\A^*$ of water is approximately $1\units{kg L\(^{-1}$}\). Hence, if the solvent is water and the solution is dilute, the numerical value of $c\B$ expressed in mol L$^{-1}$ is approximately equal to the numerical value of $m\B$ expressed in mol kg$^{-1}$.
9.1.5 The composition of a mixture
We can describe the composition of a phase with the amounts of each species, or with any of the composition variables defined earlier: mole fraction, mass fraction, concentration, or molality. If we use mole fractions or mass fractions to describe the composition, we need the values for all but one of the species, since the sum of all fractions is unity.
Other composition variables are sometimes used, such as volume fraction, mole ratio, and mole percent. To describe the composition of a gas mixture, partial pressures can be used (Sec. 9.3.1).
When the composition of a mixture is said to be fixed or constant during changes of temperature, pressure, or volume, this means there is no change in the relative amounts or masses of the various species. A mixture of fixed composition has fixed values of mole fractions, mass fractions, and molalities, but not necessarily of concentrations and partial pressures. Concentrations will change if the volume changes, and partial pressures in a gas mixture will change if the pressure changes. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/09%3A_Mixtures/9.01%3A_Composition_Variables.txt |
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$\newcommand{\C}{_{\text{C}}} % subscript C$
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$\newcommand{\mbB}{_{m,\text{B}}} % m basis, B$
$\newcommand{\kHi}{k_{\text{H},i}} % Henry's law constant, x basis, i$
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$\newcommand{\pha}{\alpha} % phase alpha$
$\newcommand{\phb}{\beta} % phase beta$
$\newcommand{\phg}{\gamma} % phase gamma$
$\newcommand{\aph}{^{\alpha}} % alpha phase superscript$
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$\newcommand{\cond}[1]{\$-2.5pt]{}\tag*{#1}}$ $\newcommand{\nextcond}[1]{\[-5pt]{}\tag*{#1}}$ $\newcommand{\R}{8.3145\units{J\,K\per\,mol\per}} % gas constant value$ $\newcommand{\Rsix}{8.31447\units{J\,K\per\,mol\per}} % gas constant value - 6 sig figs$ $\newcommand{\jn}{\hspace3pt\lower.3ex{\Rule{.6pt}{2ex}{0ex}}\hspace3pt}$ $\newcommand{\ljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}} \hspace3pt}$ $\newcommand{\lljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace1.4pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace3pt}$ The symbol $X_i$, where $X$ is an extensive property of a homogeneous mixture and the subscript $i$ identifies a constituent species of the mixture, denotes the partial molar quantity of species $i$ defined by \begin{gather} \s{ X_i \defn \Pd{X}{n_i}{T,p,n_{j \ne i}} } \tag{9.2.1} \cond{(mixture)} \end{gather} This is the rate at which property $X$ changes with the amount of species $i$ added to the mixture as the temperature, the pressure, and the amounts of all other species are kept constant. A partial molar quantity is an intensive state function. Its value depends on the temperature, pressure, and composition of the mixture. Keep in mind that as a practical matter, a macroscopic amount of a charged species (i.e., an ion) cannot be added by itself to a phase because of the huge electric charge that would result. Thus if species $i$ is charged, $X_i$ as defined by Eq. 9.2.1 is a theoretical concept whose value cannot be determined experimentally. An older notation for a partial molar quantity uses an overbar: $\overline{X}_i$. The notation $X_i'$ was suggested in the first edition of the IUPAC Green Book (Ian Mills et al, Quantities, Units and Symbols in Physical Chemistry, Blackwell, Oxford, 1988, p. 44), but is not mentioned in later editions. Example $1$ (a) $40.75 \mathrm{~cm}^3$ (one mole) of methanol is placed in a narrow tube above a much greater volume of a mixture (shaded) of composition $x_{\mathrm{B}}=0.307$. The dashed line indicates the level of the upper meniscus. (b) After the two liquid phases have mixed by diffusion, the volume of the mixture has increased by only $38.8 \mathrm{~cm}^3$. 9.2.1 Partial molar volume In order to gain insight into the significance of a partial molar quantity as defined by Eq. 9.2.1, let us first apply the concept to the volume of an open single-phase system. Volume has the advantage for our example of being an extensive property that is easily visualized. Let the system be a binary mixture of water (substance A) and methanol (substance B), two liquids that mix in all proportions. The partial molar volume of the methanol, then, is the rate at which the system volume changes with the amount of methanol added to the mixture at constant temperature and pressure: $V\B=\pd{V}{n\B}{T,p,n\A}$. At $25\units{\(\degC$}\) and $1\br$, the molar volume of pure water is $V\mA^* = 18.07\units{cm\(^3$ mol$^{-1}$}\) and that of pure methanol is $V\mB^* = 40.75\units{cm\(^3$ mol$^{-1}$}\). If we mix $100.0\units{cm\(^3$}\) of water at $25\units{\(\degC$}\) with $100.0\units{cm\(^3$}\) of methanol at $25\units{\(\degC$}\), we find the volume of the resulting mixture at $25\units{\(\degC$}\) is not the sum of the separate volumes, $200.0\units{cm\(^3$}\), but rather the slightly smaller value $193.1\units{cm\(^3$}\). The difference is due to new intermolecular interactions in the mixture compared to the pure liquids. Let us calculate the mole fraction composition of this mixture: \[ $n\A = \frac{V\A^*}{V\mA^*} = \frac{100.0\units{cm$^3$}}{18.07\units{cm$^3$ mol$^{-1}$}} = 5.53\mol \tag{9.2.2}$$
$$n\B = \frac{V\B^*}{V\mB^*} = \frac{100.0\units{cm$^3$}}{40.75\units{cm$^3$ mol$^{-1}$}} = 2.45\mol \tag{9.2.3}$$
$$x\B = \frac{n\B}{n\A+n\B} = \frac{2.45\mol}{5.53\mol + 2.45\mol} = 0.307 \tag{9.2.4}$$
Now suppose we prepare a large volume of a mixture of this composition $\left(x_{\mathrm{B}}=0.307\right)$ and add an additional $40.75 \mathrm{~cm}^3$ (one mole) of pure methanol, as shown in Fig. 9.1(a). If the initial volume of the mixture at $25^{\circ} \mathrm{C}$ was $10,000.0 \mathrm{~cm}^3$, we find the volume of the new mixture at the same temperature is $10,038.8 \mathrm{~cm}^3$, an increase of $38.8 \mathrm{~cm}^3$ - see Fig. 9.1(b). The amount of methanol added is not infinitesimal, but it is small enough compared to the amount of initial mixture to cause very little change in the mixture composition: $x_{\mathrm{B}}$
increases by only $0.5 \%$. Treating the mixture as an open system, we see that the addition of one mole of methanol to the system at constant $T, p$, and $n_{\mathrm{A}}$ causes the system volume to increase by $38.8 \mathrm{~cm}^3$. To a good approximation, then, the partial molar volume of methanol in the mixture, $V_{\mathrm{B}}=\left(\partial V / \partial n_{\mathrm{B}}\right)_{T, p, n_{\mathrm{A}}}$, is given by $\Delta V / \Delta n_{\mathrm{B}}=38.8 \mathrm{~cm}^3 \mathrm{~mol}^{-1}$.
The volume of the mixture to which we add the methanol does not matter as long as it is large. We would have observed practically the same volume increase, $38.8 \mathrm{~cm}^3$, if we had mixed one mole of pure methanol with $100,000.0 \mathrm{~cm}^3$ of the mixture instead of only $10,000.0 \mathrm{~cm}^3$.
Thus, we may interpret the partial molar volume of B as the volume change per amount of B added at constant $T$ and $p$ when B is mixed with such a large volume of mixture that the composition is not appreciably affected. We may also interpret the partial molar volume as the volume change per amount when an infinitesimal amount is mixed with a finite volume of mixture.
The partial molar volume of $\mathrm{B}$ is an intensive property that is a function of the composition of the mixture, as well as of $T$ and $p$. The limiting value of $V_{\mathrm{B}}$ as $x_{\mathrm{B}}$ approaches 1 (pure B) is $V_{\mathrm{m}, \mathrm{B}}^*$, the molar volume of pure $\mathrm{B}$. We can see this by writing $V=n_{\mathrm{B}} V_{\mathrm{m}, \mathrm{B}}^*$ for pure $\mathrm{B}$, giving us $V_{\mathrm{B}}\left(x_{\mathrm{B}}=1\right)=\left(\partial n_{\mathrm{B}} V_{\mathrm{m}, \mathrm{B}}^* / \partial n_{\mathrm{B}}\right)_{T, p, n_{\mathrm{A}}}=V_{\mathrm{m}, \mathrm{B}}^*$.
If the mixture is a binary mixture of $\mathrm{A}$ and $\mathrm{B}$, and $x_{\mathrm{B}}$ is small, we may treat the mixture as a dilute solution of solvent $\mathrm{A}$ and solute $\mathrm{B}$. As $x_{\mathrm{B}}$ approaches 0 in this solution, $V_{\mathrm{B}}$ approaches a certain limiting value that is the volume increase per amount of B mixed with a large amount of pure A. In the resulting mixture, each solute molecule is surrounded only by solvent molecules. We denote this limiting value of $V_{\mathrm{B}}$ by $V_{\mathrm{B}}^{\infty}$, the partial molar volume of solute B at infinite dilution.
It is possible for a partial molar volume to be negative. Magnesium sulfate, in aqueous solutions of molality less than $0.07 \mathrm{~mol} \mathrm{~kg}^{-1}$, has a negative partial molar volume. Physically, this means that when a small amount of crystalline $\mathrm{MgSO}_4$ dissolves at constant temperature in water, the liquid phase contracts. This unusual behavior is due to strong attractive water-ion interactions.
9.2.2 The total differential of the volume in an open system
Consider an open single-phase system consisting of a mixture of nonreacting substances. How many independent variables does this system have?
We can prepare the mixture with various amounts of each substance, and we are able to adjust the temperature and pressure to whatever values we wish (within certain limits that prevent the formation of a second phase). Each choice of temperature, pressure, and amounts results in a definite value of every other property, such as volume, density, and mole fraction composition. Thus, an open single-phase system of $C$ substances has $2+C$ independent variables. ${ }^3$
Footnote
3. C in this kind of system is actually the number of components. The number of components is usually the same as the number of substances, but is less if certain constraints exist, such as reaction equilibrium or a fixed mixture composition. The general meaning of C will be discussed in Sec. 13.
For a binary mixture $(C=2)$, the number of independent variables is four. We may choose these variables to be $T, p, n_{\mathrm{A}}$, and $n_{\mathrm{B}}$, and write the total differential of $V$ in the general form
\begin{aligned} \mathrm{d} V= & \left(\frac{\partial V}{\partial T}\right)_{p, n_{\mathrm{A}}, n_{\mathrm{B}}} \mathrm{d} T+\left(\frac{\partial V}{\partial p}\right)_{T, n_{\mathrm{A}}, n_{\mathrm{B}}} \mathrm{d} p \ & +\left(\frac{\partial V}{\partial n_{\mathrm{A}}}\right)_{T, p, n_{\mathrm{B}}} \mathrm{d} n_{\mathrm{A}}+\left(\frac{\partial V}{\partial n_{\mathrm{B}}}\right)_{T, p, n_{\mathrm{A}}} \mathrm{d} n_{\mathrm{B}} \end{aligned}
(binary mixture)
We know the first two partial derivatives on the right side are given by ${ }^4$
Footnote
4. See Eqs. 7.1.1 and 7.1.2, which are for closed syste
$\left(\frac{\partial V}{\partial T}\right)_{p, n_{\mathrm{A}}, n_{\mathrm{B}}}=\alpha V \quad\left(\frac{\partial V}{\partial p}\right)_{T, n_{\mathrm{A}}, n_{\mathrm{B}}}=-\kappa_T V$
We identify the last two partial derivatives on the right side of Eq. 9.2.5 as the partial molar volumes $V_{\mathrm{A}}$ and $V_{\mathrm{B}}$. Thus, we may write the total differential of $V$ for this open system in the compact form
$\mathrm{d} V=\alpha V \mathrm{~d} T-\kappa_T V \mathrm{~d} p+V_{\mathrm{A}} \mathrm{d} n_{\mathrm{A}}+V_{\mathrm{B}} \mathrm{d} n_{\mathrm{B}}$
(binary mixture)
If we compare this equation with the total differential of $V$ for a one-component closed system, $\mathrm{d} V=\alpha V \mathrm{~d} T-\kappa_T V \mathrm{~d} p$ (Eq. 7.1.6), we see that an additional term is required for each constituent of the mixture to allow the system to be open and the composition to vary.
When $T$ and $p$ are held constant, Eq. 9.2.7 becomes
$\mathrm{d} V=V_{\mathrm{A}} \mathrm{d} n_{\mathrm{A}}+V_{\mathrm{B}} \mathrm{d} n_{\mathrm{B}}$
(binary mixture, constant $T$ and $p$ )
We obtain an important relation between the mixture volume and the partial molar volumes by imagining the following process. Suppose we continuously pour pure water and pure methanol at constant but not necessarily equal volume rates into a stirred, thermostatted container to form a mixture of increasing volume and constant composition, as shown schematically in Fig. 9.2. If this mixture remains at constant $T$ and $p$ as it is formed, none of its intensive properties change during the process, and the partial molar volumes $V_{\mathrm{A}}$ and $V_{\mathrm{B}}$ remain constant. Under these conditions, we can integrate Eq. 9.2.8 to obtain the additivity
rule for volume: ${ }^5$
$V=V_{\mathrm{A}} n_{\mathrm{A}}+V_{\mathrm{B}} n_{\mathrm{B}}$
(binary mixture)
This equation allows us to calculate the mixture volume from the amounts of the constituents and the appropriate partial molar volumes for the particular temperature, pressure, and composition.
For example, given that the partial molar volumes in a water-methanol mixture of composition $x_{\mathrm{B}}=0.307$ are $V_{\mathrm{A}}=17.74 \mathrm{~cm}^3 \mathrm{~mol}^{-1}$ and $V_{\mathrm{B}}=38.76 \mathrm{~cm}^3 \mathrm{~mol}^{-1}$, we calculate the volume of the water-methanol mixture described at the beginning of Sec. 9.2.1 as follows:
\begin{aligned} V & =\left(17.74 \mathrm{~cm}^3 \mathrm{~mol}^{-1}\right)(5.53 \mathrm{~mol})+\left(38.76 \mathrm{~cm}^3 \mathrm{~mol}^{-1}\right)(2.45 \mathrm{~mol}) \ & =193.1 \mathrm{~cm}^3 \end{aligned}
We can differentiate Eq. 9.2.9 to obtain a general expression for $\mathrm{d} V$ under conditions of constant $T$ and $p$ :
$\mathrm{d} V=V_{\mathrm{A}} \mathrm{d} n_{\mathrm{A}}+V_{\mathrm{B}} \mathrm{d} n_{\mathrm{B}}+n_{\mathrm{A}} \mathrm{d} V_{\mathrm{A}}+n_{\mathrm{B}} \mathrm{d} V_{\mathrm{B}}$
But this expression for $\mathrm{d} V$ is consistent with Eq. 9.2.8 only if the sum of the last two terms on the right is zero:
$n_{\mathrm{A}} \mathrm{d} V_{\mathrm{A}}+n_{\mathrm{B}} \mathrm{d} V_{\mathrm{B}}=0$
(binary mixture, constant $T$ and $p$ )
Equation 9.2.12 is the Gibbs-Duhem equation for a binary mixture, applied to partial molar volumes. (Section 9.2.4 will give a general version of this equation.) Dividing both sides of the equation by $n_{\mathrm{A}}+n_{\mathrm{B}}$ gives the equivalent form
$x_{\mathrm{A}} \mathrm{d} V_{\mathrm{A}}+x_{\mathrm{B}} \mathrm{d} V_{\mathrm{B}}=0$
(binary mixture, constant $T$ and $p$ )
Equation 9.2.12 shows that changes in the values of $V_{\mathrm{A}}$ and $V_{\mathrm{B}}$ are related when the composition changes at constant $T$ and $p$. If we rearrange the equation to the form
$\mathrm{d} V_{\mathrm{A}}=-\frac{n_{\mathrm{B}}}{n_{\mathrm{A}}} \mathrm{d} V_{\mathrm{B}}$
(binary mixture, constant $T$ and $p$ )
we see that a composition change that increases $V_{\mathrm{B}}$ (so that $\mathrm{d} V_{\mathrm{B}}$ is positive) must make $V_{\mathrm{A}}$ decrease.
9.2.3 Evaluation of partial molar volumes in binary mixtures
The partial molar volumes $V_{\mathrm{A}}$ and $V_{\mathrm{B}}$ in a binary mixture can be evaluated by the method of intercepts. To use this method, we plot experimental values of the quantity $V / n$ (where $n$ is $n_{\mathrm{A}}+n_{\mathrm{B}}$ ) versus the mole fraction $x_{\mathrm{B}} . V / n$ is called the mean molar volume.
See Fig. 9.3(a) for an example. In this figure, the tangent to the curve drawn at the point on the curve at the composition of interest (the composition used as an illustration in Sec. 9.2.1) intercepts the vertical line where $x\B$ equals $0$ at $V/n = V\A = 17.7\units{cm\(^3$ mol$^{-1}$}\), and intercepts the vertical line where $x\B$ equals $1$ at $V/n = V\B = 38.8\units{cm\(^3$ mol$^{-1}$}\).
To derive this property of a tangent line for the plot of $V/n$ versus $x\B$, we use Eq. 9.2.9 to write $\begin{split} (V/n) & = \frac{ V\A n\A + V\B n\B}{n} = V\A x\A + V\B x\B \cr & =V\A(1-x\B) + V\B x\B = (V\B - V\A)x\B + V\A \end{split} \tag{9.2.15}$
Footnote
${ }^5$ The equation is an example of the result of applying Euler's theorem on homogeneous functions to $V$ treated as a function of $n_{\mathrm{A}}$ and $n_{\mathrm{B}}$.
When we differentiate this expression for $V/n$ with respect to $x\B$, keeping in mind that $V\A$ and $V\B$ are functions of $x\B$, we obtain $\begin{split} \frac{\dif (V/n)}{\dx\B} & = \frac{\dif[(V\B - V\A)x\B + V\A]}{\dx\B}\cr & = V\B - V\A + \left( \frac{\dif V\B}{\dx\B} - \frac{\dif V\A}{\dx\B} \right)x\B + \frac{\dif V\A}{\dx\B} \cr & = V\B - V\A + \left(\frac{\dif V\A}{\dx\B}\right)(1-x\B) + \left(\frac{\dif V\B}{\dx\B}\right)x\B \cr & = V\B - V\A + \left(\frac{\dif V\A}{\dx\B}\right)x\A + \left(\frac{\dif V\B}{\dx\B}\right)x\B \end{split} \tag{9.2.16}$
The differentials $\dif V\A$ and $\dif V\B$ are related to one another by the Gibbs–Duhem equation (Eq. 9.2.13): $x\A\dif V\A + x\B\dif V\B = 0$. We divide both sides of this equation by $\dx\B$ to obtain $\left(\frac{\dif V\A}{\dx\B}\right)x\A + \left(\frac{\dif V\B}{\dx\B}\right)x\B = 0 \tag{9.2.17}$ and substitute in Eq. 9.2.16 to obtain $\frac{\dif(V/n)}{\dx\B} = V\B - V\A \tag{9.2.18}$
Let the partial molar volumes of the constituents of a binary mixture of arbitrary composition $x'\B$ be $V'\A$ and $V'\B$. Equation 9.2.15 shows that the value of $V/n$ at the point on the curve of $V/n$ versus $x\B$ where the composition is $x'\B$ is $(V'\B-V'\A)x'\B+V'\A$. Equation 9.2.18 shows that the tangent to the curve at this point has a slope of $V'\B - V'\A$. The equation of the line that passes through this point and has this slope, and thus is the tangent to the curve at this point, is $y=(V'\B - V'\A)x\B + V'\A$, where $y$ is the vertical ordinate on the plot of $(V/n)$ versus $x\B$. The line has intercepts $y{=}V'\A$ at $x\B{=}0$ and $y{=}V'\B$ at $x\B{=}1$.
A variant of the method of intercepts is to plot the molar integral volume of mixing given by $\Del V\m\mix = \frac{\Del V\mix}{n} = \frac{V-n\A V\mA^*-n\B V\mB^*}{n} \tag{9.2.19}$ versus $x\B$, as illustrated in Fig. 9.3(b). $\Del V\mix$ is the integral volume of mixing—the volume change at constant $T$ and $p$ when solvent and solute are mixed to form a mixture of volume $V$ and total amount $n$ (see Sec. 11.1.1). The tangent to the curve at the composition of interest has intercepts $V\A-V\mA^*$ at $x\B{=}0$ and $V\B-V\mB^*$ at $x\B{=}1$.
To see this, we write $\begin{split} \Del V\m\mix & = (V/n) - x\A V\mA^* - x\B V\mB^* \cr & = (V/n) - (1- x\B)V\mA^* - x\B V\mB^* \end{split} \tag{9.2.20}$ We make the substitution $(V/n)=(V\B-V\A)x\B+V\A$ from Eq. 9.2.15 and rearrange: $\Del V\m\mix = \left[\left( V\B-V\mB^* \right) - \left( V\A-V\mA^* \right) \right]x\B + \left( V\A-V\mA^* \right) \tag{9.2.21}$ Differentiation with respect to $x\B$ yields $\begin{split} \frac{\dif \Del V\m\mix}{\dx\B} & = \left( V\B-V\mB^* \right) - \left( V\A-V\mA^* \right) + \left( \frac{\dif V\B}{\dx\B}-\frac{\dif V\A}{\dx\B} \right)x\B + \frac{\dif V\A}{\dx\B} \cr & = \left( V\B-V\mB^* \right) - \left( V\A-V\mA^* \right) + \left(\frac{\dif V\A}{\dx\B}\right)(1-x\B) + \left(\frac{\dif V\B}{\dx\B}\right)x\B \cr & = \left( V\B-V\mB^* \right) - \left( V\A-V\mA^* \right) + \left(\frac{\dif V\A}{\dx\B}\right)x\A + \left(\frac{\dif V\B}{\dx\B}\right)x\B \end{split} \tag{9.2.22}$ With a substitution from Eq. 9.2.17, this becomes $\frac{\dif \Del V\m\mix}{\dx\B} = \left( V\B-V\mB^* \right) - \left( V\A-V\mA^* \right) \tag{9.2.23}$ Equations 9.2.21 and 9.2.23 are analogous to Eqs. 9.2.15 and 9.2.18, with $V/n$ replaced by $\Del V\m\mix$, $V\A$ by $(V\A-V\mA^*)$, and $V\B$ by $(V\B-V\mB^*)$. Using the same reasoning as for a plot of $V/n$ versus $x\B$, we find the intercepts of the tangent to a point on the curve of $\Del V\m\mix$ versus $x\B$ are at $V\A-V\mA^*$ and $V\B-V\mB^*$.
Figure 9.3 shows smoothed experimental data for water–methanol mixtures plotted in both kinds of graphs, and the resulting partial molar volumes as functions of composition. Note in Fig. 9.3(c) how the $V\A$ curve mirrors the $V\B$ curve as $x\B$ varies, as predicted by the Gibbs–Duhem equation. The minimum in $V\B$ at $x\B {\approx} 0.09$ is mirrored by a maximum in $V\A$ in agreement with Eq. 9.2.14; the maximum is much attenuated because $n\B/n\A$ is much less than unity.
Macroscopic measurements are unable to provide unambiguous information about molecular structure. Nevertheless, it is interesting to speculate on the implications of the minimum observed for the partial molar volume of methanol. One interpretation is that in a mostly aqueous environment, there is association of methanol molecules, perhaps involving the formation of dimers.
9.2.4 General relations
The discussion above of partial molar volumes used the notation $V\mA^*$ and $V\mB^*$ for the molar volumes of pure A and B. The partial molar volume of a pure substance is the same as the molar volume, so we can simplify the notation by using $V\A^*$ and $V\B^*$ instead. Hereafter, this e-book will denote molar quantities of pure substances by such symbols as $V\A^*$, $H\B^*$, and $S_i^*$.
The relations derived above for the volume of a binary mixture may be generalized for any extensive property $X$ of a mixture of any number of constituents. The partial molar quantity of species $i$, defined by $X_i \defn \Pd{X}{n_i}{T,p,n_{j\ne i}} \tag{9.2.24}$ is an intensive property that depends on $T$, $p$, and the composition of the mixture. The additivity rule for property $X$ is \begin{gather} \s{ X = \sum_i n_i X_i } \tag{9.2.25} \cond{(mixture)} \end{gather} and the Gibbs–Duhem equation applied to $X$ can be written in the equivalent forms \begin{gather} \s{ \sum_i n_i \dif X_i = 0 } \tag{9.2.26} \cond{(constant $T$ and $p$)} \end{gather} and \begin{gather} \s{ \sum_i x_i \dif X_i = 0 } \tag{9.2.27} \cond{(constant $T$ and $p$)} \end{gather} These relations can be applied to a mixture in which each species $i$ is a nonelectrolyte substance, an electrolyte substance that is dissociated into ions, or an individual ionic species. In Eq. 9.2.27, the mole fraction $x_i$ must be based on the different species considered to be present in the mixture. For example, an aqueous solution of NaCl could be treated as a mixture of components A=H$_2$O and B=NaCl, with $x\B$ equal to $n\B/(n\A+n\B)$; or the constituents could be taken as H$_2$O, Na$^+$, and Cl$^-$, in which case the mole fraction of Na$^+$ would be $x_+=n_+/(n\A+n_++n_-)$.
A general method to evaluate the partial molar quantities $X\A$ and $X\B$ in a binary mixture is based on the variant of the method of intercepts described in Sec. 9.2.3. The molar mixing quantity $\Del X\mix/n$ is plotted versus $x\B$, where $\Del X\mix$ is $(X{-}n\A X\A^*{-}n\B X\B^*)$. On this plot, the tangent to the curve at the composition of interest has intercepts equal to $X\A{-}X\A^*$ at $x\B{=}0$ and $X\B{-}X\B^*$ at $x\B{=}1$.
We can obtain experimental values of such partial molar quantities of an uncharged species as $V_i$, $C_{p,i}$, and $S_i$. It is not possible, however, to evaluate the partial molar quantities $U_i$, $H_i$, $A_i$, and $G_i$ because these quantities involve the internal energy brought into the system by the species, and we cannot evaluate the absolute value of internal energy (Sec. 2.6.2). For example, while we can evaluate the difference $H_i-H_i^*$ from calorimetric measurements of enthalpies of mixing, we cannot evaluate the partial molar enthalpy $H_i$ itself. We can, however, include such quantities as $H_i$ in useful theoretical relations.
A partial molar quantity of a charged species is something else we cannot evaluate. It is possible, however, to obtain values relative to a reference ion. Consider an aqueous solution of a fully-dissociated electrolyte solute with the formula $\tx{M}_{\nu_+}\tx{X}_{\nu_-}$, where $\nu_+$ and $\nu_-$ are the numbers of cations and anions per solute formula unit. The partial molar volume $V\B$ of the solute, which can be determined experimentally, is related to the (unmeasurable) partial molar volumes $V_+$ and $V_-$ of the constituent ions by $V\B = \nu_+ V_+ + \nu_- V_- \tag{9.2.28}$ For aqueous solutions, the usual reference ion is H$^+$, and the partial molar volume of this ion at infinite dilution is arbitrarily set equal to zero: $V\subs{H\(^+$}^{\infty}=0\).
For example, given the value (at $298.15\K$ and $1\br$) of the partial molar volume at infinite dilution of aqueous hydrogen chloride $V\subs{HCl}^{\infty}=17.82\units{cm$^3$ mol$^{-1}$} \tag{9.2.29}$ we can find the so-called “conventional” partial molar volume of Cl$^-$ ion: $V\subs{Cl$^-$}^{\infty}=V\subs{HCl}^{\infty}-V\subs{H$^+$}^{\infty}=17.82\units{cm$^3$ mol$^{-1}$} \tag{9.2.30}$ Going one step further, the measured value $V\subs{NaCl}^{\infty}=16.61\units{cm\(^3$ mol$^{-1}$}\) gives, for Na$^+$ ion, the conventional value $V\subs{Na$^+$}^{\infty}=V\subs{NaCl}^{\infty}-V\subs{Cl$^-$}^{\infty} = (16.61-17.82)\units{cm$^3$ mol$^{-1}$} = -1.21\units{cm$^3$ mol$^{-1}$} \tag{9.2.31}$
9.2.5 Partial specific quantities
A partial specific quantity of a substance is the partial molar quantity divided by the molar mass, and has dimensions of volume divided by mass. For example, the partial specific volume $v\B$ of solute B in a binary solution is given by $v\B=\frac{V\B}{M\B} = \bPd{V}{m(\tx{B})}{T,p,m(\tx{A})} \tag{9.2.32}$ where $m(\tx{A})$ and $m(\tx{B})$ are the masses of solvent and solute.
Although this e-book makes little use of specific quantities and partial specific quantities, in some applications they have an advantage over molar quantities and partial molar quantities because they can be evaluated without knowledge of the molar mass. For instance, the value of a solute’s partial specific volume is used to determine its molar mass by the method of sedimentation equilibrium (Sec. 9.8.2).
The general relations in Sec. 9.2.4 involving partial molar quantities may be turned into relations involving partial specific quantities by replacing amounts by masses, mole fractions by mass fractions, and partial molar quantities by partial specific quantities. Using volume as an example, we can write an additivity relation $V=\sum_i m(i)v_i$, and Gibbs–Duhem relations $\sum_i m(i)\dif v_i=0$ and $\sum_i w_i\dif v_i=0$. For a binary mixture of A and B, we can plot the specific volume $v$ versus the mass fraction $w\B$; then the tangent to the curve at a given composition has intercepts equal to $v\A$ at $w\B{=}0$ and $v\B$ at $w\B{=}1$. A variant of this plot is $\left(v-w\A v\A^*-w\B v\B^*\right)$ versus $w\B$; the intercepts are then equal to $v\A-v\A^*$ and $v\B-v\B^*$.
9.2.6 The chemical potential of a species in a mixture
Just as the molar Gibbs energy of a pure substance is called the chemical potential and given the special symbol $\mu$, the partial molar Gibbs energy $G_i$ of species $i$ in a mixture is called the chemical potential of species $i$, defined by \begin{gather} \s{ \mu_i \defn \Pd{G}{n_i}{T,p,n_{j \ne i}} } \tag{9.2.33} \cond{(mixture)} \end{gather} If there are work coordinates for nonexpansion work, the partial derivative is taken at constant values of these coordinates.
The chemical potential of a species in a phase plays a crucial role in equilibrium problems, because it is a measure of the escaping tendency of the species from the phase. Although we cannot determine the absolute value of $\mu_i$ for a given state of the system, we are usually able to evaluate the difference between the value in this state and the value in a defined reference state.
In an open single-phase system containing a mixture of $s$ different nonreacting species, we may in principle independently vary $T$, $p$, and the amount of each species. This is a total of $2 + s$ independent variables. The total differential of the Gibbs energy of this system is given by Eq. 5.5.9, often called the Gibbs fundamental equation: \begin{gather} \s{ \dif G = -S\dif T + V\difp + \sum_{i=1}^{s} \mu_i \dif n_i } \tag{9.2.34} \cond{(mixture)} \end{gather}
Consider the special case of a mixture containing charged species, for example an aqueous solution of the electrolyte KCl. We could consider the constituents to be either the substances H$_2$O and KCl, or else H$_2$O and the species K$^+$ and Cl$^-$. Any mixture we can prepare in the laboratory must remain electrically neutral, or virtually so. Thus, while we are able to independently vary the amounts of H$_2$O and KCl, we cannot in practice independently vary the amounts of K$^+$ and Cl$^-$ in the mixture. The chemical potential of the K$^+$ ion is defined as the rate at which the Gibbs energy changes with the amount of K$^+$ added at constant $T$ and $p$ while the amount of Cl$^-$ is kept constant. This is a hypothetical process in which the net charge of the mixture increases. The chemical potential of a ion is therefore a valid but purely theoretical concept. Let A stand for H$_2$O, B for KCl, $+$ for K$^+$, and $-$ for Cl$^-$. Then it is theoretically valid to write the total differential of $G$ for the KCl solution either as $\dif G = -S\dif T + V\difp + \mu\A\dif n\A + \mu\B\dif n\B \tag{9.2.35}$ or as $\dif G = -S\dif T + V\difp + \mu\A\dif n\A + \mu_+\dif n_+ + \mu_-\dif n_- \tag{9.2.36}$
9.2.7 Equilibrium conditions in a multiphase, multicomponent system
This section extends the derivation described in Sec. 8.1.2, which was for equilibrium conditions in a multiphase system containing a single substance, to a more general kind of system: one with two or more homogeneous phases containing mixtures of nonreacting species. The derivation assumes there are no internal partitions that could prevent transfer of species and energy between the phases, and that effects of gravity and other external force fields are negligible.
The system consists of a reference phase, $\pha'$, and other phases labeled by $\pha{\ne}\pha'$. Species are labeled by subscript $i$. Following the procedure of Sec. 8.1.1, we write for the total differential of the internal energy $\begin{split} \dif U & = \dif U\aphp + \sum_{\pha\ne\pha'}\dif U\aph \cr & = T\aphp\dif S\aphp - p\aphp\dif V\aphp + \sum_i\mu_i\aphp\dif n_i\aphp \cr & \quad + \sum_{\pha\ne\pha'}\left(T\aph\dif S\aph - p\aph\dif V\aph + \sum_i\mu_i\aph\dif n_i\aph\right) \end{split} \tag{9.2.37}$ The conditions of isolation are $\dif U = 0 \qquad \tx{(constant internal energy)} \tag{9.2.38}$ $\dif V\aphp + \sum_{\pha\ne\pha'}\dif V\aph = 0 \qquad \tx{(no expansion work)} \tag{9.2.39}$ $\begin{split} &\tx{For each species $i$:} \cr &\dif n_i\aphp + \sum_{\pha\ne\pha'}\dif n_i\aph = 0 \qquad \tx{(closed system)} \end{split} \tag{9.2.40}$ We use these relations to substitute for $\dif U$, $\dif V\aphp$, and $\dif n_i\aphp$ in Eq. 9.2.37. After making the further substitution $\dif S\aphp = \dif S - \sum_{\pha\ne\pha'}\dif S\aph$ and solving for $\dif S$, we obtain $\dif S = \sum_{\pha\ne\pha'}\frac{T\aphp-T\aph}{T\aphp}\dif S\aph - \sum_{\pha\ne\pha'}\frac{p\aphp-p\aph}{T\aphp}\dif V\aph + \sum_i\sum_{\pha\ne\pha'}\frac{\mu_i\aphp-\mu_i\aph}{T\aphp}\dif n_i\aph \tag{9.2.41}$ This equation is like Eq. 8.1.6 with provision for more than one species.
In the equilibrium state of the isolated system, $S$ has the maximum possible value, $\dif S$ is equal to zero for an infinitesimal change of any of the independent variables, and the coefficient of each term on the right side of Eq. 9.2.41 is zero. We find that in this state each phase has the same temperature and the same pressure, and for each species the chemical potential is the same in each phase.
Suppose the system contains a species $i'$ that is effectively excluded from a particular phase, $\pha''$. For instance, sucrose molecules dissolved in an aqueous phase are not accommodated in the crystal structure of an ice phase, and a nonpolar substance may be essentially insoluble in an aqueous phase. We can treat this kind of situation by setting $\dif n^{\pha''}_{i'}$ equal to zero. Consequently there is no equilibrium condition involving the chemical potential of this species in phase $\pha''$.
To summarize these conclusions: In an equilibrium state of a multiphase, multicomponent system without internal partitions, the temperature and pressure are uniform throughout the system, and each species has a uniform chemical potential except in phases where it is excluded.
This statement regarding the uniform chemical potential of a species applies to both a substance and an ion, as the following argument explains. The derivation in this section begins with Eq. 9.2.37, an expression for the total differential of $U$. Because it is a total differential, the expression requires the amount $n_i$ of each species $i$ in each phase to be an independent variable. Suppose one of the phases is the aqueous solution of KCl used as an example at the end of the preceding section. In principle (but not in practice), the amounts of the species H$_2$O, K$^+$, and Cl$^-$ can be varied independently, so that it is valid to include these three species in the sums over $i$ in Eq. 9.2.37. The derivation then leads to the conclusion that K$^+$ has the same chemical potential in phases that are in transfer equilibrium with respect to K$^+$, and likewise for Cl$^-$. This kind of situation arises when we consider a Donnan membrane equilibrium (Sec. 12.7.3) in which transfer equilibrium of ions exists between solutions of electrolytes separated by a semipermeable membrane.
9.2.8 Relations involving partial molar quantities
Here we derive several useful relations involving partial molar quantities in a single-phase system that is a mixture. The independent variables are $T$, $p$, and the amount $n_i$ of each constituent species $i$.
From Eqs. 9.2.26 and 9.2.27, the Gibbs–Duhem equation applied to the chemical potentials can be written in the equivalent forms \begin{gather} \s{ \sum_i n_i \dif\mu_i = 0 } \tag{9.2.42} \cond{(constant $T$ and $p$)} \end{gather} and \begin{gather} \s{ \sum_i x_i \dif\mu_i = 0 } \tag{9.2.43} \cond{(constant $T$ and $p$)} \end{gather} These equations show that the chemical potentials of different species cannot be varied independently at constant $T$ and $p$.
A more general version of the Gibbs–Duhem equation, without the restriction of constant $T$ and $p$, is $S\dif T - V\difp + \sum_i n_i \dif\mu_i = 0 \tag{9.2.44}$ This version is derived by comparing the expression for $\dif G$ given by Eq. 9.2.34 with the differential $\dif G {=} \sum_i \mu_i\dif n_i {+} \sum_i n_i\dif\mu_i$ obtained from the additivity rule $G {=} \sum_i \mu_i n_i$.
The Gibbs energy is defined by $G=H-TS$. Taking the partial derivatives of both sides of this equation with respect to $n_i$ at constant $T$, $p$, and $n_{j \ne i}$ gives us $\Pd{G}{n_i}{T,p,n_{j \ne i}} = \Pd{H}{n_i}{T,p,n_{j \ne i}} - T\Pd{S}{n_i}{T,p,n_{j \ne i}} \tag{9.2.45}$ We recognize each partial derivative as a partial molar quantity and rewrite the equation as $\mu_i = H_i -TS_i \tag{9.2.46}$ This is analogous to the relation $\mu=G/n=H\m-TS\m$ for a pure substance.
From the total differential of the Gibbs energy, $\dif G = -S\dif T + V\difp + \sum_i \mu_i\dif n_i$ (Eq. 9.2.34), we obtain the following reciprocity relations: $\Pd{\mu_i}{T}{p,\allni} = -\Pd{S}{n_i}{T,p,n_{j \ne i}} \qquad \Pd{\mu_i}{p}{T,\allni} = \Pd{V}{n_i}{T,p,n_{j \ne i}} \tag{9.2.47}$ The symbol $\allni$ stands for the set of amounts of all species, and subscript $\allni$ on a partial derivative means the amount of each species is constant—that is, the derivative is taken at constant composition of a closed system. Again we recognize partial derivatives as partial molar quantities and rewrite these relations as follows: $\Pd{\mu_i}{T}{p,\allni} = -S_i \tag{9.2.48}$ $\Pd{\mu_i}{p}{T,\allni} = V_i \tag{9.2.49}$ These equations are the equivalent for a mixture of the relations $\pd{\mu}{T}{p} = -S\m$ and $\pd{\mu}{p}{T} = V\m$ for a pure phase (Eqs. 7.8.3 and 7.8.4).
Taking the partial derivatives of both sides of $U=H-pV$ with respect to $n_i$ at constant $T$, $p$, and $n_{j \ne i}$ gives $U_i = H_i - pV_i \tag{9.2.50}$
Finally, we can obtain a formula for $C_{p,i}$, the partial molar heat capacity at constant pressure of species $i$, by writing the total differential of $H$ in the form $\begin{split} \dif H & = \Pd{H}{T}{p,\allni}\dif T + \Pd{H}{p}{T,\allni}\difp + \sum_i \Pd{H}{n_i}{T,p,n_{j \ne i}}\dif n_i \cr & = C_p\dif T + \Pd{H}{p}{T,\allni}\difp + \sum_i H_i\dif n_i \end{split} \tag{9.2.51}$ from which we have the reciprocity relation $\pd{C_p}{n_i}{T,p,n_{j \ne i}} = \pd{H_i}{T}{p,\allni}$, or $C_{p,i} = \Pd{H_i}{T}{p,\allni} \tag{9.2.52}$ | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/09%3A_Mixtures/9.02%3A_Partial_Molar_Quantities.txt |
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$\newcommand{\bpht}{\small\bph} % beta phase tiny superscript$
$\newcommand{\gpht}{\small\gph} % gamma phase tiny superscript$
$\newcommand{\upOmega}{\Omega}$
$\newcommand{\dif}{\mathop{}\!\mathrm{d}} % roman d in math mode, preceded by space$
$\newcommand{\Dif}{\mathop{}\!\mathrm{D}} % roman D in math mode, preceded by space$
$\newcommand{\df}{\dif\hspace{0.05em} f} % df$
$\newcommand{\dBar}{\mathop{}\!\mathrm{d}\hspace-.3em\raise1.05ex{\Rule{.8ex}{.125ex}{0ex}}} % inexact differential$
$\newcommand{\dq}{\dBar q} % heat differential$
$\newcommand{\dw}{\dBar w} % work differential$
$\newcommand{\dQ}{\dBar Q} % infinitesimal charge$
$\newcommand{\dx}{\dif\hspace{0.05em} x} % dx$
$\newcommand{\dt}{\dif\hspace{0.05em} t} % dt$
$\newcommand{\difp}{\dif\hspace{0.05em} p} % dp$
$\newcommand{\Del}{\Delta}$
$\newcommand{\Delsub}[1]{\Delta_{\text{#1}}}$
$\newcommand{\pd}[3]{(\partial #1 / \partial #2 )_{#3}} % \pd{}{}{} - partial derivative, one line$
$\newcommand{\Pd}[3]{\left( \dfrac {\partial #1} {\partial #2}\right)_{#3}} % Pd{}{}{} - Partial derivative, built-up$
$\newcommand{\bpd}[3]{[ \partial #1 / \partial #2 ]_{#3}}$
$\newcommand{\bPd}[3]{\left[ \dfrac {\partial #1} {\partial #2}\right]_{#3}}$
$\newcommand{\dotprod}{\small\bullet}$
$\newcommand{\fug}{f} % fugacity$
$\newcommand{\g}{\gamma} % solute activity coefficient, or gamma in general$
$\newcommand{\G}{\varGamma} % activity coefficient of a reference state (pressure factor)$
$\newcommand{\ecp}{\widetilde{\mu}} % electrochemical or total potential$
$\newcommand{\Eeq}{E\subs{cell, eq}} % equilibrium cell potential$
$\newcommand{\Ej}{E\subs{j}} % liquid junction potential$
$\newcommand{\mue}{\mu\subs{e}} % electron chemical potential$
$\newcommand{\defn}{\,\stackrel{\mathrm{def}}{=}\,} % "equal by definition" symbol$
$\newcommand{\D}{\displaystyle} % for a line in built-up$
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$\newcommand{\R}{8.3145\units{J\,K\per\,mol\per}} % gas constant value$
$\newcommand{\Rsix}{8.31447\units{J\,K\per\,mol\per}} % gas constant value - 6 sig figs$
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The gas mixtures described in this chapter are assumed to be mixtures of nonreacting gaseous substances.
9.3.1 Partial pressure
The partial pressure $p_i$ of substance $i$ in a gas mixture is defined as the product of its mole fraction in the gas phase and the pressure of the phase: \begin{gather} \s{ p_i \defn y_i p } \tag{9.3.1} \cond{(gas mixture)} \end{gather} The sum of the partial pressures of all substances in a gas mixture is $\sum_i p_i = \sum_i y_i p = p\sum_i y_i$. Since the sum of the mole fractions of all substances in a mixture is $1$, this sum becomes \begin{gather} \s{ \sum_i p_i = p } \tag{9.3.2} \cond{(gas mixture)} \end{gather} Thus, the sum of the partial pressures equals the pressure of the gas phase. This statement is known as Dalton’s Law. It is valid for any gas mixture, regardless of whether or not the gas obeys the ideal gas equation.
9.3.2 The ideal gas mixture
As discussed in Sec. 3.5.1, an ideal gas (whether pure or a mixture) is a gas with negligible intermolecular interactions. It obeys the ideal gas equation $p = nRT/V$ (where $n$ in a mixture is the sum $\sum_i n_i$) and its internal energy in a closed system is a function only of temperature. The partial pressure of substance $i$ in an ideal gas mixture is $p_i = y_i p = y_i nRT/V$; but $y_i n$ equals $n_i$, giving \begin{gather} \s{ p_i = \frac{n_i RT}{V} } \tag{9.3.3} \cond{(ideal gas mixture)} \end{gather}
Equation 9.3.3 is the ideal gas equation with the partial pressure of a constituent substance replacing the total pressure, and the amount of the substance replacing the total amount. The equation shows that the partial pressure of a substance in an ideal gas mixture is the pressure the substance by itself, with all others removed from the system, would have at the same $T$ and $V$ as the mixture. Note that this statement is only true for an ideal gas mixture. The partial pressure of a substance in a real gas mixture is in general different from the pressure of the pure substance at the same $T$ and $V$, because the intermolecular interactions are different.
9.3.3 Partial molar quantities in an ideal gas mixture
We need to relate the chemical potential of a constituent of a gas mixture to its partial pressure. We cannot measure the absolute value of a chemical potential, but we can evaluate its value relative to the chemical potential in a particular reference state called the standard state.
The standard state of substance $i$ in a gas mixture is the same as the standard state of the pure gas described in Sec. 7.7: It is the hypothetical state in which pure gaseous $i$ has the same temperature as the mixture, is at the standard pressure $p\st$, and behaves as an ideal gas. The standard chemical potential $\mu_i\st\gas$ of gaseous $i$ is the chemical potential of $i$ in this gas standard state, and is a function of temperature.
By combining Eqs. 9.3.12 and 9.3.16, we obtain \begin{gather} \s{ \mu_i(p')=\mu_i\st\gas +RT\ln\frac{p'_i}{p\st} +\int_0^{p'}\!\!\left( V_i-\frac{RT}{p}\right)\difp } \tag{9.3.19} \cond{(gas mixture,} \nextcond{constant $T$)} \end{gather} which is the analogue for a gas mixture of Eq. 7.9.2 for a pure gas. Section 7.9 describes the procedure needed to obtain formulas for various molar quantities of a pure gas from Eq. 7.9.2. By following a similar procedure with Eq. 9.3.19, we obtain the formulas for differences between partial molar and standard molar quantities of a constituent of a gas mixture shown in the second column of Table 9.1. These formulas are obtained with the help of Eqs. 9.2.46, 9.2.48, 9.2.50, and 9.2.52.
Equation of state
The equation of state of a real gas mixture can be written as the virial equation $pV/n=RT\left[ 1+\frac{B}{(V/n)}+\frac{C}{(V/n)^2}+\cdots \right] \tag{9.3.20}$ This equation is the same as Eq. 2.2.2 for a pure gas, except that the molar volume $V\m$ is replaced by the mean molar volume $V/n$, and the virial coefficients $B, C, \ldots$ depend on composition as well as temperature.
At low to moderate pressures, the simple equation of state $V/n=\frac{RT}{p}+B \tag{9.3.21}$ describes a gas mixture to a sufficiently high degree of accuracy (see Eq. 2.2.8). This is equivalent to a compression factor given by $Z \defn \frac{pV}{nRT} = 1 + \frac{Bp}{RT} \tag{9.3.22}$
From statistical mechanical theory, the dependence of the second virial coefficient $B$ of a binary gas mixture on the mole fraction composition is given by \begin{gather} \s{ B = y\A^2 B\subs{AA} + 2y\A y\B B\subs{AB} + y\B^2 B\subs{BB} } \tag{9.3.23} \cond{(binary gas mixture)} \end{gather} where $B\subs{AA}$ and $B\subs{BB}$ are the second virial coefficients of pure A and B, and $B\subs{AB}$ is a mixed second virial coefficient. $B\subs{AA}$, $B\subs{BB}$, and $B\subs{AB}$ are functions of $T$ only. For a gas mixture with any number of constituents, the composition dependence of $B$ is given by \begin{gather} \s{ B = \sum_i \sum_j y_i y_j B_{ij} } \tag{9.3.24} \cond{(gas mixture, $B_{ij}{=}B_{ji}$)} \end{gather} Here $B_{ij}$ is the second virial of $i$ if $i$ and $j$ are the same, or a mixed second virial coefficient if $i$ and $j$ are different.
If a gas mixture obeys the equation of state of Eq. 9.3.21, the partial molar volume of constituent $i$ is given by $V_i = \frac{RT}{p} + B'_i \tag{9.3.25}$ where the quantity $B'_i$, in order to be consistent with $V_i=\pd{V}{n_i}{T,p,n_{j\ne i}}$, is found to be given by $B'_i = 2\sum_j y_j B_{ij} - B \tag{9.3.26}$ For the constituents of a binary mixture of A and B, Eq. 9.3.26 becomes \begin{gather} \s{ B\A' = B\subs{AA}+(-B\subs{AA}+2B\subs{AB}-B\subs{BB})y\B^2 } \tag{9.3.27} \cond{(binary gas mixture)} \end{gather} \begin{gather} \s{ B\B' = B\subs{BB}+(-B\subs{AA}+2B\subs{AB}-B\subs{BB})y\A^2 } \tag{9.3.28} \cond{(binary gas mixture)} \end{gather}
When we substitute the expression of Eq. 9.3.25 for $V_i$ in Eq. 9.3.18, we obtain a relation between the fugacity coefficient of constituent $i$ and the function $B'_i$: $\ln\phi_i = \frac{B'_i p}{RT} \tag{9.3.29}$
The third column of Table 9.1 gives formulas for various partial molar quantities of constituent $i$ in terms of $B'_i$ and its temperature derivative. The formulas are the same as the approximate formulas in the third column of Table 7.5 for molar quantities of a pure gas, with $B'_i$ replacing the second virial coefficient $B$. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/09%3A_Mixtures/9.03%3A_Gas_Mixtures.txt |
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$\newcommand{\C}{_{\text{C}}} % subscript C$
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$\newcommand{\el}{\subs{el}} % electrical$
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Homogeneous liquid and solid mixtures are condensed phases of variable composition. Most of the discussion of condensed-phase mixtures in this section focuses on liquids. The same principles, however, apply to homogeneous solid mixtures, often called solid solutions. These solid mixtures include most metal alloys, many gemstones, and doped semiconductors.
The relations derived in this section apply to mixtures of nonelectrolytes—substances that do not dissociate into charged species. Solutions of electrolytes behave quite differently in many ways, and will be discussed in the next chapter.
9.4.1 Raoult’s law
In 1888, the French physical chemist François Raoult published his finding that when a dilute liquid solution of a volatile solvent and a nonelectrolyte solute is equilibrated with a gas phase, the partial pressure $p\A$ of the solvent in the gas phase is proportional to the mole fraction $x\A$ of the solvent in the solution: $p\A=x\A p\A^* \tag{9.4.1}$ Here $p\A^*$ is the saturation vapor pressure of the pure solvent (the pressure at which the pure liquid and pure gas phases are in equilibrium).
Consider the solvent, A, of a solution that is dilute enough to be in the ideal-dilute range. In this range, the solvent fugacity obeys Raoult’s law, and the partial molar quantities of the solvent are the same as those in an ideal mixture. Formulas for these quantities were given in Eqs. 9.4.8–9.4.13 and are collected in the first column of Table 9.2. The formulas show that the chemical potential and partial molar entropy of the solvent, at constant $T$ and $p$, vary with the solution composition and, in the limit of infinite dilution ($x\A\ra 1$), approach the values for the pure solvent. The partial molar enthalpy, volume, internal energy, and heat capacity, on the other hand, are independent of composition in the ideal-dilute region and are equal to the corresponding molar quantities for the pure solvent.
Next consider a solute, B, of a binary ideal-dilute solution. The solute obeys Henry’s law, and its chemical potential is given by $\mu\B = \mu\xbB\rf + RT\ln x\B$ (Eq. 9.4.24) where $\mu\xbB\rf$ is a function of $T$ and $p$, but not of composition. $\mu\B$ varies with the composition and goes to $-\infty$ as the solution becomes infinitely dilute ($x\A\ra 1$ and $x\B\ra 0$).
For the partial molar entropy of the solute, we use $S\B=-\pd{\mu\B}{T}{p,\allni}$ (Eq. 9.2.48) and obtain $S\B = -\Pd{\mu\xbB\rf}{T}{\!p} - R \ln x\B \tag{9.4.36}$ The term $-\pd{\mu\xbB\rf}{T}{p}$ represents the partial molar entropy $S\xbB\rf$ of B in the fictitious reference state of unit solute mole fraction. Thus, we can write Eq. 9.4.36 in the form \begin{gather} \s{ S\B = S\xbB\rf - R \ln x\B } \tag{9.4.37} \cond{(ideal-dilute solution} \nextcond{of a nonelectrolyte)} \end{gather} This equation shows that the partial molar entropy varies with composition and goes to $+\infty$ in the limit of infinite dilution. From the expressions of Eqs. 9.4.27 and 9.4.28, we can derive similar expressions for $S\B$ in terms of the solute reference states on a concentration or molality basis.
The relation $H\B = \mu\B + TS\B$ (from Eq. 9.2.46), combined with Eqs. 9.4.24 and 9.4.37, yields $H\B = \mu\xbB\rf + TS\xbB\rf = H\xbB\rf \tag{9.4.38}$ showing that at constant $T$ and $p$, the partial molar enthalpy of the solute is constant throughout the ideal-dilute solution range. Therefore, we can write \begin{gather} \s{ H\B = H\B^{\infty} } \tag{9.4.39} \cond{(ideal-dilute solution} \nextcond{of a nonelectrolyte)} \end{gather} where $H\B^{\infty}$ is the partial molar enthalpy at infinite dilution. By similar reasoning, using Eqs. 9.2.49–9.2.52, we find that the partial molar volume, internal energy, and heat capacity of the solute are constant in the ideal-dilute range and equal to the values at infinite dilution. The expressions are listed in the second column of Table 9.2.
When the pressure is equal to the standard pressure $p\st$, the quantities $H\B^{\infty}$, $V\B^{\infty}$, $U\B^{\infty}$, and $C_{p,\tx{B}}^{\infty}$ are the same as the standard values $H\B\st$, $V\B\st$, $U\B\st$, and $C_{p,\tx{B}}\st$. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/09%3A_Mixtures/9.04%3A_Liquid_and_Solid_Mixtures_of_Nonelectrolytes.txt |
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An activity coefficient of a species is a kind of adjustment factor that relates the actual behavior to ideal behavior at the same temperature and pressure. The ideal behavior is based on a reference state for the species.
We begin by describing reference states for nonelectrolytes. The thermodynamic behavior of an electrolyte solution is more complicated than that of a mixture of nonelectrolytes, and will be discussed in the next chapter.
9.5.1 Reference states and standard states
A reference state of a constituent of a mixture has the same temperature and pressure as the mixture. When species $i$ is in its reference state, its chemical potential $\mu_i\rf$ depends only on the temperature and pressure of the mixture.
If the pressure is the standard pressure $p\st$, the reference state of species $i$ becomes its standard state. In the standard state, the chemical potential is the standard chemical potential $\mu_i\st$, which is a function only of temperature.
Reference states are useful for derivations involving processes taking place at constant $T$ and $p$ when the pressure is not necessarily the standard pressure.
Table 9.3 describes the reference states of nonelectrolytes used in this e-book, and lists symbols for chemical potentials of substances in these states. The symbols for solutes include $x$, $c$, or $m$ in the subscript to indicate the basis of the reference state.
9.5.2 Ideal mixtures
Since the activity coefficient of a species relates its actual behavior to its ideal behavior at the same $T$ and $p$, let us begin by examining behavior in ideal mixtures.
Consider first an ideal gas mixture at pressure $p$. The chemical potential of substance $i$ in this ideal gas mixture is given by Eq. 9.3.5 (the superscript “id” stands for ideal): $\mu_i\id\gas = \mu_i\st\gas + RT\ln\frac{p_i}{p\st} \tag{9.5.1}$ The reference state of gaseous substance $i$ is pure $i$ acting as an ideal gas at pressure $p$. Its chemical potential is given by $\mu_i\rf\gas = \mu_i\st\gas + RT\ln\frac{p}{p\st} \tag{9.5.2}$ Subtracting Eq. 9.5.2 from Eq. 9.5.1, we obtain $\mu_i\id\gas - \mu_i\rf\gas = RT\ln\frac{p_i}{p} \tag{9.5.3}$
Consider the following expressions for chemical potentials in ideal mixtures and ideal-dilute solutions of nonelectrolytes. The first equation is a rearrangement of Eq. 9.5.3, and the others are from earlier sections of this chapter (in order of occurrence, Eqs. 9.4.8, 9.4.35, 9.4.24, 9.4.27, and 9.4.28). $\tx{Constituent of an ideal gas mixture} \quad \mu_i\id\gas = \s{\mu_i\rf\gas + RT\ln\frac{p_i}{p}} \tag{9.5.4}$ $\tx{Constituent of an ideal liquid or solid mixture} \quad \mu_i\id = \mu_i^* + RT\ln x_i \tag{9.5.5}$ $\tx{Solvent of an ideal-dilute solution} \quad \mu\A\id = \mu\A^* + RT\ln x\A \tag{9.5.6}$ $\tx{Solute, ideal-dilute solution, mole fraction basis} \quad \mu\B\id = \mu\xbB\rf + RT\ln x\B \tag{9.5.7}$ $\tx{Solute, ideal-dilute solution, concentration basis} \quad \mu\B\id = \mu\cbB\rf + RT\ln\frac{c\B}{c\st} \tag{9.5.8}$ $\tx{Solute, ideal-dilute solution, molality basis} \quad \mu\B\id = \mu\mbB\rf + RT\ln\frac{m\B}{m\st} \tag{9.5.9}$
Note that the equations for the condensed phases have the general form $\mu_i\id = \mu_i\rf + RT\ln \left( \frac{\tx{composition variable}} {\tx{standard composition}} \right) \tag{9.5.10}$ where $\mu_i\rf$ is the chemical potential of component $i$ in an appropriate reference state. (The standard composition on a mole fraction basis is $x\st{=}1$.)
9.5.3 Real mixtures
If a mixture is not ideal, we can write an expression for the chemical potential of each component that includes an activity coefficient. The expression is like one of those for the ideal case (Eqs. 9.5.4–9.5.9) with the activity coefficient multiplying the quantity within the logarithm.
Consider constituent $i$ of a gas mixture. If we eliminate $\mu_i\st\gas$ from Eqs. 9.3.12 and 9.5.2, we obtain $\begin{split} \mu_i & = \mu_i\rf\gas + RT\ln\frac{\fug_i}{p} \cr & = \mu_i\rf\gas + RT\ln\frac{\phi_i p_i}{p} \end{split} \tag{9.5.11}$ where $\fug_i$ is the fugacity of constituent $i$ and $\phi_i$ is its fugacity coefficient. Here the activity coefficient is the fugacity coefficient $\phi_i$.
For components of a condensed-phase mixture, we write expressions for the chemical potential having a form similar to that in Eq. 9.5.10, with the composition variable now multiplied by an activity coefficient: $\mu_i = \mu_i\rf + RT\ln \left[ (\tx{activity coefficient of $i$}) \times \left( \frac{\tx{composition variable}} {\tx{standard composition}} \right) \right] \tag{9.5.12}$
The activity coefficient of a species is a dimensionless quantity whose value depends on the temperature, the pressure, the mixture composition, and the choice of the reference state for the species. Under conditions in which the mixture behaves ideally, the activity coefficient is unity and the chemical potential is given by one of the expressions of Eqs. 9.5.4–9.5.9; otherwise, the activity coefficient has the value that gives the actual chemical potential.
This e-book will use various symbols for activity coefficients, as indicated in the following list of expressions for the chemical potentials of nonelectrolytes: $\tx{Constituent of a gas mixture} \quad \mu_i = \mu_i\rf\gas + RT\ln\left(\phi_i\frac{p_i}{p}\right) \tag{9.5.13}$ $\tx{Constituent of a liquid or solid mixture} \quad \mu_i = \mu_i^* + RT\ln\left(\g_i x_i\right) \tag{9.5.14}$ $\tx{Solvent of a solution} \quad \mu\A = \mu\A^* + RT\ln\left(\g\A x\A\right) \tag{9.5.15}$ $\tx{Solute of a solution, mole fraction basis} \quad \mu\B = \mu\xbB\rf + RT\ln\left(\g\xbB x\B\right) \tag{9.5.16}$ $\tx{Solute of a solution, concentration basis} \quad \mu\B = \mu\cbB\rf + RT\ln\left(\g\cbB\frac{c\B}{c\st}\right) \tag{9.5.17}$ $\tx{Solute of a solution, molality basis} \quad \mu\B = \mu\mbB\rf + RT\ln\left(\g\mbB\frac{m\B}{m\st}\right) \tag{9.5.18}$
Equation 9.5.14 refers to a component of a liquid or solid mixture of substances that mix in all proportions. Equation 9.5.15 refers to the solvent of a solution. The reference states of these components are the pure liquid or solid at the temperature and pressure of the mixture. For the activity coefficients of these components, this e-book uses the symbols $\g_i$ and $\g\A$.
The IUPAC Green Book (E. Richard Cohen et al, Quantities, Units and Symbols in Physical Chemistry, 3rd edition, RSC Publishing, Cambridge, 2007, p. 59) recommends the symbol $f_i$ for the activity coefficient of component $i$ when the reference state is the pure liquid or solid. This e-book instead uses symbols such as $\g_i$ and $\g\A$, in order to avoid confusion with the symbol usually used for fugacity, $\fug_i$.
In Eqs. 9.5.16–9.5.18, the symbols $\g\xbB$, $\g\cbB$, and $\g\mbB$ for activity coefficients of a nonelectrolyte solute include $x$, $c$, or $m$ in the subscript to indicate the choice of the solute reference state. Although three different expressions for $\mu\B$ are shown, for a given solution composition they must all represent the same value of $\mu\B$, equal to the rate at which the Gibbs energy increases with the amount of substance B added to the solution at constant $T$ and $p$. The value of a solute activity coefficient, on the other hand, depends on the choice of the solute reference state.
You may find it helpful to interpret products appearing on the right sides of Eqs. 9.5.13–9.5.18 as follows.
• In other words, the value of one of these products is the value of a partial pressure or composition variable that would give the same chemical potential in an ideal mixture as the actual chemical potential in the real mixture. These effective composition variables are an alternative way to express the escaping tendency of a substance from a phase; they are related exponentially to the chemical potential, which is also a measure of escaping tendency.
A change in pressure or composition that causes a mixture to approach the behavior of an ideal mixture or ideal-dilute solution must cause the activity coefficient of each mixture constituent to approach unity:
9.5.4 Nonideal dilute solutions
How would we expect the activity coefficient of a nonelectrolyte solute to behave in a dilute solution as the solute mole fraction increases beyond the range of ideal-dilute solution behavior?
The following argument is based on molecular properties at constant $T$ and $p$.
We focus our attention on a single solute molecule. This molecule has interactions with nearby solute molecules. Each interaction depends on the intermolecular distance and causes a change in the internal energy compared to the interaction of the solute molecule with solvent at the same distance.
In Sec. 11.1.5, it will be shown that roughly speaking the internal energy change is negative if the average of the attractive forces between two solute molecules and two solvent molecules is greater than the attractive force between a solute molecule and a solvent molecule at the same distance, and is positive for the opposite situation.
The number of solute molecules in a volume element at a given distance from the solute molecule we are focusing on is proportional to the local solute concentration. If the solution is dilute and the interactions weak, we expect the local solute concentration to be proportional to the macroscopic solute mole fraction. Thus, the partial molar quantities $U\B$ and $V\B$ of the solute should be approximately linear functions of $x\B$ in a dilute solution at constant $T$ and $p$.
From Eqs. 9.2.46 and 9.2.50, the solute chemical potential is given by $\mu\B=U\B+pV\B-TS\B$. In the dilute solution, we assume $U\B$ and $V\B$ are linear functions of $x\B$ as explained above. We also assume the dependence of $S\B$ on $x\B$ is approximately the same as in an ideal mixture; this is a prediction from statistical mechanics for a mixture in which all molecules have similar sizes and shapes. Thus we expect the deviation of the chemical potential from ideal-dilute behavior, $\mu\B = \mu\xbB\rf + RT\ln x\B$, can be described by adding a term proportional to $x\B$: $\mu\B = \mu\xbB\rf + RT\ln x\B + k_x x\B$, where $k_x$ is a positive or negative constant related to solute-solute interactions.
If we equate this expression for $\mu\B$ with the one that defines the activity coefficient, $\mu\B = \mu\xbB\rf + RT\ln(\g\xbB x\B)$ (Eq. 9.5.16), and solve for the activity coefficient, we obtain the relation $\g\xbB = \exp (k_x x\B/RT)$. (This is essentially the result of the McMillan–Mayer solution theory from statistical mechanics.) An expansion of the exponential in powers of $x\B$ converts this to $\g\xbB = 1 + (k_x/RT)x\B + \cdots \tag{9.5.25}$ Thus we predict that at constant $T$ and $p$, $\g\xbB$ is a linear function of $x\B$ at low $x\B$. An ideal-dilute solution, then, is one in which $x\B$ is much smaller than $RT/k_x$ so that $\g\xbB$ is approximately 1. An ideal mixture requires the interaction constant $k_x$ to be zero.
By similar reasoning, we reach analogous conclusions for solute activity coefficients on a concentration or molality basis. For instance, at low $m\B$ the chemical potential of B should be approximately $\mu\mbB\rf + RT\ln (m\B/m\st) + k_m m\B$, where $k_m$ is a constant at a given $T$ and $p$; then the activity coefficient at low $m\B$ is given by $\g\mbB = \exp (k_m m\B/RT) = 1 + (k_m/RT)m\B + \cdots \tag{9.5.26}$
The prediction from the theoretical argument above, that a solute activity coefficient in a dilute solution is a linear function of the composition variable, is borne out experimentally as illustrated in Fig. 9.10. This prediction applies only to a nonelectrolyte solute; for an electrolyte, the slope of activity coefficient versus molality approaches $-\infty$ at low molality. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/09%3A_Mixtures/9.05%3A_Activity_Coefficients_in_Mixtures_of_Nonelectrolytes.txt |
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This section describes several methods by which activity coefficients of nonelectrolyte substances may be evaluated. Section 9.6.3 describes an osmotic coefficient method that is also suitable for electrolyte solutes, as will be explained in Sec. 10.6.
9.6.1 Activity coefficients from gas fugacities
Suppose we equilibrate a liquid mixture with a gas phase. If component $i$ of the liquid mixture is a volatile nonelectrolyte, and we are able to evaluate its fugacity $\fug_i$ in the gas phase, we have a convenient way to evaluate the activity coefficient $\g_i$ in the liquid. The relation between $\g_i$ and $\fug_i$ will now be derived.
When component $i$ is in transfer equilibrium between two phases, its chemical potential is the same in both phases. Equating expressions for $\mu_i$ in the liquid mixture and the equilibrated gas phase (from Eqs. 9.5.14 and 9.5.11, respectively), and then solving for $\g_i$, we have $\mu_i^* + RT\ln\left(\g_i x_i\right) = \mu_i\rf\gas + RT\ln (\fug_i/p) \tag{9.6.1}$ $\g_i = \exp \left[ \frac{\mu_i\rf\gas -\mu_i^*}{RT} \right] \times \frac{\fug_i}{x_i p} \tag{9.6.2}$ On the right side of Eq. 9.6.2, only $\fug_i$ and $x_i$ depend on the liquid composition. We can therefore write $\g_i = C_i\frac{\fug_i}{x_i} \tag{9.6.3}$ where $C_i$ is a factor whose value depends on $T$ and $p$, but not on the liquid composition. Solving Eq. 9.6.3 for $C_i$ gives $C_i=\g_i x_i/\fug_i$.
Now consider Eq. 9.5.20. It says that as $x_i$ approaches 1 at constant $T$ and $p$, $\g_i$ also approaches 1. We can use this limit to evaluate $C_i$: $C_i = \lim_{x_i \ra 1}\frac{\g_i x_i}{\fug_i} = \frac{1}{\fug_i^*} \tag{9.6.4}$ Here $\fug_i^*$ is the fugacity of $i$ in a gas phase equilibrated with pure liquid $i$ at the temperature and pressure of the mixture. Then substitution of this value of $C_i$ (which is independent of $x_i$) in Eq. 9.6.3 gives us an expression for $\g_i$ at any liquid composition: $\g_i=\frac{\fug_i}{x_i\fug_i^*} \tag{9.6.5}$
We can follow the same procedure for a solvent or solute of a liquid solution. We replace the left side of Eq. 9.6.1 with an expression from among Eqs. 9.5.15–9.5.18, then derive an expression analogous to Eq. 9.6.3 for the activity coefficient with a composition-independent factor, and finally apply the limiting conditions that cause the activity coefficient to approach unity (Eqs. 9.5.21–9.5.24) and allow us to evaluate the factor. When we take the limits that cause the solute activity coefficients to approach unity, the ratios $\fug\B/x\B$, $\fug\B/c\B$, and $\fug\B/m\B$ become Henry’s law constants (Eqs. 9.4.19–9.4.21). The resulting expressions for activity coefficients as functions of fugacity are listed in Table 9.4.
Examples
Figure 9.11(a) shows the function $(\phi_m - 1)/m\B$ for aqueous sucrose solutions over a wide range of molality. The dependence of the solute activity coefficient on molality, generated from Eq. 9.6.20, is shown in Fig. 9.11(b). Figure 9.11(c) is a plot of the effective sucrose molality $\g\mbB m\B$ as a function of composition. Note how the activity coefficient becomes greater than unity beyond the ideal-dilute region, and how in consequence the effective molality $\g\mbB m\B$ becomes considerably greater than the actual molality $m\B$.
9.6.4 Fugacity measurements
Section 9.6.1 described the evaluation of the activity coefficient of a constituent of a liquid mixture from its fugacity in a gas phase equilibrated with the mixture. Section 9.6.3 mentioned the use of solvent fugacities in gas phases equilibrated with pure solvent and with a solution, in order to evaluate the osmotic coefficient of the solution.
Various experimental methods are available for measuring a partial pressure in a gas phase equilibrated with a liquid mixture. A correction for gas nonideality, such as that given by Eq. 9.3.16, can be used to convert the partial pressure to fugacity.
If the solute of a solution is nonvolatile, we may pump out the air above the solution and use a manometer to measure the pressure, which is the partial pressure of the solvent. Dynamic methods involve passing a stream of inert gas through a liquid mixture and analyzing the gas mixture to evaluate the partial pressures of volatile components. For instance, we could pass dry air successively through an aqueous solution and a desiccant and measure the weight gained by the desiccant.
The isopiestic vapor pressure technique is one of the most useful methods for determining the fugacity of H$_2$O in a gas phase equilibrated with an aqueous solution. This is a comparative method using a binary solution of the solute of interest, B, and a nonvolatile reference solute of known properties. Some commonly used reference solutes for which data are available are sucrose, NaCl, and CaCl$_2$.
In this method, solute B can be either a nonelectrolyte or electrolyte. Dishes, each containing water and an accurately weighed sample of one of the solutes, are placed in wells drilled in a block made of metal for good thermal equilibration. The assembly is placed in a gas-tight chamber, the air is evacuated, and the apparatus is gently rocked in a thermostat for a period of up to several days, or even weeks. During this period, H$_2$O is transferred among the dishes through the vapor space until the chemical potential of the water becomes the same in each solution. The solutions are then said to be isopiestic. Finally, the dishes are removed from the apparatus and weighed to establish the molality of each solution. The H$_2$O fugacity is known as a function of the molality of the reference solute, and is the same as the H$_2$O fugacity in equilibrium with the solution of solute B at its measured molality.
The isopiestic vapor pressure method can also be used for nonaqueous solutions. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/09%3A_Mixtures/9.06%3A_Evaluation_of_Activity_Coefficients.txt |
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The activity $a_i$ of uncharged species $i$ (i.e., a substance) is defined by the relation \begin{gather} \s{ a_i \defn \exp\left(\frac{\mu_i-\mu_i\st}{RT}\right) } \tag{9.7.1} \cond{(uncharged species)} \end{gather} or \begin{gather} \s{ \mu_i = \mu_i\st + RT\ln a_i } \tag{9.7.2} \cond{(uncharged species)} \end{gather} where $\mu_i\st$ is the standard chemical potential of the species. The activity of a species in a given phase is a dimensionless quantity whose value depends on the choice of the standard state and on the intensive properties of the phase: temperature, pressure, and composition.
Some chemists define the activity by $\mu_i = \mu_i\rf + RT\ln a_i$. The activity defined this way is not the same as the activity used in this e-book unless the phase is at the standard pressure.
The quantity $a_i$ is sometimes called the relative activity of $i$, because it depends on the chemical potential relative to a standard chemical potential. An important application of the activity concept is the definition of equilibrium constants (Sec. 11.8.1).
For convenience in later applications, we specify that the value of $a_i$ is the same in phases that have the same temperature, pressure, and composition but are at different elevations in a gravitational field, or are at different electric potentials. Section 9.8 10.1 will describe a modification of the defining equation $\mu_i = \mu_i\st + RT\ln a_i$ for a system with phases of different elevations, and Sec. 10.1 will describe the modification needed for a charged species.
9.7.1 Standard states
The standard states of different kinds of mixture components have the same definitions as those for reference states (Table 9.3), with the additional stipulation in each case that the pressure is equal to the standard pressure $p\st$.
When component $i$ is in its standard state, its chemical potential is the standard chemical potential $\mu\st_i$. It is important to note from Eq. 9.7.2 that when $\mu_i$ equals $\mu_i\st$, the logarithm of $a_i$ is zero and the activity in the standard state is therefore unity.
The following equations in the form of Eq. 9.7.2 show the notation used in this e-book for the standard chemical potentials and activities of various kinds of uncharged mixture components: $\tx{Substance $i$ in a gas mixture} \qquad \mu_i = \mu_i\st\gas + RT\ln a_i\gas \tag{9.7.3}$ $\tx{Substance $i$ in a liquid or solid mixture} \qquad \mu_i = \mu_i\st + RT\ln a_i \tag{9.7.4}$ $\tx{Solvent A of a solution} \qquad \mu\A = \mu\A\st + RT\ln a\A \tag{9.7.5}$ $\tx{Solute B, mole fraction basis} \qquad \mu\B = \mu\xbB\st + RT\ln a\xbB \tag{9.7.6}$ $\tx{Solute B, concentration basis} \qquad \mu\B = \mu\cbB\st + RT\ln a\cbB \tag{9.7.7}$ $\tx{Solute B, molality basis} \qquad \mu\B = \mu\mbB\st + RT\ln a\mbB \tag{9.7.8}$
9.7.2 Activities and composition
We need to be able to relate the activity of component $i$ to the mixture composition. We can do this by finding the relation between the chemical potential of component $i$ in its reference state and in its standard state, both at the same temperature. These two chemical potentials, $\mu_i\rf$ and $\mu_i\st$, are equal only if the mixture is at the standard pressure $p\st$.
It will be useful to define the following dimensionless quantity: $\G_i \defn \exp\left(\frac{\mu_i\rf-\mu_i\st}{RT}\right) \tag{9.7.9}$ The symbol $\G_i$ for this quantity was introduced by Pitzer and Brewer (Thermodynamics, 2nd edition, McGraw-Hill, New York, 1961, p. 249). They called it the activity in a reference state. To see why, compare the definition of activity given by $\mu_i = \mu_i\st + RT\ln a_i$ with a rearrangement of Eq. 9.7.9: $\mu_i\rf = \mu_i\st + RT\ln\G_i$.
At a given temperature, the difference $\mu_i\rf-\mu_i\st$ depends only on the pressure $p$ of the mixture, and is zero when $p$ is equal to $p\st$. Thus $\G_i$ is a function of $p$ with a value of 1 when $p$ is equal to $p\st$. This e-book will call $\G_i$ the pressure factor of species $i$.
To understand how activity is related to composition, let us take as an example the activity $a\mbB$ of solute B based on molality. From Eqs. 9.5.18 and 9.7.8, we have $\begin{split} \mu\B & = \mu\mbB\rf + RT\ln\left(\g\mbB\frac{m\B}{m\st}\right) \cr & = \mu\mbB\st + RT\ln a\mbB \end{split} \tag{9.7.10}$ The activity is then given by $\begin{split} \ln a\mbB & = \frac{\mu\mbB\rf-\mu\mbB\st}{RT} + \ln\left(\g\mbB\frac{m\B}{m\st}\right)\cr & = \ln\G\mbB + \ln\left(\g\mbB\frac{m\B}{m\st}\right) \end{split} \tag{9.7.11}$ $a\mbB = \G\mbB \g\mbB\frac{m\B}{m\st}\hspace{2.28cm} \tag{9.7.12}$ The activity of a constituent of a condensed-phase mixture is in general equal to the product of the pressure factor, the activity coefficient, and the composition variable divided by the standard composition.
We are now able to write explicit formulas for $\G_i$ for each kind of mixture component. They are collected in Table 9.6.
Considering a constituent of a condensed-phase mixture, by how much is the pressure factor likely to differ from unity? If we use the values $p\st=1\br$ and $T=300\K$, and assume the molar volume of pure $i$ is $V_i^*=100\units{cm\(^3$ mol$^{-1}$}\) at all pressures, we find that $\G_i$ is $0.996$ in the limit of zero pressure, unity at $1\br$, $1.004$ at $2\br$, $1.04$ at $10\br$, and $1.49$ at $100\br$. For a solution with $V\B^{\infty}=100\units{cm\(^3$ mol$^{-1}$}\), we obtain the same values as these for $\G\xbB$, $\G\mbB$, and $\G\cbB$. These values demonstrate that it is only at high pressures that the pressure factor differs appreciably from unity. For this reason, it is common to see expressions for activity in which this factor is omitted: $a_i=\g_i x_i$, $a\mbB=\g\mbB m\B/m\st$, and so on.
In principle, we can specify any convenient value for the standard pressure $p\st$. For a chemist making measurements at high pressures, it would be convenient to specify a value of $p\st$ within the range of the experimental pressures, for example $p\st=1\units{kbar}$, in order that the value of each pressure factor be close to unity. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/09%3A_Mixtures/9.07%3A_Activity_of_an_Uncharged_Species.txt |
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A tall column of a gas mixture in a gravitational field, and a liquid solution in the cell of a spinning centrifuge rotor, are systems with equilibrium states that are nonuniform in pressure and composition. This section derives the ways in which pressure and composition vary spatially within these kinds of systems at equilibrium.
9.8.1 Gas mixture in a gravitational field
Consider a tall column of a gas mixture in an earth-fixed lab frame. Our treatment will parallel that for a tall column of a pure gas in Sec. 8.1.4. We imagine the gas to be divided into many thin slab-shaped phases at different elevations in a rigid container, as in Fig. 8.1. We want to find the equilibrium conditions reached spontaneously when the system is isolated from its surroundings.
The derivation is the same as that in Sec. 9.2.7, with the additional constraint that for each phase $\pha$, $\dif V\aph$ is zero in order that each phase stays at a constant elevation. The result is the relation $\dif S = \sum_{\pha\ne\pha'}\frac{T\aphp-T\aph}{T\aphp}\dif S\aph + \sum_i\sum_{\pha\ne\pha'}\frac{\mu_i\aphp-\mu_i\aph}{T\aphp}\dif n_i\aph \tag{9.8.1}$ In an equilibrium state, $S$ is at a maximum and $\dif S$ is zero for an infinitesimal change of any of the independent variables. This requires the coefficient of each term in the sums on the right side of Eq. 9.8.1 to be zero. The equation therefore tells that at equilibrium the temperature and the chemical potential of each constituent are uniform throughout the gas mixture. The equation says nothing about the pressure.
Just as the chemical potential of a pure substance at a given elevation is defined in this e-book as the molar Gibbs energy at that elevation (Sec. 8.1.4), the chemical potential of substance $i$ in a mixture at elevation $h$ is the partial molar Gibbs energy at that elevation.
We define the standard potential $\mu_i\st\gas$ of component $i$ of the gas mixture as the chemical potential of $i$ under standard state conditions at the reference elevation $h{=}0$. At this elevation, the chemical potential and fugacity are related by $\mu_i(0) = \mu_i\st\gas + RT\ln\frac{\fug_i(0)}{p\st} \tag{9.8.2}$
If we reversibly raise a small sample of mass $m$ of the gas mixture by an infinitesimal distance $\dif h$, without heat and at constant $T$ and $V$, the fugacity $\fug_i$ remains constant. The gravitational work during the elevation process is $\dw'=mg\dif h$. This work contributes to the internal energy change: $\dif U=T\dif S-p\dif V+\sum_i\mu_i\dif n_i+mg\dif h$. The total differential of the Gibbs energy of the sample is $\begin{split} \dif G & = \dif(U-TS+pV) \cr & = -S\dif T + V\difp + \sum_i\mu_i\dif n_i + mg\dif h \end{split} \tag{9.8.3}$ From this total differential, we write the reciprocity relation $\Pd{\mu_i}{h}{T,p,\allni} = \Pd{mg}{n_i}{T,p,n_{j\ne i},h} \tag{9.8.4}$ With the substitution $m=\sum_i n_i M_i$ in the partial derivative on the right side, the partial derivative becomes $M_i g$. At constant $T$, $p$, and composition, therefore, we have $\dif\mu_i=M_i g\dif h$. Integrating over a finite elevation change from $h=0$ to $h=h'$, we obtain \begin{gather} \s{ \mu_i(h') - \mu_i(0) = \int_0^{h'}\!\!\!M_i g\dif h = M_i gh' } \tag{9.8.5} \cond{($\fug_i(h'){=}\fug_i(0)$ )} \end{gather}
The general relation between $\mu_i$, $\fug_i$, and $h$ that agrees with Eqs. 9.8.2 and 9.8.5 is $\mu_i(h) = \mu_i\st\gas + RT\ln\frac{\fug_i(h)}{p\st}+M_i gh \tag{9.8.6}$ In the equilibrium state of the tall column of gas, $\mu_i(h)$ is equal to $\mu_i(0)$. Equation 9.8.6 shows that this is only possible if $\fug_i$ decreases as $h$ increases. Equating the expressions given by this equation for $\mu_i(h)$ and $\mu_i(0)$, we have $\mu_i\st\gas + RT\ln\frac{\fug_i(h)}{p\st} + M_i gh = \mu_i\st\gas + RT\ln\frac{\fug_i(0)}{p\st} \tag{9.8.7}$ Solving for $\fug_i(h)$ gives \begin{gather} \s{ \fug_i(h) = \fug_i(0)e^{-M_i gh/RT}} \tag{9.8.8} \cond{(gas mixture at equilibrium)} \end{gather}
If the gas is an ideal gas mixture, $\fug_i$ is the same as the partial pressure $p_i$: \begin{gather} \s{ p_i(h) = p_i(0)e^{-M_i gh/RT}} \tag{9.8.9} \cond{(ideal gas mixture at equilibrium)} \end{gather} Equation 9.8.9 shows that each constituent of an ideal gas mixture individually obeys the barometric formula given by Eq. 8.1.13.
The pressure at elevation $h$ is found from $p(h)=\sum_i p_i(h)$. If the constituents have different molar masses, the mole fraction composition changes with elevation. For example, in a binary ideal gas mixture the mole fraction of the constituent with the greater molar mass decreases with increasing elevation, and the mole fraction of the other constituent increases.
9.8.2 Liquid solution in a centrifuge cell
This section derives equilibrium conditions of a dilute binary solution confined to a cell embedded in a spinning centrifuge rotor.
The system is the solution. The rotor’s angle of rotation with respect to a lab frame is not relevant to the state of the system, so we use a local reference frame fixed in the rotor as shown in Fig. 9.12(a). The values of heat, work, and energy changes measured in this rotating frame are different from those in a lab frame (Sec. G.9 in Appendix G). Nevertheless, the laws of thermodynamics and the relations derived from them are obeyed in the local frame when we measure the heat, work, and state functions in this frame (Sec. G.6).
Note that an equilibrium state can only exist relative to the rotating local frame; an observer fixed in this frame would see no change in the state of the isolated solution over time. While the rotor rotates, however, there is no equilibrium state relative to the lab frame, because the system’s position in the frame constantly changes.
We assume the centrifuge rotor rotates about the vertical $z$ axis at a constant angular velocity $\omega$. As shown in Fig. 9.12(a), the elevation of a point within the local frame is given by $z$ and the radial distance from the axis of rotation is given by $r$.
In the rotating local frame, a body of mass $m$ has exerted on it a centrifugal force $F\sups{centr}=m\omega^2 r$ directed horizontally in the outward $+r$ radial direction (Sec. G.9). The gravitational force in this frame, directed in the downward $-z$ direction, is the same as the gravitational force in a lab frame. Because the height of a typical centrifuge cell is usually no greater than about one centimeter, in an equilibrium state the variation of pressure and composition between the top and bottom of the cell at any given distance from the axis of rotation is completely negligible—all the measurable variation is along the radial direction.
There is also a Coriolis force that vanishes as the body’s velocity in the rotating local frame approaches zero. The centrifugal and Coriolis forces are apparent or fictitious forces, in the sense that they are caused by the acceleration of the rotating frame rather than by interactions between particles. When we treat these forces as if they are real forces, we can use Newton’s second law of motion to relate the net force on a body and the body’s acceleration in the rotating frame (see Sec. G.6).
To find conditions for equilibrium, we imagine the solution to be divided into many thin curved volume elements at different distances from the axis of rotation as depicted in Fig. 9.12(b). We treat each volume element as a uniform phase held at constant volume so that it is at a constant distance from the axis of rotation. The derivation is the same as the one used in the preceding section to obtain Eq. 9.8.1, and leads to the same conclusion: in an equilibrium state the temperature and the chemical potential of each substance (solvent and solute) are uniform throughout the solution.
We find the dependence of pressure on $r$ as follows. Consider one of the thin slab-shaped volume elements of Fig. 9.12(b). The volume element is located at radial position $r$ and its faces are perpendicular to the direction of increasing $r$. The thickness of the volume element is $\delta r$, the surface area of each face is $\As$, and the mass of the solution in the volume element is $m=\rho \As\delta r$. Expressed as components in the direction of increasing $r$ of the forces exerted on the volume element, the force at the inner face is $p\As$, the force at the outer face is $-(p+\delta p)\As$, and the centrifugal force is $m\omega^2 r = \rho \As\omega^2 r\delta r$. From Newton’s second law, the sum of these components is zero at equilibrium: $p \As-(p+\delta p)\As+\rho \As\omega^2 r\delta r = 0 \tag{9.8.10}$ or $\delta p = \rho\omega^2 r\delta r$. In the limit as $\delta r$ and $\delta p$ are made infinitesimal, this becomes $\difp = \rho\omega^2 r\dif r \tag{9.8.11}$ We will assume the density $\rho$ is uniform throughout the solution. (In the centrifugal field, this assumption is strictly true only if the solution is incompressible and its density is independent of composition.) Then integration of Eq. 9.8.11 yields $p''-p' = \int_{p'}^{p''}\!\difp = \rho\omega^2\!\int_{r'}^{r''}\!\!\!r\dif r = \frac{\rho \omega^2}{2}\left[\left(r''\right)^2-\left(r'\right)^2\right] \tag{9.8.12}$ where the superscripts $'$ and $''$ denote positions at two different values of $r$ in the cell. The pressure is seen to increase with increasing distance from the axis of rotation.
Next we investigate the dependence of the solute concentration $c\B$ on $r$ in the equilibrium state of the binary solution. Consider a small sample of the solution of mass $m$. Assume the extent of this sample in the radial direction is small enough for the variation of the centrifugal force field to be negligible. The reversible work in the local frame needed to move this small sample an infinitesimal distance $\dif r$ at constant $z$, $T$, and $p$, using an external force $-F\sups{centr}$ that opposes the centrifugal force, is $\dw' = F\sur\dif r = (-F\sups{centr})\dif r = -m\omega^2 r\dif r \tag{9.8.13}$ This work is a contribution to the change $\dif U$ of the internal energy. The Gibbs energy of the small sample in the local frame is a function of the independent variables $T$, $p$, $n\A$, $n\B$, and $r$, and its total differential is $\begin{split} \dif G & = \dif(U-TS+pV) \cr & = -S\dif T + V\difp + \mu\A\dif n\A + \mu\B\dif n\B - m\omega^2 r\dif r \end{split} \tag{9.8.14}$ We use Eq. 9.8.14 to write the reciprocity relation $\Pd{\mu\B}{r}{T,p,n\A,n\B} = -\omega^2 r \Pd{m}{n\B}{T,p,n\A,r} \tag{9.8.15}$ Then, using $m=n\A M\A+n\B M\B$, we obtain $\Pd{\mu\B}{r}{T,p,n\A,n\B} = -M\B \omega^2 r \tag{9.8.16}$ Thus at constant $T$, $p$, and composition, which are the conditions that allow the activity $a\cbB$ to remain constant, $\mu\B$ for the sample varies with $r$ according to $\dif\mu\B=-M\B \omega^2 r\dif r$. We integrate from radial position $r'$ to position $r''$ to obtain \begin{gather} \s{ \begin{split} \mu\B(r'') - \mu\B(r') & = - M\B \omega^2 \int_{r'}^{r''}\!\!\!r\dif r \cr & = -\onehalf M\B\omega^2\left[\left(r''\right)^2-\left(r'\right)^2\right] \end{split} } \tag{9.8.17} \cond{($a\cbB(r''){=}a\cbB(r')$ )} \end{gather}
Let us take $r'$ as a reference position, such as the end of the centrifuge cell farthest from the axis of rotation. We define the standard chemical potential $\mu\st\cbB$ as the solute chemical potential under standard state conditions on a concentration basis at this position. The solute chemical potential and activity at this position are related by $\mu\B(r')=\mu\cbB\st+RT\ln a\cbB(r') \tag{9.8.18}$ From Eqs. 9.8.17 and 9.8.18, we obtain the following general relation between $\mu\B$ and $a\cbB$ at an arbitrary radial position $r''$: $\mu\B(r'') = \mu\cbB\st + RT\ln a\cbB(r'') -\onehalf M\B\omega^2\left[\left(r''\right)^2-\left(r'\right)^2\right] \tag{9.8.19}$
We found earlier that when the solution is in an equilibrium state, $\mu\B$ is independent of $r$—that is, $\mu\B(r'')$ is equal to $\mu\B(r')$ for any value of $r''$. When we equate expressions given by Eq. 9.8.19 for $\mu\B(r'')$ and $\mu\B(r')$ and rearrange, we obtain the following relation between the activities at the two radial positions: \begin{gather} \s{\ln\frac{a\cbB(r'')}{a\cbB(r')} = \frac{M\B \omega^2}{2RT}\left[\left(r''\right)^2-\left(r'\right)^2\right]} \tag{9.8.20} \cond{(solution in centrifuge} \nextcond{cell at equilibrium)} \end{gather}
The solute activity is related to the concentration $c\B$ by $a\cbB = \G\cbB \g\cbB c\B/c\st$. We assume the solution is sufficiently dilute for the activity coefficient $\g\cbB$ to be approximated by $1$. The pressure factor is given by $\G\cbB \approx \exp\left[ V\B^{\infty}(p-p\st)/RT \right]$ (Table 9.6). These relations give us another expression for the logarithm of the ratio of activities: $\ln \frac{a\cbB(r'')}{a\cbB(r')} = \frac{V\B^{\infty}(p''-p')}{RT} + \ln\frac{c\B(r'')}{c\B(r')} \tag{9.8.21}$ We substitute for $p''-p'$ from Eq. 9.8.12. It is also useful to make the substitution $V\B^{\infty}=M\B v\B^{\infty}$, where $v\B^{\infty}$ is the partial specific volume of the solute at infinite dilution.
When we equate the two expressions for $\ln[a\cbB(r'')/a\cbB(r')]$, we obtain finally \begin{gather} \s{ \ln\frac{c\B(r'')}{c\B(r')} = \frac{M\B\left(1-v\B^{\infty}\rho\right)\omega^2}{2RT} \left[\left(r''\right)^2-\left(r'\right)^2\right] } \tag{9.8.22} \cond{(solution in centrifuge} \nextcond{cell at equilibrium)} \end{gather} This equation shows that if the solution density $\rho$ is less than the effective solute density $1/v\B^{\infty}$, so that $v\B^{\infty}\rho$ is less than 1, the solute concentration increases with increasing distance from the axis of rotation in the equilibrium state. If, however, $\rho$ is greater than $1/v\B^{\infty}$, the concentration decreases with increasing $r$. The factor $\left(1-v\B^{\infty}\rho\right)$ is like a buoyancy factor for the effect of the centrifugal field on the solute.
Equation 9.8.22 is needed for sedimentation equilibrium, a method of determining the molar mass of a macromolecule. A dilute solution of the macromolecule is placed in the cell of an analytical ultracentrifuge, and the angular velocity is selected to produce a measurable solute concentration gradient at equilibrium. The solute concentration is measured optically as a function of $r$. The equation predicts that a plot of $\ln\left(c\B/c\st\right)$ versus $r^2$ will be linear, with a slope equal to $M\B\left(1-v\B^{\infty}\rho\right)\omega^2/2RT$. The partial specific volume $v\B^{\infty}$ is found from measurements of solution density as a function of solute mass fraction (Sec. 9.2.5). By this means, the molar mass $M\B$ of the macromolecule is evaluated. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/09%3A_Mixtures/9.08%3A_Mixtures_in_Gravitational_and_Centrifugal_Fields.txt |
An underlined problem number or problem-part letter indicates that the numerical answer appears in Appendix I.
9.1 For a binary solution, find expressions for the mole fractions $x_{\mathrm{B}}$ and $x_{\mathrm{A}}$ as functions of the solute molality $m_{\mathrm{B}}$.
9.2 Consider a binary mixture of two liquids, $\mathrm{A}$ and $\mathrm{B}$. The molar volume of mixing, $\Delta V(\mathrm{mix}) / n$, is given by Eq. 9.2.19.
(a) Find a formula for calculating the value of $\Delta V(\mathrm{mix}) / n$ of a binary mixture from values of $x_{\mathrm{A}}, x_{\mathrm{B}}, M_{\mathrm{A}}, M_{\mathrm{B}}, \rho, \rho_{\mathrm{A}}^{*}$, and $\rho_{\mathrm{B}}^{*} .$
Table $9.7$ Molar volumes of mixing of binary mixtures of 1-hexanol (A) and 1 -octene $(\mathrm{B})$ at $25^{\circ} \mathrm{C} .^{a}$
\begin{tabular}{lccc}
\hline$x_{\mathrm{B}}$ & {$[\Delta V(\mathrm{mix}) / n] / \mathrm{cm}^{3} \mathrm{~mol}^{-1}$} & $x_{\mathrm{B}}$ & {$[\Delta V(\mathrm{mix}) / n] / \mathrm{cm}^{3} \mathrm{~mol}^{-1}$} \
\hline 0 & 0 & $0.555$ & $0.005$ \
$0.049$ & $-0.027$ & $0.597$ & $0.011$ \
$0.097$ & $-0.050$ & $0.702$ & $0.029$ \
$0.146$ & $-0.063$ & $0.716$ & $0.035$ \
$0.199$ & $-0.077$ & $0.751$ & $0.048$ \
$0.235$ & $-0.073$ & $0.803$ & $0.056$ \
$0.284$ & $-0.074$ & $0.846$ & $0.058$ \
$0.343$ & $-0.065$ & $0.897$ & $0.057$ \
$0.388$ & $-0.053$ & $0.944$ & $0.049$ \
$0.448$ & $-0.032$ & 1 & 0 \
$0.491$ & $-0.016$ & & \
\hline
\end{tabular}
${ }^{a}$ Ref. [170].
(b) The molar volumes of mixing for liquid binary mixtures of 1-hexanol (A) and 1-octene (B) at $25^{\circ} \mathrm{C}$ have been calculated from their measured densities. The data are in Table 9.7. The molar volumes of the pure constituents are $V_{\mathrm{A}}^{*}=125.31 \mathrm{~cm}^{3} \mathrm{~mol}^{-1}$ and $V_{\mathrm{B}}^{*}=$ $157.85 \mathrm{~cm}^{3} \mathrm{~mol}^{-1}$. Use the method of intercepts to estimate the partial molar volumes of both constituents in an equimolar mixture $\left(x_{\mathrm{A}}=x_{\mathrm{B}}=0.5\right)$, and the partial molar volume $V_{\mathrm{B}}^{\infty}$ of B at infinite dilution.
9.3 Extend the derivation of Prob. 8.1, concerning a liquid droplet of radius $r$ suspended in a gas, to the case in which the liquid and gas are both mixtures. Show that the equilibrium conditions are $T^{\mathrm{g}}=T^{\mathrm{l}}, \mu_{i}^{\mathrm{g}}=\mu_{i}^{1}$ (for each species $i$ that can equilibrate between the two phases), and $p^{1}=p^{g}+2 \gamma / r$, where $\gamma$ is the surface tension. (As in Prob. 8.1, the last relation is the Laplace equation.)
9.4 Consider a gaseous mixture of $4.0000 \times 10^{-2} \mathrm{~mol}$ of $\mathrm{N}_{2}$ (A) and $4.0000 \times 10^{-2} \mathrm{~mol}$ of $\mathrm{CO}_{2}$ (B) in a volume of $1.0000 \times 10^{-3} \mathrm{~m}^{3}$ at a temperature of $298.15 \mathrm{~K}$. The second virial coefficients at this temperature have the values ${ }^{14}$
\begin{aligned} B_{\mathrm{AA}} &=-4.8 \times 10^{-6} \mathrm{~m}^{3} \mathrm{~mol}^{-1} \ B_{\mathrm{BB}} &=-124.5 \times 10^{-6} \mathrm{~m}^{3} \mathrm{~mol}^{-1} \ B_{\mathrm{AB}} &=-47.5 \times 10^{-6} \mathrm{~m}^{3} \mathrm{~mol}^{-1} \end{aligned}
Compare the pressure of the real gas mixture with that predicted by the ideal gas equation. See Eqs. 9.3.20 and 9.3.23.
${ }^{14}$ Refs. [3], [49], and [50].
9.5 At $25^{\circ} \mathrm{C}$ and 1 bar, the Henry's law constants of nitrogen and oxygen dissolved in water are $k_{\mathrm{H}, \mathrm{N}_{2}}=8.64 \times 10^{4}$ bar and $k_{\mathrm{H}, \mathrm{O}_{2}}=4.41 \times 10^{4}$ bar. ${ }^{15}$ The vapor pressure of water at this temperature and pressure is $p_{\mathrm{H}_{2} \mathrm{O}}=0.032$ bar. Assume that dry air contains only $\mathrm{N}_{2}$ and $\mathrm{O}_{2}$ at mole fractions $y_{\mathrm{N}_{2}}=0.788$ and $y_{\mathrm{O}_{2}}=0.212$. Consider liquid-gas systems formed by equilibrating liquid water and air at $25^{\circ} \mathrm{C}$ and $1.000 \mathrm{bar}$, and assume that the gas phase behaves as an ideal gas mixture.
Hint: The sum of the partial pressures of $\mathrm{N}_{2}$ and $\mathrm{O}_{2}$ must be $(1.000-0.032)$ bar $=0.968$ bar. If the volume of one of the phases is much larger than that of the other, then almost all of the $\mathrm{N}_{2}$ and $\mathrm{O}_{2}$ will be in the predominant phase and the ratio of their amounts in this phase must be practically the same as in dry air.
Determine the mole fractions of $\mathrm{N}_{2}$ and $\mathrm{O}_{2}$ in both phases in the following limiting cases:
(a) A large volume of air is equilibrated with just enough water to leave a small drop of liquid.
(b) A large volume of water is equilibrated with just enough air to leave a small bubble of gas.
9.6 Derive the expression for $\gamma_{m, \mathrm{~B}}$ given in Table 9.4, starting with Eq. 9.5.18.
9.7 Consider a nonideal binary gas mixture with the simple equation of state $V=n R T / p+n B$ (Eq. $9.3 .21$ ).
(a) The rule of Lewis and Randall states that the value of the mixed second virial coefficient $B_{\mathrm{AB}}$ is the average of $B_{\mathrm{AA}}$ and $B_{\mathrm{BB}}$. Show that when this rule holds, the fugacity coefficient of $\mathrm{A}$ in a binary gas mixture of any composition is given by $\ln \phi_{\mathrm{A}}=B_{\mathrm{AA}} p / R T$. By comparing this expression with Eq. $7.8 .18$ for a pure gas, express the fugacity of $\mathrm{A}$ in the mixture as a function of the fugacity of pure $A$ at the same temperature and pressure as the mixture.
(b) The rule of Lewis and Randall is not accurately obeyed when constituents A and B are chemically dissimilar. For example, at $298.15 \mathrm{~K}$, the second virial coefficients of $\mathrm{H}_{2} \mathrm{O}$ (A) and $\mathrm{N}_{2}$ (B) are $B_{\mathrm{AA}}=-1158 \mathrm{~cm}^{3} \mathrm{~mol}^{-1}$ and $B_{\mathrm{BB}}=-5 \mathrm{~cm}^{3} \mathrm{~mol}^{-1}$, respectively, whereas the mixed second virial coefficient is $B_{\mathrm{AB}}=-40 \mathrm{~cm}^{3} \mathrm{~mol}^{-1}$.
When liquid water is equilibrated with nitrogen at $298.15 \mathrm{~K}$ and 1 bar, the partial pressure of $\mathrm{H}_{2} \mathrm{O}$ in the gas phase is $p_{\mathrm{A}}=0.03185$ bar. Use the given values of $B_{\mathrm{AA}}, B_{\mathrm{BB}}$, and $B_{\mathrm{AB}}$ to calculate the fugacity of the gaseous $\mathrm{H}_{2} \mathrm{O}$ in this binary mixture. Compare this fugacity with the fugacity calculated with the value of $B_{\mathrm{AB}}$ predicted by the rule of Lewis and Randall.
Table $9.8$ Activity coefficient of benzene (A) in mixtures of benzene and 1 -octanol at $20^{\circ} \mathrm{C}$. The reference state
is the pure liquid.
\begin{tabular}{lccc}
\hline$x_{\mathrm{A}}$ & $\gamma_{\mathrm{A}}$ & $x_{\mathrm{A}}$ & $\gamma_{\mathrm{A}}$ \
\hline 0 & $2.0^{a}$ & $0.7631$ & $1.183$ \
$0.1334$ & $1.915$ & $0.8474$ & $1.101$ \
$0.2381$ & $1.809$ & $0.9174$ & $1.046$ \
$0.4131$ & $1.594$ & $0.9782$ & $1.005$ \
$0.5805$ & $1.370$ & & \
\hline \multicolumn{3}{l}{$a_{\text {extrapolated }}$}
\end{tabular}
${ }^{15}$ Ref. [184].
9.8 Benzene and 1-octanol are two liquids that mix in all proportions. Benzene has a measurable vapor pressure, whereas 1-octanol is practically nonvolatile. The data in Table $9.8$ on the preceding page were obtained by Platford ${ }^{16}$ using the isopiestic vapor pressure method.
(a) Use numerical integration to evaluate the integral on the right side of Eq. $9.6 .10$ at each of the values of $x_{\mathrm{A}}$ listed in the table, and thus find $\gamma_{\mathrm{B}}$ at these compositions.
(b) Draw two curves on the same graph showing the effective mole fractions $\gamma_{\mathrm{A}} x_{\mathrm{A}}$ and $\gamma_{\mathrm{B}} x_{\mathrm{B}}$ as functions of $x_{\mathrm{A}}$. Are the deviations from ideal-mixture behavior positive or negative?
Table $9.9$ Liquid and gas compositions in the two-phase system of methanol (A) and benzene (B) at $45^{\circ} \mathrm{C}^{a}$
\begin{tabular}{llllll}
\hline$x_{\mathrm{A}}$ & $y_{\mathrm{A}}$ & $p / \mathrm{kPa}$ & $x_{\mathrm{A}}$ & $y_{\mathrm{A}}$ & $p / \mathrm{kPa}$ \
\hline 0 & 0 & $29.894$ & $0.4201$ & $0.5590$ & $60.015$ \
$0.0207$ & $0.2794$ & $40.962$ & $0.5420$ & $0.5783$ & $60.416$ \
$0.0314$ & $0.3391$ & $44.231$ & $0.6164$ & $0.5908$ & $60.416$ \
$0.0431$ & $0.3794$ & $46.832$ & $0.7259$ & $0.6216$ & $59.868$ \
$0.0613$ & $0.4306$ & $50.488$ & $0.8171$ & $0.6681$ & $58.321$ \
$0.0854$ & $0.4642$ & $53.224$ & $0.9033$ & $0.7525$ & $54.692$ \
$0.1811$ & $0.5171$ & $57.454$ & $0.9497$ & $0.8368$ & $51.009$ \
$0.3217$ & $0.5450$ & $59.402$ & 1 & 1 & $44.608$ \
\hline
\end{tabular}
${ }^{a}$ Ref. [169].
9.9 Table $9.9$ lists measured values of gas-phase composition and total pressure for the binary two-phase methanol-benzene system at constant temperature and varied liquid-phase composition. $x_{\mathrm{A}}$ is the mole fraction of methanol in the liquid mixture, and $y_{\mathrm{A}}$ is the mole fraction of methanol in the equilibrated gas phase.
(a) For each of the 16 different liquid-phase compositions, tabulate the partial pressures of $\mathrm{A}$ and $\mathrm{B}$ in the equilibrated gas phase.
(b) Plot $p_{\mathrm{A}}$ and $p_{\mathrm{B}}$ versus $x_{\mathrm{A}}$ on the same graph. Notice that the behavior of the mixture is far from that of an ideal mixture. Are the deviations from Raoult's law positive or negative?
(c) Tabulate and plot the activity coefficient $\gamma_{\mathrm{B}}$ of the benzene as a function of $x_{\mathrm{A}}$ using a pure-liquid reference state. Assume that the fugacity $f_{\mathrm{B}}$ is equal to $p_{\mathrm{B}}$, and ignore the effects of variable pressure.
(d) Estimate the Henry's law constant $k_{\mathrm{H}, \mathrm{A}}$ of methanol in the benzene environment at $45^{\circ} \mathrm{C}$ by the graphical method suggested in Fig. 9.7(b). Again assume that $f_{\mathrm{A}}$ and $p_{\mathrm{A}}$ are equal, and ignore the effects of variable pressure.
9.10 Consider a dilute binary nonelectrolyte solution in which the dependence of the chemical potential of solute B on composition is given by
$\mu_{\mathrm{B}}=\mu_{m, \mathrm{~B}}^{\mathrm{ref}}+R T \ln \frac{m_{\mathrm{B}}}{m^{\circ}}+k_{m} m_{\mathrm{B}}$
where $\mu_{m, \mathrm{~B}}^{\mathrm{ref}}$ and $k_{m}$ are constants at a given $T$ and $p$. (The derivation of this equation is sketched in Sec. 9.5.4.) Use the Gibbs-Duhem equation in the form $\mathrm{d} \mu_{\mathrm{A}}=-\left(n_{\mathrm{B}} / n_{\mathrm{A}}\right) \mathrm{d} \mu_{\mathrm{B}}$ to obtain an expression for $\mu_{\mathrm{A}}-\mu_{\mathrm{A}}^{*}$ as a function of $m_{\mathrm{B}}$ in this solution.
${ }^{16}$ Ref. [145].
9.11 By means of the isopiestic vapor pressure technique, the osmotic coefficients of aqueous solutions of urea at $25^{\circ} \mathrm{C}$ have been measured at molalities up to the saturation limit of about $20 \mathrm{~mol} \mathrm{~kg}^{-1} .{ }^{17}$ The experimental values are closely approximated by the function
$\phi_{m}=1.00-\frac{0.050 m_{\mathrm{B}} / m^{\circ}}{1.00+0.179 m_{\mathrm{B}} / m^{\circ}}$
where $m^{\circ}$ is $1 \mathrm{~mol} \mathrm{~kg}^{-1}$. Calculate values of the solvent and solute activity coefficients $\gamma_{\mathrm{A}}$ and $\gamma_{m, \mathrm{~B}}$ at various molalities in the range $0-20 \mathrm{~mol} \mathrm{~kg}^{-1}$, and plot them versus $m_{\mathrm{B}} / m^{\circ}$. Use enough points to be able to see the shapes of the curves. What are the limiting slopes of these curves as $m_{\mathrm{B}}$ approaches zero?
9.12 Use Eq. $9.2 .49$ to derive an expression for the rate at which the logarithm of the activity coefficient of component $i$ of a liquid mixture changes with pressure at constant temperature and composition: $\left(\partial \ln \gamma_{i} / \partial p\right)_{T,\left\{n_{i}\right\}}=$ ?
9.13 Assume that at sea level the atmosphere has a pressure of $1.00$ bar and a composition given by $y_{\mathrm{N}_{2}}=0.788$ and $y_{\mathrm{O}_{2}}=0.212$. Find the partial pressures and mole fractions of $\mathrm{N}_{2}$ and $\mathrm{O}_{2}$, and the total pressure, at an altitude of $10.0 \mathrm{~km}$, making the (drastic) approximation that the atmosphere is an ideal gas mixture in an equilibrium state at $0^{\circ} \mathrm{C}$. For $g$ use the value of the standard acceleration of free fall listed in Appendix B.
9.14 Consider a tall column of a dilute binary liquid solution at equilibrium in a gravitational field.
(a) Derive an expression for $\ln \left[c_{\mathrm{B}}(h) / c_{\mathrm{B}}(0)\right]$, where $c_{\mathrm{B}}(h)$ and $c_{\mathrm{B}}(0)$ are the solute concentrations at elevations $h$ and 0 . Your expression should be a function of $h, M_{\mathrm{B}}, T, \rho$, and the partial specific volume of the solute at infinite dilution, $v_{\mathrm{B}}^{\infty}$. For the dependence of pressure on elevation, you may use the hydrostatic formula $\mathrm{d} p=-\rho g \mathrm{~d} h$ (Eq. $8.1 .14$ on page 200) and assume the solution density $\rho$ is the same at all elevations. Hint: use the derivation leading to Eq. $9.8 .22$ as a guide.
(b) Suppose you have a tall vessel containing a dilute solution of a macromolecule solute of molar mass $M_{\mathrm{B}}=10.0 \mathrm{~kg} \mathrm{~mol}^{-1}$ and partial specific volume $v_{\mathrm{B}}^{\infty}=0.78 \mathrm{~cm}^{3} \mathrm{~g}^{-1}$. The solution density is $\rho=1.00 \mathrm{~g} \mathrm{~cm}^{-3}$ and the temperature is $T=300 \mathrm{~K}$. Find the height $h$ from the bottom of the vessel at which, in the equilibrium state, the concentration $c_{\mathrm{B}}$ has decreased to 99 percent of the concentration at the bottom.
9.15 FhuA is a protein found in the outer membrane of the Escherichia coli bacterium. From the known amino acid sequence, its molar mass is calculated to be $78.804 \mathrm{~kg} \mathrm{~mol}^{-1}$. In aqueous solution, molecules of the detergent dodecyl maltoside bind to a FhuA molecule to form an aggregate that behaves as a single solute species. Figure $9.13$ on the next page shows data collected in a sedimentation equilibrium experiment with a dilute solution of the aggregate. ${ }^{18}$ In the graph, $A$ is the absorbance measured at a wavelength of $280 \mathrm{~nm}$ (a property that is a linear function of the aggregate concentration) and $r$ is the radial distance from the axis of rotation of the centrifuge rotor. The experimental points fall very close to the straight line shown in the graph. The sedimentation conditions were $\omega=838 \mathrm{~s}^{-1}$ and $T=293 \mathrm{~K}$. The authors used the values $v_{\mathrm{B}}^{\infty}=0.776 \mathrm{~cm}^{3} \mathrm{~g}^{-1}$ and $\rho=1.004 \mathrm{~g} \mathrm{~cm}^{-3}$.
(a) The values of $r$ at which the absorbance was measured range from $6.95 \mathrm{~cm}$ to $7.20 \mathrm{~cm}$. Find the difference of pressure in the solution between these two positions.
(b) Find the molar mass of the aggregate solute species, and use it to estimate the mass binding ratio (the mass of bound detergent divided by the mass of protein).
${ }^{17}$ Ref. [160]. ${ }^{18}$ Ref. [18]. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/09%3A_Mixtures/9.09%3A_Chapter_9_Problems.txt |
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The thermodynamic properties of electrolyte solutions differ in significant ways from the properties of mixtures of nonelectrolytes.
Figure 10.1 Partial pressure of HCl in a gas phase equilibrated with aqueous HCl at $25\units{\(\degC$}\) and $1\br$. Open circles: experimental data from Stuart J. Bates and H. Darwin Kirschman, J. Am. Chem. Soc., 41, 1991–2001, 1919.
Here is an example. Pure HCl (hydrogen chloride) is a gas that is very soluble in water. A plot of the partial pressure of gaseous HCl in equilibrium with aqueous HCl, as a function of the solution molality (Fig. 10.1), shows that the limiting slope at infinite dilution is not finite, but zero. What is the reason for this non-Henry’s law behavior? It must be because HCl is an electrolyte—it dissociates (ionizes) in the aqueous environment.
It is customary to use a molality basis for the reference and standard states of electrolyte solutes. This is the only basis used in this chapter, even when not explicitly indicated for ions. The symbol $\mu_{+}\st$, for instance, denotes the chemical potential of a cation in a standard state based on molality.
In dealing with an electrolyte solute, we can refer to the solute (a substance) as a whole and to the individual charged ions that result from dissociation. We can apply the same general definitions of chemical potential, activity coefficient, and activity to these different species, but only the activity coefficient and activity of the solute as a whole can be evaluated experimentally.
10: Electrolyte Solutions
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Consider a solution of an electrolyte solute that dissociates completely into a cation species and an anion species. Subscripts $+$ and $-$ will be used to denote the cation and anion, respectively. The solute molality $m\B$ is defined as the amount of solute formula unit divided by the mass of solvent.
We first need to investigate the relation between the chemical potential of an ion species and the electric potential of the solution phase.
The electric potential $\phi$ in the interior of a phase is called the inner electric potential, or Galvani potential. It is defined as the work needed to reversibly move an infinitesimal test charge into the phase from a position infinitely far from other charges, divided by the value of the test charge. The electrical potential energy of a charge in the phase is the product of $\phi$ and the charge.
Consider a hypothetical process in which an infinitesimal amount $\dif n_+$ of the cation is transferred into a solution phase at constant $T$ and $p$. The quantity of charge transferred is $\delta Q=z_+F\dif n_+$, where $z_+$ is the charge number ($+1$, $+2$, etc.) of the cation, and $F$ is the Faraday constant. (The Faraday constant is the charge per amount of protons.) If the phase is at zero electric potential, the process causes no change in its electrical potential energy. However, if the phase has a finite electric potential $\phi$, the transfer process changes its electrical potential energy by $\phi \delta Q=z_+F\phi\dif n_+$. Consequently, the internal energy change depends on $\phi$ according to $\dif U(\phi) = \dif U(0) + z_+F\phi\dif n_+ \tag{10.1.1}$ where the electric potential is indicated in parentheses. The change in the Gibbs energy of the phase is given by $\dif G = \dif(U-TS+pV)$, where $T$, $S$, $p$, and $V$ are unaffected by the value of $\phi$. The dependence of $\dif G$ on $\phi$ is therefore $\dif G(\phi) = \dif G(0) + z_+F\phi\dif n_+ \tag{10.1.2}$
The Gibbs fundamental equation for an open system, $\dif G=-S\dif T+V\difp+\sum_i\mu_i\dif n_i$ (Eq. 9.2.34), assumes the electric potential is zero. From this equation and Eq. 10.1.2, the Gibbs energy change during the transfer process at constant $T$ and $p$ is found to depend on $\phi$ according to $\dif G(\phi) = \left[ \mu_+(0) + z_+F\phi \right] \dif n_+ \tag{10.1.3}$ The chemical potential of the cation in a phase of electric potential $\phi$, defined by the partial molar Gibbs energy $\bpd{G(\phi)}{n_+}{T,p}$, is therefore given by $\mu_+(\phi)=\mu_+(0)+z_+F\phi \tag{10.1.4}$ The corresponding relation for an anion is $\mu_-(\phi)=\mu_-(0)+z_-F\phi \tag{10.1.5}$ where $z_-$ is the charge number of the anion ($-1$, $-2$, etc.). For a charged species in general, we have $\mu_i(\phi)=\mu_i(0)+z_iF\phi \tag{10.1.6}$
We define the standard state of an ion on a molality basis in the same way as for a nonelectrolyte solute, with the additional stipulation that the ion is in a phase of zero electric potential. Thus, the standard state is a hypothetical state in which the ion is at molality $m\st$ with behavior extrapolated from infinite dilution on a molality basis, in a phase of pressure $p=p\st$ and electric potential $\phi{=}0$.
The standard chemical potential $\mu_+\st$ or $\mu_-\st$ of a cation or anion is the chemical potential of the ion in its standard state. Single-ion activities $a_+$ and $a_-$ in a phase of zero electric potential are defined by relations having the form of Eq. 9.7.8: $\mu_+(0)=\mu_+\st+RT\ln a_+ \qquad \mu_-(0)=\mu_-\st+RT\ln a_- \tag{10.1.7}$ As explained in Sec. 9.7, $a_+$ and $a_-$ should depend on the temperature, pressure, and composition of the phase, and not on the value of $\phi$.
From Eqs. 10.1.4, 10.1.5, and 10.1.7, the relations between the chemical potential of a cation or anion, its activity, and the electric potential of its phase, are found to be $\mu_+=\mu_+\st + RT\ln a_+ + z_+ F\phi \qquad \mu_-=\mu_-\st + RT\ln a_- + z_i F\phi \tag{10.1.8}$ These relations are definitions of single-ion activities in a phase of electric potential $\phi$.
For a charged species in general, we can write $\mu_i=\mu_i\st + RT\ln a_i + z_i F\phi \tag{10.1.9}$ Note that we can also apply this equation to an uncharged species, because the charge number $z_i$ is then zero and Eq. 10.1.9 becomes the same as Eq. 9.7.2.
Some thermodynamicists call the quantity $(\mu_i\st+RT\ln a_i)$, which depends only on $T$, $p$, and composition, the chemical potential of ion $i$, and the quantity $(\mu_i\st+RT\ln a_i+z_iF\phi)$ the electrochemical potential with symbol $\tilde{\mu}_i$.
Of course there is no experimental way to evaluate either $\mu_+$ or $\mu_-$ relative to a reference state or standard state, because it is impossible to add cations or anions by themselves to a solution. We can nevertheless write some theoretical relations involving $\mu_+$ and $\mu_-$.
For a given temperature and pressure, we can write the dependence of the chemical potentials of the ions on their molalities in the same form as that given by Eq. 9.5.18 for a nonelectrolyte solute: $\mu_+=\mu_+\rf + RT\ln\left(\g_+\frac{m_+}{m\st}\right) \qquad \mu_-=\mu_-\rf + RT\ln\left(\g_-\frac{m_-}{m\st}\right) \tag{10.1.10}$ Here $\mu_+\rf$ and $\mu_-\rf$ are the chemical potentials of the cation and anion in solute reference states. Each reference state is defined as a hypothetical solution with the same temperature, pressure, and electric potential as the solution under consideration; in this solution, the molality of the ion has the standard value $m\st$, and the ion behaves according to Henry’s law based on molality. $\g_+$ and $\g_-$ are single-ion activity coefficients on a molality basis.
The single-ion activity coefficients approach unity in the limit of infinite dilution: \begin{gather} \s{ \g_+ \ra 1 \quad \tx{and} \quad \g_- \ra 1 \quad \tx{as} \quad m\B \ra 0} \tag{10.1.11} \cond{(constant $T$, $p$, and $\phi$)} \end{gather} In other words, we assume that in an extremely dilute electrolyte solution each individual ion behaves like a nonelectrolyte solute species in an ideal-dilute solution. At a finite solute molality, the values of $\g_+$ and $\g_-$ are the ones that allow Eq. 10.1.10 to give the correct values of the quantities $(\mu_+-\mu_+\rf)$ and $(\mu_- -\mu_-\rf)$. We have no way to actually measure these quantities experimentally, so we cannot evaluate either $\g_+$ or $\g_-$.
We can define single-ion pressure factors $\G_+$ and $\G_-$ as follows: $\G_+\defn\exp\left(\frac{\mu_+\rf-\mu_+\st}{RT}\right) \approx \exp\left[ \frac{V_+^{\infty}(p-p\st)}{RT} \right] \tag{10.1.12}$ $\G_-\defn\exp\left(\frac{\mu_-\rf-\mu_-\st}{RT}\right) \approx \exp\left[ \frac{V_-^{\infty}(p-p\st)}{RT} \right] \tag{10.1.13}$ The approximations in these equations are like those in Table 9.6 for nonelectrolyte solutes; they are based on the assumption that the partial molar volumes $V_+$ and $V_-$ are independent of pressure.
From Eqs. 10.1.7, 10.1.10, 10.1.12, and 10.1.13, the single-ion activities are related to the solution composition by $a_+=\G_+\g_+\frac{m_+}{m\st} \qquad a_-=\G_-\g_-\frac{m_-}{m\st} \tag{10.1.14}$ Then, from Eq. 10.1.9, we have the following relations between the chemical potentials and molalities of the ions: $\mu_+=\mu_+\st + RT\ln(\G_+\g_+m_+/m\st) + z_+F\phi \tag{10.1.15}$ $\mu_-=\mu_-\st + RT\ln(\G_-\g_-m_-/m\st) + z_-F\phi \tag{10.1.16}$
Like the values of $\g_+$ and $\g_-$, values of the single-ion quantities $a_+$, $a_-$, $\G_+$, and $\G_-$ cannot be determined by experiment. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/10%3A_Electrolyte_Solutions/10.01%3A_Single-ion_Quantities.txt |
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Let us consider properties of an electrolyte solute as a whole. The simplest case is that of a binary solution in which the solute is a symmetrical strong electrolyte—a substance whose formula unit has one cation and one anion that dissociate completely. This condition will be indicated by $\nu = 2$, where $\nu$ is the number of ions per formula unit. In an aqueous solution, the solute with $\nu$ equal to 2 might be a 1:1 salt such as NaCl, a 2:2 salt such as MgSO$_4$, or a strong monoprotic acid such as HCl.
In this binary solution, the chemical potential of the solute as a whole is defined in the usual way as the partial molar Gibbs energy $\mu\B \defn \Pd{G}{n\B}{T,p,n\A} \tag{10.2.1}$ and is a function of $T$, $p$, and the solute molality $m\B$. Although $\mu\B$ under given conditions must in principle have a definite value, we are not able to actually evaluate it because we have no way to measure precisely the energy brought into the system by the solute. This energy contributes to the internal energy and thus to $G$. We can, however, evaluate the differences $\mu\B - \mu\mbB\rf$ and $\mu\B - \mu\mbB\st$.
We can write the additivity rule (Eq. 9.2.25) for $G$ as either $G = n\A\mu\A + n\B\mu\B \tag{10.2.2}$ or $G = n\A\mu\A + n_{+}\mu_{+}+n_{-}\mu_{-} \tag{10.2.3}$ A comparison of these equations for a symmetrical electrolyte ($n\B = n_+ = n_-$) gives us the relation \begin{gather} \s{ \mu\B = \mu_{+} + \mu_{-} } \tag{10.2.4} \cond{($\nu{=}2$)} \end{gather} We see that the solute chemical potential in this case is the sum of the single-ion chemical potentials.
The solution is a phase of electric potential $\phi$. From Eqs. 10.1.4 and 10.1.5, the sum $\mu_{+}+\mu_{-}$ appearing in Eq. 10.2.4 is $\mu_+(\phi) + \mu_-(\phi)=\mu_+(0) + \mu_-(0) + (z_+ + z_-)F\phi \tag{10.2.5}$ For the symmetrical electrolyte, the sum $(z_+ + z_-)$ is zero, so that $\mu\B$ is equal to $\mu_+(0) + \mu_-(0)$. We substitute the expressions of Eq. 10.1.10, use the relation $\mu\mbB\rf=\mu_+\rf+\mu_-\rf$ with reference states at $\phi{=}0$, set the ion molalities $m_+$ and $m_-$ equal to $m\B$, and obtain \begin{gather} \s{ \mu\B=\mu\mbB\rf +RT\ln\left[\g_+\g_-\left(\frac{m\B}{m\st}\right)^2\right] } \tag{10.2.6} \cond{($\nu{=}2$)} \end{gather} The important feature of this relation is the appearance of the second power of $m\B/m\st$, instead of the first power as in the case of a nonelectrolyte. Also note that $\mu\B$ does not depend on $\phi$, unlike $\mu_+$ and $\mu_-$.
Although we cannot evaluate $\g_+$ or $\g_-$ individually, we can evaluate the product $\g_+\g_-$. This product is the square of the mean ionic activity coefficient $\g_{\pm}$, defined for a symmetrical electrolyte by \begin{gather} \s{ \g_{\pm} \defn \sqrt{\g_{+}\g_{-}} } \tag{10.2.7} \cond{($\nu{=}2$)} \end{gather} With this definition, Eq. 10.2.6 becomes \begin{gather} \s{ \mu\B=\mu\mbB\rf +RT\ln\left[\left(\g_{\pm}\right)^2 \left(\frac{m\B}{m\st}\right)^2\right] } \tag{10.2.8} \cond{($\nu{=}2$)} \end{gather} Since it is possible to determine the value of $\mu\B-\mu\mbB\rf$ for a solution of known molality, $\g_{\pm}$ is a measurable quantity.
If the electrolyte (e.g., HCl) is sufficiently volatile, its mean ionic activity coefficient in a solution can be evaluated from partial pressure measurements of an equilibrated gas phase. Section 10.6 will describe a general method by which $\g_{\pm}$ can be found from osmotic coefficients. Section 14.5 describes how, in favorable cases, it is possible to evaluate $\g_{\pm}$ from the equilibrium cell potential of a galvanic cell.
The activity $a\mbB$ of a solute substance on a molality basis is defined by Eq. 9.7.8: $\mu\B = \mu\mbB\st + RT\ln a\mbB \tag{10.2.9}$ Here $\mu\mbB\st$ is the chemical potential of the solute in its standard state, which is the solute reference state at the standard pressure. By equating the expressions for $\mu\B$ given by Eqs. 10.2.8 and 10.2.9 and solving for the activity, we obtain \begin{gather} \s{ a\mbB = \G\mbB \left(\g_{\pm}\right)^2 \left(\frac{m\B}{m\st}\right)^2 } \tag{10.2.10} \cond{($\nu{=}2$)} \end{gather} where $\G\mbB$ is the pressure factor defined by $\G\mbB \defn \exp\left(\frac{\mu\mbB\rf-\mu\mbB\st}{RT}\right) \tag{10.2.11}$ We can use the appropriate expression in Table 9.6 to evaluate $\G\mbB$ at an arbitrary pressure $p'$: $\G\mbB(p')=\exp\left(\int_{p\st}^{p'}\frac{V\B^{\infty}}{RT}\difp\right) \approx \exp\left[\frac{V\B^{\infty}(p'-p\st)}{RT}\right] \tag{10.2.12}$ The value of $\G\mbB$ is $1$ at the standard pressure, and close to $1$ at any reasonably low pressure. For this reason it is common to see Eq. 10.2.10 written as $a\mbB=\g_{\pm}^2(m\B/m\st)^2$, with $\G\mbB$ omitted.
Equation 10.2.10 predicts that the activity of HCl in aqueous solutions is proportional, in the limit of infinite dilution, to the square of the HCl molality. In contrast, the activity of a nonelectrolyte solute is proportional to the first power of the molality in this limit. This predicted behavior of aqueous HCl is consistent with the data plotted in Fig. 10.1, and is confirmed by the data for dilute HCl solutions shown in Fig. 10.2(a). The dashed line in Fig. 10.2(a) is the extrapolation of the ideal-dilute behavior given by $a\mbB=(m\B/m\st)^2$. The extension of this line to $m\B = m\st$ establishes the hypothetical solute reference state based on molality, indicated by a filled circle in Fig. 10.2(b). (Since the data are for solutions at the standard pressure of $1\br$, the solute reference state shown in the figure is also the solute standard state.)
The solid curve of Fig. 10.2(c) shows how the mean ionic activity coefficient of HCl varies with molality in approximately the same range of molalities as the data shown in Fig. 10.2(b). In the limit of infinite dilution, $\g_{\pm}$ approaches unity. The slope of the curve approaches $-\infty$ in this limit, quite unlike the behavior described in Sec. 9.5.4 for the activity coefficient of a nonelectrolyte solute.
For a symmetrical strong electrolyte, $\g_{\pm}$ is the geometric average of the single-ion activity coefficients $\g_+$ and $\g_-$. We have no way of evaluating $\g_+$ or $\g_-$ individually, even if we know the value of $\g_{\pm}$. For instance, we cannot assume that $\g_+$ and $\g_-$ are equal. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/10%3A_Electrolyte_Solutions/10.02%3A_Solution_of_a_Symmetrical_Electrolyte.txt |
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The formula unit of a nonsymmetrical electrolyte solute has more than two ions. General formulas for the solute as a whole are more complicated than those for the symmetrical case treated in the preceding section, but are derived by the same reasoning.
Again we assume the solute dissociates completely into its constituent ions. We define the following symbols:
$\nu_+ =$ the number of cations per solute formula unit
$\nu_- =$ the number of anions per solute formula unit
$\nu =$ the sum $\nu_+ + \nu_-$
For example, if the solute formula is Al$_2$(SO$_4$)$_3$, the values are $\nu_+ {=} 2$, $\nu_- {=} 3$, and $\nu {=} 5$.
10.3.1 Solution of a single electrolyte
In a solution of a single electrolyte solute that is not necessarily symmetrical, the ion molalities are related to the overall solute molality by $m_+ = \nu_+m\B \qquad m_- = \nu_-m\B \tag{10.3.1}$ From the additivity rule for the Gibbs energy, we have $\begin{split} G & =n\A\mu\A + n\B\mu\B\cr & = n\A\mu\A + \nu_+n\B\mu_{+}+\nu_-n\B\mu_{-} \end{split} \tag{10.3.2}$ giving the relation $\mu\B=\nu_+\mu_+ + \nu_-\mu_- \tag{10.3.3}$ in place of Eq. 10.2.4. The cations and anions are in the same phase of electric potential $\phi$. We use Eqs. 10.1.4 and 10.1.5 to obtain $\nu_+\mu_+(\phi) + \nu_-\mu_-(\phi) = \nu_+\mu_+(0) + \nu_-\mu_-(0) + (\nu_+z_+ + \nu_-z_-)F\phi \tag{10.3.4}$ Electrical neutrality requires that $(\nu_+z_+ + \nu_-z_-)$ be zero, giving $\mu\B = \nu_+\mu_+(0) + \nu_-\mu_-(0) \tag{10.3.5}$
By combining Eq. 10.3.5 with Eqs. 10.1.10, 10.3.1, and 10.3.3, we obtain $\mu\B = \mu\B\rf + RT \ln \left[ \left(\nu_+^{\nu_+}\nu_-^{\nu_-}\right) \left(\g_+^{\nu_+}\right)\left(\g_-^{\nu_-}\right) \left( \frac{m\B}{m\st} \right)^{\nu} \right] \tag{10.3.6}$ where $\mu\B\rf=\nu_+\mu_+\rf+\nu_-\mu_-\rf$ is the chemical potential of the solute in the hypothetical reference state at $\phi{=}0$ in which B is at the standard molality and behaves as at infinite dilution. Equation 10.3.6 is the generalization of Eq. 10.2.6. It shows that although $\mu_+$ and $\mu_-$ depend on $\phi$, $\mu\B$ does not.
The mean ionic activity coefficient $\g_{\pm}$ is defined in general by $\g_{\pm}^{\nu} = \left( \g_{+}^{\nu_{+}} \right) \left( \g_{-}^{\nu_{-}} \right) \tag{10.3.7}$ or $\g_{\pm} = \left( \g_{+}^{\nu_{+}}\g_{-}^{\nu_{-}} \right)^{1/\nu} \tag{10.3.8}$ Thus $\g_{\pm}$ is a geometric average of $\g_{+}$ and $\g_{-}$ weighted by the numbers of the cations and anions in the solute formula unit. With a substitution from Eq. 10.3.7, Eq. 10.3.6 becomes $\mu\B = \mu\B\rf + RT \ln \left[ \left(\nu_+^{\nu_+}\nu_-^{\nu_-}\right) \g_{\pm}^{\nu} \left( \frac{m\B}{m\st} \right)^{\nu} \right] \tag{10.3.9}$ Since $\mu\B-\mu\B\rf$ is a measurable quantity, so also is $\g_{\pm}$.
The solute activity, defined by $\mu\B=\mu\mbB\st+RT\ln a\mbB$, is $a\mbB = \left( \nu_{+}^{\nu_{+}} \nu_{-}^{\nu_{-}} \right) \G\mbB \g_{\pm}^{\nu} \left( \frac{m\B}{m\st} \right)^{\nu} \tag{10.3.10}$ where $\G\mbB$ is the pressure factor that we can evaluate with Eq. 10.2.12. Equation 10.3.10 is the generalization of Eq. 10.2.10. From Eqs. 10.1.12, 10.1.13, and 10.2.11 and the relations $\mu\B\rf=\nu_+\mu_+\rf+\nu_-\mu_-\rf$ and $\mu\B\st=\nu_+\mu_+\st+\nu_-\mu_-\st$, we obtain the relation $\G\mbB = \G_+^{\nu_+}\G_-^{\nu_-} \tag{10.3.11}$
10.3.2 Multisolute solution
Equation 10.3.3 relates the chemical potential of electrolyte B in a binary solution to the single-ion chemical potentials of its constituent ions: $\mu\B=\nu_+\mu_+ + \nu_-\mu_- \tag{10.3.12}$ This relation is valid for each individual solute substance in a multisolute solution, even when two or more of the electrolyte solutes have an ion species in common.
As an illustration of this principle, consider a solution prepared by dissolving amounts $n\B$ of BaI$_2$ and $n\C$ of CsI in an amount $n\A$ of H$_2$O. Assume the dissolved salts are completely dissociated into ions, with the I$^-$ ion common to both. The additivity rule for the Gibbs energy of this solution can be written in the form $G = n\A\mu\A + n\B\mu\B + n\C\mu\C \tag{10.3.13}$ and also, using single-ion quantities, in the form $G = n\A\mu\A + n\B\mu(\tx{Ba$^{2+}$}) + 2n\B\mu(\tx{I$^-$}) + n\C\mu(\tx{Cs$^+$}) + n\C\mu(\tx{I$^-$}) \tag{10.3.14}$ Comparing Eqs. 10.3.13 and 10.3.14, we find the following relations must exist between the chemical potentials of the solute substances and the ion species: $\mu\B = \mu(\tx{Ba$^{2+}$})+2\mu(\tx{I$^-$}) \qquad \mu\C = \mu(\tx{Cs$^+$})+\mu(\tx{I$^-$}) \tag{10.3.15}$ These relations agree with Eq. 10.3.12. Note that $\mu(\tx{I\(^-$})\), the chemical potential of the ion common to both salts, appears in both relations.
The solute activity $a\mbB$ is defined by the relation $\mu\B = \mu\B\st + RT\ln a\mbB$ (Eq. 10.2.9). Using this relation together with Eqs. 10.1.7 and 10.1.14, we find that the solute activity is related to ion molalities by $a\mbB = \G\mbB \g_{\pm}^\nu \left( \frac{m_+}{m\st} \right)^{\nu_+} \left( \frac{m_-}{m\st} \right)^{\nu_-} \tag{10.3.16}$ where the pressure factor $\G\mbB$ is defined in Eq. 10.2.11. The ion molalities in this expression refer to the constituent ions of solute B, which in a multisolute solution are not necessarily present in the same stoichiometric ratio as in the solute substance.
For instance, suppose we apply Eq. 10.3.16 to the solution of BaI$_2$ and CsI used above as an illustration of a multisolute solution, letting $a\mbB$ be the activity of solute substance BaI$_2$. The quantities $m_+$ and $m_-$ in the equation are then the molalities of the Ba$^{2+}$ and I$^-$ ions, and $\g_{\pm}$ is the mean ionic activity coefficient of the dissolved BaI$_2$. Note that in this solution the Ba$^{2+}$ and I$^-$ ions are not present in the 1:2 ratio found in BaI$_2$, because I$^-$ is a constituent of both solutes.
10.3.3 Incomplete dissociation
In the preceding sections of this chapter, the electrolyte solute or solutes have been assumed to be completely dissociated into their constituent ions at all molalities. Some solutions, however, contain ion pairs—closely associated ions of opposite charge. Furthermore, in solutions of some electrolytes (often called “weak” electrolytes), an equilibrium is established between ions and electrically-neutral molecules. In these kinds of solutions, the relations between solute molality and ion molalities given by Eq. 10.3.1 are no longer valid. When dissociation is not complete, the expression for $\mu\B$ given by Eq. 10.3.9 can still be used. However, the quantity $\g_{\pm}$ appearing in the expression no longer has the physical significance of being the geometric average of the activity coefficients of the actual dissociated ions, and is called the stoichiometric activity coefficient of the electrolyte. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/10%3A_Electrolyte_Solutions/10.03%3A_Electrolytes_in_General.txt |
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The theory of Peter Debye and Erich Hückel (1923) provides theoretical expressions for single-ion activity coefficients and mean ionic activity coefficients in electrolyte solutions. The expressions in one form or another are very useful for extrapolation of quantities that include mean ionic activity coefficients to low solute molality or infinite dilution.
The only interactions the theory considers are the electrostatic interactions between ions. These interactions are much stronger than those between uncharged molecules, and they die off more slowly with distance. If the positions of ions in an electrolyte solution were completely random, the net effect of electrostatic ion–ion interactions would be zero, because each cation–cation or anion–anion repulsion would be balanced by a cation–anion attraction. The positions are not random, however: each cation has a surplus of anions in its immediate environment, and each anion has a surplus of neighboring cations. Each ion therefore has a net attractive interaction with the surrounding ion atmosphere. The result for a cation species at low electrolyte molality is a decrease of $\mu_+$ compared to the cation at same molality in the absence of ion–ion interactions, meaning that the single-ion activity coefficient $\g_+$ becomes less than $1$ as the electrolyte molality is increased beyond the ideal-dilute range. Similarly, $\g_-$ also becomes less than $1$.
According to the Debye–Hückel theory, the single-ion activity coefficient $\g_i$ of ion $i$ in a solution of one or more electrolytes is given by $\ln\g_i = -\frac{A\subs{DH}z_i^2\sqrt{I_m}}{1 + B\subs{DH}a\sqrt{I_m}} \tag{10.4.1}$ where
• The definitions of the quantities $A\subs{DH}$ and $B\subs{DH}$ appearing in Eq. 10.4.1 are $A\subs{DH} \defn \left(N\subs{A}^2 e^3/8\pi\right)\left(2\rho\A^*\right)^{1/2} \left(\epsilon\subs{r}\epsilon_0 RT\right)^{-3/2} \tag{10.4.3}$ $B\subs{DH} \defn N\subs{A}e\left(2\rho\A^*\right)^{1/2} \left(\epsilon\subs{r}\epsilon_0 RT\right)^{-1/2} \tag{10.4.4}$ where $N\subs{A}$ is the Avogadro constant, $e$ is the elementary charge (the charge of a proton), $\rho\A^*$ and $\epsilon\subs{r}$ are the density and relative permittivity (dielectric constant) of the solvent, and $\epsilon_0$ is the electric constant (or permittivity of vacuum).
Lewis and Randall (J. Am. Chem. Soc., 1112–1154, 1921) introduced the term ionic strength, defined by Eq. 10.4.2, two years before the Debye–Hückel theory was published. They found empirically that in dilute solutions, the mean ionic activity coefficient of a given strong electrolyte is the same in all solutions having the same ionic strength.
From Eqs. 10.3.8 and 10.4.1 and the electroneutrality condition $\nu_+ z_+ {=} \nu_- z_-$, we obtain the following expression for the logarithm of the mean ionic activity coefficient of an electrolyte solute: $\ln\g_{\pm} = -\frac{A\subs{DH} \left| z_{+}z_{-} \right| \sqrt{I_m}} {1 + B\subs{DH}a\sqrt{I_m}} \tag{10.4.7}$ In this equation, $z_+$ and $z_-$ are the charge numbers of the cation and anion of the solute. Since the right side of Eq. 10.4.7 is negative at finite solute molalities, and zero at infinite dilution, the theory predicts that $\g_{\pm}$ is less than $1$ at finite solute molalities and approaches $1$ at infinite dilution.
Figure 10.4 shows $\ln\g_{\pm}$ as a function of $\sqrt{I_m}$ for aqueous HCl and CaCl$_2$. The experimental curves have the limiting slopes predicted by the Debye–Hückel limiting law (Eq. 10.4.8), but at a low ionic strength the curves begin to deviate significantly from the linear relations predicted by that law. The full Debye–Hückel equation (Eq. 10.4.7) fits the experimental curves over a wider range of ionic strength. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/10%3A_Electrolyte_Solutions/10.04%3A_The_Debye-Huckel_Theory.txt |
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Debye and Hückel derived Eq. 10.4.1 using a combination of electrostatic theory, statistical mechanical theory, and thermodynamics. This section gives a brief outline of their derivation.
The derivation starts by focusing on an individual ion of species $i$ as it moves through the solution; call it the central ion. Around this central ion, the time-average spatial distribution of any ion species $j$ is not random, on account of the interaction of these ions of species $j$ with the central ion. (Species $i$ and $j$ may be the same or different.) The distribution, whatever it is, must be spherically symmetric about the central ion; that is, a function only of the distance $r$ from the center of the ion. The local concentration, $c'_j$, of the ions of species $j$ at a given value of $r$ depends on the ion charge $z_j e$ and the electric potential $\phi$ at that position. The time-average electric potential in turn depends on the distribution of all ions and is symmetric about the central ion, so expressions must be found for $c'_j$ and $\phi$ as functions of $r$ that are mutually consistent.
Debye and Hückel assumed that $c'_j$ is given by the Boltzmann distribution $c'_j = c_j e^{-z_j e \phi/ kT} \tag{10.5.1}$ where $z_j e\phi$ is the electrostatic energy of an ion of species $j$, and $k$ is the Boltzmann constant ($k = R/N\subs{A}$). As $r$ becomes large, $\phi$ approaches zero and $c'_j$ approaches the macroscopic concentration $c_j$. As $T$ increases, $c'_j$ at a fixed value of $r$ approaches $c_j$ because of the randomizing effect of thermal energy. Debye and Hückel expanded the exponential function in powers of $1/T$ and retained only the first two terms: $c'_j \approx c_j(1 - z_j e\phi/kT)$. The distribution of each ion species is assumed to follow this relation. The electric potential function consistent with this distribution and with the electroneutrality of the solution as a whole is $\phi = (z_i e / 4\pi\epsilon\subs{r}\epsilon_0 r) e^{\kappa(a-r)} / (1 + \kappa a) \tag{10.5.2}$ Here $\kappa$ is defined by $\kappa^2 = 2N\subs{A}^2 e^2 I_c/\epsilon_r \epsilon_0 RT$, where $I_c$ is the ionic strength on a concentration basis defined by $I_c = (1/2)\sum_i c_i z_i^2$.
The electric potential $\phi$ at a point is assumed to be a sum of two contributions: the electric potential the central ion would cause at infinite dilution, $z_i e/4\pi \epsilon_r \epsilon_0 r$, and the electric potential due to all other ions, $\phi'$. Thus, $\phi'$ is equal to $\phi - z_i e/4\pi \epsilon_r \epsilon_0 r$, or $\phi' = (z_i e/4\pi\epsilon\subs{r}\epsilon_0 r) [e^{\kappa(a-r)}/(1+\kappa a)-1] \tag{10.5.3}$ This expression for $\phi'$ is valid for distances from the center of the central ion down to $a$, the distance of closest approach of other ions. At smaller values of $r$, $\phi'$ is constant and equal to the value at $r = a$, which is $\phi'(a) = -(z_i e/4\pi \epsilon_r \epsilon_0)\kappa/(1 + \kappa a)$. The interaction energy between the central ion and the surrounding ions (the ion atmosphere) is the product of the central ion charge and $\phi'(a)$.
The last step of the derivation is the calculation of the work of a hypothetical reversible process in which the surrounding ions stay in their final distribution, and the charge of the central ion gradually increases from zero to its actual value $z_i e$. Let $\alpha z_i e$ be the charge at each stage of the process, where $\alpha$ is a fractional advancement that changes from $0$ to $1$. Then the work $w'$ due to the interaction of the central ion with its ion atmosphere is $\phi'(a)$ integrated over the charge: $\begin{split} w' & = -\int_{\alpha=0}^{\alpha=1} [(\alpha z_i e / 4\pi\epsilon\subs{r}\epsilon_0)\kappa /(1+\kappa a)]\dif(\alpha z_i \epsilon) \cr & = -(z_i^2 e^2/8\pi\epsilon\subs{r}\epsilon_0) \kappa/(1+\kappa a) \end{split} \tag{10.5.4}$ Since the infinitesimal Gibbs energy change in a reversible process is given by $\dif G = -S\dif T + V\difp + \dw'$ (Eq. 5.8.6), this reversible nonexpansion work at constant $T$ and $p$ is equal to the Gibbs energy change. The Gibbs energy change per amount of species $i$ is $w'N\subs{A} = -(z_i^2 e^2 N\subs{A}/8\pi \epsilon\subs{r}\epsilon_0)\kappa/(1 + \kappa a)$. This quantity is $\Del G/n_i$ for the process in which a solution of fixed composition changes from a hypothetical state lacking ion–ion interactions to the real state with ion–ion interactions present. $\Del G/n_i$ may be equated to the difference of the chemical potentials of $i$ in the final and initial states. If the chemical potential without ion–ion interactions is taken to be that for ideal-dilute behavior on a molality basis, $\mu_i=\mu_{m,i}\rf + RT\ln(m_i/m\st)$, then $-(z_i^2 e^2 N\subs{A}/8\pi \epsilon\subs{r}\epsilon_0)\kappa/(1 + \kappa a)$ is equal to $\mu_i - [\mu_{m,i}\rf + RT\ln(m_i/m\st)] = RT\ln\g_{m,i}$. In a dilute solution, $c_i$ can with little error be set equal to $\rho\A^* m_i$, and $I_c$ to $\rho\A^*I_m$. Equation 10.4.1 follows. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/10%3A_Electrolyte_Solutions/10.05%3A_Derivation_of_the_Debye-Huckel_Theory.txt |
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Recall that $\g_{\pm}$ is the mean ionic activity coefficient of a strong electrolyte, or the stoichiometric activity coefficient of an electrolyte that does not dissociate completely.
The general procedure described in this section for evaluating $\g_{\pm}$ requires knowledge of the osmotic coefficient $\phi_m$ as a function of molality. $\phi_m$ is commonly evaluated by the isopiestic method (Sec. 9.6.4) or from measurements of freezing-point depression (Sec. 12.2).
The osmotic coefficient of a binary solution of an electrolyte is defined by \begin{gather} \s{ \phi_m \defn \frac{\mu\A^*-\mu\A}{RTM\A\nu m\B} } \tag{10.6.1} \cond{(binary electrolyte solution)} \end{gather} That is, for an electrolyte the sum $\sum_{i\neq \tx{A}}m_i$ appearing in the definition of $\phi_m$ for a nonelectrolyte solution (Eq. 9.6.11) is replaced by $\nu m\B$, the sum of the ion molalities assuming complete dissociation. It will now be shown that $\phi_m$ defined this way can be used to evaluate $\g_{\pm}$.
The derivation is like that described in Sec. 9.6.3 for a binary solution of a nonelectrolyte. Solving Eq. 10.6.1 for $\mu\A$ and taking the differential of $\mu\A$ at constant $T$ and $p$, we obtain $\dif\mu\A = -RTM\A\nu(\phi_m\dif m\B + m\B\dif\phi_m) \tag{10.6.2}$ From Eq. 10.3.9, we obtain $\dif\mu\B = RT\nu\left(\dif\ln\g_{\pm} + \frac{\dif m\B}{m\B}\right) \tag{10.6.3}$ Substitution of these expressions in the Gibbs–Duhem equation $n\A \dif\mu\A + n\B \dif\mu\B = 0$, together with the substitution $n\A M\A = n\B/m\B$, yields $\dif\ln\g_{\pm} = \dif\phi_m + \frac{\phi_m - 1}{m\B}\dif m\B \tag{10.6.4}$ Then integration from $m\B = 0$ to any desired molality $m'\B$ gives the result $\ln\g_{\pm}(m'\B) = \phi_m (m'\B) - 1 + \int_{0}^{m'\B}\frac{\phi_m - 1}{m\B}\dif m\B \tag{10.6.5}$ The right side of this equation is the same expression as derived for $\ln\g\mbB$ for a nonelectrolyte (Eq. 9.6.20).
The integrand of the integral on the right side of Eq. 10.6.5 approaches $-\infty$ as $m\B$ approaches zero, making it difficult to evaluate the integral by numerical integration starting at $m\B = 0$. (This difficulty does not exist when the solute is a nonelectrolyte.) Instead, we can split the integral into two parts $\int_{0}^{m'\B}\frac{\phi_m - 1}{m\B}\dif m\B = \int_{0}^{m''\B}\frac{\phi_m - 1}{m\B}\dif m\B + \int_{m''\B}^{m'\B}\frac{\phi_m - 1}{m\B}\dif m\B \tag{10.6.6}$ where the integration limit $m''\B$ is a low molality at which the value of $\phi_m$ is available and at which $\g_{\pm}$ can either be measured or estimated from the Debye–Hückel equation.
We next rewrite Eq. 10.6.5 with $m\B'$ replaced with $m\B''$: $\ln\g_{\pm}(m''\B) = \phi_m (m''\B) - 1 + \int_{0}^{m''\B}\frac{\phi_m - 1}{m\B}\dif m\B \tag{10.6.7}$ By eliminating the integral with an upper limit of $m\B''$ from Eqs. 10.6.6 and 10.6.7, we obtain $\int_{0}^{m'\B}\frac{\phi_m - 1}{m\B}\dif m\B = \ln\g_{\pm}(m''\B) - \phi_m (m''\B) + 1 + \int_{m''\B}^{m'\B}\frac{\phi_m - 1}{m\B}\dif m\B \tag{10.6.8}$ Equation 10.6.5 becomes $\ln\g_{\pm}(m'\B) = \phi_m(m'\B) - \phi_m(m''\B) + \ln\g_{\pm}(m''\B) + \int_{m''\B}^{m'\B}\frac{\phi_m - 1}{m\B}\dif m\B \tag{10.6.9}$ The integral on the right side of this equation can easily be evaluated by numerical integration. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/10%3A_Electrolyte_Solutions/10.06%3A_Mean_Ionic_Activity_Coefficients_from_Osmotic_Coefficients.txt |
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An underlined problem number or problem-part letter indicates that the numerical answer appears in Appendix I.
10.1
The mean ionic activity coefficient of NaCl in a 0.100 molal aqueous solution at $298.15\K$ has been evaluated with measurements of equilibrium cell potentials, with the result $\ln\g_{\pm}=-0.2505$ (R. A. Robinson and R. H. Stokes, Electrolyte Solutions, 2nd edition, Butterworths, London, 1959, Table 9.3). Use this value in Eq. 10.6.9, together with the values of osmotic coefficients in Table 10.1, to evaluate $\g_{\pm}$ at each of the molalities shown in the table; then plot $\g_{\pm}$ as a function of $m\B$. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/10%3A_Electrolyte_Solutions/10.07%3A_Chapter_10_Problems.txt |
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$\newcommand{\CVm}{C_{V,\text{m}}} % molar heat capacity at const.V$
$\newcommand{\Cpm}{C_{p,\text{m}}} % molar heat capacity at const.p$
$\newcommand{\kT}{\kappa_T} % isothermal compressibility$
$\newcommand{\A}{_{\text{A}}} % subscript A for solvent or state A$
$\newcommand{\B}{_{\text{B}}} % subscript B for solute or state B$
$\newcommand{\bd}{_{\text{b}}} % subscript b for boundary or boiling point$
$\newcommand{\C}{_{\text{C}}} % subscript C$
$\newcommand{\f}{_{\text{f}}} % subscript f for freezing point$
$\newcommand{\mA}{_{\text{m},\text{A}}} % subscript m,A (m=molar)$
$\newcommand{\mB}{_{\text{m},\text{B}}} % subscript m,B (m=molar)$
$\newcommand{\mi}{_{\text{m},i}} % subscript m,i (m=molar)$
$\newcommand{\fA}{_{\text{f},\text{A}}} % subscript f,A (for fr. pt.)$
$\newcommand{\fB}{_{\text{f},\text{B}}} % subscript f,B (for fr. pt.)$
$\newcommand{\xbB}{_{x,\text{B}}} % x basis, B$
$\newcommand{\xbC}{_{x,\text{C}}} % x basis, C$
$\newcommand{\cbB}{_{c,\text{B}}} % c basis, B$
$\newcommand{\mbB}{_{m,\text{B}}} % m basis, B$
$\newcommand{\kHi}{k_{\text{H},i}} % Henry's law constant, x basis, i$
$\newcommand{\kHB}{k_{\text{H,B}}} % Henry's law constant, x basis, B$
$\newcommand{\arrow}{\,\rightarrow\,} % right arrow with extra spaces$
$\newcommand{\arrows}{\,\rightleftharpoons\,} % double arrows with extra spaces$
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$\newcommand{\eq}{\subs{eq}} % equilibrium state$
$\newcommand{\onehalf}{\textstyle\frac{1}{2}\D} % small 1/2 for display equation$
$\newcommand{\sys}{\subs{sys}} % system property$
$\newcommand{\sur}{\sups{sur}} % surroundings$
$\renewcommand{\in}{\sups{int}} % internal$
$\newcommand{\lab}{\subs{lab}} % lab frame$
$\newcommand{\cm}{\subs{cm}} % center of mass$
$\newcommand{\rev}{\subs{rev}} % reversible$
$\newcommand{\irr}{\subs{irr}} % irreversible$
$\newcommand{\fric}{\subs{fric}} % friction$
$\newcommand{\diss}{\subs{diss}} % dissipation$
$\newcommand{\el}{\subs{el}} % electrical$
$\newcommand{\cell}{\subs{cell}} % cell$
$\newcommand{\As}{A\subs{s}} % surface area$
$\newcommand{\E}{^\mathsf{E}} % excess quantity (superscript)$
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$\newcommand{\dBar}{\mathop{}\!\mathrm{d}\hspace-.3em\raise1.05ex{\Rule{.8ex}{.125ex}{0ex}}} % inexact differential$
$\newcommand{\dq}{\dBar q} % heat differential$
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$\newcommand{\dotprod}{\small\bullet}$
$\newcommand{\fug}{f} % fugacity$
$\newcommand{\g}{\gamma} % solute activity coefficient, or gamma in general$
$\newcommand{\G}{\varGamma} % activity coefficient of a reference state (pressure factor)$
$\newcommand{\ecp}{\widetilde{\mu}} % electrochemical or total potential$
$\newcommand{\Eeq}{E\subs{cell, eq}} % equilibrium cell potential$
$\newcommand{\Ej}{E\subs{j}} % liquid junction potential$
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This chapter discusses the thermodynamics of mixing processes and processes described by reaction equations (chemical equations). It introduces the important concepts of molar mixing and reaction quantities, advancement, and the thermodynamic equilibrium constant. The focus is on chemical processes that take place in closed systems at constant pressure, with no work other than expansion work. Under these conditions, the enthalpy change is equal to the heat (Eq. 5.3.7). The processes either take place at constant temperature, or have initial and final states of the same temperature.
Most of the processes to be described involve mixtures and have intermediate states that are nonequilibrium states. At constant temperature and pressure, these processes proceed spontaneously with decreasing Gibbs energy (Sec. 5.8). (Processes in which $G$ decreases are sometimes called exergonic.) When the rates of change are slow enough for thermal and mechanical equilibrium to be maintained, the spontaneity is due to lack of transfer equilibrium or reaction equilibrium. An equilibrium phase transition of a pure substance, however, is a special case: it is a reversible process of constant Gibbs energy (Sec. 8.3).
11: Reactions and Other Chemical Processes
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$\newcommand{\mA}{_{\text{m},\text{A}}} % subscript m,A (m=molar)$
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$\newcommand{\cbB}{_{c,\text{B}}} % c basis, B$
$\newcommand{\mbB}{_{m,\text{B}}} % m basis, B$
$\newcommand{\kHi}{k_{\text{H},i}} % Henry's law constant, x basis, i$
$\newcommand{\kHB}{k_{\text{H,B}}} % Henry's law constant, x basis, B$
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$\renewcommand{\in}{\sups{int}} % internal$
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$\newcommand{\cm}{\subs{cm}} % center of mass$
$\newcommand{\rev}{\subs{rev}} % reversible$
$\newcommand{\irr}{\subs{irr}} % irreversible$
$\newcommand{\fric}{\subs{fric}} % friction$
$\newcommand{\diss}{\subs{diss}} % dissipation$
$\newcommand{\el}{\subs{el}} % electrical$
$\newcommand{\cell}{\subs{cell}} % cell$
$\newcommand{\As}{A\subs{s}} % surface area$
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$\newcommand{\bPd}[3]{\left[ \dfrac {\partial #1} {\partial #2}\right]_{#3}}$
$\newcommand{\dotprod}{\small\bullet}$
$\newcommand{\fug}{f} % fugacity$
$\newcommand{\g}{\gamma} % solute activity coefficient, or gamma in general$
$\newcommand{\G}{\varGamma} % activity coefficient of a reference state (pressure factor)$
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$\newcommand{\Eeq}{E\subs{cell, eq}} % equilibrium cell potential$
$\newcommand{\Ej}{E\subs{j}} % liquid junction potential$
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$\newcommand{\cond}[1]{\$-2.5pt]{}\tag*{#1}}$ $\newcommand{\nextcond}[1]{\[-5pt]{}\tag*{#1}}$ $\newcommand{\R}{8.3145\units{J\,K\per\,mol\per}} % gas constant value$ $\newcommand{\Rsix}{8.31447\units{J\,K\per\,mol\per}} % gas constant value - 6 sig figs$ $\newcommand{\jn}{\hspace3pt\lower.3ex{\Rule{.6pt}{2ex}{0ex}}\hspace3pt}$ $\newcommand{\ljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}} \hspace3pt}$ $\newcommand{\lljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace1.4pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace3pt}$ Many of the processes of interest to chemists can be described by balanced reaction equations, or chemical equations, for the conversion of reactants into products. Thus, for the vaporization of water we write \[ \ce{H2O}\tx{(l)} \arrow \ce{H2O}\tx{(g)}$ For the dissolution of sodium chloride in water, we write $\ce{NaCl}\tx{(s)} \arrow \ce{Na+}\tx{(aq)} + \ce{Cl-}\tx{(aq)}$ For the Haber synthesis of ammonia, the reaction equation can be written \begin{equation*} \ce{N2}\tx{(g)} + \ce{3H2}\tx{(g)} \arrow \ce{2NH3}\tx{(g)} \end{equation*}
The essential feature of a reaction equation is that equal amounts of each element and equal net charges appear on both sides; the equation is said to be balanced. Thus, matter and charge are conserved during the process, and the process can take place in a closed system. The species to the left of a single arrow are called reactants, the species to the right are called products, and the arrow indicates the forward direction of the process.
A reaction equation is sometimes written with right and left arrows $\ce{N2}\tx{(g)} + \ce{3H2}\tx{(g)} \arrows \ce{2NH3}\tx{(g)}$ to indicate that the process is at reaction equilibrium. It can also be written as a stoichiometric equation with an equal sign: $\ce{N2}\tx{(g)} + \ce{3H2}\tx{(g)} = \ce{2NH3}\tx{(g)}$
A reaction equation shows stoichiometric relations among the reactants and products. It is important to keep in mind that it specifies neither the initial and final states of a chemical process, nor the change in the amount of a reactant or product during the process. For example, the reaction equation N$_2$ + 3 H$_2$ $\ra$ 2 NH$_3$ does not imply that the system initially contains only N$_2$ and H$_2$, or that only NH$_3$ is present in the final state; and it does not mean that the process consists of the conversion of exactly one mole of N$_2$ and three moles of H$_2$ to two moles of NH$_3$ (although this is a possibility). Instead, the reaction equation tells us that a change in the amount of N$_2$ is accompanied by three times this change in the amount of H$_2$ and by twice this change, with the opposite sign, in the amount of NH$_3$.
11.2.1 An example: ammonia synthesis
It is convenient to indicate the progress of a chemical process with a variable called the advancement. The reaction equation
N$_2$ + 3 H$_2$ $\ra$ 2 NH$_3$ for the synthesis of ammonia synthesis will serve to illustrate this concept. Let the system be a gaseous mixture of N$_2$, H$_2$, and NH$_3$.
If the system is open and the intensive properties remain uniform throughout the gas mixture, there are five independent variables. We can choose them to be $T$, $p$, and the amounts of the three substances. We can write the total differential of the enthalpy, for instance, as $\begin{split} \dif H & = \Pd{H}{T}{p,\allni}\dif T + \Pd{H}{p}{T,\allni}\difp \cr & \quad + H\subs{N$_2$}\dif n\subs{N$_2$} + H\subs{H$_2$}\dif n\subs{H$_2$} + H\subs{NH$_3$}\dif n\subs{NH$_3$} \end{split} \tag{11.2.1}$ The notation $\allni$ stands for the set of amounts of all substances in the mixture, and the quantities $H\subs{N\(_2$}\), $H\subs{H\(_2$}\), and $H\subs{NH\(_3$}\) are partial molar enthalpies. For example, $H\subs{N\(_2$}\) is defined by $H\subs{N$_2$} = \Pd{H}{n\subs{N$_2$}}{T, p, n\subs{H$_2$}, n\subs{NH$_3$}} \tag{11.2.2}$
If the system is closed, the amounts of the three substances can still change because of the reaction N$_2$ + 3 H$_2$ $\ra$ 2 NH$_3$, and the number of independent variables is reduced from five to three. We can choose them to be $T$, $p$, and a variable called advancement.
The advancement (or extent of reaction), $\xi$, is the amount by which the reaction defined by the reaction equation has advanced in the forward direction from specified initial conditions. The quantity $\xi$ has dimensions of amount of substance, the usual unit being the mole.
Let the initial amounts be $n_{\tx{N}_2,0}$, $n_{\tx{H}_2,0}$, and $n_{\tx{NH}_3,0}$. Then at any stage of the reaction process in the closed system, the amounts are given by $n\subs{N$_2$} = n_{\tx{N}_2,0} - \xi \qquad n\subs{H$_2$} = n_{\tx{H}_2,0} - 3\xi \qquad n\subs{NH$_3$} = n_{\tx{NH}_3,0} + 2\xi \tag{11.2.3}$ These relations come from the stoichiometry of the reaction as expressed by the stoichiometric coefficients in the reaction equation. The second relation, for example, expresses the fact that when one mole of reaction has occurred ($\xi=1\mol$), the amount of H$_2$ in the closed system has decreased by three moles.
Taking the differentials of Eqs. 11.2.3, we find that infinitesimal changes in the amounts are related to the change of $\xi$ as follows: $\dif n\subs{N$_2$} = - \dif\xi \qquad \dif n\subs{H$_2$} = - 3\dif\xi \qquad \dif n\subs{NH$_3$} = 2\dif\xi \tag{11.2.4}$ These relations show that in a closed system, the changes in the various amounts are not independent. Substitution in Eq. 11.2.1 of the expressions for $\dif n\subs{N\(_2$}\), $\dif n\subs{H\(_2$}\), and $\dif n\subs{NH\(_3$}\) gives \begin{gather} \s{ \begin{split} \dif H & = \Pd{H}{T}{p, \xi}\dif T + \Pd{H}{p}{T, \xi}\difp \cr & \quad + \left( -H\subs{N$_2$} - 3H\subs{H$_2$} + 2H\subs{NH$_3$} \right)\dif\xi \end{split} } \tag{11.2.5} \cond{(closed system)} \end{gather} (The subscript $\allni$ on the partial derivatives has been replaced by $\xi$ to indicate the same thing: that the derivative is taken with the amount of each species held constant.)
Equation 11.2.5 gives an expression for the total differential of the enthalpy with $T$, $p$, and $\xi$ as the independent variables. The coefficient of $\dif \xi$ in this equation is called the molar reaction enthalpy, or molar enthalpy of reaction, $\Delsub{r}H$: $\Delsub{r}H = -H\subs{N$_2$} - 3H\subs{H$_2$} + 2H\subs{NH$_3$} \tag{11.2.6}$ We identify this coefficient as the partial derivative $\Delsub{r}H = \Pd{H}{\xi}{T, p} \tag{11.2.7}$ That is, the molar reaction enthalpy is the rate at which the enthalpy changes with the advancement as the reaction proceeds in the forward direction at constant $T$ and $p$.
The partial molar enthalpy of a species is the enthalpy change per amount of the species added to an open system. To see why the particular combination of partial molar enthalpies on the right side of Eq. 11.2.6 is the rate at which enthalpy changes with advancement in the closed system, we can imagine the following process at constant $T$ and $p$: An infinitesimal amount $\dif n$ of N$_2$ is removed from an open system, three times this amount of H$_2$ is removed from the same system, and twice this amount of NH$_3$ is added to the system. The total enthalpy change in the open system is $\dif H = (-H\subs{N\(_2$} - 3H\subs{H$_2$} + 2H\subs{NH$_3$})\dif n\). The net change in the state of the system is equivalent to an advancement $\dif\xi = \dif n$ in a closed system, so $\dif H/\dif\xi$ in the closed system is equal to $(-H\subs{N\(_2$} - 3H\subs{H$_2$} + 2H\subs{NH$_3$})\) in agreement with Eqs. 11.2.6 and 11.2.7.
Note that because the advancement is defined by how we write the reaction equation, the value of $\Delsub{r}H$ also depends on the reaction equation. For instance, if we change the reaction equation for ammonia synthesis from N$_2$ + 3 H$_2$ $\ra$ 2 NH$_3$ to $\ce{1/2N2} + \ce{3/2H2} \arrow \ce{NH3}$ then the value of $\Delsub{r}H$ is halved.
11.2.2 Molar reaction quantities in general
Now let us generalize the relations of the preceding section for any chemical process in a closed system. Suppose the stoichiometric equation has the form $a\tx{A} + b\tx{B} = d\tx{D} + e\tx{E} \tag{11.2.8}$ where A and B are reactant species, D and E are product species, and $a$, $b$, $d$, and $e$ are the corresponding stoichiometric coefficients. We can rearrange this equation to $0 = - a\tx{A} - b\tx{B} + d\tx{D} + e\tx{E} \tag{11.2.9}$
In general, the stoichiometric relation for any chemical process is $0 = \sum_i\nu_i \tx{A}_i \tag{11.2.10}$ where $\nu_i$ is the stoichiometric number of species A$_i$, a dimensionless quantity taken as negative for a reactant and positive for a product. In the ammonia synthesis example of the previous section, the stoichiometric relation is $0 = -\tx{N}_2 - 3\tx{H}_2 + 2\tx{NH}_3$ and the stoichiometric numbers are $\nu\subs{N\(_2$} = -1\), $\nu\subs{H\(_2$} = -3\), and $\nu\subs{NH\(_3$} = +2\). In other words, each stoichiometric number is the same as the stoichiometric coefficient in the reaction equation, except that the sign is negative for a reactant.
The amount of reactant or product species $i$ present in the closed system at any instant depends on the advancement at that instant, and is given by \begin{gather} \s{ n_i = n_{i,0} + \nu_i\xi } \tag{11.2.11} \cond{(closed system)} \end{gather} The infinitesimal change in the amount due to an infinitesimal change in the advancement is \begin{gather} \s{ \dif n_i = \nu_i\dif\xi } \tag{11.2.12} \cond{(closed system)} \end{gather}
In an open system, the total differential of extensive property $X$ is $\dif X = \Pd{X}{T}{p, \allni}\dif T + \Pd{X}{p}{T, \allni}\difp + \sum_i X_i\dif n_i \tag{11.2.13}$ where $X_i$ is a partial molar quantity. We restrict the system to a closed one with $T$, $p$, and $\xi$ as the independent variables. Then, with the substitution $\dif n_i = \nu_i\dif\xi$ from Eq. 11.2.12, the total differential of $X$ becomes \begin{gather} \s{ \dif X = \Pd{X}{T}{p, \xi}\dif T + \Pd{X}{p}{T, \xi}\difp + \Delsub{r}X\dif\xi } \tag{11.2.14} \cond{(closed system)} \end{gather} where the coefficient $\Delsub{r}X$ is the molar reaction quantity defined by $\Delsub{r}X \defn \sum_i\nu_i X_i \tag{11.2.15}$ Equation 11.2.14 allows us to identify the molar reaction quantity as a partial derivative: \begin{gather} \s{ \Delsub{r}X = \Pd{X}{\xi}{T, p} } \tag{11.2.16} \cond{(closed system)} \end{gather}
It is important to observe the distinction between the notations $\Del X$, the finite change of $X$ during a process, and $\Delsub{r}X$, a differential quantity that is a property of the system in a given state. The fact that both notations use the symbol $\Del$ can be confusing. Equation 11.2.16 shows that we can think of $\Delsub{r}$ as an operator.
In dealing with the change of an extensive property $X$ as $\xi$ changes, we must distinguish between molar integral and molar differential reaction quantities.
• $\Del X/\Del\xi$ is a molar integral reaction quantity, the ratio of two finite differences between the final and initial states of a process. These states are assumed to have the same temperature and the same pressure. This e-book will use a notation such as $\Del H\m\rxn$ for a molar integral reaction enthalpy: \begin{gather} \s{ \Del H\m\rxn = \frac{\Del H\rxn}{\Del\xi} = \frac{H(\xi_2)-H(\xi_1)}{\xi_2 - \xi_1} } \tag{11.2.17} \cond{($T_2{=}T_1,p_2{=}p_1$)} \end{gather}
• $\Delsub{r}X$ is a molar differential reaction quantity. Equation 11.2.16 shows that $\Delsub{r}X$ is the rate at which the extensive property $X$ changes with the advancement in a closed system at constant $T$ and $p$. The value of $\Delsub{r}X$ is in general a function of the independent variables $T$, $p$, and $\xi$.
The notation for a molar differential reaction quantity such as $\Delsub{r}H$ includes a subscript following the $\Del$ symbol to indicate the kind of chemical process. The subscript “r” denotes a reaction or process in general. The meanings of “vap,” “sub,” “fus,” and “trs” were described in Sec. 8.3.1. Subscripts for specific kinds of reactions and processes are listed in Sec. D.2 of Appendix D and are illustrated in sections to follow.
For certain kinds of processes, it may happen that a partial molar quantity $X_i$ remains constant for each species $i$ as the process advances at constant $T$ and $p$. If $X_i$ remains constant for each $i$, then according to Eq. 11.2.15 the value of $\Delsub{r}X$ must also remain constant as the process advances. Since $\Delsub{r}X$ is the rate at which $X$ changes with $\xi$, in such a situation $X$ is a linear function of $\xi$. This means that the molar integral reaction quantity $\Del X\m\rxn$ defined by $\Del X/\Del\xi$ is equal, for any finite change of $\xi$, to $\Delsub{r}X$.
Figure 11.6 Enthalpy and entropy as functions of advancement at constant $T$ and $p$. The curves are for a reaction A$\rightarrow$2B with positive $\Delsub{r}H$ taking place in an ideal gas mixture with initial amounts $n_{\tx{A},0}=1\mol$ and $n_{\tx{B},0}=0$.
An example is the partial molar enthalpy $H_i$ of a constituent of an ideal gas mixture, an ideal condensed-phase mixture, or an ideal-dilute solution. In these ideal mixtures, $H_i$ is independent of composition at constant $T$ and $p$ (Secs. 9.3.3, 9.4.3, and 9.4.7). When a reaction takes place at constant $T$ and $p$ in one of these mixtures, the molar differential reaction enthalpy $\Delsub{r}H$ is constant during the process, $H$ is a linear function of $\xi$, and $\Delsub{r}H$ and $\Del H\m\rxn$ are equal. Figure 11.6(a) illustrates this linear dependence for a reaction in an ideal gas mixture.
In contrast, Fig. 11.6(b) shows the nonlinearity of the entropy as a function of $\xi$ during the same reaction. The nonlinearity is a consequence of the dependence of the partial molar entropy $S_i$ on the mixture composition (Eq. 11.1.24). In the figure, the slope of the curve at each value of $\xi$ equals $\Delsub{r}S$ at that point; its value changes as the reaction advances and the composition of the reaction mixture changes. Consequently, the molar integral reaction entropy $\Del S\m\rxn=\Del S\rxn/\Del\xi$ approaches the value of $\Delsub{r}S$ only in the limit as $\Del\xi$ approaches zero.
11.2.3 Standard molar reaction quantities
If a chemical process takes place at constant temperature while each reactant and product remains in its standard state of unit activity, the molar reaction quantity $\Delsub{r}X$ is called the standard molar reaction quantity and is denoted by $\Delsub{r}X\st$. For instance, $\Delsub{vap}H\st$ is a standard molar enthalpy of vaporization (already discussed in Sec. 8.3.3), and $\Delsub{r}G\st$ is the standard molar Gibbs energy of a reaction.
From Eq. 11.2.15, the relation between a standard molar reaction quantity and the standard molar quantities of the reactants and products at the same temperature is $\Delsub{r}X\st \defn \sum_i\nu_i X_i\st \tag{11.2.18}$
Two comments are in order.
1. Whereas a molar reaction quantity is usually a function of $T$, $p$, and $\xi$, a standard molar reaction quantity is a function only of $T$. This is evident because standard-state conditions imply that each reactant and product is in a separate phase of constant defined composition and constant pressure $p\st$.
2. Since the value of a standard molar reaction quantity is independent of $\xi$, the standard molar integral and differential quantities are identical (Sec. 11.2.2): $\Del X\m\st\rxn = \Delsub{r}X\st \tag{11.2.19}$
These general concepts will now be applied to some specific chemical processes. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/11%3A_Reactions_and_Other_Chemical_Processes/11.02%3A_The_Advancement_and_Molar_Reaction_Quantities.txt |
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$\newcommand{\cond}[1]{\$-2.5pt]{}\tag*{#1}}$ $\newcommand{\nextcond}[1]{\[-5pt]{}\tag*{#1}}$ $\newcommand{\R}{8.3145\units{J\,K\per\,mol\per}} % gas constant value$ $\newcommand{\Rsix}{8.31447\units{J\,K\per\,mol\per}} % gas constant value - 6 sig figs$ $\newcommand{\jn}{\hspace3pt\lower.3ex{\Rule{.6pt}{2ex}{0ex}}\hspace3pt}$ $\newcommand{\ljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}} \hspace3pt}$ $\newcommand{\lljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace1.4pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace3pt}$ Recall that $\Del H\m\rxn$ is a molar integral reaction enthalpy equal to $\Del H\rxn/\Del\xi$, and that $\Delsub{r}H$ is a molar differential reaction enthalpy defined by $\sum_i\!\nu_i H_i$ and equal to $\pd{H}{\xi}{T,p}$. 11.3.1 Molar reaction enthalpy and heat During a process in a closed system at constant pressure with expansion work only, the enthalpy change equals the energy transferred across the boundary in the form of heat: $\dif H=\dq$ (Eq. 5.3.7). Thus for the molar reaction enthalpy $\Delsub{r}H = \pd{H}{\xi}{T,p}$, which refers to a process not just at constant pressure but also at constant temperature, we can write \begin{gather} \s{ \Delsub{r}H = \frac{\dq}{\dif\xi} } \tag{11.3.1} \cond{(constant $T$ and $p$, $\dw'{=}0$)} \end{gather} Note that when there is nonexpansion work ($w'$), such as electrical work, the enthalpy change is not equal to the heat. For example, if we compare a reaction taking place in a galvanic cell with the same reaction in a reaction vessel, the heats at constant $T$ and $p$ for a given change of $\xi$ are different, and may even have opposite signs. The value of $\Delsub{r}H$ is the same in both systems, but the ratio of heat to advancement, $\dq/\dif\xi$, is different. An exothermic reaction is one for which $\Delsub{r}H$ is negative, and an endothermic reaction is one for which $\Delsub{r}H$ is positive. Thus in a reaction at constant temperature and pressure with expansion work only, heat is transferred out of the system during an exothermic process and into the system during an endothermic process. If the process takes place at constant pressure in a system with thermally-insulated walls, the temperature increases during an exothermic process and decreases during an endothermic process. These comments apply not just to chemical reactions, but to the other chemical processes at constant temperature and pressure discussed in this chapter. 11.3.2 Standard molar enthalpies of reaction and formation A standard molar reaction enthalpy, $\Delsub{r}H\st$, is the same as the molar integral reaction enthalpy $\Del H\m\rxn$ for the reaction taking place under standard state conditions (each reactant and product at unit activity) at constant temperature. At constant temperature, partial molar enthalpies depend only mildly on pressure. It is therefore usually safe to assume that unless the experimental pressure is much greater than $p\st$, the reaction is exothermic if $\Delsub{r}H\st$ is negative and endothermic if $\Delsub{r}H\st$ is positive. The formation reaction of a substance is the reaction in which the substance, at a given temperature and in a given physical state, is formed from the constituent elements in their reference states at the same temperature. The reference state of an element is usually chosen to be the standard state of the element in the allotropic form and physical state that is stable at the given temperature and the standard pressure. For instance, at $298.15\K$ and $1\br$ the stable allotrope of carbon is crystalline graphite rather than diamond. Phosphorus is an exception to the rule regarding reference states of elements. Although red phosphorus is the stable allotrope at $298.15\K$, it is not well characterized. Instead, the reference state is white phosphorus (crystalline P$_4$) at $1\br$. At $298.15\K$, the reference states of the elements are the following: • The standard molar enthalpy of formation (or standard molar heat of formation), $\Delsub{f}H\st$, of a substance is the enthalpy change per amount of substance produced in the formation reaction of the substance in its standard state. Thus, the standard molar enthalpy of formation of gaseous methyl bromide at $298.15\K$ is the molar reaction enthalpy of the reaction \[ \textstyle \tx{C(s, graphite, $p\st$)} + \frac{3}{2}\tx{H$_2$(ideal gas, $p\st$)} + \frac{1}{2}\tx{Br$_2$(l, $p\st$)} \arrow \tx{CH$_3$Br(ideal gas, $p\st$)}$ The value of $\Delsub{f}H\st$ for a given substance depends only on $T$. By definition, $\Delsub{f}H\st$ for the reference state of an element is zero.
A principle called Hess’s law can be used to calculate the standard molar enthalpy of formation of a substance at a given temperature from standard molar reaction enthalpies at the same temperature, and to calculate a standard molar reaction enthalpy from tabulated values of standard molar enthalpies of formation. The principle is an application of the fact that enthalpy is a state function. Therefore, $\Del H$ for a given change of the state of the system is independent of the path and is equal to the sum of $\Del H$ values for any sequence of changes whose net result is the given change. (We may apply the same principle to a change of any state function.)
This value is one of the many standard molar enthalpies of formation to be found in compilations of thermodynamic properties of individual substances, such as the table in Appendix H. We may use the tabulated values to evaluate the standard molar reaction enthalpy $\Delsub{r}H\st$ of a reaction using a formula based on Hess’s law. Imagine the reaction to take place in two steps: First each reactant in its standard state changes to the constituent elements in their reference states (the reverse of a formation reaction), and then these elements form the products in their standard states. The resulting formula is \begin{gather} \s{ \Delsub{r}H\st = \sum_i\nu_i \Delsub{f}H\st(i) } \tag{11.3.3} \cond{(Hess’s law)} \end{gather} where $\Delsub{f}H\st(i)$ is the standard molar enthalpy of formation of substance $i$. Recall that the stoichiometric number $\nu_i$ of each reactant is negative and that of each product is positive, so according to Hess’s law the standard molar reaction enthalpy is the sum of the standard molar enthalpies of formation of the products minus the sum of the standard molar enthalpies of formation of the reactants. Each term is multiplied by the appropriate stoichiometric coefficient from the reaction equation.
A standard molar enthalpy of formation can be defined for a solute in solution to use in Eq. 11.3.3. For instance, the formation reaction of aqueous sucrose is $\textstyle \tx{12 C(s, graphite)} + \tx{11 H$_2$(g)} + \frac{11}{2}\tx{O$_2$(g)} \arrow \tx{C$_{12}$H$_{22}$O$_{11}$(aq)}$ and $\Delsub{f}H\st$ for C$_{12}$H$_{22}$O$_{11}$(aq) is the enthalpy change per amount of sucrose formed when the reactants and product are in their standard states. Note that this formation reaction does not include the formation of the solvent H$_2$O from H$_2$ and O$_2$. Instead, the solute once formed combines with the amount of pure liquid water needed to form the solution. If the aqueous solute is formed in its standard state, the amount of water needed is very large so as to have the solute exhibit infinite-dilution behavior.
There is no ordinary reaction that would produce an individual ion in solution from its element or elements without producing other species as well. We can, however, prepare a consistent set of standard molar enthalpies of formation of ions by assigning a value to a single reference ion. ({This procedure is similar to that described in Sec. 9.2.4 for partial molar volumes of ions.) We can use these values for ions in Eq. 11.3.3 just like values of $\Delsub{f}H\st$ for substances and nonionic solutes. Aqueous hydrogen ion is the usual reference ion, to which is assigned the arbitrary value $\Delsub{f}H\st\tx{(H$^+$, aq)} = 0 \qquad \tx{(at all temperatures)} \tag{11.3.4}$
To see how we can use this reference value, consider the reaction for the formation of aqueous HCl (hydrochloric acid): \begin{equation*} \ce{1/2H2}\tx{(g)} + \ce{1/2Cl2}\tx{(g)} \arrow \ce{H+}\tx{(aq)} + \ce{Cl-}\tx{(aq)} \end{equation*} The standard molar reaction enthalpy at $298.15\K$ for this reaction is known, from reaction calorimetry, to have the value $\Delsub{r}H\st = -167.08\units{kJ mol\(^{-1}$}\). The standard states of the gaseous H$_2$ and Cl$_2$ are, of course, the pure gases acting ideally at pressure $p\st$, and the standard state of each of the aqueous ions is the ion at the standard molality and standard pressure, acting as if its activity coefficient on a molality basis were $1$. From Eq. 11.3.3, we equate the value of $\Delsub{r}H\st$ to the sum $-\onehalf\Delsub{f}H\st\tx{(H$_2$, g)} -\onehalf\Delsub{f}H\st\tx{(Cl$_2$, g)} + \Delsub{f}H\st\tx{(H$^+$, aq)} + \Delsub{f}H\st\tx{(Cl$^-$, aq)}$ But the first three terms of this sum are zero. Therefore, the value of $\Delsub{f}H\st$(Cl$^-$, aq) is $-167.08\units{kJ mol\(^{-1}$}\).
Next we can combine this value of $\Delsub{f}H\st$(Cl$^-$, aq) with the measured standard molar enthalpy of formation of aqueous sodium chloride $\ce{Na}\tx{(s)} + \ce{1/2Cl2}\tx{(g)} \arrow \ce{Na+}\tx{(aq)} + \ce{Cl-}\tx{(aq)}$ to evaluate the standard molar enthalpy of formation of aqueous sodium ion. By continuing this procedure with other reactions, we can build up a consistent set of $\Delsub{f}H\st$ values of various ions in aqueous solution.
11.3.3 Molar reaction heat capacity
The molar reaction enthalpy $\Delsub{r}H$ is in general a function of $T$, $p$, and $\xi$. Using the relations $\Delsub{r}H=\sum_i\!\nu_i H_i$ (from Eq. 11.2.15) and $C_{p,i}=\pd{H_i}{T}{p, \xi}$ (Eq. 9.2.52), we can write $\Pd{\Delsub{r}H}{T}{p, \xi} = \Pd{\sum_i\nu_i H_i}{T}{p, \xi} = \sum_i\nu_i C_{p,i} = \Delsub{r}C_p \tag{11.3.5}$ where $\Delsub{r}C_p$ is the molar reaction heat capacity at constant pressure, equal to the rate at which the heat capacity $C_p$ changes with $\xi$ at constant $T$ and $p$.
Under standard state conditions, Eq. 11.3.5 becomes $\dif\Delsub{r}H\st/\dif T = \Delsub{r}C_p\st \tag{11.3.6}$
11.3.4 Effect of temperature on reaction enthalpy
Consider a reaction occurring with a certain finite change of the advancement in a closed system at temperature $T'$ and at constant pressure. The reaction is characterized by a change of the advancement from $\xi_1$ to $\xi_2$, and the integral reaction enthalpy at this temperature is denoted $\Del H\tx{(rxn, \(T'$)}\). We wish to find an expression for the reaction enthalpy $\Del H\tx{(rxn, \(T''$)}\) for the same values of $\xi_1$ and $\xi_2$ at the same pressure but at a different temperature, $T''$.
The heat capacity of the system at constant pressure is related to the enthalpy by Eq. 5.6.3: $C_p=\pd{H}{T}{p, \xi}$. We integrate $\dif H=C_p\dif T$ from $T'$ to $T''$ at constant $p$ and $\xi$, for both the final and initial values of the advancement: $H(\xi_2, T'') = H(\xi_2, T') + \int_{T'}^{T''}\!\!C_p(\xi_2)\dif T \tag{11.3.7}$ $H(\xi_1, T'') = H(\xi_1, T') + \int_{T'}^{T''}\!\!C_p(\xi_1)\dif T \tag{11.3.8}$ Subtracting Eq. 11.3.8 from Eq. 11.3.7, we obtain $\Del H\tx{(rxn, $T''$)} = \Del H\tx{(rxn, $T'$)} + \int_{T'}^{T''}\!\!\!\Del C_p\dif T \tag{11.3.9}$ where $\Del C_p$ is the difference between the heat capacities of the system at the final and initial values of $\xi$, a function of $T$: $\Del C_p = C_p(\xi_2)-C_p(\xi_1)$. Equation 11.3.9 is the Kirchhoff equation.
When $\Del C_p$ is essentially constant in the temperature range from $T'$ to $T''$, the Kirchhoff equation becomes $\Del H\tx{(rxn, $T''$)} = \Del H\tx{(rxn, $T'$)} + \Del C_p(T''-T') \tag{11.3.10}$
Figure 11.7 illustrates the principle of the Kirchhoff equation as expressed by Eq. 11.3.10. $\Del C_p$ equals the difference in the slopes of the two dashed lines in the figure, and the product of $\Del C_p$ and the temperature difference $T''-T'$ equals the change in the value of $\Del H\rxn$. The figure illustrates an exothermic reaction with negative $\Del C_p$, resulting in a more negative value of $\Del H\rxn$ at the higher temperature.
We can also find the effect of temperature on the molar differential reaction enthalpy $\Delsub{r}H$. From Eq. 11.3.5, we have $\pd{\Delsub{r}H}{T}{p, \xi} = \Delsub{r}C_p$. Integration from temperature $T'$ to temperature $T''$ yields the relation $\Delsub{r}H(T''\!,\xi)=\Delsub{r}H(T'\!,\xi) + \int_{T'}^{T''}\!\!\Delsub{r}C_p(T,\xi)\dif T \tag{11.3.11}$ This relation is analogous to Eq. 11.3.9, using molar differential reaction quantities in place of integral reaction quantities. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/11%3A_Reactions_and_Other_Chemical_Processes/11.03%3A_Molar_Reaction_Enthalpy.txt |
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$\newcommand{\V}{\units{V}} % volts$
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$\newcommand{\per}{^{-1}} % minus one power$
$\newcommand{\m}{_{\text{m}}} % subscript m for molar quantity$
$\newcommand{\CVm}{C_{V,\text{m}}} % molar heat capacity at const.V$
$\newcommand{\Cpm}{C_{p,\text{m}}} % molar heat capacity at const.p$
$\newcommand{\kT}{\kappa_T} % isothermal compressibility$
$\newcommand{\A}{_{\text{A}}} % subscript A for solvent or state A$
$\newcommand{\B}{_{\text{B}}} % subscript B for solute or state B$
$\newcommand{\bd}{_{\text{b}}} % subscript b for boundary or boiling point$
$\newcommand{\C}{_{\text{C}}} % subscript C$
$\newcommand{\f}{_{\text{f}}} % subscript f for freezing point$
$\newcommand{\mA}{_{\text{m},\text{A}}} % subscript m,A (m=molar)$
$\newcommand{\mB}{_{\text{m},\text{B}}} % subscript m,B (m=molar)$
$\newcommand{\mi}{_{\text{m},i}} % subscript m,i (m=molar)$
$\newcommand{\fA}{_{\text{f},\text{A}}} % subscript f,A (for fr. pt.)$
$\newcommand{\fB}{_{\text{f},\text{B}}} % subscript f,B (for fr. pt.)$
$\newcommand{\xbB}{_{x,\text{B}}} % x basis, B$
$\newcommand{\xbC}{_{x,\text{C}}} % x basis, C$
$\newcommand{\cbB}{_{c,\text{B}}} % c basis, B$
$\newcommand{\mbB}{_{m,\text{B}}} % m basis, B$
$\newcommand{\kHi}{k_{\text{H},i}} % Henry's law constant, x basis, i$
$\newcommand{\kHB}{k_{\text{H,B}}} % Henry's law constant, x basis, B$
$\newcommand{\arrow}{\,\rightarrow\,} % right arrow with extra spaces$
$\newcommand{\arrows}{\,\rightleftharpoons\,} % double arrows with extra spaces$
$\newcommand{\ra}{\rightarrow} % right arrow (can be used in text mode)$
$\newcommand{\eq}{\subs{eq}} % equilibrium state$
$\newcommand{\onehalf}{\textstyle\frac{1}{2}\D} % small 1/2 for display equation$
$\newcommand{\sys}{\subs{sys}} % system property$
$\newcommand{\sur}{\sups{sur}} % surroundings$
$\renewcommand{\in}{\sups{int}} % internal$
$\newcommand{\lab}{\subs{lab}} % lab frame$
$\newcommand{\cm}{\subs{cm}} % center of mass$
$\newcommand{\rev}{\subs{rev}} % reversible$
$\newcommand{\irr}{\subs{irr}} % irreversible$
$\newcommand{\fric}{\subs{fric}} % friction$
$\newcommand{\diss}{\subs{diss}} % dissipation$
$\newcommand{\el}{\subs{el}} % electrical$
$\newcommand{\cell}{\subs{cell}} % cell$
$\newcommand{\As}{A\subs{s}} % surface area$
$\newcommand{\E}{^\mathsf{E}} % excess quantity (superscript)$
$\newcommand{\allni}{\{n_i \}} % set of all n_i$
$\newcommand{\sol}{\hspace{-.1em}\tx{(sol)}}$
$\newcommand{\solmB}{\tx{(sol,\,m\B)}}$
$\newcommand{\dil}{\tx{(dil)}}$
$\newcommand{\sln}{\tx{(sln)}}$
$\newcommand{\mix}{\tx{(mix)}}$
$\newcommand{\rxn}{\tx{(rxn)}}$
$\newcommand{\expt}{\tx{(expt)}}$
$\newcommand{\solid}{\tx{(s)}}$
$\newcommand{\liquid}{\tx{(l)}}$
$\newcommand{\gas}{\tx{(g)}}$
$\newcommand{\pha}{\alpha} % phase alpha$
$\newcommand{\phb}{\beta} % phase beta$
$\newcommand{\phg}{\gamma} % phase gamma$
$\newcommand{\aph}{^{\alpha}} % alpha phase superscript$
$\newcommand{\bph}{^{\beta}} % beta phase superscript$
$\newcommand{\gph}{^{\gamma}} % gamma phase superscript$
$\newcommand{\aphp}{^{\alpha'}} % alpha prime phase superscript$
$\newcommand{\bphp}{^{\beta'}} % beta prime phase superscript$
$\newcommand{\gphp}{^{\gamma'}} % gamma prime phase superscript$
$\newcommand{\apht}{\small\aph} % alpha phase tiny superscript$
$\newcommand{\bpht}{\small\bph} % beta phase tiny superscript$
$\newcommand{\gpht}{\small\gph} % gamma phase tiny superscript$
$\newcommand{\upOmega}{\Omega}$
$\newcommand{\dif}{\mathop{}\!\mathrm{d}} % roman d in math mode, preceded by space$
$\newcommand{\Dif}{\mathop{}\!\mathrm{D}} % roman D in math mode, preceded by space$
$\newcommand{\df}{\dif\hspace{0.05em} f} % df$
$\newcommand{\dBar}{\mathop{}\!\mathrm{d}\hspace-.3em\raise1.05ex{\Rule{.8ex}{.125ex}{0ex}}} % inexact differential$
$\newcommand{\dq}{\dBar q} % heat differential$
$\newcommand{\dw}{\dBar w} % work differential$
$\newcommand{\dQ}{\dBar Q} % infinitesimal charge$
$\newcommand{\dx}{\dif\hspace{0.05em} x} % dx$
$\newcommand{\dt}{\dif\hspace{0.05em} t} % dt$
$\newcommand{\difp}{\dif\hspace{0.05em} p} % dp$
$\newcommand{\Del}{\Delta}$
$\newcommand{\Delsub}[1]{\Delta_{\text{#1}}}$
$\newcommand{\pd}[3]{(\partial #1 / \partial #2 )_{#3}} % \pd{}{}{} - partial derivative, one line$
$\newcommand{\Pd}[3]{\left( \dfrac {\partial #1} {\partial #2}\right)_{#3}} % Pd{}{}{} - Partial derivative, built-up$
$\newcommand{\bpd}[3]{[ \partial #1 / \partial #2 ]_{#3}}$
$\newcommand{\bPd}[3]{\left[ \dfrac {\partial #1} {\partial #2}\right]_{#3}}$
$\newcommand{\dotprod}{\small\bullet}$
$\newcommand{\fug}{f} % fugacity$
$\newcommand{\g}{\gamma} % solute activity coefficient, or gamma in general$
$\newcommand{\G}{\varGamma} % activity coefficient of a reference state (pressure factor)$
$\newcommand{\ecp}{\widetilde{\mu}} % electrochemical or total potential$
$\newcommand{\Eeq}{E\subs{cell, eq}} % equilibrium cell potential$
$\newcommand{\Ej}{E\subs{j}} % liquid junction potential$
$\newcommand{\mue}{\mu\subs{e}} % electron chemical potential$
$\newcommand{\defn}{\,\stackrel{\mathrm{def}}{=}\,} % "equal by definition" symbol$
$\newcommand{\D}{\displaystyle} % for a line in built-up$
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$\newcommand{\cond}[1]{\[-2.5pt]{}\tag*{#1}}$
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$\newcommand{\R}{8.3145\units{J\,K\per\,mol\per}} % gas constant value$
$\newcommand{\Rsix}{8.31447\units{J\,K\per\,mol\per}} % gas constant value - 6 sig figs$
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The processes of solution (dissolution) and dilution are related. The IUPAC Green Book (E. Richard Cohen et al, Quantities, Units and Symbols in Physical Chemistry, 3rd edition, RSC Publishing, Cambridge, 2007, Sec. 2.11.1) recommends the abbreviations sol and dil for these processes.
For an electrolyte solute, a plot of $\Del H\m\solmB$ versus $m\B$ has a limiting slope of $+\infty$ at $m\B{=}0$, whereas the limiting slope of $\Del H\m\solmB$ versus $\sqrt{m\B}$ is finite and can be predicted from the Debye–Hückel limiting law. Accordingly, a satisfactory procedure is to plot $\Del H\m\solmB$ versus $\sqrt{m\B}$, perform a linear extrapolation of the experimental points to $\sqrt{m\B}{=}0$, and then shift the origin to the extrapolated intercept. The result is a plot of $\varPhi_L$ versus $\sqrt{m\B}$. An example for aqueous NaCl solutions is shown in Fig. 11.10(a).
We can also evaluate $\varPhi_L$ from experimental enthalpies of dilution. From Eqs. 11.4.10 and 11.4.22, we obtain the relation $\varPhi_L(m\B'')-\varPhi_L(m\B') = \Del H\m(\tx{dil, $m\B'{\ra}m\B''$}) \tag{11.4.25}$ We can measure the enthalpy changes for diluting a solution of initial molality $m\B'$ to various molalities $m\B''$, plot the values of $\Del H\m(\tx{dil, \(m\B'{\ra}m\B''$})\) versus $\sqrt{m\B}$, extrapolate the curve to $\sqrt{m\B}{=}0$, and shift the origin to the extrapolated intercept, resulting in a plot of $\varPhi_L$ versus $\sqrt{m\B}$.
In order to be able to use Eq. 11.4.23, we need to relate the derivative $\dif\varPhi_L/\dif m\B$ to the slope of the curve of $\varPhi_L$ versus $\sqrt{m\B}$. We write $\dif \sqrt{m\B} = \frac{1}{2\sqrt{m\B}}\dif m\B \qquad \dif m\B = 2\sqrt{m\B} \dif\sqrt{m\B} \tag{11.4.26}$ Substituting this expression for $\dif m\B$ into Eq. 11.4.23, we obtain the following operational equation for evaluating $L\B$ from the plot of $\varPhi_L$ versus $\sqrt{m\B}$: \begin{gather} \s{ L\B = \varPhi_L + \frac{\sqrt{m\B}}{2} \frac{\dif\varPhi_L}{\dif\sqrt{m\B}} } \tag{11.4.27} \cond{(constant $T$ and $p$)} \end{gather}
The value of $\varPhi_L$ goes to zero at infinite dilution. When the solute is an electrolyte, the dependence of $\varPhi_L$ on $m\B$ in solutions dilute enough for the Debye–Hückel limiting law to apply is given by \begin{gather} \s{ \varPhi_L = C_{\varPhi_L}\sqrt{m\B} } \tag{11.4.28} \cond{(very dilute solution)} \end{gather} For aqueous solutions of a 1:1 electrolyte at $25\units{\(\degC$}\), the coefficient $C_{\varPhi_L}$ has the value $C_{\varPhi_L} = 1.988\timesten{3}\units{J kg$^{1/2}$ mol$^{-3/2}$} \tag{11.4.29}$ (The fact that $C_{\varPhi_L}$ is positive means, according to Eq. 11.4.25, that dilution of a very dilute electrolyte solution is an exothermic process.) $C_{\varPhi_L}$ is equal to the limiting slope of $\varPhi_L$ versus $\sqrt{m\B}$, of $\Del H\m\solmB$ versus $\sqrt{m\B}$, and of $\Del H\m(\tx{dil, \(m\B'{\ra}m\B''$})\) versus $\sqrt{m'\B}$. The value given by Eq. 11.4.29 can be used for extrapolation of measurements at $25\units{\(\degC$}\) and low molality to infinite dilution.
Equation 11.4.28 can be derived as follows. For simplicity, we assume the pressure is the standard pressure $p\st$. At this pressure $H\B^\infty$ is the same as $H\B\st$, and Eq. 11.4.17 becomes $L\B=H\B-H\B\st$. From Eqs. 12.1.3 and 12.1.6 in the next chapter, we can write the relations $H\B=-T^2\bPd{(\mu\B/T)}{T}{p,\allni} \qquad H\B\st=-T^2\frac{\dif(\mu\mbB\st/T)}{\dif T} \tag{11.4.30}$ Subtracting the second of these relations from the first, we obtain $H\B-H\B\st = -T^2\bPd{(\mu\B-\mu\mbB\st)/T}{T}{p,\allni} \tag{11.4.31}$ The solute activity on a molality basis, $a\mbB$, is defined by $\mu\B-\mu\mbB\st=RT\ln a\mbB$. The activity of an electrolyte solute at the standard pressure, from Eq. 10.3.10, is given by $a\mbB = (\nu_{+}^{\nu_{+}} \nu_{-}^{\nu_{-}}) \g_{\pm}^{\nu} (m\B/m\st)^{\nu}$. Accordingly, the relative partial molar enthalpy of the solute is related to the mean ionic activity coefficient by $L\B=-RT^2\nu\Pd{\ln\g_{\pm}}{T}{\!\!p,\allni} \tag{11.4.32}$
We assume the solution is sufficiently dilute for the mean ionic activity coefficient to be adequately described by the Debye–Hückel limiting law, Eq. 10.4.8: $\ln\g_{\pm} = -A\subs{DH}\left|z_+z_-\right|\sqrt{I_m}$, where $A\subs{DH}$ is a temperature-dependent quantity defined in Sec. 10.4. Then Eq. 11.4.32 becomes \begin{gather} \s{ L\B=RT^2\nu\left|z_+z_-\right|\sqrt{I_m}\Pd{A\subs{DH}}{T}{\!\!p,\allni} } \tag{11.4.33} \cond{(very dilute solution)} \end{gather} Substitution of the expression given by Eq. 10.4.9 for $I_m$ in a solution of a single completely-dissociated electrolyte converts Eq. 11.4.33 to \begin{gather} \s{ L\B = \left[ \frac{RT^2}{\sqrt{2}}\Pd{\rho\A^*A\subs{DH}}{T}{p,\allni} \left(\nu\left|z_+z_-\right|\right)^{3/2} \right]\sqrt{m\B} = C_{L\B}\sqrt{m\B} } \tag{11.4.34} \cond{(very dilute solution)} \end{gather} The coefficient $C_{L\B}$ (the quantity in brackets) depends on $T$, the kind of solvent, and the ion charges and number of ions per solute formula unit, but not on the solute molality.
Let $C_{\varPhi_L}$ represent the limiting slope of $\varPhi_L$ versus $\sqrt{m\B}$. In a very dilute solution we have $\varPhi_L = C_{\varPhi_L}\sqrt{m\B}$, and Eq. 11.4.27 becomes $L\B = \varPhi_L + \frac{\sqrt{m\B}}{2} \frac{\dif\varPhi_L}{\dif\sqrt{m\B}} = C_{\varPhi_L}\sqrt{m\B} + \frac{\sqrt{m\B}}{2} C_{\varPhi_L} \tag{11.4.35}$ By equating this expression for $L\B$ with the one given by Eq. 11.4.34 and solving for $C_{\varPhi_L}$, we obtain $C_{\varPhi_L}=(2/3)C_{L\B}$ and $\varPhi_L = (2/3)C_{L\B}\sqrt{m\B}$. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/11%3A_Reactions_and_Other_Chemical_Processes/11.04%3A__Enthalpies_of_Solution_and_Dilution.txt |
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$\newcommand{\mol}{\units{mol}} % mole$
$\newcommand{\V}{\units{V}} % volts$
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$\newcommand{\CVm}{C_{V,\text{m}}} % molar heat capacity at const.V$
$\newcommand{\Cpm}{C_{p,\text{m}}} % molar heat capacity at const.p$
$\newcommand{\kT}{\kappa_T} % isothermal compressibility$
$\newcommand{\A}{_{\text{A}}} % subscript A for solvent or state A$
$\newcommand{\B}{_{\text{B}}} % subscript B for solute or state B$
$\newcommand{\bd}{_{\text{b}}} % subscript b for boundary or boiling point$
$\newcommand{\C}{_{\text{C}}} % subscript C$
$\newcommand{\f}{_{\text{f}}} % subscript f for freezing point$
$\newcommand{\mA}{_{\text{m},\text{A}}} % subscript m,A (m=molar)$
$\newcommand{\mB}{_{\text{m},\text{B}}} % subscript m,B (m=molar)$
$\newcommand{\mi}{_{\text{m},i}} % subscript m,i (m=molar)$
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$\newcommand{\fB}{_{\text{f},\text{B}}} % subscript f,B (for fr. pt.)$
$\newcommand{\xbB}{_{x,\text{B}}} % x basis, B$
$\newcommand{\xbC}{_{x,\text{C}}} % x basis, C$
$\newcommand{\cbB}{_{c,\text{B}}} % c basis, B$
$\newcommand{\mbB}{_{m,\text{B}}} % m basis, B$
$\newcommand{\kHi}{k_{\text{H},i}} % Henry's law constant, x basis, i$
$\newcommand{\kHB}{k_{\text{H,B}}} % Henry's law constant, x basis, B$
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$\newcommand{\sur}{\sups{sur}} % surroundings$
$\renewcommand{\in}{\sups{int}} % internal$
$\newcommand{\lab}{\subs{lab}} % lab frame$
$\newcommand{\cm}{\subs{cm}} % center of mass$
$\newcommand{\rev}{\subs{rev}} % reversible$
$\newcommand{\irr}{\subs{irr}} % irreversible$
$\newcommand{\fric}{\subs{fric}} % friction$
$\newcommand{\diss}{\subs{diss}} % dissipation$
$\newcommand{\el}{\subs{el}} % electrical$
$\newcommand{\cell}{\subs{cell}} % cell$
$\newcommand{\As}{A\subs{s}} % surface area$
$\newcommand{\E}{^\mathsf{E}} % excess quantity (superscript)$
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$\newcommand{\solid}{\tx{(s)}}$
$\newcommand{\liquid}{\tx{(l)}}$
$\newcommand{\gas}{\tx{(g)}}$
$\newcommand{\pha}{\alpha} % phase alpha$
$\newcommand{\phb}{\beta} % phase beta$
$\newcommand{\phg}{\gamma} % phase gamma$
$\newcommand{\aph}{^{\alpha}} % alpha phase superscript$
$\newcommand{\bph}{^{\beta}} % beta phase superscript$
$\newcommand{\gph}{^{\gamma}} % gamma phase superscript$
$\newcommand{\aphp}{^{\alpha'}} % alpha prime phase superscript$
$\newcommand{\bphp}{^{\beta'}} % beta prime phase superscript$
$\newcommand{\gphp}{^{\gamma'}} % gamma prime phase superscript$
$\newcommand{\apht}{\small\aph} % alpha phase tiny superscript$
$\newcommand{\bpht}{\small\bph} % beta phase tiny superscript$
$\newcommand{\gpht}{\small\gph} % gamma phase tiny superscript$
$\newcommand{\upOmega}{\Omega}$
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$\newcommand{\Dif}{\mathop{}\!\mathrm{D}} % roman D in math mode, preceded by space$
$\newcommand{\df}{\dif\hspace{0.05em} f} % df$
$\newcommand{\dBar}{\mathop{}\!\mathrm{d}\hspace-.3em\raise1.05ex{\Rule{.8ex}{.125ex}{0ex}}} % inexact differential$
$\newcommand{\dq}{\dBar q} % heat differential$
$\newcommand{\dw}{\dBar w} % work differential$
$\newcommand{\dQ}{\dBar Q} % infinitesimal charge$
$\newcommand{\dx}{\dif\hspace{0.05em} x} % dx$
$\newcommand{\dt}{\dif\hspace{0.05em} t} % dt$
$\newcommand{\difp}{\dif\hspace{0.05em} p} % dp$
$\newcommand{\Del}{\Delta}$
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$\newcommand{\pd}[3]{(\partial #1 / \partial #2 )_{#3}} % \pd{}{}{} - partial derivative, one line$
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$\newcommand{\bpd}[3]{[ \partial #1 / \partial #2 ]_{#3}}$
$\newcommand{\bPd}[3]{\left[ \dfrac {\partial #1} {\partial #2}\right]_{#3}}$
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$\newcommand{\fug}{f} % fugacity$
$\newcommand{\g}{\gamma} % solute activity coefficient, or gamma in general$
$\newcommand{\G}{\varGamma} % activity coefficient of a reference state (pressure factor)$
$\newcommand{\ecp}{\widetilde{\mu}} % electrochemical or total potential$
$\newcommand{\Eeq}{E\subs{cell, eq}} % equilibrium cell potential$
$\newcommand{\Ej}{E\subs{j}} % liquid junction potential$
$\newcommand{\mue}{\mu\subs{e}} % electron chemical potential$
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Reaction calorimetry is used to evaluate the molar integral reaction enthalpy $\Del H\m\rxn$ of a reaction or other chemical process at constant temperature and pressure. The measurement actually made, however, is a temperature change.
Sections 11.5.1 and 11.5.2 will describe two common types of calorimeters designed for reactions taking place at either constant pressure or constant volume. The constant-pressure type is usually called a reaction calorimeter, and the constant-volume type is known as a bomb calorimeter or combustion calorimeter.
In either type of calorimeter, the chemical process takes place in a reaction vessel surrounded by an outer jacket. The jacket may be of either the adiabatic type or the isothermal-jacket type described in Sec. 7.3.2 in connection with heat capacity measurements. A temperature-measuring device is immersed either in the vessel or in a phase in thermal contact with it. The measured temperature change is caused by the chemical process, instead of by electrical work as in the determination of heat capacity. One important way in which these calorimeters differ from ones used for heat capacity measurements is that work is kept deliberately small, in order to minimize changes of internal energy and enthalpy during the experimental process.
11.5.1 The constant-pressure reaction calorimeter
The contents of a constant-pressure calorimeter are usually open to the atmosphere, so this type of calorimeter is unsuitable for processes involving gases. It is, however, a convenient apparatus in which to study a liquid-phase chemical reaction, the dissolution of a solid or liquid solute in a liquid solvent, or the dilution of a solution with solvent.
The process is initiated in the calorimeter by allowing the reactants to come into contact. The temperature in the reaction vessel is measured over a period of time starting before the process initiation and ending after the advancement has reached a final value with no further change.
The heating or cooling curve (temperature as a function of time) is observed over a period of time that includes the period during which the advancement $\xi$ changes. For an exothermic reaction occurring in an adiabatic calorimeter, the heating curve may resemble that shown in Fig. 7.3, and the heating curve in an isothermal-jacket calorimeter may resemble that shown in Fig. 7.4. Two points are designated on the heating or cooling curve: one at temperature $T_1$, before the reaction is initiated, and the other at $T_2$, after $\xi$ has reached its final value. These points are indicated by open circles in Figs. 7.3 and 7.4.
The relations derived here parallel those of Sec. 11.5.1 for a constant-pressure calorimeter. The three paths depicted in Fig. 11.13 are similar to those in Fig. 11.11, except that instead of being at constant pressure they are at constant volume. We shall assume the combustion reaction is exothermic, with $T_2$ being greater than $T_1$.
The internal energy change of the experimental process that actually occurs in the calorimeter between times $t_1$ and $t_2$ is denoted $\Del U\expt$ in the figure. Conceptually, the overall change of state during this process would be duplicated by a path in which the temperature of the system with the reactants present increases from $T_1$ to $T_2$, followed by the isothermal bomb process at temperature $T_2$. (When one investigates a combustion reaction, the path in which temperature changes without reaction is best taken with reactants rather than products present because the reactants are more easily characterized.) In the figure these paths are labeled with the internal energy changes $\Del U(\tx{R})$ and $\Del U(\tx{IBP},T_2)$, and we can write $\Del U\expt = \Del U(\tx{R}) + \Del U(\tx{IBP},T_2) \tag{11.5.4}$
To evaluate $\Del U(\tx{R})$, we can use the energy equivalent $\epsilon\subs{R}$ of the calorimeter with reactants present in the bomb vessel. $\epsilon\subs{R}$ is the average heat capacity of the system between $T_1$ and $T_2$—that is, the ratio $q/(T_2 - T_1)$, where $q$ is the heat that would be needed to change the temperature from $T_1$ to $T_2$. From the first law, with expansion work assumed negligible, the internal energy change equals this heat, giving us the relation $\Del U(\tx{R}) = \epsilon\subs{R}(T_2 - T_1) \tag{11.5.5}$ The initial and final states of the path are assumed to be equilibrium states, and there may be some transfer of reactants or H$_2$O from one phase to another within the bomb vessel during the heating process.
The value of $\epsilon\subs{R}$ is obtained in a separate calibration experiment. The calibration is usually carried out with the combustion of a reference substance, such as benzoic acid, whose internal energy of combustion under controlled conditions is precisely known from standardization based on electrical work. If the bomb vessel is immersed in the same mass of water in both experiments and other conditions are similar, the difference in the values of $\epsilon\subs{R}$ in the two experiments is equal to the known difference in the heat capacities of the initial contents (reactants, water, etc.) of the bomb vessel in the two experiments.
The internal energy change we wish to find is $\Del U(\tx{IBP},T_2)$, that of the isothermal bomb process in which reactants change to products at temperature $T_2$, accompanied perhaps by some further transfer of substances between phases. From Eqs. 11.5.4 and 11.5.5, we obtain $\Del U(\tx{IBP},T_2) = -\epsilon (T_2 - T_1)+\Del U\expt \tag{11.5.6}$
The value of $\Del U\expt$ is small. To evaluate it, we must look in detail at the possible sources of energy transfer between the system and the surroundings during the experimental process. These sources are
1. The ignition work occurs during only a short time interval at the beginning of the process, and its value is known. The effects of heat transfer, stirring work, and temperature measurement continue throughout the course of the experiment. With these considerations, Eq. 11.5.6 becomes $\Del U(\tx{IBP},T_2) = - \epsilon(T_2 - T_1) + w\subs{ign} + \Del U'\expt \tag{11.5.7}$ where $\Del U'\expt$ is the internal energy change due to heat, stirring, and temperature measurement. $\Del U'\expt$ can be evaluated from the energy equivalent and the observed rates of temperature change at times $t_1$ and $t_2$; the relevant relations for an isothermal jacket are Eq. 7.3.24 (with $w\el$ set equal to zero) and Eq. 7.3.32.
Reduction to standard states
We want to obtain the value of $\Delsub{c}U\st(T\subs{ref})$, the molar internal energy change for the main combustion reaction at the reference temperature under standard-state conditions. Once we have this value, it is an easy matter to find the molar enthalpy change under standard-state conditions, our ultimate goal.
Consider a hypothetical process with the following three isothermal steps carried out at the reference temperature $T\subs{ref}$:
1. The net change is a decrease in the amount of each reactant in its standard state and an increase in the amount of each product in its standard state. The internal energy change of step 2 is $\Del U(\tx{IBP},T\subs{ref})$, whose value is found from Eq. 11.5.8. The internal energy changes of steps 1 and 3 are called Washburn corrections (Edward W. Washburn, J. Res. Natl. Bur. Stand. (U.S.), 10, 525–558, 1933).
Thus, we calculate the standard internal energy change of the main combustion reaction at temperature $T\subs{ref}$ from $\Del U\st(\tx{cmb},T\subs{ref}) = \Del U(\tx{IBP},T\subs{ref}) + \tx{(Washburn corrections)} - \sum_i \Del\xi_i \Delsub{r}U\st(i) \tag{11.5.9}$ where the sum over $i$ is for side reactions and auxiliary reactions if present. Finally, we calculate the standard molar internal energy of combustion from $\Delsub{c}U\st(T\subs{ref}) = \frac{\Del U\st(\tx{cmb},T\subs{ref})}{\Del\xi\subs{c}} \tag{11.5.10}$ where $\Del\xi\subs{c}$ is the advancement of the main combustion reaction in the bomb vessel.
Washburn corrections
The Washburn corrections needed in Eq. 11.5.9 are internal energy changes for certain hypothetical physical processes occurring at the reference temperature $T\subs{ref}$ involving the substances present in the bomb vessel. In these processes, substances change from their standard states to the initial state of the isothermal bomb process, or change from the final state of the isothermal bomb process to their standard states.
For example, consider the complete combustion of a solid or liquid compound of carbon, hydrogen, and oxygen in which the combustion products are CO$_2$ and H$_2$O and there are no side reactions or auxiliary reactions. In the initial state of the isothermal bomb process, the bomb vessel contains the pure reactant, liquid water with O$_2$ dissolved in it, and a gaseous mixture of O$_2$ and H$_2$O, all at a high pressure $p_1$. In the final state, the bomb vessel contains liquid water with O$_2$ and CO$_2$ dissolved in it and a gaseous mixture of O$_2$, H$_2$O, and CO$_2$, all at pressure $p_2$. In addition, the bomb vessel contains internal parts of constant mass such as the sample holder and ignition wires.
In making Washburn corrections, we must use a single standard state for each substance in order for Eq. 11.5.9 to correctly give the standard internal energy of combustion. In the present example we choose the following standard states: pure solid or liquid for the reactant compound, pure liquid for the H$_2$O, and pure ideal gases for the O$_2$ and CO$_2$, each at pressure $p\st=1\br$.
We can calculate the amount of each substance in each phase, in both the initial state and final state of the isothermal bomb process, from the following information: the internal volume of the bomb vessel; the mass of solid or liquid reactant initially placed in the vessel; the initial amount of H$_2$O; the initial O$_2$ pressure; the water vapor pressure; the solubilities (estimated from Henry’s law constants) of O$_2$ and CO$_2$ in the water; and the stoichiometry of the combustion reaction. Problem 11.7 guides you through these calculations.
11.5.3 Other calorimeters
Experimenters have used great ingenuity in designing calorimeters to measure reaction enthalpies and to improve their precision. In addition to the constant-pressure reaction calorimeter and bomb calorimeter described above, three additional types will be briefly mentioned.
A phase-change calorimeter has two coexisting phases of a pure substance in thermal contact with the reaction vessel and an adiabatic outer jacket. The two coexisting phases constitute a univariant subsystem that at constant pressure is at the fixed temperature of the equilibrium phase transition. The thermal energy released or absorbed by the reaction, instead of changing the temperature, is transferred isothermally to or from the coexisting phases and can be measured by the volume change of the phase transition. A reaction enthalpy, of course, can only be measured by this method at the temperature of the equilibrium phase transition. The well-known Bunsen ice calorimeter uses the ice–water transition at $0\units{\(\degC$}\). The solid–liquid transition of diphenyl ether has a relatively large volume change and is useful for measurements at $26.9\units{\(\degC$}\). Phase-transition calorimeters are especially useful for slow reactions.
A heat-flow calorimeter is a variation of an isothermal-jacket calorimeter. It uses a thermopile (Fig. 2.7) to continuously measure the temperature difference between the reaction vessel and an outer jacket acting as a constant-temperature heat sink. The heat transfer takes place mostly through the thermocouple wires, and to a high degree of accuracy is proportional to the temperature difference integrated over time. This is the best method for an extremely slow reaction, and it can also be used for rapid reactions.
A flame calorimeter is a flow system in which oxygen, fluorine, or another gaseous oxidant reacts with a gaseous fuel. The heat transfer between the flow tube and a heat sink can be measured with a thermopile, as in a heat-flow calorimeter. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/11%3A_Reactions_and_Other_Chemical_Processes/11.05%3A_Reaction_Calorimetry.txt |
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With a few simple approximations, we can estimate the temperature of a flame formed in a flowing gas mixture of oxygen or air and a fuel. We treat a moving segment of the gas mixture as a closed system in which the temperature increases as combustion takes place. We assume that the reaction occurs at a constant pressure equal to the standard pressure, and that the process is adiabatic and the gas is an ideal-gas mixture.
The principle of the calculation is similar to that used for a constant-pressure calorimeter as explained by the paths shown in Fig. 11.11. When the combustion reaction in the segment of gas reaches reaction equilibrium, the advancement has changed by $\Del\xi$ and the temperature has increased from $T_1$ to $T_2$. Because the reaction is assumed to be adiabatic at constant pressure, $\Del H\expt$ is zero. Therefore, the sum of $\Del H(\tx{rxn},T_1)$ and $\Del H(\tx{P})$ is zero, and we can write $\Del\xi\Delsub{c}H\st(T_1) + \int_{T_1}^{T_2}\!C_p(\tx{P})\dif T = 0 \tag{11.6.1}$ where $\Delsub{c}H\st(T_1)$ is the standard molar enthalpy of combustion at the initial temperature, and $C_p(\tx{P})$ is the heat capacity at constant pressure of the product mixture.
The value of $T_2$ that satisfies Eq. 11.6.1 is the estimated flame temperature. Problem 11.9 presents an application of this calculation. Several factors cause the actual temperature in a flame to be lower: the process is never completely adiabatic, and in the high temperature of the flame there may be product dissociation and other reactions in addition to the main combustion reaction. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/11%3A_Reactions_and_Other_Chemical_Processes/11.06%3A_Adiabatic_Flame_Temperature.txt |
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$\newcommand{\cond}[1]{\$-2.5pt]{}\tag*{#1}}$ $\newcommand{\nextcond}[1]{\[-5pt]{}\tag*{#1}}$ $\newcommand{\R}{8.3145\units{J\,K\per\,mol\per}} % gas constant value$ $\newcommand{\Rsix}{8.31447\units{J\,K\per\,mol\per}} % gas constant value - 6 sig figs$ $\newcommand{\jn}{\hspace3pt\lower.3ex{\Rule{.6pt}{2ex}{0ex}}\hspace3pt}$ $\newcommand{\ljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}} \hspace3pt}$ $\newcommand{\lljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace1.4pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace3pt}$ This section begins by examining the way in which the Gibbs energy changes as a chemical process advances in a closed system at constant $T$ and $p$ with expansion work only. A universal criterion for reaction equilibrium is derived involving the molar reaction Gibbs energy. The molar reaction Gibbs energy Applying the general definition of a molar differential reaction quantity (Eq. 11.2.15) to the Gibbs energy of a closed system with $T$, $p$, and $\xi$ as the independent variables, we obtain the definition of the molar reaction Gibbs energy or molar Gibbs energy of reaction, $\Delsub{r}G$: \[$\Delsub{r}G \defn \sum_i\nu_i \mu_i \label{11.7.1}$$
Equation 11.2.16 shows that this quantity is also given by the partial derivative
$\begin{gather} \s{ \Delsub{r}G = \Pd{G}{\xi}{T,p} } \label{11.7.2} \cond{(closed system)} \end{gather}$
The total differential of $G$ is then
$\begin{gather} \s{ \dif G = -S\dif T + V\difp + \Delsub{r}G\dif\xi } \label{11.7.3} \cond{(closed system)} \end{gather}$
Spontaneity and reaction equilibrium
In Sec. 5.8, we found that the spontaneous direction of a process taking place in a closed system at constant $T$ and $p$, with expansion work only, is the direction of decreasing $G$. In the case of a chemical process occurring at constant $T$ and $p$, $\Delsub{r}G$ is the rate at which $G$ changes with $\xi$. Thus if $\Delsub{r}G$ is positive, $\xi$ spontaneously decreases; if $\Delsub{r}G$ is negative, $\xi$ spontaneously increases. During a spontaneous process $\dif\xi$ and $\Delsub{r}G$ have opposite signs.
Sometimes reaction spontaneity at constant $T$ and $p$ is ascribed to the “driving force” of a quantity called the affinity of reaction, defined as the negative of $\Delsub{r}G$. $\xi$ increases spontaneously if the affinity is positive and decreases spontaneously if the affinity is negative; the system is at equilibrium when the affinity is zero.
Note how the equality of Equation \ref{11.7.3} agrees with the inequality $\dif G<-S\dif T+V\difp$, a criterion of spontaneity in a closed system with expansion work only (Eq. 5.8.6). When $\dif\xi$ and $\Delsub{r}G$ have opposite signs, $\Delsub{r}G\dif\xi$ is negative and $\dif G=(-S\dif T+V\difp+\Delsub{r}G\dif\xi)$ is less than $(-S\dif T+V\difp)$.
If the system is closed and contains at least one phase that is a mixture, a state of reaction equilibrium can be approached spontaneously at constant $T$ and $p$ in either direction of the reaction; that is, by both positive and negative changes of $\xi$. In this equilibrium state, therefore, $G$ has its minimum value for the given $T$ and $p$. Since $G$ is a smooth function of $\xi$, its rate of change with respect to $\xi$ is zero in the equilibrium state. The condition for reaction equilibrium, then, is that $\Delsub{r}G$ must be zero: \begin{gather} \s{ \Delsub{r}G=\sum_i\nu_i\mu_i=0 } \tag{11.7.4} \cond{(reaction equilibrium)} \end{gather}
It is important to realize that this condition is independent of whether or not reaction equilibrium is approached at constant temperature and pressure. It is a universal criterion of reaction equilibrium. The value of $\Delsub{r}G$ is equal to $\sum_i\!\nu_i\mu_i$ and depends on the state of the system. If the state is such that $\Delsub{r}G$ is positive, the direction of spontaneous change is one that, under the existing constraints, allows $\Delsub{r}G$ to decrease. If $\Delsub{r}G$ is negative, the spontaneous change increases the value of $\Delsub{r}G$. When the system reaches reaction equilibrium, whatever the path of the spontaneous process, the value of $\Delsub{r}G$ becomes zero.
General derivation
We can obtain the condition of reaction equilibrium given by Eq. 11.7.4 in a more general and rigorous way by an extension of the derivation of Sec. 9.2.7, which was for equilibrium conditions in a multiphase, multicomponent system.
Consider a system with a reference phase, $\pha'$, and optionally other phases labeled by $\pha\ne\pha'$. Each phase contains one or more species labeled by subscript $i$, and some or all of the species are the reactants and products of a reaction.
The total differential of the internal energy is given by Eq. 9.2.37: $\begin{split} \dif U & = T\aphp\dif S\aphp - p\aphp\dif V\aphp + \sum_i\mu_i\aphp\dif n_i\aphp \cr & \quad + \sum_{\pha\ne\pha'}\left(T\aph\dif S\aph - p\aph\dif V\aph + \sum_i\mu_i\aph\dif n_i\aph\right) \end{split} \tag{11.7.5}$ The conditions of isolation are $\dif U = 0 \qquad \tx{(constant internal energy)} \tag{11.7.6}$ $\dif V\aphp + \sum_{\pha\ne\pha'}\dif V\aph = 0 \qquad \tx{(no expansion work)} \tag{11.7.7}$ $\begin{split} &\tx{For each species $i$:} \cr &\dif n_i\aphp + \sum_{\pha\ne\pha'}\dif n_i\aph = \nu_i\dif\xi \qquad \tx{(closed system)} \end{split} \tag{11.7.8}$ In Eq. 11.7.8, $\dif n^{\pha''}_{i'}$ should be set equal to zero for a species $i'$ that is excluded from phase $\pha''$, and $\nu_{i''}$ should be set equal to zero for a species $i''$ that is not a reactant or product of the reaction.
We use these conditions of isolation to substitute for $\dif U$, $\dif V\aphp$, and $\dif n_i\aphp$ in Eq. 11.7.5, and make the further substitution $\dif S\aphp = \dif S - \sum_{\pha\ne\pha'}\dif S\aph$. Solving for $\dif S$, we obtain $\begin{split} \dif S & = \sum_{\pha\ne\pha'}\frac{(T\aphp-T\aph)}{T\aphp}\dif S\aph - \sum_{\pha\ne\pha'}\frac{(p\aphp-p\aph)}{T\aphp}\dif V\aph \cr & \quad + \sum_i\sum_{\pha\ne\pha'}\frac{(\mu_i\aphp-\mu_i\aph)}{T\aphp}\dif n_i\aph - \frac{ \sum_i\nu_i\mu_i\aphp}{T\aphp}\dif\xi \end{split} \tag{11.7.9}$ The equilibrium condition is that the coefficient multiplying each differential on the right side of Eq. 11.7.9 must be zero. We conclude that at equilibrium the temperature of each phase is equal to that of phase $\pha'$; the pressure of each phase is equal to that of phase $\pha'$; the chemical potential of each species, in each phase containing that species, is equal to the chemical potential of the species in phase $\pha'$; and the quantity $\sum_i\!\nu_i\mu_i\aphp$ (which is equal to $\Delsub{r}G$) is zero.
In short, in an equilibrium state each phase has the same temperature and the same pressure, each species has the same chemical potential in the phases in which it is present, and the molar reaction Gibbs energy of each phase is zero.
Pure phases
Consider a chemical process in which each reactant and product is in a separate pure phase. For example, the decomposition of calcium carbonate,
$\ce{CaCO3(s) -> CaO(s) + CO2(g)}$
involves three pure phases if no other gas is allowed to mix with the $\ce{CO2}$.
The situation is different when the number of molecules changes during the reaction. Consider the reaction A$\arrow$2 B in an ideal gas mixture. As this reaction proceeds to the right at constant $T$, the volume increases if the pressure is held constant and the pressure increases if the volume is held constant. Figure 11.17 shows how $G$ depends on both $p$ and $V$ for this reaction. Movement along the horizontal dashed line in the figure corresponds to reaction at constant $T$ and $p$. The minimum of $G$ along this line is at the volume indicated by the open circle. At this volume, $G$ has an even lower minimum at the pressure indicated by the filled circle, where the vertical dashed line is tangent to one of the contours of constant $G$. The condition needed for reaction equilibrium, however, is that $\Delsub{r}G$ must be zero. This condition is satisfied along the vertical dashed line only at the position of the open circle.
This example demonstrates that for a reaction occurring at constant temperature and volume in which the pressure changes, the point of reaction equilibrium is not the point of minimum $G$. Instead, the point of reaction equilibrium in this case is at the minimum of the Helmholtz energy $A$ (Sec. 11.7.5). | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/11%3A_Reactions_and_Other_Chemical_Processes/11.07%3A_Gibbs_Energy_and_Reaction_Equilibrium.txt |
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$\newcommand{\id}{^{\text{id}}} % ideal$
$\newcommand{\rf}{^{\text{ref}}} % reference state$
$\newcommand{\units}[1]{\mbox{\thinspace#1}}$
$\newcommand{\K}{\units{K}} % kelvins$
$\newcommand{\degC}{^\circ\text{C}} % degrees Celsius$
$\newcommand{\br}{\units{bar}} % bar (\bar is already defined)$
$\newcommand{\Pa}{\units{Pa}}$
$\newcommand{\mol}{\units{mol}} % mole$
$\newcommand{\V}{\units{V}} % volts$
$\newcommand{\timesten}[1]{\mbox{\,\times\,10^{#1}}}$
$\newcommand{\per}{^{-1}} % minus one power$
$\newcommand{\m}{_{\text{m}}} % subscript m for molar quantity$
$\newcommand{\CVm}{C_{V,\text{m}}} % molar heat capacity at const.V$
$\newcommand{\Cpm}{C_{p,\text{m}}} % molar heat capacity at const.p$
$\newcommand{\kT}{\kappa_T} % isothermal compressibility$
$\newcommand{\A}{_{\text{A}}} % subscript A for solvent or state A$
$\newcommand{\B}{_{\text{B}}} % subscript B for solute or state B$
$\newcommand{\bd}{_{\text{b}}} % subscript b for boundary or boiling point$
$\newcommand{\C}{_{\text{C}}} % subscript C$
$\newcommand{\f}{_{\text{f}}} % subscript f for freezing point$
$\newcommand{\mA}{_{\text{m},\text{A}}} % subscript m,A (m=molar)$
$\newcommand{\mB}{_{\text{m},\text{B}}} % subscript m,B (m=molar)$
$\newcommand{\mi}{_{\text{m},i}} % subscript m,i (m=molar)$
$\newcommand{\fA}{_{\text{f},\text{A}}} % subscript f,A (for fr. pt.)$
$\newcommand{\fB}{_{\text{f},\text{B}}} % subscript f,B (for fr. pt.)$
$\newcommand{\xbB}{_{x,\text{B}}} % x basis, B$
$\newcommand{\xbC}{_{x,\text{C}}} % x basis, C$
$\newcommand{\cbB}{_{c,\text{B}}} % c basis, B$
$\newcommand{\mbB}{_{m,\text{B}}} % m basis, B$
$\newcommand{\kHi}{k_{\text{H},i}} % Henry's law constant, x basis, i$
$\newcommand{\kHB}{k_{\text{H,B}}} % Henry's law constant, x basis, B$
$\newcommand{\arrow}{\,\rightarrow\,} % right arrow with extra spaces$
$\newcommand{\arrows}{\,\rightleftharpoons\,} % double arrows with extra spaces$
$\newcommand{\ra}{\rightarrow} % right arrow (can be used in text mode)$
$\newcommand{\eq}{\subs{eq}} % equilibrium state$
$\newcommand{\onehalf}{\textstyle\frac{1}{2}\D} % small 1/2 for display equation$
$\newcommand{\sys}{\subs{sys}} % system property$
$\newcommand{\sur}{\sups{sur}} % surroundings$
$\renewcommand{\in}{\sups{int}} % internal$
$\newcommand{\lab}{\subs{lab}} % lab frame$
$\newcommand{\cm}{\subs{cm}} % center of mass$
$\newcommand{\rev}{\subs{rev}} % reversible$
$\newcommand{\irr}{\subs{irr}} % irreversible$
$\newcommand{\fric}{\subs{fric}} % friction$
$\newcommand{\diss}{\subs{diss}} % dissipation$
$\newcommand{\el}{\subs{el}} % electrical$
$\newcommand{\cell}{\subs{cell}} % cell$
$\newcommand{\As}{A\subs{s}} % surface area$
$\newcommand{\E}{^\mathsf{E}} % excess quantity (superscript)$
$\newcommand{\allni}{\{n_i \}} % set of all n_i$
$\newcommand{\sol}{\hspace{-.1em}\tx{(sol)}}$
$\newcommand{\solmB}{\tx{(sol,\,m\B)}}$
$\newcommand{\dil}{\tx{(dil)}}$
$\newcommand{\sln}{\tx{(sln)}}$
$\newcommand{\mix}{\tx{(mix)}}$
$\newcommand{\rxn}{\tx{(rxn)}}$
$\newcommand{\expt}{\tx{(expt)}}$
$\newcommand{\solid}{\tx{(s)}}$
$\newcommand{\liquid}{\tx{(l)}}$
$\newcommand{\gas}{\tx{(g)}}$
$\newcommand{\pha}{\alpha} % phase alpha$
$\newcommand{\phb}{\beta} % phase beta$
$\newcommand{\phg}{\gamma} % phase gamma$
$\newcommand{\aph}{^{\alpha}} % alpha phase superscript$
$\newcommand{\bph}{^{\beta}} % beta phase superscript$
$\newcommand{\gph}{^{\gamma}} % gamma phase superscript$
$\newcommand{\aphp}{^{\alpha'}} % alpha prime phase superscript$
$\newcommand{\bphp}{^{\beta'}} % beta prime phase superscript$
$\newcommand{\gphp}{^{\gamma'}} % gamma prime phase superscript$
$\newcommand{\apht}{\small\aph} % alpha phase tiny superscript$
$\newcommand{\bpht}{\small\bph} % beta phase tiny superscript$
$\newcommand{\gpht}{\small\gph} % gamma phase tiny superscript$
$\newcommand{\upOmega}{\Omega}$
$\newcommand{\dif}{\mathop{}\!\mathrm{d}} % roman d in math mode, preceded by space$
$\newcommand{\Dif}{\mathop{}\!\mathrm{D}} % roman D in math mode, preceded by space$
$\newcommand{\df}{\dif\hspace{0.05em} f} % df$
$\newcommand{\dBar}{\mathop{}\!\mathrm{d}\hspace-.3em\raise1.05ex{\Rule{.8ex}{.125ex}{0ex}}} % inexact differential$
$\newcommand{\dq}{\dBar q} % heat differential$
$\newcommand{\dw}{\dBar w} % work differential$
$\newcommand{\dQ}{\dBar Q} % infinitesimal charge$
$\newcommand{\dx}{\dif\hspace{0.05em} x} % dx$
$\newcommand{\dt}{\dif\hspace{0.05em} t} % dt$
$\newcommand{\difp}{\dif\hspace{0.05em} p} % dp$
$\newcommand{\Del}{\Delta}$
$\newcommand{\Delsub}[1]{\Delta_{\text{#1}}}$
$\newcommand{\pd}[3]{(\partial #1 / \partial #2 )_{#3}} % \pd{}{}{} - partial derivative, one line$
$\newcommand{\Pd}[3]{\left( \dfrac {\partial #1} {\partial #2}\right)_{#3}} % Pd{}{}{} - Partial derivative, built-up$
$\newcommand{\bpd}[3]{[ \partial #1 / \partial #2 ]_{#3}}$
$\newcommand{\bPd}[3]{\left[ \dfrac {\partial #1} {\partial #2}\right]_{#3}}$
$\newcommand{\dotprod}{\small\bullet}$
$\newcommand{\fug}{f} % fugacity$
$\newcommand{\g}{\gamma} % solute activity coefficient, or gamma in general$
$\newcommand{\G}{\varGamma} % activity coefficient of a reference state (pressure factor)$
$\newcommand{\ecp}{\widetilde{\mu}} % electrochemical or total potential$
$\newcommand{\Eeq}{E\subs{cell, eq}} % equilibrium cell potential$
$\newcommand{\Ej}{E\subs{j}} % liquid junction potential$
$\newcommand{\mue}{\mu\subs{e}} % electron chemical potential$
$\newcommand{\defn}{\,\stackrel{\mathrm{def}}{=}\,} % "equal by definition" symbol$
$\newcommand{\D}{\displaystyle} % for a line in built-up$
$\newcommand{\s}{\smash[b]} % use in equations with conditions of validity$
$\newcommand{\cond}[1]{\$-2.5pt]{}\tag*{#1}}$ $\newcommand{\nextcond}[1]{\[-5pt]{}\tag*{#1}}$ $\newcommand{\R}{8.3145\units{J\,K\per\,mol\per}} % gas constant value$ $\newcommand{\Rsix}{8.31447\units{J\,K\per\,mol\per}} % gas constant value - 6 sig figs$ $\newcommand{\jn}{\hspace3pt\lower.3ex{\Rule{.6pt}{2ex}{0ex}}\hspace3pt}$ $\newcommand{\ljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}} \hspace3pt}$ $\newcommand{\lljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace1.4pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace3pt}$ 11.8.1 Activities and the definition of $K$ Equation 10.1.9 gives the general relation between the chemical potential $\mu_i$ and the activity $a_i$ of species $i$ in a phase of electric potential $\phi$: $\mu_i=\mu_i\st + RT\ln a_i + z_i F\phi \tag{11.8.1}$ The electric potential affects $\mu_i$ only if the charge number $z_i$ is nonzero, i.e., only if species $i$ is an ion. Consider a reaction in which any reactants and products that are ions are in a single phase of electric potential $\phi'$, or in several phases of equal electric potential $\phi'$. Under these conditions, substitution of the expression above for $\mu_i$ in $\Delsub{r}G = \sum_i\!\nu_i\mu_i$ gives \begin{gather} \s{ \Delsub{r}G = \sum_i\nu_i\mu_i\st + RT \sum_i\nu_i\ln a_i + F\phi'\sum_i\nu_i z_i } \tag{11.8.2} \cond{(all ions at $\phi{=}\phi'$)} \end{gather} The first term on the right side of Eq. 11.8.2 is the standard molar reaction Gibbs energy, or standard molar Gibbs energy of reaction: $\Delsub{r}G\st\defn \sum_i\nu_i \mu_i\st \tag{11.8.3}$ Since the standard chemical potential $\mu\st_i$ of each species $i$ is a function only of $T$, the value of $\Delsub{r}G\st$ for a given reaction as defined by the reaction equation depends only on $T$ and on the choice of a standard state for each reactant and product. The last term on the right side of Eq. 11.8.2 is the sum $\sum_i\!\nu_i z_i$. Because charge is conserved during the advancement of a reaction in a closed system, this sum is zero. With these substitutions, Eq. 11.8.2 becomes \begin{gather} \s{ \Delsub{r}G = \Delsub{r}G\st + RT\sum_i\nu_i\ln a_i } \tag{11.8.4} \cond{(all ions at same $\phi$)} \end{gather} This relation enables us to say that for a reaction at a given temperature in which any charged reactants or products are all in the same phase, or in phases of equal electric potential, the value of $\Delsub{r}G$ and $\sum_i\!\nu_i\mu_i$ depends only on the activities of the reactants and products and is independent of what the electric potentials of any of the phases might happen to be. Unless a reaction involving ions is carried out in a galvanic cell, the ions are usually present in a single phase, and this will not be shown as a condition of validity in the rest of this chapter. The special case of a reaction in a galvanic cell will be discussed in Sec. 14.3. We may use properties of logarithms to write the sum on the right side of Eq. 11.8.4 as follows: $\sum_i\nu_i\ln a_i = \sum_i\ln \left( a_i^{\nu_i} \right) = \ln \prod_i a_i^{\nu_i} \tag{11.8.5}$ The symbol $\prod$ stands for a continued product. If, for instance, there are three species, $\prod_i a_i^{\nu_i}$ is the product $(a_1^{\nu_1})(a_2^{\nu_2})(a_3^{\nu_3})$. The product $\prod_i a_i^{\nu_i}$ is called the reaction quotient or activity quotient, $Q\subs{rxn}$: $Q\subs{rxn} \defn \prod_i a_i ^{\nu_i} \tag{11.8.6}$ $Q\subs{rxn}$ consists of a factor for each reactant and product. Each factor is the activity raised to the power of the stoichiometric number $\nu_i$. Since the value of $\nu_i$ is positive for a product and negative for a reactant, $Q\subs{rxn}$ is a quotient in which the activities of the products appear in the numerator and those of the reactants appear in the denominator, with each activity raised to a power equal to the corresponding stoichiometric coefficient in the reaction equation. Such a quotient, with quantities raised to these powers, is called a proper quotient. The reaction quotient is a proper quotient of activities. For instance, for the ammonia synthesis reaction N$_2$(g) + 3 H$_2$(g)$\arrow$2 NH$_3$(g) the reaction quotient is given by $Q\subs{rxn} = \frac{a\subs{NH$_3$}^2}{a\subs{N$_2$}a\subs{H$_2$}^3} \tag{11.8.7}$ $Q\subs{rxn}$ is a dimensionless quantity. It is a function of $T$, $p$, and the mixture composition, so its value changes as the reaction advances. The expression for the molar reaction Gibbs energy given by Eq. 11.8.4 can now be written $\Delsub{r}G = \Delsub{r}G\st + RT\ln Q\subs{rxn} \tag{11.8.8}$ The value of $Q\subs{rxn}$ under equilibrium conditions is the thermodynamic equilibrium constant, $K$. The general definition of $K$ is $K \defn \prod_i (a_i)\eq^{\nu_i} \tag{11.8.9}$ where the subscript eq indicates an equilibrium state. Note that $K$, like $Q\subs{rxn}$, is dimensionless. The IUPAC Green Book (E. Richard Cohen et al, Quantities, Units and Symbols in Physical Chemistry, 3rd edition, RSC Publishing, Cambridge, 2007, p. 58) gives $K^{\small \unicode{x29B5}}$ as an alternative symbol for the thermodynamic equilibrium constant, the appended superscript denoting “standard.” An IUPAC Commission on Thermodynamics (M. B. Ewing et al, Pure Appl. Chem., 66, 533–552, 1994) has furthermore recommended the name “standard equilibrium constant,” apparently because its value depends on the choice of standard states. Using this alternative symbol and name could cause confusion, since the quantity defined by Eq. 11.8.9 does not refer to reactants and products in their standard states but rather to reactants and products in an equilibrium state. Substituting the equilibrium conditions $\Delsub{r}G = 0$ and $Q\subs{rxn} = K$ in Eq. 11.8.8 gives an important relation between the standard molar reaction Gibbs energy and the thermodynamic equilibrium constant: $\Delsub{r}G\st = -RT\ln K \tag{11.8.10}$ We can solve this equation for $K$ to obtain the equivalent relation $K = \exp \left( -\frac{\Delsub{r}G\st}{RT} \right) \tag{11.8.11}$ We have seen that the value of $\Delsub{r}G\st$ depends only on $T$ and the choice of the standard states of the reactants and products. This being so, Eq. 11.8.11 shows that the value of $K$ for a given reaction depends only on $T$ and the choice of standard states. No other condition, neither pressure nor composition, can affect the value of $K$. We also see from Eq. 11.8.11 that $K$ is less than $1$ if $\Delsub{r}G\st$ is positive and greater than $1$ if $\Delsub{r}G\st$ is negative. At a fixed temperature, reaction equilibrium is attained only if and only if the value of $Q\subs{rxn}$ becomes equal to the value of $K$ at that temperature. The thermodynamic equilibrium constant $K$ is the proper quotient of the activities of species in reaction equilibrium. At typical temperatures and pressures, an activity cannot be many orders of magnitude greater than $1$. For instance, a partial pressure cannot be greater than the total pressure, so at a pressure of $10\br$ the activity of a gaseous constituent cannot be greater than about $10$. The molarity of a solute is rarely much greater than $10\units{mol dm\(^{-3}$}\), corresponding to an activity (on a concentration basis) of about $10$. Activities can, however, be extremely small. These considerations lead us to the conclusion that in an equilibrium state of a reaction with a very large value of $K$, the activity of at least one of the reactants must be very small. That is, if $K$ is very large then the reaction goes practically to completion and at equilibrium a limiting reactant is essentially entirely exhausted. The opposite case, a reaction with a very small value of $K$, must have at equilibrium one or more products with very small activities. These two cases are the two extremes of the trends shown in Fig. 11.16. Equation 11.8.10 correctly relates $\Delsub{r}G\st$ and $K$ only if they are both calculated with the same standard states. For instance, if we base the standard state of a particular solute species on molality in calculating $\Delsub{r}G\st$, the activity of that species appearing in the expression for $K$ (Eq. 11.8.9) must also be based on molality. 11.8.2 Reaction in a gas phase If a reaction takes place in a gaseous mixture, the standard state of each reactant and product is the pure gas behaving ideally at the standard pressure $p\st$ (Sec. 9.3.3). In this case, each activity is given by $a_i\gas = \fug_i/p\st = \phi_i p_i/p\st$ where $\phi_i$ is a fugacity coefficient (Table 9.5). When we substitute this expression into Eq. 11.8.9, we find we can express the thermodynamic equilibrium constant as the product of three factors: \begin{gather} \s{ K = \left[\prod_i (\phi_i)\eq^{\nu_i}\right] \left[\prod_i (p_i)\eq^{\nu_i}\right] \left[(p\st)^{- \sum_i\nu_i}\right] } \tag{11.8.12} \cond{(gas mixture)} \end{gather} On the right side of this equation, the first factor is the proper quotient of fugacity coefficients in the mixture at reaction equilibrium, the second factor is the proper quotient of partial pressures in this mixture, and the third factor is the power of $p\st$ needed to make $K$ dimensionless. The proper quotient of equilibrium partial pressures is an equilibrium constant on a pressure basis, $K_p$: \begin{gather} \s{ K_p = \prod_i (p_i)\eq^{\nu_i} } \tag{11.8.13} \cond{(gas mixture)} \end{gather} Note that $K_p$ is dimensionless only if $\sum_i\!\nu_i$ is equal to zero. The value of $K_p$ can vary at constant temperature, so $K_p$ is not a thermodynamic equilibrium constant. For instance, consider what happens when we take an ideal gas mixture at reaction equilibrium and compress it isothermally. As the gas pressure increases, the fugacity coefficient of each constituent changes from its low pressure value of $1$ and the gas mixture becomes nonideal. In order for the mixture to remain in reaction equilibrium, and the product of factors on the right side of Eq. 11.8.12 to remain constant, there must be a change in the value of $K_p$. In other words, the reaction equilibrium shifts as we increase $p$ at constant $T$, an effect that will be considered in more detail in Sec. 11.9. As an example of the difference between $K$ and $K_p$, consider again the ammonia synthesis $\ce{N2}\tx{(g)} + \ce{3H2}\tx{(g)} \arrow \ce{2NH3}\tx{(g)}$ in which the sum $\sum_i\!\nu_i$ equals $-2$. For this reaction, the expression for the thermodynamic equilibrium constant is $K = \left( \frac{\phi\subs{NH$_3$}^2}{\phi\subs{N$_2$}\phi\subs{H$_2$}^3} \right)\eq K_p (p\st)^2 \tag{11.8.14}$ where $K_p$ is given by $K_p = \left( \frac{p\subs{NH$_3$}^2}{p\subs{N$_2$}p\subs{H$_2$}^3} \right)\eq \tag{11.8.15}$ 11.8.3 Reaction in solution If any of the reactants or products are solutes in a solution, the value of $K$ depends on the choice of the solute standard state. For a given reaction at a given temperature, we can derive relations between values of $K$ that are based on different solute standard states. In the limit of infinite dilution, each solute activity coefficient is unity, and at the standard pressure each pressure factor is unity. Under these conditions of infinite dilution and standard pressure, the activities of solute B on a mole fraction, concentration, and molality basis are therefore $a\xbB=x\B\qquad a\cbB=c\B/c\st \qquad a\mbB=m\B/m\st \tag{11.8.16}$ In the limit of infinite dilution, the solute composition variables approach values given by the relations in Eq. 9.1.14: $x\B = V\A^*c\B = M\A m\B$. Combining these with $a\xbB=x\B$ from Eq. 11.8.16, we write $a\xbB = V\A^*c\B = M\A m\B \tag{11.8.17}$ Then, using the relations for $a\cbB$ and $a\mbB$ in Eq. 11.8.16, we find that the activities of solute B at infinite dilution and pressure $p\st$ are related by $a\xbB = V\A^* c\st a\cbB = M\A m\st a\mbB \tag{11.8.18}$ The expression $K=\prod_i(a_i)\eq^{\nu_i}$ has a factor $(a\B)\eq^{\nu\B}$ for each solute B that is a reactant or product. From Eq. 11.8.18, we see that for solutes at infinite dilution at pressure $p\st$, the relations between the values of $K$ based on different solute standard states are $K\tx{($x$ basis)} = \prod_\tx{B}(V\A^*c\st)^{\nu\B} K\tx{($c$ basis)} = \prod_\tx{B}(M\A m\st)^{\nu\B} K\tx{($m$ basis)} \tag{11.8.19}$ For a given reaction at a given temperature, and with a given choice of solute standard state, the value of $K$ is not affected by pressure or dilution. The relations of Eq. 11.8.19 are therefore valid under all conditions. 11.8.4 Evaluation of $K$ The relation $K=\exp (-\Delsub{r}G\st/RT)$ (Eq. 11.8.11) gives us a way to evaluate the thermodynamic equilibrium constant $K$ of a reaction at a given temperature from the value of the standard molar reaction Gibbs energy $\Delsub{r}G\st$ at that temperature. If we know the value of $\Delsub{r}G\st$, we can calculate the value of $K$. One method is to calculate $\Delsub{r}G\st$ from values of the standard molar Gibbs energy of formation $\Delsub{f}G\st$ of each reactant and product. These values are the standard molar reaction Gibbs energies for the formation reactions of the substances. To relate $\Delsub{f}G\st$ to measurable quantities, we make the substitution $\mu_i = H_i - TS_i$ (Eq. 9.2.46) in $\Delsub{r}G = \sum_i\!\nu_i\mu_i$ to give $\Delsub{r}G = \sum_i\!\nu_i H_i - T \sum_i\!\nu_i S_i$, or $\Delsub{r}G = \Delsub{r}H - T\Delsub{r}S \tag{11.8.20}$ When we apply this equation to a reaction with each reactant and product in its standard state, it becomes $\Delsub{r}G\st = \Delsub{r}H\st - T\Delsub{r}S\st \tag{11.8.21}$ where the standard molar reaction entropy is given by $\Delsub{r}S\st = \sum_i\nu_i S_i\st \tag{11.8.22}$ If the reaction is the formation reaction of a substance, we have $\Delsub{f}G\st = \Delsub{f}H\st - T\sum_i\nu_i S_i\st \tag{11.8.23}$ where the sum over $i$ is for the reactants and product of the formation reaction. We can evaluate the standard molar Gibbs energy of formation of a substance, then, from its standard molar enthalpy of formation and the standard molar entropies of the reactants and product. Extensive tables are available of values of $\Delsub{f}G\st$ for substances and ions. An abbreviated version at the single temperature $298.15\K$ is given in Appendix H. For a reaction of interest, the tabulated values enable us to evaluate $\Delsub{r}G\st$, and then $K$, from the expression (analogous to Hess’s law) $\Delsub{r}G\st = \sum_i\nu_i\Delsub{f}G\st(i) \tag{11.8.24}$ The sum over $i$ is for the reactants and products of the reaction of interest. Recall that the standard molar enthalpies of formation needed in Eq. 11.8.23 can be evaluated by calorimetric methods (Sec. 11.3.2). The absolute molar entropy values $S_i\st$ come from heat capacity data or statistical mechanical theory by methods discussed in Sec. 6.2. Thus, it is entirely feasible to use nothing but calorimetry to evaluate an equilibrium constant, a goal sought by thermodynamicists during the first half of the 20th century. (Another method, for a reaction that can be carried out reversibly in a galvanic cell, is described in Sec. 14.3.3.) For ions in aqueous solution, the values of $S\m\st$ and $\Delsub{f}G\st$ found in Appendix H are based on the reference values $S\m\st=0$ and $\Delsub{f}G\st = 0$ for H$^+$(aq) at all temperatures, similar to the convention for $\Delsub{f}H\st$ values discussed in Sec. 11.3.2. For a reaction with aqueous ions as reactants or products, these values correctly give $\Delsub{r}S\st$ using Eq. 11.8.22, or $\Delsub{r}G\st$ using Eq. 11.8.24. Note that the values of $S\m\st$ in Appendix H for some ions, unlike the values for substances, are negative; this simply means that the standard molar entropies of these ions are less than that of H$^+$(aq). The relation of Eq. 11.8.23 does not apply to an ion, because we cannot write a formation reaction for a single ion. Instead, the relation between $\Delsub{f}G\st$, $\Delsub{f}H\st$ and $S\m\st$ is more complicated. Consider first a hypothetical reaction in which hydrogen ions and one or more elements form H$_2$ and a cation M$^{z_+}$ with charge number $z_+$: \[ z_+\tx{H$^+$(aq)}+\tx{elements} \arrow (z_+/2)\tx{H$_2$(g)}+\tx{M$^{z_+}$(aq)}$ For this reaction, using the convention that $\Delsub{f}H\st$, $S\m\st$, and $\Delsub{f}G\st$ are zero for the aqueous H$^+$ ion and the fact that $\Delsub{f}H\st$ and $\Delsub{f}G\st$ are zero for the elements, we can write the following expressions for standard molar reaction quantities: $\Delsub{r}H\st = \Delsub{f}H\st(\tx{M$^{z_+}$}) \tag{11.8.25}$ $\Delsub{r}S\st = (z_+/2)S\m\st(\tx{H$_2$}) +S\m\st(\tx{M$^{z_+}$})- \! \sum_{\tx{elements}}\!\!\!S_i\st \tag{11.8.26}$ $\Delsub{r}G\st = \Delsub{f}G\st(\tx{M$^{z_+}$}) \tag{11.8.27}$ Then, from $\Delsub{r}G\st=\Delsub{r}H\st-T\Delsub{r}S\st$, we find $\Delsub{f}G\st(\tx{M$^{z_+}$}) = \Delsub{f}H\st(\tx{M$^{z_+}$}) \quad -T\left[ S\m\st(\tx{M$^{z_+}$}) -\sum_{\tx{elements}}\!\!\!S_i\st + (z_+/2)S\m\st(\tx{H$_2$}) \right] \tag{11.8.28}$ For example, the standard molar Gibbs energy of the aqueous mercury(I) ion is found from $\textstyle \Delsub{f}G\st(\tx{Hg$_2$$^{2+}$}) = \Delsub{f}H\st(\tx{Hg$_2$$^{2+}$}) - TS\m\st(\tx{Hg$_2$$^{2+}$}) + 2TS\m\st(\tx{Hg}) - \frac{2}{2}TS\m\st(\tx{H$_2$}) \tag{11.8.29}$
For an anion X$^{z_-}$ with negative charge number $z_-$, using the hypothetical reaction $|z_-/2| \tx{H$_2$(g)}+\tx{elements} \arrow |z_-| \tx{H$^+$(aq)}+\tx{X$^{z_-}$(aq)}$ we find by the same method $\Delsub{f}G\st(\tx{X$^{z_-}$}) = \Delsub{f}H\st(\tx{X$^{z_-}$}) -T\left[ S\m\st(\tx{X$^{z_-}$})- \! \sum_{\tx{elements}}\!\!\!S_i\st -|z_-/2| S\m\st(\tx{H$_2$}) \right] \tag{11.8.30}$ For example, the calculation for the nitrate ion is $\textstyle \Delsub{f}G\st(\tx{NO$_3$$^-$}) = \Delsub{f}H\st(\tx{NO$_3$$^-$}) - TS\m\st(\tx{NO$_3$$^-$}) + \frac{1}{2}TS\m\st(\tx{N$_2$}) + \frac{3}{2}TS\m\st(\tx{O$_2$}) + \frac{1}{2}TS\m\st(\tx{H$_2$}) \tag{11.8.31}$ | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/11%3A_Reactions_and_Other_Chemical_Processes/11.08%3A_The_Thermodynamic_Equilibrium_Constant.txt |
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The advancement $\xi$ of a chemical reaction in a closed system describes the changes in the amounts of the reactants and products from specified initial values of these amounts. We have seen that if the system is maintained at constant temperature and pressure, $\xi$ changes spontaneously in the direction that decreases the Gibbs energy. The change continues until the system reaches a state of reaction equilibrium at the minimum of $G$. The value of the advancement in this equilibrium state will be denoted $\xi\eq$, as shown in Fig. 11.15. The value of $\xi\eq$ depends in general on the values of $T$ and $p$. Thus when we change the temperature or pressure of a closed system that is at equilibrium, $\xi\eq$ usually changes also and the reaction spontaneously shifts to a new equilibrium position.
To investigate this effect, we write the total differential of $G$ with $T$, $p$, and $\xi$ as independent variables $\dif G = -S\dif T + V\difp + \Delsub{r}G\dif\xi \tag{11.9.1}$ and obtain the reciprocity relations $\Pd{\Delsub{r}G}{T}{p, \xi} = -\Pd{S}{\xi}{T,p} \qquad \Pd{\Delsub{r}G}{p}{T, \xi} = \Pd{V}{\xi}{T,p} \tag{11.9.2}$ We recognize the partial derivative on the right side of each of these relations as a molar differential reaction quantity: $\Pd{\Delsub{r}G}{T}{p, \xi} = -\Delsub{r}S \qquad \Pd{\Delsub{r}G}{p}{T, \xi} = \Delsub{r}V \tag{11.9.3}$ We use these expressions for two of the coefficients in an expression for the total differential of $\Delsub{r}G$: \begin{gather} \s{ \dif\Delsub{r}G = -\Delsub{r}S\dif T + \Delsub{r}V\difp + \Pd{\Delsub{r}G}{\xi}{T,p}\dif\xi } \tag{11.9.4} \cond{(closed system)} \end{gather} Since $\Delsub{r}G$ is the partial derivative of $G$ with respect to $\xi$ at constant $T$ and $p$, the coefficient $\pd{\Delsub{r}G}{\xi}{T,p}$ is the partial second derivative of $G$ with respect to $\xi$: $\Pd{\Delsub{r}G}{\xi}{T,p} = \Pd{^2 G}{\xi^2}{T,p} \tag{11.9.5}$ We know that at a fixed $T$ and $p$, a plot of $G$ versus $\xi$ has a slope at each point equal to $\Delsub{r}G$ and a minimum at the position of reaction equilibrium where $\xi$ is $\xi\eq$. At the minimum of the plotted curve, the slope $\Delsub{r}G$ is zero and the second derivative is positive (see Fig. 11.15). By setting $\Delsub{r}G$ equal to zero in the general relation $\Delsub{r}G = \Delsub{r}H - T\Delsub{r}S$, we obtain the equation $\Delsub{r}S = \Delsub{r}H/T$ which is valid only at reaction equilibrium where $\xi$ equals $\xi\eq$. Making this substitution in Eq. 11.9.4, and setting $\dif\Delsub{r}G$ equal to zero and $\dif\xi$ equal to $\dif\xi\eq$, we obtain \begin{gather} \s{ 0 = -\frac{\Delsub{r}H}{T}\dif T + \Delsub{r}V\difp + \Pd{^2 G}{\xi^2}{T,p}\dif\xi\eq } \tag{11.9.6} \cond{(closed system)} \end{gather} which shows how infinitesimal changes in $T$, $p$, and $\xi\eq$ are related.
Now we are ready to see how $\xi\eq$ is affected by changes in $T$ or $p$. Solving Eq. 11.9.6 for $\dif\xi\eq$ gives \begin{gather} \s{ \dif\xi\eq = \frac{\displaystyle \frac{\Delsub{r}H}{T}\dif T - \Delsub{r}V\difp} { \Pd{^2 G}{\xi^2}{T,p}} } \tag{11.9.7} \cond{(closed system)} \nextcond{} \end{gather}
The right side of Eq. 11.9.7 is the expression for the total differential of $\xi$ in a closed system at reaction equilibrium, with $T$ and $p$ as the independent variables. Thus, at constant pressure the equilibrium shifts with temperature according to \begin{gather} \s{ \Pd{\xi\eq}{T}{p} = \frac{\Delsub{r}H} { T\Pd{^2 G}{\xi^2}{T,p}} } \tag{11.9.8} \cond{(closed system)} \nextcond{} \end{gather}
and at constant temperature the equilibrium shifts with pressure according to \begin{gather} \s{ \Pd{\xi\eq}{p}{T} = - \frac{\Delsub{r}V} { \Pd{^2 G}{\xi^2}{T,p}} } \tag{11.9.9} \cond{(closed system)} \nextcond{} \end{gather}
Because the partial second derivative $\pd{^2G}{\xi^2}{T,p}$ is positive, Eqs. 11.9.8 and 11.9.9 show that $\pd{\xi\eq}{T}{p}$ and $\Delsub{r}H$ have the same sign, whereas $\pd{\xi\eq}{p}{T}$ and $\Delsub{r}V$ have opposite signs.
These statements express the application to temperature and pressure changes of what is known as Le Chatelier’s principle: When a change is made to a closed system at equilibrium, the equilibrium shifts in the direction that tends to oppose the change. Here are two examples.
1. It is easy to misuse or to be misled by Le Chatelier’s principle. Consider the solution process B$^*$(s)$\arrow$B(sln) for which $\pd{\xi\eq}{T}{p}$, the rate of change of solubility with $T$, has the same sign as the molar differential enthalpy of solution $\Delsub{sol}H$ at saturation. The sign of $\Delsub{sol}H$ at saturation may be different from the sign of the molar integral enthalpy of solution, $\Del H\m\sol$. This is the situation for the dissolution of sodium acetate shown in Fig. 11.9. The equilibrium position (saturation) with one kilogram of water is at $\xi\subs{sol} \approx 15\mol$, indicated in the figure by an open circle. At this position, $\Delsub{sol}H$ is positive and $\Del H\m\sol$ is negative. So, despite the fact that the dissolution of 15 moles of sodium acetate in one kilogram of water to form a saturated solution is an exothermic process, the solubility of sodium acetate actually increases with increasing temperature, contrary to what one might predict from Le Chatelier’s principle (L. K. Brice, J. Chem. Educ., 60, 387–389, 1983).
Another kind of change for which Le Chatelier’s principle gives an incorrect prediction is the addition of an inert gas to a gas mixture of constant volume. Adding the inert gas at constant $V$ increases the pressure, but has little effect on the equilibrium position of a gas-phase reaction regardless of the value of $\Delsub{r}V$. This is because the inert gas affects the activities of the reactants and products only slightly, and not at all if the gas mixture is ideal, so there is little or no effect on the value of $Q\subs{rxn}$. (Note that the dependence of $\xi\eq$ on $p$ expressed by Eq. 11.9.9 does not apply to an open system.) | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/11%3A_Reactions_and_Other_Chemical_Processes/11.09%3A_Effects_of_Temperature_and_Pressure_on_Equilibrium_Position.txt |
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$\newcommand{\g}{\gamma} % solute activity coefficient, or gamma in general$
$\newcommand{\G}{\varGamma} % activity coefficient of a reference state (pressure factor)$
$\newcommand{\ecp}{\widetilde{\mu}} % electrochemical or total potential$
$\newcommand{\Eeq}{E\subs{cell, eq}} % equilibrium cell potential$
$\newcommand{\Ej}{E\subs{j}} % liquid junction potential$
$\newcommand{\mue}{\mu\subs{e}} % electron chemical potential$
$\newcommand{\defn}{\,\stackrel{\mathrm{def}}{=}\,} % "equal by definition" symbol$
$\newcommand{\D}{\displaystyle} % for a line in built-up$
$\newcommand{\s}{\smash[b]} % use in equations with conditions of validity$
$\newcommand{\cond}[1]{\$-2.5pt]{}\tag*{#1}}$ $\newcommand{\nextcond}[1]{\[-5pt]{}\tag*{#1}}$ $\newcommand{\R}{8.3145\units{J\,K\per\,mol\per}} % gas constant value$ $\newcommand{\Rsix}{8.31447\units{J\,K\per\,mol\per}} % gas constant value - 6 sig figs$ $\newcommand{\jn}{\hspace3pt\lower.3ex{\Rule{.6pt}{2ex}{0ex}}\hspace3pt}$ $\newcommand{\ljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}} \hspace3pt}$ $\newcommand{\lljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace1.4pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace3pt}$ An underlined problem number or problem-part letter indicates that the numerical answer appears in Appendix I. 11.1 Use values of $\Delsub{f}H\st$ and $\Delsub{f}G\st$ in Appendix H to evaluate the standard molar reaction enthalpy and the thermodynamic equilibrium constant at $298.15\K$ for the oxidation of nitrogen to form aqueous nitric acid: \[ \ce{1/2N2}\tx{(g)} + \ce{5/4O2}\tx{(g)} + \ce{1/2H2O}\tx{(l)} \arrow \ce{H+}\tx{(aq)} + \ce{NO3-}\tx{(aq)}$
11.2
In 1982, the International Union of Pure and Applied Chemistry recommended that the value of the standard pressure $p\st$ be changed from $1\units{atm}$ to $1\br$. This change affects the values of some standard molar quantities of a substance calculated from experimental data.
(a) Find the changes in $H\m\st$, $S\m\st$, and $G\m\st$ for a gaseous substance when the standard pressure is changed isothermally from $1.01325\br$ ($1\units{atm}$) to exactly $1\br$. (Such a small pressure change has an entirely negligible effect on these quantities for a substance in a condensed phase.)
(b) What are the values of the corrections that need to be made to the standard molar enthalpy of formation, the standard molar entropy of formation, and the standard molar Gibbs energy of formation of N$_2$O$_4$(g) at $298.15\K$ when the standard pressure is changed from $1.01325\br$ to $1\br$?
11.3
From data for mercury listed in Appendix H, calculate the saturation vapor pressure of liquid mercury at both $298.15\K$ and $273.15\K$. You may need to make some reasonable approximations.
11.4
Given the following experimental values at $T = 298.15\K$, $p=1\br$: \begin{alignat*}{2} & \tx{H$^+$(aq)} + \tx{OH$^-$(aq)} \arrow \tx{H$_2$O(l)} & & \Delsub{r}H\st = -55.82\units{kJ mol$^{-1}$} \cr & \tx{Na(s)} + \tx{H$_2$O(l)} \arrow \tx{Na$^+$(aq}) + \tx{OH$^-$(aq)} + \textstyle \frac{1}{2}\tx{H$_2$(g)} & \qquad & \Delsub{r}H\st = -184.52\units{kJ mol$^{-1}$} \cr & \tx{NaOH(s)} \arrow \tx{NaOH(aq)} & & \Delsub{sol}H^{\infty} = -44.75\units{kJ mol$^{-1}$} \cr & \tx{NaOH in 5 H$_2$O} \arrow \tx{NaOH in $\infty$ H$_2$O} & & \Del H\m\dil = -4.93\units{kJ mol$^{-1}$} \cr & \tx{NaOH(s)} & & \Delsub{f}H\st = -425.61\units{kJ mol$^{-1}$} \end{alignat*}
Using only these values, calculate:
(a) $\Delsub{f}H\st$ for Na$^+$(aq), NaOH(aq), and OH$^-$(aq);
(b) $\Delsub{f}H$ for NaOH in 5 H$_2$O;
(c) $\Del H\m\sol$ for the dissolution of $1\mol$ NaOH(s) in $5\mol$ H$_2$O.
States 1 and 2 referred to in this problem are the initial and final states of the isothermal bomb process. The temperature is the reference temperature of $298.15\K$.
(a) Parts (a)–(c) consist of simple calculations of some quantities needed in later parts of the problem. Begin by using the masses of C$_6$H$_{14}$ and H$_2$O placed in the bomb vessel, and their molar masses, to calculate the amounts (moles) of C$_6$H$_{14}$ and H$_2$O present initially in the bomb vessel. Then use the stoichiometry of the combustion reaction to find the amount of O$_2$ consumed and the amounts of H$_2$O and CO$_2$ present in state 2. (There is not enough information at this stage to allow you to find the amount of O$_2$ present, just the change.) Also find the final mass of H$_2$O. Assume that oxygen is present in excess and the combustion reaction goes to completion.
(b) From the molar masses and the densities of liquid C$_6$H$_{14}$ and H$_2$O, calculate their molar volumes.
(c) From the amounts present initially in the bomb vessel and the internal volume, find the volumes of liquid C$_6$H$_{14}$, liquid H$_2$O, and gas in state 1 and the volumes of liquid H$_2$O and gas in state 2. For this calculation, you can neglect the small change in the volume of liquid H$_2$O due to its vaporization.
(d) When the bomb vessel is charged with oxygen and before the inlet valve is closed, the pressure at $298.15\K$ measured on an external gauge is found to be $p_1 = 30.00\br$. To a good approximation, the gas phase of state 1 has the equation of state of pure O$_2$ (since the vapor pressure of water is only $0.1\units{\(\%$}\) of $30.00\br$). Assume that this equation of state is given by $V\m=RT/p+B\subs{BB}$ (Eq. 2.2.8), where $B\subs{BB}$ is the second virial coefficient of O$_2$ listed in Table 11.3. Solve for the amount of O$_2$ in the gas phase of state 1. The gas phase of state 2 is a mixture of O$_2$ and CO$_2$, again with a negligible partial pressure of H$_2$O. Assume that only small fractions of the total amounts of O$_2$ and CO$_2$ dissolve in the liquid water, and find the amount of O$_2$ in the gas phase of state 2 and the mole fractions of O$_2$ and CO$_2$ in this phase.
(e) You now have the information needed to find the pressure in state 2, which cannot be measured directly. For the mixture of O$_2$ and CO$_2$ in the gas phase of state 2, use Eq. 9.3.23 to calculate the second virial coefficient. Then solve the equation of state of Eq. 9.3.21 for the pressure. Also calculate the partial pressures of the O$_2$ and CO$_2$ in the gas mixture.
(f) Although the amounts of H$_2$O in the gas phases of states 1 and 2 are small, you need to know their values in order to take the energy of vaporization into account. In this part, you calculate the fugacities of the H$_2$O in the initial and final gas phases, in part (g) you use gas equations of state to evaluate the fugacity coefficients of the H$_2$O (as well as of the O$_2$ and CO$_2$), and then in part (h) you find the amounts of H$_2$O in the initial and final gas phases.
The pressure at which the pure liquid and gas phases of H$_2$O are in equilibrium at $298.15\K$ (the saturation vapor pressure of water) is $0.03169\br$. Use Eq. 7.8.18 to estimate the fugacity of H$_2$O(g) in equilibrium with pure liquid water at this temperature and pressure. The effect of pressure on fugacity in a one-component liquid–gas system is discussed in Sec. 12.8.1; use Eq. 12.8.3 to find the fugacity of H$_2$O in gas phases equilibrated with liquid water at the pressures of states 1 and 2 of the isothermal bomb process. (The mole fraction of O$_2$ dissolved in the liquid water is so small that you can ignore its effect on the chemical potential of the water.)
(g) Calculate the fugacity coefficients of H$_2$O and O$_2$ in the gas phase of state 1 and of H$_2$O, O$_2$, and CO$_2$ in the gas phase of state 2.
For state 1, in which the gas phase is practically-pure O$_2$, you can use Eq. 7.8.18 to calculate $\phi\subs{O\(_2$}\). The other calculations require Eq. 9.3.29, with the value of $B_i'$ found from the formulas of Eq. 9.3.26 or Eqs. 9.3.27 and 9.3.28 ($y\A$ is so small that you can set it equal to zero in these formulas).
Use the fugacity coefficient and partial pressure of O$_2$ to evaluate its fugacity in states 1 and 2; likewise, find the fugacity of CO$_2$ in state 2. [You calculated the fugacity of the H$_2$O in part (f).]
(h) From the values of the fugacity and fugacity coefficient of a constituent of a gas mixture, you can calculate the partial pressure with Eq. 9.3.17, then the mole fraction with $y_i=p_i/p$, and finally the amount with $n_i=y_i n$. Use this method to find the amounts of H$_2$O in the gas phases of states 1 and 2, and also calculate the amounts of H$_2$O in the liquid phases of both states.
(i) Next, consider the O$_2$ dissolved in the water of state 1 and the O$_2$ and CO$_2$ dissolved in the water of state 2. Treat the solutions of these gases as ideal dilute with the molality of solute $i$ given by $m_i=\fug_i/k_{m,i}$ (Eq. 9.4.21). The values of the Henry’s law constants of these gases listed in Table 11.3 are for the standard pressure of $1\br$. Use Eq. 12.8.35 to find the appropriate values of $k_{m,i}$ at the pressures of states 1 and 2, and use these values to calculate the amounts of the dissolved gases in both states.
(j) At this point in the calculations, you know the values of all properties needed to describe the initial and final states of the isothermal bomb process. You are now able to evaluate the various Washburn corrections. These corrections are the internal energy changes, at the reference temperature of $298.15\K$, of processes that connect the standard states of substances with either state 1 or state 2 of the isothermal bomb process.
First, consider the gaseous H$_2$O. The Washburn corrections should be based on a pure-liquid standard state for the H$_2$O. Section 7.9 shows that the molar internal energy of a pure gas under ideal-gas conditions (low pressure) is the same as the molar internal energy of the gas in its standard state at the same temperature. Thus, the molar internal energy change when a substance in its pure-liquid standard state changes isothermally to an ideal gas is equal to the standard molar internal energy of vaporization, $\Delsub{vap}U\st$. Using the value of $\Delsub{vap}U\st$ for H$_2$O given in Table 11.3, calculate $\Del U$ for the vaporization of liquid H$_2$O at pressure $p\st$ to ideal gas in the amount present in the gas phase of state 1. Also calculate $\Del U$ for the condensation of ideal gaseous H$_2$O in the amount present in the gas phase of state 2 to liquid at pressure $p\st$.
(k) Next, consider the dissolved O$_2$ and CO$_2$, for which gas standard states are used. Assume that the solutions are sufficiently dilute to have infinite-dilution behavior; then the partial molar internal energy of either solute in the solution at the standard pressure $p\st=1\br$ is equal to the standard partial molar internal energy based on a solute standard state (Sec. 9.7.1). Values of $\Delsub{sol}U\st$ are listed in Table 11.3. Find $\Del U$ for the dissolution of O$_2$ from its gas standard state to ideal-dilute solution at pressure $p\st$ in the amount present in the aqueous phase of state 1. Find $\Del U$ for the desolution (transfer from solution to gas phase) of O$_2$ and of CO$_2$ from ideal-dilute solution at pressure $p\st$, in the amounts present in the aqueous phase of state 2, to their gas standard states.
(l) Calculate the internal energy changes when the liquid phases of state 1 (n-hexane and aqueous solution) are compressed from $p\st$ to $p_1$ and the aqueous solution of state 2 is decompressed from $p_2$ to $p\st$. Use an approximate expression from Table 7.4, and treat the cubic expansion coefficient of the aqueous solutions as being the same as that of pure water.
(m) The final Washburn corrections are internal energy changes of the gas phases of states 1 and 2. H$_2$O has such low mole fractions in these phases that you can ignore H$_2$O in these calculations; that is, treat the gas phase of state 1 as pure O$_2$ and the gas phase of state 2 as a binary mixture of O$_2$ and CO$_2$.
One of the internal energy changes is for the compression of gaseous O$_2$, starting at a pressure low enough for ideal-gas behavior ($U\m=U\m\st$) and ending at pressure $p_1$ to form the gas phase present in state 1. Use the approximate expression for $U\m-U\m\st\gas$ in Table 7.5 to calculate $\Del U = U(p_1) - nU\m\st\gas$; a value of $\dif B/\dif T$ for pure O$_2$ is listed in Table 11.3.
The other internal energy change is for a process in which the gas phase of state 2 at pressure $p_2$ is expanded until the pressure is low enough for the gas to behave ideally, and the mixture is then separated into ideal-gas phases of pure O$_2$ and CO$_2$. The molar internal energies of the separated low-pressure O$_2$ and CO$_2$ gases are the same as the standard molar internal energies of these gases. The internal energy of unmixing ideal gases is zero (Eq. 11.1.11). The dependence of the internal energy of the gas mixture is given, to a good approximation, by $U = \sum_i U_i\st\gas - npT\dif B/\dif T$, where $B$ is the second virial coefficient of the gas mixture; this expression is the analogy for a gas mixture of the approximate expression for $U\m-U\m\st\gas$ in Table 7.5. Calculate the value of $\dif B/\dif T$ for the mixture of O$_2$ and CO$_2$ in state 2 (you need Eq. 9.3.23 and the values of $\dif B_{ij}/\dif T$ in Table 11.3) and evaluate $\Del U = \sum_i n_i U_i\st\gas -U(p_2)$ for the gas expansion.
(n) Add the internal energy changes you calculated in parts (j)–(m) to find the total internal energy change of the Washburn corrections. Note that most of the corrections occur in pairs of opposite sign and almost completely cancel one another. Which contributions are the greatest in magnitude?
(o) The internal energy change of the isothermal bomb process in the bomb vessel, corrected to the reference temperature of $298.15\K$, is found to be $\Del U(\tx{IBP},T\subs{ref}) = -32.504\units{kJ}$. Assume there are no side reactions or auxiliary reactions. From Eqs. 11.5.9 and 11.5.10, calculate the standard molar internal energy of combustion of n-hexane at $298.15\K$.
(p) From Eq. 11.5.13, calculate the standard molar enthalpy of combustion of n-hexane at $298.15\K$.
11.8
By combining the results of Prob. 11.7(p) with the values of standard molar enthalpies of formation from Appendix H, calculate the standard molar enthalpy of formation of liquid n-hexane at $298.15\K$.
11.9
Consider the combustion of methane: $\ce{CH4}\tx{(g)} + \ce{2O2}\tx{(g)} \arrow \ce{CO2}\tx{(g)} + \ce{2H2O}\tx{(g)}$ Suppose the reaction occurs in a flowing gas mixture of methane and air. Assume that the pressure is constant at $1\br$, the reactant mixture is at a temperature of $298.15\K$ and has stoichiometric proportions of methane and oxygen, and the reaction goes to completion with no dissociation. For the quantity of gaseous product mixture containing $1\mol$ CO$_2$, $2\mol$ H$_2$O, and the nitrogen and other substances remaining from the air, you may use the approximate formula $C_p(\tx{P})=a+bT$, where the coefficients have the values $a=297.0\units{J K\(^{-1}$}\) and $b=8.520\timesten{-2}\units{J K\(^{-2}$}\). Solve Eq. 11.6.1 for $T_2$ to estimate the flame temperature to the nearest kelvin.
11.10
The standard molar Gibbs energy of formation of crystalline mercury(II) oxide at $600.00\K$ has the value $\Delsub{f}G\st=-26.386\units{kJ mol\(^{-1}$}\). Estimate the partial pressure of O$_2$ in equilibrium with HgO at this temperature: $\ce{2HgO}\tx{(s)} \arrows \ce{2Hg}\tx{(l)} + \ce{O2}\tx{(g)}$.
11.11
The combustion of hydrogen is a reaction that is known to “go to completion.”
(a) Use data in Appendix H to evaluate the thermodynamic equilibrium constant at $298.15\K$ for the reaction $\ce{H2}\tx{(g)}+\ce{1/2O2}\tx{(g)} \arrow \ce{H2O}\tx{(l)}$
(b) Assume that the reaction is at equilibrium at $298.15\K$ in a system in which the partial pressure of O$_2$ is $1.0\br$. Assume ideal-gas behavior and find the equilibrium partial pressure of H$_2$ and the number of H$_2$ molecules in $1.0\units{m\(^3$}\) of the gas phase.
(c) In the preceding part, you calculated a very small value (a fraction) for the number of H$_2$ molecules in $1.0\units{m\(^3$}\). Statistically, this fraction can be interpreted as the fraction of a given length of time during which one molecule is present in the system. Take the age of the universe as $1.0\timesten{10}$ years and find the total length of time in seconds, during the age of the universe, that a H$_2$ molecule is present in the equilibrium system. (This hypothetical value is a dramatic demonstration of the statement that the limiting reactant is essentially entirely exhausted during a reaction with a large value of $K$.)
11.12
Let G represent carbon in the form of graphite and D represent the diamond crystal form. At $298.15\K$, the thermodynamic equilibrium constant for G$\rightleftharpoons$D, based on a standard pressure $p\st=1\br$, has the value $K=0.31$. The molar volumes of the two crystal forms at this temperature are $V\m(\tx{G})=5.3\timesten{-6}\units{m\(^3$ mol$^{-1}$}\) and $V\m(\tx{D})=3.4\timesten{-6}\units{m\(^3$ mol$^{-1}$}\).
(a) Write an expression for the reaction quotient $Q\subs{rxn}$ as a function of pressure. Use the approximate expression of the pressure factor given in Table 9.6.
(b) Use the value of $K$ to estimate the pressure at which the D and G crystal forms are in equilibrium with one another at $298.15\K$. (This is the lowest pressure at which graphite could in principle be converted to diamond at this temperature.)
11.13
Consider the dissociation reaction $\ce{N2O4}\tx{(g)} \arrow \ce{2NO2}\tx{(g)}$ taking place at a constant temperature of $298.15\K$ and a constant pressure of $0.0500\br$. Initially (at $\xi=0$) the system contains $1.000\mol$ of N$_2$O$_4$ and no NO$_2$. Other needed data are found in Appendix H. Assume ideal-gas behavior.
(a) For values of the advancement $\xi$ ranging from 0 to $1\mol$, at an interval of $0.1\mol$ or less, calculate $[ G(\xi)-G(0) ]$ to the nearest $0.01\units{kJ}$. A computer spreadsheet would be a convenient way to make the calculations.
(b) Plot your values of $G(\xi)-G(0)$ as a function of $\xi$, and draw a smooth curve through the points.
(c) On your curve, indicate the estimated position of $\xi\eq$. Calculate the activities of N$_2$O$_4$ and NO$_2$ for this value of $\xi$, use them to estimate the thermodynamic equilibrium constant $K$, and compare your result with the value of $K$ calculated from Eq. 11.8.11. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/11%3A_Reactions_and_Other_Chemical_Processes/11.10%3A_Chapter_11_Problems.txt |
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This chapter applies equilibrium theory to a variety of chemical systems of more than one component. Two different approaches will be used as appropriate: one based on the relation $\mu_i\aph=\mu_i\bph$ for transfer equilibrium, the other based on $\sum_i\!\nu_i\mu_i=0$ or $K=\prod_i a_i^{\nu_i}$ for reaction equilibrium.
12: Equilibrium Conditions in Multicomponent Systems
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$\newcommand{\rf}{^{\text{ref}}} % reference state$
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$\newcommand{\degC}{^\circ\text{C}} % degrees Celsius$
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$\newcommand{\mol}{\units{mol}} % mole$
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$\newcommand{\bd}{_{\text{b}}} % subscript b for boundary or boiling point$
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$\newcommand{\mB}{_{\text{m},\text{B}}} % subscript m,B (m=molar)$
$\newcommand{\mi}{_{\text{m},i}} % subscript m,i (m=molar)$
$\newcommand{\fA}{_{\text{f},\text{A}}} % subscript f,A (for fr. pt.)$
$\newcommand{\fB}{_{\text{f},\text{B}}} % subscript f,B (for fr. pt.)$
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$\newcommand{\xbC}{_{x,\text{C}}} % x basis, C$
$\newcommand{\cbB}{_{c,\text{B}}} % c basis, B$
$\newcommand{\mbB}{_{m,\text{B}}} % m basis, B$
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$\newcommand{\cm}{\subs{cm}} % center of mass$
$\newcommand{\rev}{\subs{rev}} % reversible$
$\newcommand{\irr}{\subs{irr}} % irreversible$
$\newcommand{\fric}{\subs{fric}} % friction$
$\newcommand{\diss}{\subs{diss}} % dissipation$
$\newcommand{\el}{\subs{el}} % electrical$
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$\newcommand{\E}{^\mathsf{E}} % excess quantity (superscript)$
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$\newcommand{\Eeq}{E\subs{cell, eq}} % equilibrium cell potential$
$\newcommand{\Ej}{E\subs{j}} % liquid junction potential$
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For some of the derivations in this chapter, we will need an expression for the rate at which the ratio $\mu_i/T$ varies with temperature in a phase of fixed composition maintained at constant pressure. This expression leads, among other things, to an important relation between the temperature dependence of an equilibrium constant and the standard molar reaction enthalpy.
12.1.1 Variation of $\mu_i/T$ with temperature
In a phase containing species $i$, either pure or in a mixture, the partial derivative of $\mu_i/T$ with respect to $T$ at constant $p$ and a fixed amount of each species is given by $\bPd{\left( \mu_i/T \right)}{T}{p,\allni} =\frac{1}{T}\Pd{\mu_i}{T}{\!\!p,\allni} - \frac{\mu_i}{T^2} \tag{12.1.1}$ This equality comes from a purely mathematical operation; no thermodynamics is involved.
The relation is obtained from the formula $\dif (uv)/\dx = u(\dif v/\dx) + v(\dif u/\dx)$ (Appendix E), where $u$ is $1/T$, $v$ is $\mu_i$, and $x$ is $T$.
The partial derivative $\pd{\mu_i}{T}{p,\allni}$ is equal to $-S_i$ (Eq. 9.2.48), so that Eq. 12.1.1 becomes $\bPd{\left( \mu_i/T \right)}{T}{p,\allni} = -\frac{S_i}{T} - \frac{\mu_i}{T^2} = -\frac{TS_i + \mu_i}{T^2} \tag{12.1.2}$ The further substitution $\mu_i = H_i - TS_i$ (Eq. 9.2.46) gives finally $\bPd{\left( \mu_i/T \right)}{T}{p,\allni} = -\frac{H_i}{T^2} \tag{12.1.3}$
For a pure substance in a closed system, Eq. 12.1.3 when multiplied by the amount $n$ becomes $\bPd{\left( G/T \right)}{T}{p} = -\frac{H}{T^2} \tag{12.1.4}$ This is the Gibbs–Helmholtz equation.
12.1.2 Variation of $\mu_i\st/T$ with temperature
If we make the substitution $\mu_i = \mu_i\st + RT\ln a_i$ in Eq. 12.1.3 and rearrange, we obtain $\frac{\dif(\mu_i\st/T)}{\dif T} = -\frac{H_i}{T^2} - R\Pd{\ln a_i}{T}{\!\!p,\allni} \tag{12.1.5}$ Because $\mu_i\st/T$ is a function only of $T$, its derivative with respect to $T$ is itself a function only of $T$. We can therefore use any convenient combination of pressure and composition in the expression on the right side of Eq. 12.1.5 in order to evaluate $\dif(\mu_i\st/T)/\dif T$ at a given temperature.
If species $i$ is a constituent of a gas mixture, we take a constant pressure of the gas that is low enough for the gas to behave ideally. Under these conditions $H_i$ is the standard molar enthalpy $H_i\st$ (Eq. 9.3.7). In the expression for activity, $a_i\gas = \G_i\gas \phi_i p_i/p$ (Table 9.5), the pressure factor $\G_i\gas$ is constant when $p$ is constant, the fugacity coefficient $\phi_i$ for the ideal gas is unity, and $p_i/p=y_i$ is constant at constant $\allni$, so that the partial derivative $\bpd{\ln a_i\gas}{T}{p,\allni}$ is zero.
For component $i$ of a condensed-phase mixture, we take a constant pressure equal to the standard pressure $p\st$, and a mixture composition in the limit given by Eqs. 9.5.20–9.5.24 in which the activity coefficient is unity. $H_i$ is then the standard molar enthalpy $H_i\st$, and the activity is given by an expression in Table 9.5 with the pressure factor and activity coefficient set equal to 1: $a_i{=}x_i$, $a\A{=}x\A$, $a\xbB{=}x\B$, $a\cbB{=}c\B/c\st$, or $a\mbB{=}m\B/m\st$. With the exception of $a\cbB$, these activities are constant as $T$ changes at constant $p$ and $\allni$.
If solute B is an electrolyte, $a\mbB$ is given instead by Eq. 10.3.10; like $a\mbB$ for a nonelectrolyte, it is constant as $T$ changes at constant $p$ and $\allni$.
Thus for a gas-phase species, or a species with a standard state based on mole fraction or molality, $\bpd{\ln a_i\gas}{T}{p,\allni}$ is zero and Eq. 12.1.5 becomes \begin{gather} \s{ \frac{\dif(\mu_i\st/T)}{\dif T} = -\frac{H_i\st}{T^2} } \tag{12.1.6} \cond{(standard state not based} \nextcond{on concentration)} \end{gather}
Equation 12.1.6, as the conditions of validity indicate, does not apply to a solute standard state based on concentration, except as an approximation. The reason is the volume change that accompanies an isobaric temperature change. We can treat this case by considering the following behavior of $\ln(c\B/c\st)$: $\begin{split} \bPd{\ln(c\B/c\st)}{T}{p,\allni} & = \frac{1}{c\B}\Pd{c\B}{T}{\!\!p,\allni} = \frac{1}{n\B/V}\bPd{(n\B/V)}{T}{p,\allni} \cr & = V\bPd{(1/V)}{T}{p,\allni} = -\frac{1}{V}\Pd{V}{T}{\!\!p,\allni} \cr & = -\alpha \end{split} \tag{12.1.7}$ Here $\alpha$ is the cubic expansion coefficient of the solution (Eq. 7.1.1). If the activity coefficient is to be unity, the solution must be an ideal-dilute solution, and $\alpha$ is then $\alpha\A^*$, the cubic expansion coefficient of the pure solvent. Eq. 12.1.5 for a nonelectrolyte becomes $\frac{\dif(\mu\cbB\st/T)}{\dif T} = -\frac{H\B\st}{T^2} + R\alpha\A^* \tag{12.1.8}$
12.1.3 Variation of $\ln K$ with temperature
The thermodynamic equilibrium constant $K$, for a given reaction equation and a given choice of reactant and product standard states, is a function of $T$ and only of $T$. By equating two expressions for the standard molar reaction Gibbs energy, $\Delsub{r}G\st= \sum_i\!\nu_i \mu_i\st$ and $\Delsub{r}G\st=-RT\ln K$ (Eqs. 11.8.3 and 11.8.10), we obtain $\ln K = -\frac{1}{RT} \sum_i\nu_i\mu_i\st \tag{12.1.9}$ The rate at which $\ln K$ varies with $T$ is then given by $\frac{\dif\ln K}{\dif T} = -\frac{1}{R} \sum_i\nu_i \frac{\dif(\mu_i\st/T)}{\dif T} \tag{12.1.10}$
Combining Eq. 12.1.10 with Eqs. 12.1.6 or 12.1.8, and recognizing that $\sum_i\!\nu_i H_i\st$ is the standard molar reaction enthalpy $\Delsub{r}H\st$, we obtain the final expression for the temperature dependence of $\ln K$: $\frac{\dif\ln K}{\dif T} = \frac{\Delsub{r}H\st}{RT^2} - \alpha\A^* \!\!\!\! \sum_{\stackrel{\tx{ solutes,}} {\tx{ conc. basis}}} \!\!\!\! \nu_i \tag{12.1.11}$ The sum on the right side includes only solute species whose standard states are based on concentration. The expression is simpler if all solute standard states are based on mole fraction or molality: \begin{gather} \s{ \frac{\dif\ln K}{\dif T} = \frac{\Delsub{r}H\st}{RT^2} } \tag{12.1.12} \cond{(no solute standard states} \nextcond{based on concentration)} \end{gather} We can rearrange Eq. 12.1.12 to \begin{gather} \s{ \Delsub{r}H\st = RT^2\frac{\dif\ln K}{\dif T} } \tag{12.1.13} \cond{(no solute standard states} \nextcond{based on concentration)} \end{gather} We can convert this expression for $\Delsub{r}H\st$ to an equivalent form by using the mathematical identity $\dif (1/T) = -(1/T^2)\dif T$: \begin{gather} \s{ \Delsub{r}H\st = -R\frac{\dif\ln K}{\dif(1/T)} } \tag{12.1.14} \cond{(no solute standard states} \nextcond{based on concentration)} \end{gather}
Equations 12.1.13 and 12.1.14 are two forms of the van’t Hoff equation. They allow us to evaluate the standard molar reaction enthalpy of a reaction by a noncalorimetric method from the temperature dependence of $\ln K$. For example, we can plot $\ln K$ versus $1/T$; then according to Eq. 12.1.14, the slope of the curve at any value of $1/T$ is equal to $-\Delsub{r}H\st/R$ at the corresponding temperature $T$.
A simple way to derive the equation for this last procedure is to substitute $\Delsub{r}G\st=\Delsub{r}H\st-T\Delsub{r}S\st$ in $\Delsub{r}G\st=-RT\ln K$ and rearrange to $\ln K = -\frac{\Delsub{r}H\st}{R}\left(\frac{1}{T}\right) + \frac{\Delsub{r}S\st}{R} \tag{12.1.15}$ Suppose we plot $\ln K$ versus $1/T$. In a small temperature interval in which $\Delsub{r}H\st$ and $\Delsub{r}S\st$ are practically constant, the curve will appear linear. According to Eq. 12.1.15, the curve in this interval has a slope of $-\Delsub{r}H\st/R$, and the tangent to a point on the curve has its intercept at $1/T{=}0$ equal to $\Delsub{r}S\st/R$.
When we apply Eq. 12.1.14 to the vaporization process A(l)$\ra$A(g) of pure A, it resembles the Clausius–Clapeyron equation for the same process (Eq. 8.4.15). These equations are not exactly equivalent, however, as the comparison in Table 12.1 shows. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/12%3A_Equilibrium_Conditions_in_Multicomponent_Systems/12.01%3A_Effects_of_Temperature.txt |
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$\newcommand{\C}{_{\text{C}}} % subscript C$
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$\newcommand{\fB}{_{\text{f},\text{B}}} % subscript f,B (for fr. pt.)$
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$\newcommand{\gph}{^{\gamma}} % gamma phase superscript$
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$\newcommand{\bphp}{^{\beta'}} % beta prime phase superscript$
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Section 9.6.3 explained how we can evaluate the activity coefficient $\g\mbB$ of a nonelectrolyte solute of a binary solution if we know the variation of the osmotic coefficient of the solution from infinite dilution to the molality of interest. A similar procedure for the mean ionic activity coefficient of an electrolyte solute was described in Sec. 10.6.
The physical measurements needed to find the osmotic coefficient $\phi_m$ of a binary solution must be directed to the calculation of the quantity $\mu\A^*-\mu\A$, the difference between the chemical potentials of the pure solvent and the solvent in the solution at the temperature and pressure of interest. This difference is positive, because the presence of the solute reduces the solvent’s chemical potential.
To calculate $\phi_m$ from $\mu\A^*-\mu\A$, we use Eq. 9.6.16 for a nonelectrolyte solute, or Eq. 10.6.1 for an electrolyte solute. Both equations are represented by $\phi_m=\frac{\mu\A^*-\mu\A}{RTM\A\nu m\B} \tag{12.2.1}$ where $\nu$ for a nonelectrolyte is $1$ and for an electrolyte is the number of ions per formula unit.
The sequence of steps, then, is (1) the determination of $\mu\A^*-\mu\A$ over a range of molality at constant $T$ and $p$, (2) the conversion of these values to $\phi_m$ using Eq. 12.2.1, and (3) the evaluation of the solute activity coefficient by a suitable integration from infinite dilution to the molality of interest.
A measurement of $\mu\A^*-\mu\A$ also gives us the solvent activity coefficient, based on the pure-solvent reference state, through the relation $\mu\A=\mu\A^*+RT\ln(\g\A x\A)$ (Eq. 9.5.15).
Sections 12.2.1 and 12.2.2 will describe freezing-point and osmotic-pressure measurements, two much-used methods for evaluating $\mu\A^*-\mu\A$ in a binary solution at a given $T$ and $p$. The isopiestic vapor-pressure method was described in Sec. 9.6.4. The freezing-point and isopiestic vapor-pressure methods are often used for electrolyte solutions, and osmotic pressure is especially useful for solutions of macromolecules.
12.2.1 Freezing-point measurements
This section explains how we can evaluate $\mu\A^*-\mu\A$ for a solution of a given composition at a given $T$ and $p$ from the freezing point of the solution combined with additional data obtained from calorimetric measurements.
Consider a binary solution of solvent A and solute B. We assume that when this solution is cooled at constant pressure and composition, the solid that first appears is pure A. For example, for a dilute aqueous solution the solid would be ice. The temperature at which solid A first appears is $T\f$, the freezing point of the solution. This temperature is lower than the freezing point $T\f^*$ of the pure solvent, a consequence of the lowering of $\mu\A$ by the presence of the solute. Both $T\f$ and $T\f^*$ can be measured experimentally.
Let $T'$ be a temperature of interest that is equal to or greater than $T\f^*$. We wish to determine the value of $\mu\A^*(\tx{l},T')-\mu\A(\tx{sln},T')$, where $\mu\A^*(\tx{l},T')$ refers to pure liquid solvent and $\mu\A(\tx{sln},T')$ refers to the solution.
A second method for evaluating $\mu\A^*-\mu\A$ uses the solution property called osmotic pressure. A simple apparatus to measure the osmotic pressure of a binary solution is shown schematically in Fig. 12.2. The system consists of two liquid phases separated by a semipermeable membrane. Phase $\pha$ is pure solvent and phase $\phb$ is a solution with the same solvent at the same temperature. The semipermeable membrane is permeable to the solvent and impermeable to the solute.
The presence of the membrane makes this system different from the multiphase, multicomponent system of Sec. 9.2.7, used there to derive conditions for transfer equilibrium. By a modification of that procedure, we can derive the conditions of equilibrium for the present system. We take phase $\phb$ as the reference phase because it includes both solvent and solute. In order to prevent expansion work in the isolated system, both pistons shown in the figure must be fixed in stationary positions. This keeps the volume of each phase constant: $\dif V\aph=\dif V\bph=0$. Equation 9.2.41, expressing the total differential of the entropy in an isolated multiphase, multicomponent system, becomes $\dif S = \frac{T\bph-T\aph}{T\bph}\dif S\aph + \frac{\mu\A\bph-\mu\A\aph}{T\bph}\dif n\A\aph \tag{12.2.6}$ In an equilibrium state, the coefficients $(T\bph-T\aph)/T\bph$ and $(\mu\A\bph-\mu\A\aph)/T\bph$ must be zero. Therefore, in an equilibrium state the temperature is the same in both phases and the solvent has the same chemical potential in both phases. The presence of the membrane, however, allows the pressures of the two phases to be unequal in the equilibrium state.
Suppose we start with both phases shown in Fig. 12.2 at the same temperature and pressure. Under these conditions, the value of $\mu\A$ is less in the solution than in the pure liquid, and a spontaneous flow of solvent will occur through the membrane from the pure solvent to the solution. This phenomenon is called osmosis (Greek for push). If we move the right-hand piston down slightly in order to increase the pressure $p''$ of the solution in phase $\phb$, $\mu\A$ increases in this phase. The osmotic pressure of the solution, $\varPi$, is defined as the additional pressure the solution must have, compared to the pressure $p'$ of the pure solvent at the same temperature, to establish an equilibrium state with no flow of solvent in either direction through the membrane: $p'' = p' + \varPi$.
In practice, the membrane may not be completely impermeable to a solute. All that is required for the establishment of an equilibrium state with different pressures on either side of the membrane is that solvent transfer equilibrium be established on a short time scale compared to the period of observation, and that the amount of solute transferred during this period be negligible.
The osmotic pressure $\varPi$ is an intensive property of a solution whose value depends on the solution’s temperature, pressure, and composition. Strictly speaking, $\varPi$ in an equilibrium state of the system shown in Fig. 12.2 refers to the osmotic pressure of the solution at pressure $p'$, the pressure of the pure solvent. In other words, the osmotic pressure of a solution at temperature $T$ and pressure $p'$ is the additional pressure that would have to be exerted on the solution to establish transfer equilibrium with pure solvent that has temperature $T$ and pressure $p'$. A solution has the property called osmotic pressure regardless of whether this additional pressure is actually present, just as a solution has a freezing point even when its actual temperature is different from the freezing point.
Because in an equilibrium state the solvent chemical potential must be the same on both sides of the semipermeable membrane, there is a relation between chemical potentials and osmotic pressure given by \begin{gather} \s{ \mu\A(p'') = \mu\A(p'+\varPi)=\mu\A^*(p') } \tag{12.2.7} \cond{(equilibrium state)} \end{gather} We can use this relation to derive an expression for $\mu\A^*(p')-\mu\A(p')$ as a function of $\varPi$. The dependence of $\mu\A$ on pressure is given according to Eq. 9.2.49 by $\Pd{\mu\A}{p}{T,\allni} = V\A \tag{12.2.8}$ where $V\A$ is the partial molar volume of the solvent in the solution. Rewriting this equation in the form $\dif\mu\A=V\A\difp$ and integrating at constant temperature and composition from $p'$ to $p'+\varPi$, we obtain $\mu\A(p'+\varPi)-\mu\A(p')=\int_{p'}^{p'+\varPi}\!V\A\difp \tag{12.2.9}$ Substitution from Eq. 12.2.7 changes this to \begin{gather} \s {\mu\A^*(p')-\mu\A(p')=\int_{p'}^{p'+\varPi}\!V\A\difp } \tag{12.2.10} \cond{(constant $T$)} \end{gather} which is the desired expression for $\mu\A^*-\mu\A$ at a single temperature and pressure. To evaluate the integral, we need an experimental value of the osmotic pressure $\varPi$ of the solution. If we assume $V\A$ is constant in the pressure range from $p'$ to $p'+\varPi$, Eq. 12.2.10 becomes simply $\mu\A^*(p')-\mu\A(p') = V\A\varPi \tag{12.2.11}$ | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/12%3A_Equilibrium_Conditions_in_Multicomponent_Systems/12.02%3A_Solvent_Chemical_Potentials_from_Phase_Equilibria.txt |
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This section considers a binary liquid mixture of components A and B in equilibrium with either pure solid A or pure gaseous A. The aim is to find general relations among changes of temperature, pressure, and mixture composition in the two-phase equilibrium system that can be applied to specific situations in later sections.
In this section, $\mu\A$ is the chemical potential of component A in the mixture and $\mu\A^*$ is for the pure solid or gaseous phase. We begin by writing the total differential of $\mu\A/T$ with $T$, $p$, and $x\A$ as the independent variables. These quantities refer to the binary liquid mixture, and we have not yet imposed a condition of equilibrium with another phase. The general expression for the total differential is $\dif(\mu\A/T) = \bPd{(\mu\A/T)}{T}{p,x\A}\!\dif T + \bPd{(\mu\A/T)}{p}{T,x\A}\!\difp + \bPd{(\mu\A/T)}{x\A}{T,p}\!\dx\A \tag{12.3.1}$ With substitutions from Eqs. 9.2.49 and 12.1.3, this becomes $\dif(\mu\A/T) = -\frac{H\A}{T^2}\dif T + \frac{V\A}{T}\difp + \bPd{(\mu\A/T)}{x\A}{T,p}\dx\A \tag{12.3.2}$
Next we write the total differential of $\mu\A^*/T$ for pure solid or gaseous A. The independent variables are $T$ and $p$; the expression is like Eq. 12.3.2 with the last term missing: $\dif(\mu\A^*/T) = -\frac{H\A^*}{T^2}\dif T + \frac{V\A^*}{T}\difp \tag{12.3.3}$
When the two phases are in transfer equilibrium, $\mu\A$ and $\mu\A^*$ are equal. If changes occur in $T$, $p$, or $x\A$ while the phases remain in equilibrium, the condition $\dif(\mu\A/T) = \dif(\mu\A^*/T)$ must be satisfied. Equating the expressions on the right sides of Eqs. 12.3.2 and 12.3.3 and combining terms, we obtain the equation $\frac{H\A-H\A^*}{T^2}\dif T - \frac{V\A-V\A^*}{T}\difp = \bPd{(\mu\A/T)}{x\A}{T,p}\dx\A \tag{12.3.4}$ which we can rewrite as \begin{gather} \s{ \frac{\Delsub{sol,A}H}{T^2}\dif T - \frac{\Delsub{sol,A}V}{T}\difp = \bPd{(\mu\A/T)}{x\A}{T,p}\dx\A } \tag{12.3.5} \cond{(phases in} \nextcond{equilibrium)} \end{gather} Here $\Delsub{sol,A}H$ is the molar differential enthalpy of solution of solid or gaseous A in the liquid mixture, and $\Delsub{sol,A}V$ is the molar differential volume of solution. Equation 12.3.5 is a relation between changes in the variables $T$, $p$, and $x\A$, only two of which are independent in the equilibrium system.
Suppose we set $\difp$ equal to zero in Eq. 12.3.5 and solve for $\dif T/\dx\A$. This gives us the rate at which $T$ changes with $x\A$ at constant $p$: \begin{gather} \s{ \Pd{T}{x\A}{\!p} = \frac{T^2}{\Delsub{sol,A}H} \bPd{(\mu\A/T)}{x\A}{T,p} } \tag{12.3.6} \cond{(phases in} \nextcond{equilibrium)} \end{gather} We can also set $\dif T$ equal to zero in Eq. 12.3.5 and find the rate at which $p$ changes with $x\A$ at constant $T$: \begin{gather} \s{ \Pd{p}{x\A}{T} = -\frac{T}{\Delsub{sol,A}V}\bPd{(\mu\A/T)}{x\A}{T,p} } \tag{12.3.7} \cond{(phases in} \nextcond{equilibrium)} \end{gather}
Equations 12.3.6 and 12.3.7 will be needed in Secs. 12.4 and 12.5. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/12%3A_Equilibrium_Conditions_in_Multicomponent_Systems/12.03%3A_Binary_Mixture_in_Equilibrium_with_a_Pure_Phase.txt |
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The colligative properties of a solution are usually considered to be:
1. Note that all four properties are defined by an equilibrium between the liquid solution and a solid, liquid, or gas phase of the pure solvent. The properties called colligative (Latin: tied together) have in common a dependence on the concentration of solute particles that affects the solvent chemical potential.
Figure 12.3 illustrates the freezing-point depression and boiling-point elevation of an aqueous solution. At a fixed pressure, pure liquid water is in equilibrium with ice at the freezing point and with steam at the boiling point. These are the temperatures at which H$_2$O has the same chemical potential in both phases at this pressure. At these temperatures, the chemical potential curves for the phases intersect, as indicated by open circles in the figure. The presence of dissolved solute in the solution causes a lowering of the H$_2$O chemical potential compared to pure water at the same temperature. Consequently, the curve for the chemical potential of H$_2$O in the solution intersects the curve for ice at a lower temperature, and the curve for steam at a higher temperature, as indicated by open triangles. The freezing point is depressed by $\Del T\subs{f}$, and the boiling point (if the solute is nonvolatile) is elevated by $\Del T\bd$.
Although these expressions provide no information about the activity coefficient of a solute, they are useful for estimating the solute molar mass. For example, from a measurement of any of the colligative properties of a dilute solution and the appropriate theoretical relation, we can obtain an approximate value of the solute molality $m\B$. (It is only approximate because, for a measurement of reasonable precision, the solution cannot be extremely dilute.) If we prepare the solution with a known amount $n\A$ of solvent and a known mass of solute, we can calculate the amount of solute from $n\B=n\A M\A m\B$; then the solute molar mass is the solute mass divided by $n\B$.
12.4.1 Freezing-point depression
As in Sec. 12.2.1, we assume the solid that forms when a dilute solution is cooled to its freezing point is pure component A.
Equation 12.3.6 gives the general dependence of temperature on the composition of a binary liquid mixture of A and B that is in equilibrium with pure solid A. We treat the mixture as a solution. The solvent is component A, the solute is B, and the temperature is the freezing point $T\f$: $\Pd{T\f}{x\A}{\!p} = \frac{T\f^2}{\Delsub{sol,A}H} \bPd{(\mu\A/T)}{x\A}{T,p} \tag{12.4.1}$
Consider the expression on the right side of this equation in the limit of infinite dilution. In this limit, $T\f$ becomes $T\f^*$, the freezing point of the pure solvent, and $\Delsub{sol,A}H$ becomes $\Delsub{fus,A}H$, the molar enthalpy of fusion of the pure solvent.
To deal with the partial derivative on the right side of Eq. 12.4.1 in the limit of infinite dilution, we use the fact that the solvent activity coefficient $\g\A$ approaches $1$ in this limit. Then the solvent chemical potential is given by the Raoult’s law relation \begin{gather} \s{ \mu\A = \mu\A^* + RT\ln x\A } \tag{12.4.2} \cond{(solution at infinite dilution)} \end{gather} where $\mu\A^*$ is the chemical potential of A in a pure-liquid reference state at the same $T$ and $p$ as the mixture. (At the freezing point of the mixture, the reference state is an unstable supercooled liquid.)
If the solute is an electrolyte, Eq. 12.4.2 can be derived by the same procedure as described in Sec. 9.4.6 for an ideal-dilute binary solution of a nonelectrolyte. We must calculate $x\A$ from the amounts of all species present at infinite dilution. In the limit of infinite dilution, any electrolyte solute is completely dissociated to its constituent ions: ion pairs and weak electrolytes are completely dissociated in this limit. Thus, for a binary solution of electrolyte B with $\nu$ ions per formula unit, we should calculate $x\A$ from $x\A = \frac{n\A}{n\A + \nu n\B} \tag{12.4.3}$ where $n\B$ is the amount of solute formula unit. (If the solute is a nonelectrolyte, we simply set $\nu$ equal to $1$ in this equation.)
From Eq. 12.4.2, we can write $\bPd{(\mu\A/T)}{x\A}{T,p} \ra R \quad \tx{as} \quad x\A \ra 1 \tag{12.4.4}$ In the limit of infinite dilution, then, Eq. 12.4.1 becomes $\lim_{x\A\rightarrow 1}\Pd{T\f}{x\A}{\!p} = \frac{R(T\f^*)^2}{\Delsub{fus,A}H} \tag{12.4.5}$
It is customary to relate freezing-point depression to the solute concentration $c\B$ or molality $m\B$. From Eq. 12.4.3, we obtain $1-x\A=\frac{\nu n\B}{n\A+\nu n\B} \tag{12.4.6}$ In the limit of infinite dilution, when $\nu n\B$ is much smaller than $n\A$, $1-x\A$ approaches the value $\nu n\B/n\A$. Then, using expressions in Eq. 9.1.14, we obtain the relations \begin{gather} \s{ \begin{split} \dx\A & = -\dif(1- x\A) = -\nu\dif(n\B/n\A) \cr & = -\nu V\A^*\dif c\B \cr & = -\nu M\A\dif m\B \end{split} } \tag{12.4.7} \cond{(binary solution at} \nextcond{infinite dilution)} \end{gather} which transform Eq. 12.4.5 into the following (ignoring a small dependence of $V\A^*$ on $T$): \begin{gather} \lim_{c\B\rightarrow 0} \tag{12.4.8} \Pd{T\f}{c\B}{\!p} = -\frac{\nu V\A^* R(T\f^*)^2}{\Delsub{fus,A}H} \cr \lim_{m\B\rightarrow 0}\Pd{T\f}{m\B}{\!p} = -\frac{\nu M\A R(T\f^*)^2}{\Delsub{fus,A}H} \end{gather} We can apply these equations to a nonelectrolyte solute by setting $\nu$ equal to $1$.
As $c\B$ or $m\B$ approaches zero, $T\f$ approaches $T\f^*$. The freezing-point depression (a negative quantity) is $\Del T\subs{f}=T\f-T\f^*$. In the range of molalities of a dilute solution in which $\pd{T\f}{m\B}{p}$ is given by the expression on the right side of Eq. 12.4.8, we can write $\Del T\subs{f} = -\frac{\nu M\A R(T\f^*)^2}{\Delsub{fus,A}H}m\B \tag{12.4.9}$
The molal freezing-point depression constant or cryoscopic constant, $K\subs{f}$, is defined for a binary solution by $K\subs{f} \defn -\lim_{m\B\rightarrow 0} \frac{\Del T\subs{f}}{\nu m\B} \tag{12.4.10}$ and, from Eq. 12.4.9, has a value given by $K\subs{f} = \frac{M\A R(T\f^*)^2}{\Delsub{fus,A}H} \tag{12.4.11}$ The value of $K\subs{f}$ calculated from this formula depends only on the kind of solvent and the pressure. For H$_2$O at $1\br$, the calculated value is $K\bd=1.860\units{K kg mol\(^{-1}$}\) (Prob. 12.4).
In the dilute binary solution, we have the relation \begin{gather} \s{ \Del T\subs{f} = - \nu K\subs{f} m\B } \tag{12.4.12} \cond{(dilute binary solution)} \end{gather} This relation is useful for estimating the molality of a dilute nonelectrolyte solution ($\nu{=}1$) from a measurement of the freezing point. The relation is of little utility for an electrolyte solute, because at any electrolyte molality that is high enough to give a measurable depression of the freezing point, the mean ionic activity coefficient deviates greatly from unity and the relation is not accurate.
12.4.2 Boiling-point elevation
We can apply Eq. 12.3.6 to the boiling point $T\bd$ of a dilute binary solution. The pure phase of A in equilibrium with the solution is now a gas instead of a solid. (We must assume the solute is nonvolatile or has negligible partial pressure in the gas phase.) Following the procedure of Sec. 12.4.1, we obtain $\lim_{m\B\rightarrow 0}\Pd{T\bd}{m\B}{\!p} = \frac{\nu M\A R(T\bd^*)^2}{\Delsub{vap,A}H} \tag{12.4.13}$ where $\Delsub{vap,A}H$ is the molar enthalpy of vaporization of pure solvent at its boiling point $T\bd^*$.
The molal boiling-point elevation constant or ebullioscopic constant, $K\bd$, is defined for a binary solution by $K\bd \defn \lim_{m\B\rightarrow 0}\frac{\Del T\bd}{\nu m\B} \tag{12.4.14}$ where $\Del T\bd=T\bd-T\bd^*$ is the boiling-point elevation. Accordingly, $K\bd$ has a value given by $K\bd = \frac{M\A R(T\bd^*)^2}{\Delsub{vap,A}H} \tag{12.4.15}$ For the boiling point of a dilute solution, the analogy of Eq. 12.4.12 is \begin{gather} \s{ \Del T\bd = \nu K\bd m\B } \tag{12.4.16} \cond{(dilute binary solution)} \end{gather} Since $K\subs{f}$ has a larger value than $K\bd$ (because $\Delsub{fus,A}H$ is smaller than $\Delsub{vap,A}H$), the measurement of freezing-point depression is more useful than that of boiling-point elevation for estimating the molality of a dilute solution.
12.4.3 Vapor-pressure lowering
In a binary two-phase system in which a solution of volatile solvent A and nonvolatile solute B is in equilibrium with gaseous A, the vapor pressure of the solution is equal to the system pressure $p$.
Equation 12.3.7 gives the general dependence of $p$ on $x\A$ for a binary liquid mixture in equilibrium with pure gaseous A. In this equation, $\Delsub{sol,A}V$ is the molar differential volume change for the dissolution of the gas in the solution. In the limit of infinite dilution, $-\Delsub{sol,A}V$ becomes $\Delsub{vap,A}V$, the molar volume change for the vaporization of pure solvent. We also apply the limiting expressions of Eqs. 12.4.4 and 12.4.7. The result is $\lim_{c\B\rightarrow 0}\Pd{p}{c\B}{T} = -\frac{\nu V\A^* RT}{\Delsub{vap,A}V} \qquad \lim_{m\B\rightarrow 0}\Pd{p}{m\B}{T} = -\frac{\nu M\A RT}{\Delsub{vap,A}V} \tag{12.4.17}$
If we neglect the molar volume of the liquid solvent compared to that of the gas, and assume the gas is ideal, then we can replace $\Delsub{vap,A}V$ in the expressions above by $V\A^*\gas =RT/p\A^*$ and obtain $\lim_{c\B\rightarrow 0}\Pd{p}{c\B}{T} \approx -\nu V\A^* p\A^* \qquad \lim_{m\B\rightarrow 0}\Pd{p}{m\B}{T} \approx -\nu M\A p\A^* \tag{12.4.18}$ where $p\A^*$ is the vapor pressure of the pure solvent at the temperature of the solution.
Thus, approximate expressions for vapor-pressure lowering in the limit of infinite dilution are $\Del p \approx -\nu V\A^* p\A^* c\B \qquad \tx{and} \qquad \Del p \approx -\nu M\A p\A^* m\B \tag{12.4.19}$ We see that the lowering in this limit depends on the kind of solvent and the solution composition, but not on the kind of solute.
12.4.4 Osmotic pressure
The osmotic pressure $\varPi$ is an intensive property of a solution and was defined in Sec. 12.2.2. In a dilute solution of low $\varPi$, the approximation used to derive Eq. 12.2.11 (that the partial molar volume $V\A$ of the solvent is constant in the pressure range from $p$ to $p+\varPi$) becomes valid, and we can write $\varPi = \frac{\mu\A^*-\mu\A}{V\A} \tag{12.4.20}$ In the limit of infinite dilution, $\mu\A^*-\mu\A$ approaches $-RT\ln x\A$ (Eq. 12.4.2) and $V\A$ becomes the molar volume $V\A^*$ of the pure solvent. In this limit, Eq. 12.4.20 becomes $\varPi = -\frac{RT\ln x\A}{V\A^*} \tag{12.4.21}$ from which we obtain the equation $\lim_{x\A\rightarrow 1}\Pd{\varPi}{x\A}{T,p} = -\frac{RT}{V\A^*} \tag{12.4.22}$ The relations in Eq. 12.4.7 transform Eq. 12.4.22 into $\lim_{c\B\rightarrow 0}\Pd{\varPi}{c\B}{T,p} = \nu RT \tag{12.4.23}$ $\lim_{m\B\rightarrow 0}\Pd{\varPi}{m\B}{T,p} = \frac{\nu RTM\A}{V\A^*} = \nu \rho\A^* RT \tag{12.4.24}$
Equations 12.4.23 and 12.4.24 show that the osmotic pressure becomes independent of the kind of solute as the solution approaches infinite dilution. The integrated forms of these equations are \begin{gather} \s{ \varPi=\nu c\B RT } \tag{12.4.25} \cond{(dilute binary solution)} \end{gather} \begin{gather} \s{ \varPi=\frac{RTM\A}{V\A^*}\nu m\B = \rho\A^* RT\nu m\B } \tag{12.4.26} \cond{(dilute binary solution)} \end{gather} Equation 12.4.25 is van’t Hoff’s equation for osmotic pressure. If there is more than one solute species, $\nu c\B$ can be replaced by $\sum_{i\ne\tx{A}}c_i$ and $\nu m\B$ by $\sum_{i\ne\tx{A}}m_i$ in these expressions.
In Sec. 9.6.3, it was stated that $\varPi/m\B$ is equal to the product of $\phi_m$ and the limiting value of $\varPi/m\B$ at infinite dilution, where $\phi_m = (\mu\A^*-\mu\A)/RTM\A\sum_{i \ne \tx{A}}m_i$ is the osmotic coefficient. This relation follows directly from Eqs. 12.2.11 and 12.4.26. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/12%3A_Equilibrium_Conditions_in_Multicomponent_Systems/12.04%3A_Colligative_Properties_of_a_Dilute_Solution.txt |
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$\newcommand{\cond}[1]{\$-2.5pt]{}\tag*{#1}}$ $\newcommand{\nextcond}[1]{\[-5pt]{}\tag*{#1}}$ $\newcommand{\R}{8.3145\units{J\,K\per\,mol\per}} % gas constant value$ $\newcommand{\Rsix}{8.31447\units{J\,K\per\,mol\per}} % gas constant value - 6 sig figs$ $\newcommand{\jn}{\hspace3pt\lower.3ex{\Rule{.6pt}{2ex}{0ex}}\hspace3pt}$ $\newcommand{\ljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}} \hspace3pt}$ $\newcommand{\lljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace1.4pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace3pt}$ A freezing-point curve (freezing point as a function of liquid composition) and a solubility curve (composition of a solution in equilibrium with a pure solid as a function of temperature) are different ways of describing the same physical situation. Thus, strange as it may sound, the composition $x\A$ of an aqueous solution at the freezing point is the mole fraction solubility of ice in the solution. 12.5.1 Freezing points of ideal binary liquid mixtures Section 12.2.1 described the use of freezing-point measurements to determine the solvent chemical potential in a solution of arbitrary composition relative to the chemical potential of the pure solvent. The way in which freezing point varies with solution composition in the limit of infinite dilution was derived in Sec. 12.4.1. Now let us consider the freezing behavior over the entire composition range of an ideal liquid mixture. Let $T'\subs{f}$ be the freezing point of a liquid mixture of composition $x'\A$ and $x'\B=1-x'\A$, and let $T''\subs{f}$ be the melting point of the solid compound of composition $x''\A=a/(a+b)$ and $x''\B=b/(a+b)$. Figure 12.7 shows an example of a molten metal mixture that solidifies to an alloy of fixed composition. The freezing-point curve of this system is closely approximated by Eq. 12.5.23. 12.5.5 Solubility of a solid electrolyte Consider an equilibrium between a crystalline salt (or other kind of ionic solid) and a solution containing the solvated ions: \[ \tx{M$_{\nu_+}$X$_{\nu_-}$(s)} \arrows \nu_+\tx{M$^{z_+}$(aq)} + \nu_-\tx{X$^{z_-}$(aq)}$ Here $\nu_+$ and $\nu_-$ are the numbers of cations and anions in the formula unit of the salt, and $z_+$ and $z_-$ are the charge numbers of these ions. The solution in equilibrium with the solid salt is a saturated solution. The thermodynamic equilibrium constant for this kind of equilibrium is called a solubility product, $K\subs{s}$.
We can readily derive a relation between $K\subs{s}$ and the molalities of the ions in the saturated solution by treating the dissolved salt as a single solute substance, B. We write the equilibrium in the form B$^*$(s)$\arrows$B(sln), and write the expression for the solubility product as a proper quotient of activities: $K\subs{s} = \frac{a\mbB}{a\B^*} \tag{12.5.24}$ From Eq. 10.3.16, we have $a\mbB= \G\mbB \g_{\pm}^\nu(m_+/m\st)^{\nu_+}(m_-/m\st)^{\nu_-}$. This expression is valid whether or not the ions M$^{z_+}$ and X$^{z_-}$ are present in solution in the same ratio as in the solid salt. When we replace $a\mbB$ with this expression, and replace $a\B^*$ with $\G\B^*$ (Table 9.5), we obtain $K\subs{s} = \left(\frac{\G\mbB}{\G\B^*}\right) \g_{\pm}^\nu \left( \frac{m_+}{m\st} \right)^{\nu_+} \left( \frac{m_-}{m\st} \right)^{\nu_-} \tag{12.5.25}$ where $\nu=\nu_+ + \nu_-$ is the total number of ions per formula unit. $\g_{\pm}$ is the mean ionic activity coefficient of the dissolved salt in the saturated solution, and the molalities $m_+$ and $m_-$ refer to the ions M$^{z_+}$ and X$^{z_-}$ in this solution.
The first factor on the right side of Eq. 12.5.25, the proper quotient of pressure factors for the reaction B$^*$(s)$\ra$B(sln), will be denoted $\G\subs{r}$ (the subscript “r” stands for reaction). The value of $\G\subs{r}$ is exactly $1$ if the system is at the standard pressure, and is otherwise approximately $1$ unless the pressure is very high.
If the aqueous solution is produced by allowing the salt to dissolve in pure water, or in a solution of a second solute containing no ions in common with the salt, then the ion molalities in the saturated solution are $m_+=\nu_+m\B$ and $m_-=\nu_-m\B$ where $m\B$ is the solubility of the salt expressed as a molality. Under these conditions, Eq. 12.5.25 becomes \begin{gather} \s{ K\subs{s} = \G\subs{r} \g_{\pm}^{\nu} \left(\nu_+^{\nu_+}\nu_-^{\nu_-}\right) \left( \frac{m\B}{m\st} \right)^{\nu} } \tag{12.5.26} \cond{(no common ion)} \end{gather} We could also have obtained this equation by using the expression of Eq. 10.3.10 for $a\mbB$.
If the ionic strength of the saturated salt solution is sufficiently low (i.e., the solubility is sufficiently low), it may be practical to evaluate the solubility product with Eq. 12.5.26 and an estimate of $\g_{\pm}$ from the Debye–Hückel limiting law (see Prob. 12.19). The most accurate method of measuring a solubility product, however, is through the standard cell potential of an appropriate galvanic cell (Sec. 14.3.3).
Since $K\subs{s}$ is a thermodynamic equilibrium constant that depends only on $T$, and $\G\subs{r}$ depends only on $T$ and $p$, Eq. 12.5.26 shows that any change in the solution composition at constant $T$ and $p$ that decreases $\g_{\pm}$ must increase the solubility. For example, the solubility of a sparingly-soluble salt increases when a second salt, lacking a common ion, is dissolved in the solution; this is a salting-in effect.
Equation 12.5.25 is a general equation that applies even if the solution saturated with one salt contains a second salt with a common ion. For instance, consider the sparingly-soluble salt M$_{\nu_+}$X$_{\nu_-}$ in transfer equilibrium with a solution containing the more soluble salt M$_{\nu'_+}$Y$_{\nu'_-}$ at molality $m\C$. The common ion in this example is the cation M$^{z_+}$. The expression for the solubility product is now \begin{gather} \s{ K\subs{s} = \G\subs{r} \g_{\pm}^{\nu} (\nu_+m\B+\nu'_+m\C)^{\nu_+}(\nu_-m\B)^{\nu_-}/(m\st)^{\nu} } \tag{12.5.27} \cond{(common cation)} \end{gather} where $m\B$ again is the solubility of the sparingly-soluble salt, and $m\C$ is the molality of the second salt. $K\subs{s}$ and $\G\subs{r}$ are constant if $T$ and $p$ do not change, so any increase in $m\C$ at constant $T$ and $p$ must cause a decrease in the solubility $m\B$. This is called the common ion effect.
From the measured solubility of a salt in pure solvent, or in an electrolyte solution with a common cation, and a known value of $K\subs{s}$, we can evaluate the mean ionic activity coefficient $\g_{\pm}$ through Eq. 12.5.26 or 12.5.27. This procedure has the disadvantage of being limited to the value of $m\B$ existing in the saturated solution.
We find the temperature dependence of $K\subs{s}$ by applying Eq. 12.1.12: $\frac{\dif\ln K\subs{s}}{\dif T} = \frac{\Delsub{sol,B}H\st}{RT^2} \tag{12.5.28}$ At the standard pressure, $\Delsub{sol,B}H\st$ is the same as the molar enthalpy of solution at infinite dilution, $\Delsub{sol,B}H^{\infty}$. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/12%3A_Equilibrium_Conditions_in_Multicomponent_Systems/12.05%3A_Solid-Liquid_Equilibria.txt |
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$\newcommand{\C}{_{\text{C}}} % subscript C$
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12.6.1 Miscibility in binary liquid systems
When two different pure liquids are unable to mix in all proportions, they are said to be partially miscible. When these liquids are placed in contact with one another and allowed to come to thermal, mechanical, and transfer equilibrium, the result is two coexisting liquid mixtures of different compositions.
Liquids are never actually completely immiscible. To take an extreme case, liquid mercury, when equilibrated with water, has some H$_2$O dissolved in it, and some mercury dissolves in the water, although the amounts may be too small to measure.
The Gibbs phase rule for a multicomponent system to be described in Sec. 13.1 shows that a two-component, two-phase system at equilibrium has only two independent intensive variables. Thus at a given temperature and pressure, the mole fraction compositions of both phases are fixed; the compositions depend only on the identity of the substances and the temperature and pressure.
Figure 13.5 shows a phase diagram for a typical binary liquid mixture that spontaneously separates into two phases when the temperature is lowered. The thermodynamic conditions for phase separation of this kind were discussed in Sec. 11.1.6. The phase separation is usually the result of positive deviations from Raoult’s law. Typically, when phase separation occurs, one of the substances is polar and the other nonpolar.
12.6.2 Solubility of one liquid in another
Suppose substances A and B are both liquids when pure. In discussing the solubility of liquid B in liquid A, we can treat B as either a solute or as a constituent of a liquid mixture. The difference lies in the choice of the standard state or reference state of B.
We can define the solubility of B in A as the maximum amount of B that can dissolve without phase separation in a given amount of A at the given temperature and pressure. Treating B as a solute, we can express its solubility as the mole fraction of B in the phase at the point of phase separation. The addition of any more B to the system will result in two coexisting liquid phases of fixed composition, one of which will have mole fraction $x\B$ equal to its solubility.
Experimentally, the solubility of B in A can be determined from the cloud point, the point during titration of A with B at which persistent turbidity is observed.
Consider a system with two coexisting liquid phases $\pha$ and $\phb$ containing components A and B. Let $\pha$ be the A-rich phase and $\phb$ be the B-rich phase. For example, A could be water and B could be benzene, a hydrophobic substance. Phase $\pha$ would then be an aqueous phase polluted with a low concentration of dissolved benzene, and phase $\phb$ would be wet benzene. $x\B\aph$ would be the solubility of the benzene in water, expressed as a mole fraction.
Below, relations are derived for this kind of system using both choices of standard state or reference state.
Solute standard state
Assume that the two components have low mutual solubilities, so that B has a low mole fraction in phase $\pha$ and a mole fraction close to 1 in phase $\phb$. It is then appropriate to treat B as a solute in phase $\pha$ and as a constituent of a liquid mixture in phase $\phb$. The value of $x\B\aph$ is the solubility of liquid B in liquid A.
The equilibrium when two liquid phases are present is B($\phb$)$\arrows$B($\pha$), and the expression for the thermodynamic equilibrium constant, with the solute standard state based on mole fraction, is $K = \frac{a\xbB\aph}{a\B\bph} = \frac{\G\xbB\aph \g\xbB\aph x\B\aph}{\G\B\bph \g\B\bph x\B\bph} \tag{12.6.1}$ The solubility of B is then given by $x\B\aph = \frac{\G\B\bph\g\B\bph x\B\bph}{\G\xbB\aph\g\xbB\aph}K \tag{12.6.2}$ The values of the pressure factors and activity coefficients are all close to $1$, so that the solubility of B in A is given by $x\B\aph \approx K$. The temperature dependence of the solubility is given by $\frac{\dif\ln x\B\aph}{\dif T} \approx \frac{\dif\ln K}{\dif T} = \frac{\Delsub{sol,B}H\st}{RT^2} \tag{12.6.3}$ where $\Delsub{sol,B}H\st$ is the molar enthalpy change for the transfer at pressure $p\st$ of pure liquid solute to the solution at infinite dilution.
H$_2$O and n-butylbenzene are two liquids with very small mutual solubilities. Figure 12.8 shows that the solubility of n-butylbenzene in water exhibits a minimum at about $12\units{\(\degC$}\). Equation 12.6.3 allows us to deduce from this behavior that $\Delsub{sol,B}H\st$ is negative below this temperature, and positive above.
Pure-liquid reference state
The condition for transfer equilibrium of component B is $\mu\B\aph=\mu\B\bph$. If we use a pure-liquid reference state for B in both phases, this condition becomes $\mu\B^* + RT\ln(\g\B\aph x\B\aph) = \mu\B^* + RT\ln(\g\B\bph x\B\bph) \tag{12.6.4}$ This results in the following relation between the compositions and activity coefficients: $\g\B\aph x\B\aph = \g\B\bph x\B\bph \tag{12.6.5}$
As before, we assume the two components have low mutual solubilities, so that the B-rich phase is almost pure liquid B. Then $x\B\bph$ is only slightly less than $1$, $\g\B\bph$ is close to $1$, and Eq. 12.6.5 becomes $x\B\aph \approx 1/\g\B\aph$. Since $x\B\aph$ is much less than $1$, $\g\B\aph$ must be much greater than $1$.
In environmental chemistry it is common to use a pure-liquid reference state for a nonpolar liquid solute that has very low solubility in water, so that the aqueous solution is essentially at infinite dilution. Let the nonpolar solute be component B, and let the aqueous phase that is equilibrated with liquid B be phase $\pha$. The activity coefficient $\g\B\aph$ is then a limiting activity coefficient or activity coefficient at infinite dilution. As explained above, the aqueous solubility of B in this case is given by $x\B\aph \approx 1/\g\B\aph$, and $\g\B\aph$ is much greater than $1$.
We can also relate the solubility of B to its Henry’s law constant $\kHB\aph$. Suppose the two liquid phases are equilibrated not only with one another but also with a gas phase. Since B is equilibrated between phase $\pha$ and the gas, we have $\g\xbB\aph=\fug\B/\kHB\aph x\B\aph$ (Table 9.4). From the equilibration of B between phase $\phb$ and the gas, we also have $\g\B\bph=\fug\B/x\B\bph \fug\B^*$. By eliminating the fugacity $\fug\B$ from these relations, we obtain the general relation $x\B\aph = \frac{\g\B\bph x\B\bph \fug\B^*}{\g\xbB\aph \kHB\aph} \tag{12.6.6}$ If we assume as before that the activity coefficients and $x\B\bph$ are close to 1, and that the gas phase behaves ideally, the solubility of B is given by $x\B\aph \approx p\B^*/\kHB\aph$, where $p\B^*$ is the vapor pressure of the pure solute.
12.6.3 Solute distribution between two partially-miscible solvents
Consider a two-component system of two equilibrated liquid phases, $\pha$ and $\phb$. If we add a small quantity of a third component, C, it will distribute itself between the two phases. It is appropriate to treat C as a solute in both phases. The thermodynamic equilibrium constant for the equilibrium $\tx{C}(\phb) \arrows \tx{C}(\pha)$, with solute standard states based on mole fraction, is $K = \frac{a\xbC\aph}{a\xbC\bph} = \frac{\G\xbC\aph \g\xbC\aph x\C\aph}{\G\xbC\bph \g\xbC\bph x\C\bph} \tag{12.6.7}$
We define $K'$ as the ratio of the mole fractions of C in the two phases at equilibrium: $K' \defn \frac{x\C\aph}{x\C\bph} = \frac{\G\xbC\bph \g\xbC\bph}{\G\xbC\aph \g\xbC\aph}K \tag{12.6.8}$ At a fixed $T$ and $p$, the pressure factors and equilibrium constant are constants. If $x\C$ is low enough in both phases for $\g\xbC\aph$ and $\g\xbC\bph$ to be close to unity, $K'$ becomes a constant for the given $T$ and $p$. The constancy of $K'$ over a range of dilute composition is the Nernst distribution law.
Since solute molality and concentration are proportional to mole fraction in dilute solutions, the ratios $m\C\aph/m\C\bph$ and $c\C\aph/c\C\bph$ also approach constant values at a given $T$ and $p$. The ratio of concentrations is called the partition coefficient or distribution coefficient.
In the limit of infinite dilution of C, the two phases have the compositions that exist when only components A and B are present. As C is added and $x\C\aph$ and $x\C\bph$ increase beyond the region of dilute solution behavior, the ratios $x\B\aph/x\A\aph$ and $x\B\bph/x\A\bph$ may change. Continued addition of C may increase the mutual solubilities of A and B, resulting, when enough C has been added, in a single liquid phase containing all three components. It is easier to understand this behavior with the help of a ternary phase diagram such as Fig. 13.17. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/12%3A_Equilibrium_Conditions_in_Multicomponent_Systems/12.06%3A_Liquid-Liquid_Equilibria.txt |
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A semipermeable membrane used to separate two liquid phases can, in principle, be permeable to certain species and impermeable to others. A membrane, however, may not be perfect in this respect over a long time period. We will assume that during the period of observation, those species to which the membrane is supposed to be permeable quickly achieve transfer equilibrium, and only negligible amounts of the other species are transferred across the membrane.
Section 12.2.2 sketched a derivation of the conditions needed for equilibrium in a two-phase system in which a membrane permeable only to solvent separates a solution from pure solvent. We can generalize the results for any system with two liquid phases separated by a semipermeable membrane: in an equilibrium state, both phases must have the same temperature, and any species to which the membrane is permeable must have the same chemical potential in both phases. The two phases, however, need not and usually do not have the same pressure.
12.7.1 Osmotic membrane equilibrium
An equilibrium state in a system with two solutions of the same solvent and different solute compositions, separated by a membrane permeable only to the solvent, is called an osmotic membrane equilibrium. We have already seen this kind of equilibrium in an apparatus that measures osmotic pressure (Fig. 12.2).
Consider a system with transfer equilibrium of the solvent across a membrane separating phases $\pha$ and $\phb$. The phases have equal solvent chemical potentials but different pressures: $\mu\A\bph(p\bph)=\mu\A\aph(p\aph) \tag{12.7.1}$ The dependence of $\mu\A$ on pressure in a phase of fixed temperature and composition is given by $\pd{\mu\A}{p}{T,\allni}=V\A$ (from Eq. 9.2.49), where $V\A$ is the partial molar volume of A in the phase. If we apply this relation to the solution of phase $\phb$, treat the partial molar volume $V\A$ as independent of pressure, and integrate at constant temperature and composition from the pressure of phase $\pha$ to that of phase $\phb$, we obtain $\mu\A\bph(p\bph)=\mu\A\bph(p\aph)+V\A\bph(p\bph-p\aph) \tag{12.7.2}$ By equating the two expressions for $\mu\A\bph(p\bph)$ and rearranging, we obtain the following expression for the pressure difference needed to achieve transfer equilibrium: $p\bph-p\aph=\frac{\mu\A\aph(p\aph)-\mu\A\bph(p\aph)}{V\A\bph} \tag{12.7.3}$
The pressure difference can be related to the osmotic pressures of the two phases. From Eq. 12.2.11, the solvent chemical potential in a solution phase can be written $\mu\A(p)=\mu\A^*(p)-V\A\varPi(p)$. Using this to substitute for $\mu\A\aph(p\aph)$ and $\mu\A\bph(p\aph)$ in Eq. 12.7.3, we obtain $p\bph-p\aph=\varPi\bph(p\aph)-\left( \frac{V\A\aph}{V\A\bph} \right) \varPi\aph(p\aph) \tag{12.7.4}$
12.7.2 Equilibrium dialysis
Equilibrium dialysis is a useful technique for studying the binding of a small uncharged solute species (a ligand) to a macromolecule. The macromolecule solution is placed on one side of a membrane through which it cannot pass, with a solution without the macromolecule on the other side, and the ligand is allowed to come to transfer equilibrium across the membrane. If the same solute standard state is used for the ligand in both solutions, at equilibrium the unbound ligand must have the same activity in both solutions. Measurements of the total ligand molality in the macromolecule solution and the ligand molality in the other solution, combined with estimated values of the unbound ligand activity coefficients, allow the amount of ligand bound per macromolecule to be calculated.
12.7.3 Donnan membrane equilibrium
If one of the solutions in a two-phase membrane equilibrium contains certain charged solute species that are unable to pass through the membrane, whereas other ions can pass through, the situation is more complicated than the osmotic membrane equilibrium described in Sec. 12.7.1. Usually if the membrane is impermeable to one kind of ion, an ion species to which it is permeable achieves transfer equilibrium across the membrane only when the phases have different pressures and different electric potentials. The equilibrium state in this case is a Donnan membrane equilibrium, and the resulting electric potential difference across the membrane is called the Donnan potential. This phenomenon is related to the membrane potentials that are important in the functioning of nerve and muscle cells (although the cells of a living organism are not, of course, in equilibrium states).
A Donnan potential can be measured electrically, with some uncertainty due to unknown liquid junction potentials, by connecting silver-silver chloride electrodes (described in Sec. 14.1) to both phases through salt bridges.
General expressions
Consider solution phases $\pha$ and $\phb$ separated by a semipermeable membrane. Both phases contain a dissolved salt, designated solute B, that has $\nu_+$ cations and $\nu_-$ anions in each formula unit. The membrane is permeable to these ions. Phase $\phb$ also contains a protein or other polyelectrolyte with a net positive or negative charge, together with counterions of the opposite charge that are the same species as the cation or anion of the salt. The presence of the counterions in phase $\phb$ prevents the cation and anion of the salt from being present in stoichiometric amounts in this phase. The membrane is impermeable to the polyelectrolyte, perhaps because the membrane pores are too small to allow the polyelectrolyte to pass through.
The condition for transfer equilibrium of solute B is $\mu\B\aph=\mu\B\bph$, or $(\mu\mbB\st)\aph+RT\ln a\mbB\aph=(\mu\mbB\st)\bph+RT\ln a\mbB\bph \tag{12.7.5}$ Solute B has the same standard state in the two phases, so that $(\mu\mbB\st)\aph$ and $(\mu\mbB\st)\bph$ are equal. The activities $a\mbB\aph$ and $a\mbB\bph$ are therefore equal at equilibrium. Using the expression for solute activity from Eq. 10.3.16, which is valid for a multisolute solution, we find that at transfer equilibrium the following relation must exist between the molalities of the salt ions in the two phases: $\G\mbB\aph\left(\g_{\pm}\aph\right)^{\nu} \left(m_+\aph\right)^{\nu_+} \left(m_-\aph\right)^{\nu_-} = \G\mbB\bph\left(\g_{\pm}\bph\right)^{\nu} \left(m_+\bph\right)^{\nu_+} \left(m_-\bph\right)^{\nu_-} \tag{12.7.6}$
To find an expression for the Donnan potential, we can equate the single-ion chemical potentials of the salt cation: $\mu_+\aph(\phi\aph)=\mu_+\bph(\phi\bph)$. When we use the expression of Eq. 10.1.15 for $\mu_+(\phi)$, we obtain \begin{gather} \s{ \phi\aph-\phi\bph = \frac{RT}{z_+F}\ln\frac{\G_+\bph \g_+\bph m_+\bph}{\G_+\aph \g_+\aph m_+\aph} } \tag{12.7.7} \cond{(Donnan potential)} \end{gather}
The condition needed for an osmotic membrane equilibrium related to the solvent can be written $\mu\A\bph(p\bph) - \mu\A\aph(p\aph) = 0 \tag{12.7.8}$ The chemical potential of the solvent is $\mu\A=\mu\A\st+RT\ln a\A=\mu\A\st+RT\ln(\G\A \g\A x\A)$. From Table 9.6, we have to a good approximation the expression $RT\ln\G\A = V\A^*(p-p\st)$. With these substitutions, Eq. 12.7.8 becomes $RT\ln\frac{\g\A\bph x\A\bph}{\g\A\aph x\A\aph} + V\A^*\left(p\bph-p\aph\right) = 0 \tag{12.7.9}$ We can use this equation to estimate the pressure difference needed to maintain an equilibrium state. For dilute solutions, with $\g\A\aph$ and $\g\A\bph$ set equal to 1, the equation becomes $p\bph-p\aph \approx \frac{RT}{V\A^*}\ln\frac{x\A\aph}{x\A\bph} \tag{12.7.10}$ In the limit of infinite dilution, $\ln x\A$ can be replaced by $-M\A\sum_{i\ne\tx{A}}m_i$ (Eq. 9.6.12), giving the relation $p\bph-p\aph \approx \frac{M\A RT}{V\A^*}\sum_{i\ne\tx{A}}\left(m_i\bph-m_i\aph\right) = \rho\A^*RT\sum_{i\ne\tx{A}}\left(m_i\bph-m_i\aph\right) \tag{12.7.11}$
Example
As a specific example of a Donnan membrane equilibrium, consider a system in which an aqueous solution of a polyelectrolyte with a net negative charge, together with a counterion M$^+$ and a salt MX of the counterion, is equilibrated with an aqueous solution of the salt across a semipermeable membrane. The membrane is permeable to the H$_2$O solvent and to the ions M$^+$ and X$^-$, but is impermeable to the polyelectrolyte. The species in phase $\pha$ are H$_2$O, M$^+$, and X$^-$; those in phase $\phb$ are H$_2$O, M$^+$, X$^-$, and the polyelectrolyte. In an equilibrium state, the two phases have the same temperature but different compositions, electric potentials, and pressures.
Because the polyelectrolyte in this example has a negative charge, the system has more M$^+$ ions than X$^-$ ions. Figure 12.9(a) is a schematic representation of an initial state of this kind of system. Phase $\phb$ is shown as a solution confined to a closed dialysis bag immersed in phase $\pha$. The number of cations and anions shown in each phase indicate the relative amounts of these ions.
For simplicity, let us assume the two phases have equal masses of water, so that the molality of an ion is proportional to its amount by the same ratio in both phases. It is clear that in the initial state shown in the figure, the chemical potentials of both M$^+$ and X$^-$ are greater in phase $\phb$ (greater amounts) than in phase $\pha$, and this is a nonequilibrium state. A certain quantity of salt MX will therefore pass spontaneously through the membrane from phase $\phb$ to phase $\pha$ until equilibrium is attained.
The equilibrium ion molalities must agree with Eq. 12.7.6. We make the approximation that the pressure factors and mean ionic activity coefficients are unity. Then for the present example, with $\nu_+=\nu_-=1$, the equation becomes $m_+\aph m_-\aph \approx m_+\bph m_-\bph \tag{12.7.12}$ There is furthermore an electroneutrality condition for each phase: $m\aph_+=m\aph_- \qquad m\bph_+ = m\bph_- + |z\subs{P}|m\subs{P} \tag{12.7.13}$ Here $z\subs{P}$ is the negative charge of the polyelectrolyte, and $m\subs{P}$ is its molality. Substitution of these expressions into Eq. 12.7.12 gives the relation $\left(m\aph_-\right)^2 \approx \left(m\bph_- + |z\subs{P}|m\subs{P}\right)m\bph_- \tag{12.7.14}$ This shows that in the equilibrium state, $m\aph_-$ is greater than $m\bph_-$. Then Eq. 12.7.12 shows that $m\aph_+$ is less than $m\bph_+$. These equilibrium molalities are depicted in Fig. 12.9(b).
The chemical potential of a cation, its activity, and the electric potential of the phase are related by Eq. 10.1.9: $\mu_+=\mu_+\st + RT\ln a_+ + z_+F\phi$. In order for M$^+$ to have the same chemical potential in both phases, despite its lower activity in phase $\pha$, the electric potential of phase $\pha$ must be greater than that of phase $\phb$. Thus the Donnan potential $\phi\aph-\phi\bph$ in the present example is positive. Its value can be estimated from Eq. 12.7.7 with the values of the single-ion pressure factors and activity coefficients approximated by 1 and with $z_+$ for this example set equal to 1: $\phi\aph-\phi\bph \approx \frac{RT}{F}\ln\frac{m_+\bph}{m_+\aph} \tag{12.7.15}$
The existence of a Donnan potential in the equilibrium state is the result of a very small departure of the phases on both sides of the membrane from exact electroneutrality. In the example, phase $\pha$ has a minute net positive charge and phase $\phb$ has a net negative charge of equal magnitude. The amount of M$^+$ ion transferred across the membrane to achieve equilibrium is slightly greater than the amount of X$^-$ ion transferred; the difference between these two amounts is far too small to be measured chemically. At equilibrium, the excess charge on each side of the membrane is distributed over the boundary surface of the solution phase on that side, and is not part of the bulk phase composition.
The pressure difference $p\bph-p\aph$ at equilibrium can be estimated with Eq. 12.7.11, and for the present example is found to be positive. Without this pressure difference, the solution in phase $\pha$ would move spontaneously through the membrane into phase $\phb$ until phase $\pha$ completely disappears. With phase $\pha$ open to the atmosphere, as in Fig. 12.9, the volume of phase $\phb$ must be constrained in order to allow its pressure to differ from atmospheric pressure. If the volume of phase $\phb$ remains practically constant, the transfer of a minute quantity of solvent across the membrane is sufficient to cause the pressure difference.
It should be clear that the existence of a Donnan membrane equilibrium introduces complications that would make it difficult to use a measured pressure difference to estimate the molar mass of the polyelectrolyte by the method of Sec. 12.4, or to study the binding of a charged ligand by equilibrium dialysis. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/12%3A_Equilibrium_Conditions_in_Multicomponent_Systems/12.07%3A_Membrane_Equilibria.txt |
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This section describes multicomponent systems in which a liquid phase is equilibrated with a gas phase.
12.8.1 Effect of liquid pressure on gas fugacity
If we vary the pressure of a liquid mixture at constant temperature and composition, there is a small effect on the fugacity of each volatile component in an equilibrated gas phase. One way to vary the pressure at essentially constant liquid composition is to change the partial pressure of a component of the gas phase that has negligible solubility in the liquid.
At transfer equilibrium, component $i$ has the same chemical potential in both phases: $\mu_i\liquid =\mu_i\gas$. Combining the relations $\bpd{\mu_i\liquid}{p}{T,\allni}=V_i\liquid$ and $\mu_i\gas=\mu_i\st\gas +RT\ln(\fug_i/p\st)$ (Eqs. 9.2.49 and 9.3.12), we obtain \begin{gather} \s{ \frac{\dif\ln(\fug_i/p\st)}{\difp}=\frac{V_i\liquid}{RT} } \tag{12.8.1} \cond{(equilibrated liquid and} \nextcond{gas mixtures, constant $T$} \nextcond{and liquid composition)} \end{gather} Equation 12.8.1 shows that an increase in pressure, at constant temperature and liquid composition, causes an increase in the fugacity of each component in the gas phase.
Integration of Eq. 12.8.1 between pressures $p_1$ and $p_2$ yields \begin{gather} \s{ \fug_i(p_2)=\fug_i(p_1) \exp\left[\int_{p_1}^{p_2}\frac{V_i\liquid}{RT} \difp\right] } \tag{12.8.2} \cond{(equilibrated liquid and} \nextcond{gas mixtures, constant $T$} \nextcond{and liquid composition)} \end{gather} The exponential on the right side is called the Poynting factor.
The integral in the Poynting factor is simplified if we make the approximation that $V_i\liquid$ is independent of pressure. Then we obtain the approximate relation \begin{gather} \s{ \fug_i(p_2) \approx \fug_i(p_1)\exp\left[ \frac{V_i\liquid (p_2-p_1)}{RT} \right] } \tag{12.8.3} \cond{(equilibrated liquid and} \nextcond{gas mixtures, constant $T$} \nextcond{and liquid composition)} \end{gather}
The effect of pressure on fugacity is usually small, and can often be neglected. For typical values of the partial molar volume $V_i\liquid$, the exponential factor is close to unity unless $|p_2{-}p_1|$ is very large. For instance, for $V_i\liquid{=}100\units{cm\(^3$ mol$^{-1}$}\) and $T{=}300\K$, we obtain a value for the ratio $\fug_i(p_2)/\fug_i(p_1)$ of $1.004$ if $p_2{-}p_1$ is $1\br$, $1.04$ if $p_2{-}p_1$ is $10\br$, and $1.5$ if $p_2{-}p_1$ is $100\br$. Thus, unless the pressure change is large, we can to a good approximation neglect the effect of total pressure on fugacity. This statement applies only to the fugacity of a substance in a gas phase that is equilibrated with a liquid phase of constant composition containing the same substance. If the liquid phase is absent, the fugacity of $i$ in a gas phase of constant composition is of course approximately proportional to the total gas pressure.
We can apply Eqs. 12.8.2 and 12.8.3 to pure liquid A, in which case $V_i\liquid$ is the molar volume $V\A^*\liquid$. Suppose we have pure liquid A in equilibrium with pure gaseous A at a certain temperature. This is a one-component, two-phase equilibrium system with one degree of freedom (Sec. 8.1.7), so that at the given temperature the value of the pressure is fixed. This pressure is the saturation vapor pressure of pure liquid A at this temperature. We can make the pressure $p$ greater than the saturation vapor pressure by adding a second substance to the gas phase that is essentially insoluble in the liquid, without changing the temperature or volume. The fugacity $\fug\A$ is greater at this higher pressure than it was at the saturation vapor pressure. The vapor pressure $p\A$, which is approximately equal to $\fug\A$, has now become greater than the saturation vapor pressure. It is, however, safe to say that the difference is negligible unless the difference between $p$ and $p\A$ is much greater than $1\br$.
As an application of these relations, consider the effect of the size of a liquid droplet on the equilibrium vapor pressure. The calculation of Prob. 12.8(b) shows that the fugacity of H$_2$O in a gas phase equilibrated with liquid water in a small droplet is slightly greater than when the liquid is in a bulk phase. The smaller the radius of the droplet, the greater is the fugacity and the vapor pressure.
12.8.2 Effect of liquid composition on gas fugacities
Consider system 1 in Fig. 9.5. A binary liquid mixture of two volatile components, A and B, is equilibrated with a gas mixture containing A, B, and a third gaseous component C of negligible solubility used to control the total pressure. In order for A and B to be in transfer equilibrium, their chemical potentials must be the same in both phases: $\mu\A\liquid =\mu\A\st\gas +RT\ln\frac{\fug\A}{p\st} \qquad \mu\B\liquid =\mu\B\st\gas +RT\ln\frac{\fug\B}{p\st} \tag{12.8.4}$
Suppose we make an infinitesimal change in the liquid composition at constant $T$ and $p$. This causes infinitesimal changes in the chemical potentials and fugacities: $\dif\mu\A\liquid=RT\frac{\dif\fug\A}{\fug\A} \qquad \dif\mu\B\liquid=RT\frac{\dif\fug\B}{\fug\B} \tag{12.8.5}$ By inserting these expressions in the Gibbs–Duhem equation $x\A\dif\mu\A = - x\B\dif\mu\B$ (Eq. 9.2.43), we obtain \begin{gather} \s{ \frac{x\A}{\fug\A}\dif\fug\A=-\frac{x\B}{\fug\B}\dif\fug\B} \tag{12.8.6} \cond{(binary liquid mixture equilibrated} \nextcond{with gas, constant $T$ and $p$)} \end{gather} This equation is a relation between changes in gas-phase fugacities caused by a change in the liquid-phase composition. It shows that a composition change at constant $T$ and $p$ that increases the fugacity of A in the equilibrated gas phase must decrease the fugacity of B.
Now let us treat the liquid mixture as a binary solution with component B as the solute. In the ideal-dilute region, at constant $T$ and $p$, the solute obeys Henry’s law for fugacity:$\fug\B = \kHB x\B \tag{12.8.7}$ For composition changes in the ideal-dilute region, we can write $\frac{\dif\fug\B}{\dx\B} = \kHB = \frac{\fug\B}{x\B} \tag{12.8.8}$ With the substitution $\dx\B=-\dx\A$ and rearrangement, Eq. 12.8.8 becomes $-\frac{x\B}{\fug\B}\dif\fug\B = \dx\A \tag{12.8.9}$ Combined with Eq. 12.8.6, this is $(x\A/\fug\A)\dif\fug\A=\dx\A$, which we can rearrange and integrate as follows within the ideal-dilute region: $\int_{\fug\A^*}^{\fug'\A}\frac{\dif\fug\A}{\fug\A} = \int_{1}^{x'\A}\frac{\dx\A}{x\A} \qquad \ln\frac{\fug'\A}{\fug\A^*}=\ln x'\A \tag{12.8.10}$ The result is \begin{gather} \s{ \fug\A=x\A\fug\A^* } \tag{12.8.11} \cond{(ideal-dilute binary solution)} \end{gather} Here $\fug\A^*$ is the fugacity of A in a gas phase equilibrated with pure liquid A at the same $T$ and $p$ as the mixture. Equation 12.8.11 is Raoult’s law for fugacity applied to component A.
If component B obeys Henry’s law at all compositions, then the Henry’s law constant $\kHB$ is equal to $\fug\B^*$ and B obeys Raoult’s law, $\fug\B=x\B\fug\B^*$, over the entire range of $x\B$.
We can draw two conclusions:
1. Figure 12.11 illustrates the case of a binary mixture in which component B has only positive deviations from Raoult’s law, whereas component A has both positive and negative deviations ($\fug\A$ is slightly less than $x\A\fug\A^*$ for $x\B$ less than 0.3). This unusual behavior is possible because both fugacity curves have two inflection points instead of the usual one. Other types of unusual nonideal behavior are possible (M. L. McGlashan, J. Chem. Educ., 40, 516–518, 1963).
12.8.3 The Duhem–Margules equation
To a good approximation, by assuming an ideal gas mixture and neglecting the effect of total pressure on fugacity, we can apply Eq. 12.8.20 to a liquid–gas system in which the total pressure is not constant, but instead is the sum of $p\A$ and $p\B$. Under these conditions, we obtain the following expression for the rate at which the total pressure changes with the liquid composition at constant $T$: $\begin{split} \frac{\difp}{\dx\A} & = \frac{\dif(p\A+p\B)}{\dx\A} = \frac{\difp\A}{\dx\A} -\frac{x\A p\B}{x\B p\A} \frac{\difp\A}{\dx\A} = \frac{\difp\A}{\dx\A}\left( 1 - \frac{x\A/x\B}{p\A/p\B} \right) \cr & = \frac{\difp\A}{\dx\A}\left( 1 - \frac{x\A/x\B}{y\A/y\B}\right) \end{split} \tag{12.8.21}$ Here $y\A$ and $y\B$ are the mole fractions of A and B in the gas phase given by $y\A=p\A/p$ and $y\B=p\B/p$.
We can use Eq. 12.8.21 to make several predictions for a binary liquid–gas system at constant $T$.
• In some binary liquid–gas systems, the total pressure at constant temperature exhibits a maximum or minimum at a particular liquid composition. At this composition, $\difp/\dx\A$ is zero but $\difp\A/\dx\A$ is positive. From Eq. 12.8.21, we see that at this composition $x\A/x\B$ must equal $y\A/y\B$, meaning that the liquid and gas phases have identical mole fraction compositions. The liquid with this composition is called an azeotrope. The behavior of systems with azeotropes will be discussed in Sec. 13.2.5.
12.8.4 Gas solubility
The activity of B in the gas phase is given by $a\B\gas =\fug\B/p\st$. If the solute is a nonelectrolyte and we choose a standard state based on mole fraction, the activity in the solution is $a\B\sln=\G\xbB \g\xbB x\B$. The equilibrium constant is then given by $K = \frac{\G\xbB \g\xbB x\B}{\fug\B/p\st} \tag{12.8.22}$ and the solubility, expressed as the equilibrium mole fraction of solute in the solution, is given by \begin{gather} \s{ x\B = \frac{K\fug\B/p\st}{\G\xbB \g\xbB} } \tag{12.8.23} \cond{(nonelectrolyte solute in} \nextcond{equilibrium with gas)} \end{gather}
At a fixed $T$ and $p$, the values of $K$ and $\G\xbB$ are constant. Therefore any change in the solution composition that increases the value of the activity coefficient $\g\xbB$ will decrease the solubility for the same gas fugacity. This solubility decrease is often what happens when a salt is dissolved in an aqueous solution, and is known as the salting-out effect (Prob. 12.11).
Unless the pressure is much greater than $p\st$, we can with negligible error set the pressure factor $\G\xbB$ equal to 1. When the gas solubility is low and the solution contains no other solutes, the activity coefficient $\g\xbB$ is close to 1. If furthermore we assume ideal gas behavior, then Eq. 12.8.23 becomes \begin{gather} \s{ x\B=K\frac{p\B}{p\st} } \tag{12.8.24} \cond{(nonelectrolyte solute in equilibrium} \nextcond{with ideal gas, $\G\xbB{=}1$, $\g\xbB{=}1$)} \end{gather} The solubility is predicted to be proportional to the partial pressure. The solubility of a gas that dissociates into ions in solution has a quite different dependence on partial pressure. An example is the solubility of gaseous HCl in water to form an electrolyte solution, shown in Fig. 10.1.
If the actual conditions are close to those assumed for Eq. 12.8.24, we can use Eq. 12.1.13 to derive an expression for the temperature dependence of the solubility for a fixed partial pressure of the gas: $\Pd{\ln x\B}{T}{\!\!p\B} = \frac{\dif\ln K}{\dif T} = \frac{\Delsub{sol,B}H\st}{RT^2} \tag{12.8.25}$ At the standard pressure, $\Delsub{sol,B}H\st$ is the same as the molar enthalpy of solution at infinite dilution.
Since the dissolution of a gas in a liquid is invariably an exothermic process, $\Delsub{sol,B}H\st$ is negative, and Eq. 12.8.25 predicts the solubility decreases with increasing temperature.
Note the similarity of Eq. 12.8.25 and the expressions derived previously for the temperature dependence of the solubilities of solids (Eq. 12.5.8) and liquids (Eq. 12.6.3). When we substitute the mathematical identity $\dif T=-T^2\dif(1/T)$, Eq. 12.8.25 becomes $\bPd{\ln x\B}{(1/T)}{p\B} = -\frac{\Delsub{sol,B}H\st}{R} \tag{12.8.26}$ We can use this form to evaluate $\Delsub{sol,B}H\st$ from a plot of $\ln x\B$ versus $1/T$.
The ideal solubility of a gas is the solubility calculated on the assumption that the dissolved gas obeys Raoult’s law for partial pressure: $p\B = x\B p\B^*$. The ideal solubility, expressed as a mole fraction, is then given as a function of partial pressure by \begin{gather} \s{ x\B = \frac{p\B}{p\B^*} } \tag{12.8.27} \cond{(ideal solubility of a gas)} \end{gather} Here $p\B^*$ is the vapor pressure of pure liquid solute at the same temperature and total pressure as the solution. If the pressure is too low for pure B to exist as a liquid at this temperature, we can with little error replace $p\B^*$ with the saturation vapor pressure of liquid B at the same temperature, because the effect of total pressure on the vapor pressure of a liquid is usually negligible (Sec. 12.8.1). If the temperature is above the critical temperature of pure B, we can estimate a hypothetical vapor pressure by extrapolating the liquid–vapor coexistence curve beyond the critical point.
We can use Eq. 12.8.27 to make several predictions regarding the ideal solubility of a gas at a fixed value of $p\B$.
1. Of course, these predictions apply only to solutions that behave approximately as ideal liquid mixtures, but even for many nonideal mixtures the predictions are found to have good agreement with experiment.
As an example of the general validity of prediction 1, Hildebrand and Scott (The Solubility of Nonelectrolytes, 3rd edition, Dover, New York, 1964, Chap. XV) list the following solubilities of gaseous Cl$_2$ in several dissimilar solvents at $0\units{\(\degC$}\) and a partial pressure of $1.01\br$: $x\B=0.270$ in heptane, $x\B=0.288$ in SiCl$_4$, and $x\B=0.298$ in CCl$_4$. These values are similar to one another and close to the ideal value $p\B/p\B^*=0.273$.
12.8.5 Effect of temperature and pressure on Henry’s law constants
At the standard pressure $p\st=1\br$, the value of $\G\xbB$ is unity, and Eqs. 12.1.13 and 12.1.14 then give the following expressions for the dependence of the dimensionless quantity $\kHB/p\st$ on temperature: \begin{gather} \s{ \frac{\dif\ln(\kHB/p\st)}{\dif T} = -\frac{\dif\ln K}{\dif T} = -\frac{\Delsub{sol,B}H\st}{RT^2}} \tag{12.8.31} \cond{($p{=}p\st$)} \end{gather} \begin{gather} \s{ \frac{\dif\ln(\kHB/p\st)}{\dif(1/T)} = -\frac{\dif\ln K}{\dif(1/T)} = \frac{\Delsub{sol,B}H\st}{R}} \tag{12.8.32} \cond{($p{=}p\st$)} \end{gather} These expressions can be used with little error at any pressure that is not much greater than $p\st$, say up to at least $2\br$, because under these conditions $\G\xbB$ does not differ appreciably from unity.
To find the dependence of $\kHB$ on pressure, we substitute $\G\xbB$ in Eq. 12.8.30 with the expression for $\G\xbB$ at pressure $p'$ found in Table 9.6: $\kHB(p') = \frac{\G\xbB(p') p\st}{K} = \frac{p\st}{K} \exp\left(\int_{p\st}^{p'}\frac{V\B^{\infty}}{RT}\difp\right) \tag{12.8.33}$ We can use Eq. 12.8.33 to compare the values of $\kHB$ at the same temperature and two different pressures, $p_1$ and $p_2$: $\kHB(p_2) = \kHB(p_1)\exp\left(\int_{p_1}^{p_2}\frac{V\B^{\infty}}{RT}\difp\right) \tag{12.8.34}$ An approximate version of this relation, found by treating $V\B^{\infty}$ as independent of pressure, is $\kHB(p_2) \approx \kHB(p_1)\exp\left[\frac{V\B^{\infty}(p_2-p_1)}{RT}\right] \tag{12.8.35}$
Unless $|p_2-p_1|$ is much greater than $1\br$, the effect of pressure on $\kHB$ is small; see Prob. 12.12 for an example. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/12%3A_Equilibrium_Conditions_in_Multicomponent_Systems/12.08%3A_Liquid-Gas_Equilibria.txt |
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$\newcommand{\CVm}{C_{V,\text{m}}} % molar heat capacity at const.V$
$\newcommand{\Cpm}{C_{p,\text{m}}} % molar heat capacity at const.p$
$\newcommand{\kT}{\kappa_T} % isothermal compressibility$
$\newcommand{\A}{_{\text{A}}} % subscript A for solvent or state A$
$\newcommand{\B}{_{\text{B}}} % subscript B for solute or state B$
$\newcommand{\bd}{_{\text{b}}} % subscript b for boundary or boiling point$
$\newcommand{\C}{_{\text{C}}} % subscript C$
$\newcommand{\f}{_{\text{f}}} % subscript f for freezing point$
$\newcommand{\mA}{_{\text{m},\text{A}}} % subscript m,A (m=molar)$
$\newcommand{\mB}{_{\text{m},\text{B}}} % subscript m,B (m=molar)$
$\newcommand{\mi}{_{\text{m},i}} % subscript m,i (m=molar)$
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$\newcommand{\fB}{_{\text{f},\text{B}}} % subscript f,B (for fr. pt.)$
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$\newcommand{\xbC}{_{x,\text{C}}} % x basis, C$
$\newcommand{\cbB}{_{c,\text{B}}} % c basis, B$
$\newcommand{\mbB}{_{m,\text{B}}} % m basis, B$
$\newcommand{\kHi}{k_{\text{H},i}} % Henry's law constant, x basis, i$
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$\renewcommand{\in}{\sups{int}} % internal$
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$\newcommand{\cm}{\subs{cm}} % center of mass$
$\newcommand{\rev}{\subs{rev}} % reversible$
$\newcommand{\irr}{\subs{irr}} % irreversible$
$\newcommand{\fric}{\subs{fric}} % friction$
$\newcommand{\diss}{\subs{diss}} % dissipation$
$\newcommand{\el}{\subs{el}} % electrical$
$\newcommand{\cell}{\subs{cell}} % cell$
$\newcommand{\As}{A\subs{s}} % surface area$
$\newcommand{\E}{^\mathsf{E}} % excess quantity (superscript)$
$\newcommand{\allni}{\{n_i \}} % set of all n_i$
$\newcommand{\sol}{\hspace{-.1em}\tx{(sol)}}$
$\newcommand{\solmB}{\tx{(sol,\,m\B)}}$
$\newcommand{\dil}{\tx{(dil)}}$
$\newcommand{\sln}{\tx{(sln)}}$
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$\newcommand{\rxn}{\tx{(rxn)}}$
$\newcommand{\expt}{\tx{(expt)}}$
$\newcommand{\solid}{\tx{(s)}}$
$\newcommand{\liquid}{\tx{(l)}}$
$\newcommand{\gas}{\tx{(g)}}$
$\newcommand{\pha}{\alpha} % phase alpha$
$\newcommand{\phb}{\beta} % phase beta$
$\newcommand{\phg}{\gamma} % phase gamma$
$\newcommand{\aph}{^{\alpha}} % alpha phase superscript$
$\newcommand{\bph}{^{\beta}} % beta phase superscript$
$\newcommand{\gph}{^{\gamma}} % gamma phase superscript$
$\newcommand{\aphp}{^{\alpha'}} % alpha prime phase superscript$
$\newcommand{\bphp}{^{\beta'}} % beta prime phase superscript$
$\newcommand{\gphp}{^{\gamma'}} % gamma prime phase superscript$
$\newcommand{\apht}{\small\aph} % alpha phase tiny superscript$
$\newcommand{\bpht}{\small\bph} % beta phase tiny superscript$
$\newcommand{\gpht}{\small\gph} % gamma phase tiny superscript$
$\newcommand{\upOmega}{\Omega}$
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$\newcommand{\Dif}{\mathop{}\!\mathrm{D}} % roman D in math mode, preceded by space$
$\newcommand{\df}{\dif\hspace{0.05em} f} % df$
$\newcommand{\dBar}{\mathop{}\!\mathrm{d}\hspace-.3em\raise1.05ex{\Rule{.8ex}{.125ex}{0ex}}} % inexact differential$
$\newcommand{\dq}{\dBar q} % heat differential$
$\newcommand{\dw}{\dBar w} % work differential$
$\newcommand{\dQ}{\dBar Q} % infinitesimal charge$
$\newcommand{\dx}{\dif\hspace{0.05em} x} % dx$
$\newcommand{\dt}{\dif\hspace{0.05em} t} % dt$
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$\newcommand{\Delsub}[1]{\Delta_{\text{#1}}}$
$\newcommand{\pd}[3]{(\partial #1 / \partial #2 )_{#3}} % \pd{}{}{} - partial derivative, one line$
$\newcommand{\Pd}[3]{\left( \dfrac {\partial #1} {\partial #2}\right)_{#3}} % Pd{}{}{} - Partial derivative, built-up$
$\newcommand{\bpd}[3]{[ \partial #1 / \partial #2 ]_{#3}}$
$\newcommand{\bPd}[3]{\left[ \dfrac {\partial #1} {\partial #2}\right]_{#3}}$
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$\newcommand{\fug}{f} % fugacity$
$\newcommand{\g}{\gamma} % solute activity coefficient, or gamma in general$
$\newcommand{\G}{\varGamma} % activity coefficient of a reference state (pressure factor)$
$\newcommand{\ecp}{\widetilde{\mu}} % electrochemical or total potential$
$\newcommand{\Eeq}{E\subs{cell, eq}} % equilibrium cell potential$
$\newcommand{\Ej}{E\subs{j}} % liquid junction potential$
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$\newcommand{\cond}[1]{\$-2.5pt]{}\tag*{#1}}$ $\newcommand{\nextcond}[1]{\[-5pt]{}\tag*{#1}}$ $\newcommand{\R}{8.3145\units{J\,K\per\,mol\per}} % gas constant value$ $\newcommand{\Rsix}{8.31447\units{J\,K\per\,mol\per}} % gas constant value - 6 sig figs$ $\newcommand{\jn}{\hspace3pt\lower.3ex{\Rule{.6pt}{2ex}{0ex}}\hspace3pt}$ $\newcommand{\ljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}} \hspace3pt}$ $\newcommand{\lljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace1.4pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace3pt}$ The definition of the thermodynamic equilibrium constant of a reaction or other chemical process is given by Eq. 11.8.9: $K = \prod_i (a_i)\eq^{\nu_i} \tag{12.9.1}$ The activity $a_i$ of each reactant or product species is based on an appropriate standard state. We can replace each activity on the right side of Eq. 12.9.1 by an expression in Table 12.2. For example, consider the following heterogeneous equilibrium that is important in the formation of limestone caverns: \[ \ce{CaCO3}\tx{(cr, calcite)} + \ce{CO2}\tx{(g)} + \ce{H2O}\tx{(sln)} \arrows \ce{Ca^2+}\tx{(aq)} + \ce{2HCO3-}\tx{(aq)}$ If we treat H$_2$O as a solvent and Ca$^{2+}$ and HCO$_3$$^-$ as the solute species, then we write the thermodynamic equilibrium constant as follows: $K = \frac{a_+ a_-^2}{a\subs{CaCO$_3$} a\subs{CO$_2$} a\subs{H$_2$O}} = \G\subs{r} \frac{\g_+\g_-^2m_+m_-^2/(m\st)^3} {\left(\fug\subs{CO$_2$}/p\st\right)\g\subs{H$_2$O} x\subs{H$_2$O}} \tag{12.9.2}$ The subscripts $+$ and $-$ refer to the Ca$^{2+}$ and HCO$_3$$^-$ ions, and all quantities are for the system at reaction equilibrium. $\G\subs{r}$ is the proper quotient of pressure factors, given for this reaction by $\G\subs{r} = \frac{\G_+\G_-^2}{\G\subs{CaCO$_3$}\G\subs{H$_2$O}} \tag{12.9.3}$ Unless the pressure is very high, we can with little error set the value of $\G\subs{r}$ equal to unity.
The product $\G_+\G_-^2$ in the numerator of Eq. 12.9.3 is the pressure factor $\G\mbB$ for the solute Ca(HCO$_3$)$_2$ (see Eq. 10.3.11).
Equation 12.9.2 is an example of a “mixed” equilibrium constant—one using more than one kind of standard state. From the definition of the mean ionic activity coefficient (Eq. 10.3.7), we can replace the product $\g_+\g_-^2$ by $\g_{\pm}^3$, where $\g_{\pm}$ is the mean ionic activity coefficient of aqueous Ca(HCO$_3$)$_2$: $K = \G\subs{r} \frac{\g_{\pm}^3 m_+m_-^2/(m\st)^3} {\left(\fug\subs{CO$_2$}/p\st\right)\g\subs{H$_2$O} x\subs{H$_2$O}} \tag{12.9.4}$ Instead of treating the aqueous Ca$^{2+}$ and HCO$_3$$^-$ ions as solute species, we can regard the dissolved Ca(HCO$_3$)$_2$ electrolyte as the solute and write $K = \frac{a\mbB}{a\subs{CaCO$_3$} a\subs{CO$_2$} a\subs{H$_2$O}} \tag{12.9.5}$ We then obtain Eq. 12.9.4 by replacing $a\mbB$ with the expression in Table 12.2 for an electrolyte solute.
The value of $K$ depends only on $T$, and the value of $\G\subs{r}$ depends only on $T$ and $p$. Suppose we dissolve some NaCl in the aqueous phase while maintaining the system at constant $T$ and $p$. The increase in the ionic strength will alter $\g_{\pm}$ and necessarily cause a compensating change in the solute molarity in order for the system to remain in reaction equilibrium.
An example of a different kind of reaction equilibrium is the dissociation (ionization) of a weak monoprotic acid such as acetic acid $\ce{HA}\tx{(aq)} \arrows \ce{H+}\tx{(aq)} + \ce{A-}\tx{(aq)}$ for which the thermodynamic equilibrium constant (the acid dissociation constant) is $K\subs{a} = \G\subs{r} \frac{\g_+\g_-m_+m_-}{\g\subs{$m$,HA} m\subs{HA}m\st} = \G\subs{r} \frac{\g_{\pm}^2 m_+m_-}{\g\subs{$m$,HA} m\subs{HA}m\st} \tag{12.9.6}$ Suppose the solution is prepared from water and the acid, and H$^+$ from the dissociation of H$_2$O is negligible compared to H$^+$ from the acid dissociation. We may then write $m_+=m_-=\alpha m\B$, where $\alpha$ is the degree of dissociation and $m\B$ is the overall molality of the acid. The molality of the undissociated acid is $m\subs{HA}=(1-\alpha)m\B$, and the dissociation constant can be written $K\subs{a}=\G\subs{r} \frac{\g_{\pm}^2 \alpha^2m\B/m\st} {\g\subs{$m$,HA}(1-\alpha)} \tag{12.9.7}$ From this equation, we see that a change in the ionic strength that decreases $\g_{\pm}$ when $T$, $p$, and $m\B$ are held constant must increase the degree of dissociation (Prob. 12.17). | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/12%3A_Equilibrium_Conditions_in_Multicomponent_Systems/12.09%3A_Reaction_Equilibria.txt |
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Some of the most useful experimentally-derived data for thermodynamic calculations are values of standard molar reaction enthalpies, standard molar reaction Gibbs energies, and standard molar reaction entropies. The values of these quantities for a given reaction are related, as we know (Eq. 11.8.21), by $\Delsub{r}G\st=\Delsub{r}H\st-T\Delsub{r}S\st \tag{12.10.1}$ and $\Delsub{r}S\st$ can be calculated from the standard molar entropies of the reactants and products using Eq. 11.8.22: $\Delsub{r}S\st=\sum_i\nu_i S_i\st \tag{12.10.2}$
The standard molar quantities appearing in Eqs. 12.10.1 and 12.10.2 can be evaluated through a variety of experimental techniques. Reaction calorimetry can be used to evaluate $\Delsub{r}H\st$ for a reaction (Sec. 11.5). Calorimetric measurements of heat capacity and phase-transition enthalpies can be used to obtain the value of $S_i\st$ for a solid or liquid (Sec. 6.2.1). For a gas, spectroscopic measurements can be used to evaluate $S_i\st$ (Sec. 6.2.2). Evaluation of a thermodynamic equilibrium constant and its temperature derivative, for any of the kinds of equilibria discussed in this chapter (vapor pressure, solubility, chemical reaction, etc.), can provide values of $\Delsub{r}G\st$ and $\Delsub{r}H\st$ through the relations $\Delsub{r}G\st=-RT\ln K$ and $\Delsub{r}H\st=-R\dif\ln K/\dif(1/T)$.
In addition to these methods, measurements of cell potentials are useful for a reaction that can be carried out reversibly in a galvanic cell. Section 14.3.3 will describe how the standard cell potential and its temperature derivative allow $\Delsub{r}H\st$, $\Delsub{r}G\st$, and $\Delsub{r}S\st$ to be evaluated for such a reaction.
An efficient way of tabulating the results of experimental measurements is in the form of standard molar enthalpies and Gibbs energies of formation. These values can be used to generate the values of standard molar reaction quantities for reactions not investigated directly. The relations between standard molar reaction and formation quantities (Sec. 11.3.2) are $\Delsub{r}H\st= \sum_i\nu_i\Delsub{f}H\st(i) \qquad \Delsub{r}G\st= \sum_i\nu_i\Delsub{f}G\st(i) \tag{12.10.3}$ and for ions the conventions used are $\Delsub{f}H\st\tx{(H$^+$, aq)}=0 \qquad \Delsub{f}G\st\tx{(H$^+$, aq)}=0 \qquad S\m\st\tx{(H$^+$, aq)}=0 \tag{12.10.4}$ Appendix H gives an abbreviated set of values of $\Delsub{f}H\st$, $S\m\st$, and $\Delsub{f}G\st$ at $298.15\K$.
For examples of the evaluation of standard molar reaction quantities and standard molar formation quantities from measurements made by various experimental techniques, see Probs. 12.18–12.20, 14.3, and 14.4. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/12%3A_Equilibrium_Conditions_in_Multicomponent_Systems/12.10%3A_Evaluation_of_Standard_Molar_Quantities.txt |
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$\newcommand{\cond}[1]{\$-2.5pt]{}\tag*{#1}}$ $\newcommand{\nextcond}[1]{\[-5pt]{}\tag*{#1}}$ $\newcommand{\R}{8.3145\units{J\,K\per\,mol\per}} % gas constant value$ $\newcommand{\Rsix}{8.31447\units{J\,K\per\,mol\per}} % gas constant value - 6 sig figs$ $\newcommand{\jn}{\hspace3pt\lower.3ex{\Rule{.6pt}{2ex}{0ex}}\hspace3pt}$ $\newcommand{\ljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}} \hspace3pt}$ $\newcommand{\lljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace1.4pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace3pt}$ An underlined problem number or problem-part letter indicates that the numerical answer appears in Appendix I. 12.1 Consider the heterogeneous equilibrium $\ce{CaCO3}\tx{(s)} \arrows \ce{CaO}\tx{(s)} + \ce{CO2}\tx{(g)}$. Table 12.3 lists pressures measured over a range of temperatures for this system. 12.14 The method described in Prob. 12.13 has been used to obtain high-precision values of the Henry’s law constant, $\kHB$, for gaseous methane dissolved in water (Timothy R. Rettich, Y. Paul Handa, Rubin Battino, and Emmerich Wilhelm, J. Phys. Chem., 85, 3230–3237, 1981). Table 12.6 lists values of $\ln (\kHB/p\st)$ at eleven temperatures in the range $275\K$–$328\K$ and at pressures close to $1\br$. Use these data to evaluate $\Delsub{sol,B}H\st$ and $\Delsub{sol,B}C\st_p$ at $T=298.15\K$. This can be done by a graphical method. Better precision will be obtained by making a least-squares fit of the data to the three-term polynomial \begin{equation*} \ln (\kHB/p\st) = a + b(1/T) + c(1/T)^2 \end{equation*} and using the values of the coefficients $a$, $b$, and $c$ for the evaluations. 12.15 Liquid water and liquid benzene have very small mutual solubilities. Equilibria in the binary water–benzene system were investigated by Tucker, Lane, and Christian (J. Solution Chem., 10, 1–20, 1981) as follows. A known amount of distilled water was admitted to an evacuated, thermostatted vessel. Part of the water vaporized to form a vapor phase. Small, precisely measured volumes of liquid benzene were then added incrementally from the sample loop of a liquid-chromatography valve. The benzene distributed itself between the liquid and gaseous phases in the vessel. After each addition, the pressure was read with a precision pressure gauge. From the known amounts of water and benzene and the total pressure, the liquid composition and the partial pressure of the benzene were calculated. The fugacity of the benzene in the vapor phase was calculated from its partial pressure and the second virial coefficient. At a fixed temperature, for mole fractions $x\B$ of benzene in the liquid phase up to about $3\timesten{-4}$ (less than the solubility of benzene in water), the fugacity of the benzene in the equilibrated gas phase was found to have the following dependence on $x\B$: \[ \frac{\fug\B}{x\B} = \kHB - Ax\B$ Here $\kHB$ is the Henry’s law constant and $A$ is a constant related to deviations from Henry’s law. At $30\units{\(\degC$}\), the measured values were $\kHB=385.5\br$ and $A=2.24\timesten{4}\br$.
(a) Treat benzene (B) as the solute and find its activity coefficient on a mole fraction basis, $\g\xbB$, at $30\units{\(\degC$}\) in the solution of composition $x\B=3.00\timesten{-4}$.
(b) The fugacity of benzene vapor in equilibrium with pure liquid benzene at $30\units{\(\degC$}\) is $\fug\B^*=0.1576\br$. Estimate the mole fraction solubility of liquid benzene in water at this temperature.
(c) The calculation of $\g\xbB$ in part (a) treated the benzene as a single solute species with deviations from infinite-dilution behavior. Tucker et al suggested a dimerization model to explain the observed negative deviations from Henry’s law. (Classical thermodynamics, of course, cannot prove such a molecular interpretation of observed macroscopic behavior.) The model assumes that there are two solute species, a monomer (M) and a dimer (D), in reaction equilibrium: $\ce{2M} \arrows \ce{D}$. Let $n\B$ be the total amount of C$_6$H$_6$ present in solution, and define the mole fractions $x\B \defn \frac{n\B}{n\A+n\B} \approx \frac{n\B}{n\A}$ $x\subs{M} \defn \frac{n\subs{M}}{n\A+n\subs{M}+n\subs{D}} \approx \frac{n\subs{M}}{n\A} \qquad x\subs{D} \defn \frac{n\subs{D}}{n\A+n\subs{M}+n\subs{D}} \approx \frac{n\subs{D}}{n\A}$ where the approximations are for dilute solution. In the model, the individual monomer and dimer particles behave as solutes in an ideal-dilute solution, with activity coefficients of unity. The monomer is in transfer equilibrium with the gas phase: $x\subs{M}=\fug\B/\kHB$. The equilibrium constant expression (using a mole fraction basis for the solute standard states and setting pressure factors equal to 1) is $K=x\subs{D}/x\subs{M}^2$. From the relation $n\B=n\subs{M}+2n\subs{D}$, and because the solution is very dilute, the expression becomes $K = \frac{x\B-x\subs{M}}{2x\subs{M}^2}$
Make individual calculations of $K$ from the values of $\fug\B$ measured at $x\B=1.00\timesten{-4}$, $x\B=2.00\timesten{-4}$, and $x\B=3.00\timesten{-4}$. Extrapolate the calculated values of $K$ to $x\B{=}0$ in order to eliminate nonideal effects such as higher aggregates. Finally, find the fraction of the benzene molecules present in the dimer form at $x\B=3.00\timesten{-4}$ if this model is correct.
12.16
Use data in Appendix H to evaluate the thermodynamic equilibrium constant at $298.15\K$ for the limestone reaction $\ce{CaCO3}\tx{(cr, calcite)} + \ce{CO2}\tx{(g)} + \ce{H2O}\tx{(l)} \arrow \ce{Ca^2+}\tx{(aq)} + \ce{2HCO3-}\tx{(aq)}$
12.17
For the dissociation equilibrium of formic acid, $\ce{HCO2H}\tx{(aq)} \arrows \ce{H+}\tx{(aq)} + \ce{HCO2-}\tx{(aq)}$, the acid dissociation constant at $298.15\K$ has the value $K\subs{a}=1.77\timesten{-4}$.
(a) Use Eq. 12.9.7 to find the degree of dissociation and the hydrogen ion molality in a 0.01000 molal formic acid solution. You can safely set $\G\subs{r}$ and $\g\subs{\(m$,HA}\) equal to $1$, and use the Debye–Hückel limiting law (Eq. 10.4.8) to calculate $\g_{\pm}$. You can do this calculation by iteration: Start with an initial estimate of the ionic strength (in this case 0), calculate $\g_{\pm}$ and $\alpha$, and repeat these steps until the value of $\alpha$ no longer changes.
(b) Estimate the degree of dissociation of formic acid in a solution that is 0.01000 molal in both formic acid and sodium nitrate, again using the Debye–Hückel limiting law for $\g_{\pm}$. Compare with the value in part (a).
12.18
Use the following experimental information to evaluate the standard molar enthalpy of formation and the standard molar entropy of the aqueous chloride ion at $298.15\K$, based on the conventions $\Delsub{f}H\st(\tx{H\(^+$, aq})=0\) and $S\m\st(\tx{H\(^+$, aq})=0\) (Secs. 11.3.2 and 11.8.4). (Your calculated values will be close to, but not exactly the same as, those listed in Appendix H, which are based on the same data combined with data of other workers.)
• 12.19
The solubility of crystalline AgCl in ultrapure water has been determined from the electrical conductivity of the saturated solution (J. A. Gledhill and G. McP. Malan, Trans. Faraday Soc., 48, 258–262, 1952). The average of five measurements at $298.15\K$ is $s\B=1.337\timesten{-5}\units{mol dm\(^{-3}$}\). The density of water at this temperature is $\rho\A^*=0.9970\units{kg dm\(^{-3}$}\).
(a) From these data and the Debye–Hückel limiting law, calculate the solubility product $K\subs{s}$ of AgCl at $298.15\K$.
12.20
The following reaction was carried out in an adiabatic solution calorimeter by Wagman and Kilday (J. Res. Natl. Bur. Stand. (U.S.), 77A, 569–579, 1973): $\tx{AgNO$_3$(s)} + \tx{KCl(aq, } m\B=0.101\units{mol kg$^{-1}$}) \arrow \tx{AgCl(s)} + \tx{KNO$_3$(aq)}$ The reaction can be assumed to go to completion, and the amount of KCl was in slight excess, so the amount of AgCl formed was equal to the initial amount of AgNO$_3$. After correction for the enthalpies of diluting the solutes in the initial and final solutions to infinite dilution, the standard molar reaction enthalpy at $298.15\K$ was found to be $\Delsub{r}H\st=-43.042\units{kJ mol\(^{-1}$}\). The same workers used solution calorimetry to obtain the molar enthalpy of solution at infinite dilution of crystalline AgNO$_3$ at $298.15\K$: $\Delsub{sol,B}H^{\infty}=22.727\units{kJ mol\(^{-1}$}\).
(a) Show that the difference of these two values is the standard molar reaction enthalpy for the precipitation reaction $\ce{Ag+}\tx{(aq)} + \ce{Cl-}\tx{(aq)} \arrow \ce{AgCl}\tx{(s)}$ and evaluate this quantity.
(b) Evaluate the standard molar enthalpy of formation of aqueous Ag$^+$ ion at $298.15\K$, using the results of part (a) and the values $\Delsub{f}H\st(\tx{Cl\(^-$, aq})=-167.08\units{kJ mol$^{-1}$}\) and $\Delsub{f}H\st(\tx{AgCl, s})=-127.01\units{kJ mol\(^{-1}$}\) from Appendix H. (These values come from calculations similar to those in Probs. 12.18 and 14.4.) The calculated value will be close to, but not exactly the same as, the value listed in Appendix H, which is based on the same data combined with data of other workers. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/12%3A_Equilibrium_Conditions_in_Multicomponent_Systems/12.11%3A_Chapter_12_Problems.txt |
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We encountered the Gibbs phase rule and phase diagrams in Chapter 8 in connection with single-substance systems. The present chapter derives the full version of the Gibbs phase rule for multicomponent systems. It then discusses phase diagrams for some representative types of multicomponent systems, and shows how they are related to the phase rule and to equilibrium concepts developed in Chapters 11 and 12.
13: The Phase Rule and Phase Diagrams
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$\newcommand{\cond}[1]{\$-2.5pt]{}\tag*{#1}}$ $\newcommand{\nextcond}[1]{\[-5pt]{}\tag*{#1}}$ $\newcommand{\R}{8.3145\units{J\,K\per\,mol\per}} % gas constant value$ $\newcommand{\Rsix}{8.31447\units{J\,K\per\,mol\per}} % gas constant value - 6 sig figs$ $\newcommand{\jn}{\hspace3pt\lower.3ex{\Rule{.6pt}{2ex}{0ex}}\hspace3pt}$ $\newcommand{\ljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}} \hspace3pt}$ $\newcommand{\lljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace1.4pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace3pt}$ In Sec. 8.1.7, the Gibbs phase rule for a pure substance was written $F = 3 - P$. We now consider a system of more than one substance and more than one phase in an equilibrium state. The phase rule assumes the system is at thermal and mechanical equilibrium. We shall assume furthermore that in addition to the temperature and pressure, the only other state functions needed to describe the state are the amounts of the species in each phase; this means for instance that surface effects are ignored. The derivations to follow will show that the phase rule may be written either in the form $F = 2 + C - P \tag{13.1.1}$ or $F = 2 + s - r - P \tag{13.1.2}$ where the symbols have the following meanings: $F=$ the number of degrees of freedom (or variance) $\hspace2ex =$ the maximum number of intensive variables that can be varied independently while the system remains in an equilibrium state; $C=$ the number of components $\hspace2ex =$ the minimum number of substances (or fixed-composition mixtures of substances) $\hspace 3ex$ that could be used to prepare each phase individually; $P=$ the number of different phases; $s=$ the number of different species; $r=$ the number of independent relations among intensive variables of individual phases other than relations $\hspace 3ex$ needed for thermal, mechanical, and transfer equilibrium. If we subdivide a phase, that does not change the number of phases $P$. That is, we treat noncontiguous regions of the system that have identical intensive properties as parts of the same phase. 13.1.1 Degrees of freedom Consider a system in an equilibrium state. In this state, the system has one or more phases; each phase contains one or more species; and intensive properties such as $T$, $p$, and the mole fraction of a species in a phase have definite values. Starting with the system in this state, we can make changes that place the system in a new equilibrium state having the same kinds of phases and the same species, but different values of some of the intensive properties. The number of different independent intensive variables that we may change in this way is the number of degrees of freedom or variance, $F$, of the system. Clearly, the system remains in equilibrium if we change the amount of a phase without changing its temperature, pressure, or composition. This, however, is the change of an extensive variable and is not counted as a degree of freedom. The phase rule, in the form to be derived, applies to a system that continues to have complete thermal, mechanical, and transfer equilibrium as intensive variables change. This means different phases are not separated by adiabatic or rigid partitions, or by semipermeable or impermeable membranes. Furthermore, every conceivable reaction among the species is either at reaction equilibrium or else is frozen at a fixed advancement during the time period we observe the system. The number of degrees of freedom is the maximum number of intensive properties of the equilibrium system we may independently vary, or fix at arbitrary values, without causing a change in the number and kinds of phases and species. We cannot, of course, change one of these properties to just any value whatever. We are able to vary the value only within a certain finite (sometimes quite narrow) range before a phase disappears or a new one appears. The number of degrees of freedom is also the number of independent intensive variables needed to specify the equilibrium state in all necessary completeness, aside from the amount of each phase. In other words, when we specify values of $F$ different independent intensive variables, then the values of all other intensive variables of the equilibrium state have definite values determined by the physical nature of the system. Just as for a one-component system, we can use the terms bivariant, univariant, and invariant depending on the value of $F$ (Sec. 8.1.7). 13.1.2 Species approach to the phase rule This section derives an expression for the number of degrees of freedom, $F$, based on species. Section 13.1.3 derives an expression based on components. Both approaches yield equivalent versions of the phase rule. Recall that a species is an entity, uncharged or charged, distinguished from other species by its chemical formula (Sec. 9.1.1). Thus, CO$_2$ and CO$_3$$^{2-}$ are different species, but CO$_2$(aq) and CO$_2$(g) is the same species in different phases. Consider an equilibrium system of $P$ phases, each of which contains the same set of species. Let the number of different species be $s$. If we could make changes while the system remains in thermal and mechanical equilibrium, but not necessarily in transfer equilibrium, we could independently vary the temperature and pressure of the system as a whole and the amount of each species in each phase; there would then be $2 + Ps$ independent variables. The equilibrium system is, however, in transfer equilibrium, which requires each species to have the same chemical potential in each phase: $\mu_i\bph = \mu_i\aph$, $\mu_i\gph = \mu_i\aph$, and so on. There are $P - 1$ independent relations like this for each species, and a total of $s(P - 1)$ independent relations for all species. Each such independent relation introduces a constraint and reduces the number of independent variables by one. Accordingly, taking transfer equilibrium into account, the number of independent variables is $2 + Ps - s(P - 1) = 2 + s$. We obtain the same result if a species present in one phase is totally excluded from another. For example, solvent molecules of a solution are not found in a pure perfectly-ordered crystal of the solute, undissociated molecules of a volatile strong acid such as HCl can exist in a gas phase but not in aqueous solution, and ions of an electrolyte solute are usually not found in a gas phase. For each such species absent from a phase, there is one fewer amount variable and also one fewer relation for transfer equilibrium; on balance, the number of independent variables is still $2 + s$. Next, we consider the possibility that further independent relations exist among intensive variables in addition to the relations needed for thermal, mechanical, and transfer equilibrium. (Relations such as $\sum_i p_i=p$ for a gas phase or $\sum_i x_i=1$ for a phase in general have already been accounted for in the derivation by the specification of $p$ and the amount of each species.) If there are $r$ of these additional relations, the total number of independent variables is reduced to $2 + s - r$. These relations may come from 1. In the case of a reaction equilibrium, the relation is $\Delsub{r}G=\sum_i\!\nu_i \mu_i = 0$, or the equivalent relation $K=\prod_i(a_i)^{\nu_i}$ for the thermodynamic equilibrium constant. Thus, $r$ is the sum of the number of independent reaction equilibria, the number of phases containing ions, and the number of independent initial conditions. Several examples will be given in Sec. 13.1.4. There is an infinite variety of possible choices of the independent variables (both extensive and intensive) for the equilibrium system, but the total number of independent variables is fixed at $2+s-r$. Keeping intensive properties fixed, we can always vary how much of each phase is present (e.g., its volume, mass, or amount) without destroying the equilibrium. Thus, at least $P$ of the independent variables, one for each phase, must be extensive. It follows that the maximum number of independent intensive variables is the difference $(2 + s - r) - P$. Since the maximum number of independent intensive variables is the number of degrees of freedom, our expression for $F$ based on species is $F = 2 + s - r - P \tag{13.1.3}$ 13.1.3 Components approach to the phase rule The derivation of the phase rule in this section uses the concept of components. The number of components, $C$, is the minimum number of substances or mixtures of fixed composition from which we could in principle prepare each individual phase of an equilibrium state of the system, using methods that may be hypothetical. These methods include the addition or removal of one or more of the substances or fixed-composition mixtures, and the conversion of some of the substances into others by means of a reaction that is at equilibrium in the actual system. It is not always easy to decide on the number of components of an equilibrium system. The number of components may be less than the number of substances present, on account of the existence of reaction equilibria that produce some substances from others. When we use a reaction to prepare a phase, nothing must remain unused. For instance, consider a system consisting of solid phases of CaCO$_3$ and CaO and a gas phase of CO$_2$. Assume the reaction CaCO$_3$(s) $\ra$ CaO(s) + CO$_2$(g) is at equilibrium. We could prepare the CaCO$_3$ phase from CaO and CO$_2$ by the reverse of this reaction, but we can only prepare the CaO and CO$_2$ phases from the individual substances. We could not use CaCO$_3$ to prepare either the CaO phase or the CO$_2$ phase, because CO$_2$ or CaO would be left over. Thus this system has three substances but only two components, namely CaO and CO$_2$. In deriving the phase rule by the components approach, it is convenient to consider only intensive variables. Suppose we have a system of $P$ phases in which each substance present is a component (i.e., there are no reactions) and each of the $C$ components is present in each phase. If we make changes to the system while it remains in thermal and mechanical equilibrium, but not necessarily in transfer equilibrium, we can independently vary the temperature and pressure of the whole system, and for each phase we can independently vary the mole fraction of all but one of the substances (the value of the omitted mole fraction comes from the relation $\sum_i x_i = 1$). This is a total of $2 + P(C - 1)$ independent intensive variables. When there also exist transfer and reaction equilibria, not all of these variables are independent. Each substance in the system is either a component, or else can be formed from components by a reaction that is in reaction equilibrium in the system. Transfer equilibria establish $P - 1$ independent relations for each component ($\mu_i\bph = \mu_i\aph$, $\mu_i\gph = \mu_i\aph$, etc.) and a total of $C(P - 1)$ relations for all components. Since these are relations among chemical potentials, which are intensive properties, each relation reduces the number of independent intensive variables by one. The resulting number of independent intensive variables is $F = [2 + P(C - 1)] - C(P - 1) = 2 + C - P \tag{13.1.4}$ If the equilibrium system lacks a particular component in one phase, there is one fewer mole fraction variable and one fewer relation for transfer equilibrium. These changes cancel in the calculation of $F$, which is still equal to $2 + C - P$. If a phase contains a substance that is formed from components by a reaction, there is an additional mole fraction variable and also the additional relation $\sum_i\!\nu_i \mu_i = 0$ for the reaction; again the changes cancel. We may need to remove a component from a phase to achieve the final composition. Note that it is not necessary to consider additional relations for electroneutrality or initial conditions; they are implicit in the definitions of the components. For instance, since each component is a substance of zero electric charge, the electrical neutrality of the phase is assured. We conclude that, regardless of the kind of system, the expression for $F$ based on components is given by $F=2+C-P$. By comparing this expression and $F=2+s-r-P$, we see that the number of components is related to the number of species by $C = s - r \tag{13.1.5}$ 13.1.4 Examples The five examples below illustrate various aspects of using the phase rule. Example 1: liquid water For a single phase of pure water, $P$ equals $1$. If we treat the water as the single species H$_2$O, $s$ is 1 and $r$ is 0. The phase rule then predicts two degrees of freedom: $\begin{split} F & = 2 + s - r - P\cr & = 2 + 1 - 0 - 1 = 2 \end{split} \tag{13.1.6}$ Since $F$ is the number of intensive variables that can be varied independently, we could for instance vary $T$ and $p$ independently, or $T$ and $\rho$, or any other pair of independent intensive variables. Next let us take into account the proton transfer equilibrium \[ \ce{2H2O}\tx{(l)} \arrows \ce{H3O+}\tx{(aq)} + \ce{OH-}\tx{(aq)}$ and consider the system to contain the three species H$_2$O, H$_3$O$^+$, and OH$^-$. Then for the species approach to the phase rule, we have $s = 3$. We can write two independent relations:
1. Thus, we have two relations involving intensive variables only. Now $s$ is 3, $r$ is 2, $P$ is 1, and the number of degrees of freedom is given by $F = 2 + s-r-P = 2 \tag{13.1.7}$ which is the same value of $F$ as before.
If we consider water to contain additional cation species (e.g., $\ce{H5O2+}$), each such species would add $1$ to $s$ and $1$ to $r$, but $F$ would remain equal to 2. Thus, no matter how complicated are the equilibria that actually exist in liquid water, the number of degrees of freedom remains $2$.
Example 2: carbon, oxygen, and carbon oxides
Consider a system containing solid carbon (graphite) and a gaseous mixture of O$_2$, CO, and CO$_2$. There are four species and two phases. If reaction equilibrium is absent, as might be the case at low temperature in the absence of a catalyst, we have $r = 0$ and $C = s - r = 4$. The four components are the four substances. The phase rule tells us the system has four degrees of freedom. We could, for instance, arbitrarily vary $T$, $p$, $y\subs{O\(_2$}\), and $y\subs{CO}$.
Now suppose we raise the temperature or introduce an appropriate catalyst to allow the following reaction equilibria to exist:
1. These equilibria introduce two new independent relations among chemical potentials and among activities. We could also consider the equilibrium $\ce{2CO}\tx{(g)} + \ce{O2}\tx{(g)} \arrows \ce{2CO2}\tx{(g)}$, but it does not contribute an additional independent relation because it depends on the other two equilibria: the reaction equation is obtained by subtracting the reaction equation for equilibrium 1 from twice the reaction equation for equilibrium 2. By the species approach, we have $s = 4$, $r = 2$, and $P=2$; the number of degrees of freedom from these values is $F = 2 + s - r - P = 2 \tag{13.1.8}$
If we wish to calculate $F$ by the components approach, we must decide on the minimum number of substances we could use to prepare each phase separately. (This does not refer to how we actually prepare the two-phase system, but to a hypothetical preparation of each phase with any of the compositions that can actually exist in the equilibrium system.) Assume equilibria 1 and 2 are present. We prepare the solid phase with carbon, and we can prepare any possible equilibrium composition of the gas phase from carbon and O$_2$ by using the reactions of both equilibria. Thus, there are two components (C and O$_2$) giving the same result of two degrees of freedom.
1. Now to introduce an additional complexity: Suppose we prepare the system by placing a certain amount of O$_2$ and twice this amount of carbon in an evacuated container, and wait for the reactions to come to equilibrium. This method of preparation imposes an initial condition on the system, and we must decide whether the number of degrees of freedom is affected. Equating the total amount of carbon atoms to the total amount of oxygen atoms in the equilibrated system gives the relation $n\subs{C}+n\subs{CO}+n\subs{CO$_2$} = 2n\subs{O$_2$} + n\subs{CO} + 2n\subs{CO$_2$} \qquad \tx{or} \qquad n\subs{C} = 2n\subs{O$_2$} + n\subs{CO$_2$} \tag{13.1.9}$ Either equation is a relation among extensive variables of the two phases. From them, we are unable to obtain any relation among intensive variables of the phases. Therefore, this particular initial condition does not change the value of $r$, and $F$ remains equal to 2.
Example 3: a solid salt and saturated aqueous solution
Applying the components approach to this system is straightforward. The solid phase is prepared from PbCl$_2$ and the aqueous phase could be prepared by dissolving solid PbCl$_2$ in H$_2$O. Thus, there are two components and two phases: $F = 2+C-P=2 \tag{13.1.10}$
For the species approach, we note that there are four species (PbCl$_2$, Pb$^{2+}$, Cl$^-$, and H$_2$O) and two independent relations among intensive variables:
1. We have $s=4$, $r=2$, and $P=2$, giving the same result as the components approach: $F = 2 + s-r-P = 2 \tag{13.1.11}$
Example 4: liquid water and water-saturated air
If there is no special relation among the total amounts of N$_2$ and O$_2$, there are three components and the phase rule gives $F = 2 + C - P = 3 \tag{13.1.12}$ Since there are three degrees of freedom, we could, for instance, specify arbitrary values of $T$, $p$, and $y\subs{N\(_2$}\) (arbitrary, that is, within the limits that would allow the two phases to coexist); then the values of other intensive variables such as the mole fractions $y\subs{H\(_2$O}\) and $x\subs{N\(_2$}\) would have definite values.
Now suppose we impose an initial condition by preparing the system with water and dry air of a fixed composition. The mole ratio of N$_2$ and O$_2$ in the aqueous solution is not necessarily the same as in the equilibrated gas phase; consequently, the air does not behave like a single substance. The number of components is still three: H$_2$O, N$_2$, and O$_2$ are all required to prepare each phase individually, just as when there was no initial condition, giving $F = 3$ as before.
The fact that the compositions of both phases depend on the relative amounts of the phases is illustrated in Prob. 9.5.
We can reach the same conclusion with the species approach. The initial condition can be expressed by an equation such as $\frac{(n\subs{N$_2$}\sups{l} + n\subs{N$_2$}\sups{g})} {(n\subs{O$_2$}\sups{l} + n\subs{O$_2$}\sups{g})} = a \tag{13.1.13}$ where $a$ is a constant equal to the mole ratio of N$_2$ and O$_2$ in the dry air. This equation cannot be changed to a relation between intensive variables such as $x\subs{N\(_2$}\) and $x\subs{O\(_2$}\), so that $r$ is zero and there are still three degrees of freedom.
Finally, let us assume that we prepare the system with dry air of fixed composition, as before, but consider the solubilities of N$_2$ and O$_2$ in water to be negligible. Then $n\subs{N\(_2$}\sups{l} \) and $n\subs{O\(_2$}\sups{l} \) are zero and Eq. 13.1.13 becomes $n\subs{N\(_2$}\sups{g} / n\subs{O$_2$}\sups{g} = a\), or $y\subs{N\(_2$} = ay\subs{O$_2$}\), which is a relation between intensive variables. In this case, $r$ is 1 and the phase rule becomes $F = 2 + s - r - P = 2 \tag{13.1.14}$ The reduction in the value of $F$ from 3 to 2 is a consequence of our inability to detect any dissolved N$_2$ or O$_2$. According to the components approach, we may prepare the liquid phase with H$_2$O and the gas phase with H$_2$O and air of fixed composition that behaves as a single substance; thus, there are only two components.
Example 5: equilibrium between two solid phases and a gas phase
Consider the following reaction equilibrium: $\ce{3CuO}\tx{(s)} + \ce{2NH3}\tx{(g)} \arrows \ce{3Cu}\tx{(s)} + \ce{3H2O}\tx{(g)} + \ce{N2}\tx{(g)}$ According to the species approach, there are five species, one relation (for reaction equilibrium), and three phases. The phase rule gives $F = 2 + s - r - P = 3 \tag{13.1.15}$
It is more difficult to apply the components approach to this example. As components, we might choose CuO and Cu (from which we could prepare the solid phases) and also NH$_3$ and H$_2$O. Then to obtain the N$_2$ needed to prepare the gas phase, we could use CuO and NH$_3$ as reactants in the reaction $\ce{3CuO} + \ce{2NH3} \arrow \ce{3Cu} + \ce{3H2O} + \ce{N2}$ and remove the products Cu and H$_2$O. In the components approach, we are allowed to remove substances from the system provided they are counted as components. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/13%3A_The_Phase_Rule_and_Phase_Diagrams/13.01%3A_The_Gibbs_Phase_Rule_for_Multicomponent_Systems.txt |
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As explained in Sec. 8.2, a phase diagram is a kind of two-dimensional map that shows which phase or phases are stable under a given set of conditions. This section discusses some common kinds of binary systems, and Sec. 13.3 will describe some interesting ternary systems.
13.2.1 Generalities
A binary system has two components; $C$ equals $2$, and the number of degrees of freedom is $F=4-P$. There must be at least one phase, so the maximum possible value of $F$ is 3. Since $F$ cannot be negative, the equilibrium system can have no more than four phases.
We can independently vary the temperature, pressure, and composition of the system as a whole. Instead of using these variables as the coordinates of a three-dimensional phase diagram, we usually draw a two-dimensional phase diagram that is either a temperature–composition diagram at a fixed pressure or a pressure–composition diagram at a fixed temperature. The position of the system point on one of these diagrams then corresponds to a definite temperature, pressure, and overall composition. The composition variable usually varies along the horizontal axis and can be the mole fraction, mass fraction, or mass percent of one of the components, as will presently be illustrated by various examples.
The way in which we interpret a two-dimensional phase diagram to obtain the compositions of individual phases depends on the number of phases present in the system.
• If the system point falls within a one-phase area of the phase diagram, the composition variable is the composition of that single phase. There are three degrees of freedom. On the phase diagram, the value of either $T$ or $p$ has been fixed, so there are two other independent intensive variables. For example, on a temperature–composition phase diagram, the pressure is fixed and the temperature and composition can be changed independently within the boundaries of the one-phase area of the diagram.
• If the system point is in a two-phase area of the phase diagram, we draw a horizontal tie line of constant temperature (on a temperature–composition phase diagram) or constant pressure (on a pressure–composition phase diagram). The lever rule applies. The position of the point at each end of the tie line, at the boundary of the two-phase area, gives the value of the composition variable of one of the phases and also the physical state of this phase: either the state of an adjacent one-phase area, or the state of a phase of fixed composition when the boundary is a vertical line. Thus, a boundary that separates a two-phase area for phases $\pha$ and $\phb$ from a one-phase area for phase $\pha$ is a curve that describes the composition of phase $\pha$ as a function of $T$ or $p$ when it is in equilibrium with phase $\phb$. The curve is called a solidus, liquidus, or vaporus depending on whether phase $\pha$ is a solid, liquid, or gas.
• A binary system with three phases has only one degree of freedom and cannot be represented by an area on a two-dimensional phase diagram. Instead, there is a horizontal boundary line between areas, with a special point along the line at the junction of several areas. The compositions of the three phases are given by the positions of this point and the points at the two ends of the line. The position of the system point on this line does not uniquely specify the relative amounts in the three phases.
The examples that follow show some of the simpler kinds of phase diagrams known for binary systems.
13.2.2 Solid–liquid systems
Figure 13.1 Temperature–composition phase diagram for a binary system exhibiting a eutectic point.
Figure 13.1 is a temperature–composition phase diagram at a fixed pressure. The composition variable $z\B$ is the mole fraction of component B in the system as a whole. The phases shown are a binary liquid mixture of A and B, pure solid A, and pure solid B.
The one-phase liquid area is bounded by two curves, which we can think of either as freezing-point curves for the liquid or as solubility curves for the solids. These curves comprise the liquidus. As the mole fraction of either component in the liquid phase decreases from unity, the freezing point decreases. The curves meet at point a, which is a eutectic point. At this point, both solid A and solid B can coexist in equilibrium with a binary liquid mixture. The composition at this point is the eutectic composition, and the temperature here (denoted $T\subs{e}$) is the eutectic temperature. (“Eutectic” comes from the Greek for easy melting.) $T\subs{e}$ is the lowest temperature for the given pressure at which the liquid phase is stable.
Suppose we combine $0.60\mol$ A and $0.40\mol$ B ($z\B=0.40$) and adjust the temperature so as to put the system point at b. This point is in the one-phase liquid area, so the equilibrium system at this temperature has a single liquid phase. If we now place the system in thermal contact with a cold reservoir, heat is transferred out of the system and the system point moves down along the isopleth (path of constant overall composition) b–h. The cooling rate depends on the temperature gradient at the system boundary and the system’s heat capacity.
At point c on the isopleth, the system point reaches the boundary of the one-phase area and is about to enter the two-phase area labeled A(s) + liquid. At this point in the cooling process, the liquid is saturated with respect to solid A, and solid A is about to freeze out from the liquid. There is an abrupt decrease (break) in the cooling rate at this point, because the freezing process involves an extra enthalpy decrease.
At the still lower temperature at point d, the system point is within the two-phase solid–liquid area. The tie line through this point is line e–f. The compositions of the two phases are given by the values of $z\B$ at the ends of the tie line: $x\B\sups{s}=0$ for the solid and $x\B\sups{l} =0.50$ for the liquid. From the general lever rule (Eq. 8.2.8), the ratio of the amounts in these phases is $\frac{n\sups{l} }{n\sups{s}} = \frac{z\B-x\B\sups{s}}{x\B\sups{l} -z\B} = \frac{0.40-0}{0.50-0.40} = 4.0 \tag{13.2.1}$ Since the total amount is $n\sups{s}+n\sups{l} =1.00\mol$, the amounts of the two phases must be $n\sups{s}=0.20\mol$ and $n\sups{l} =0.80\mol$.
When the system point reaches the eutectic temperature at point g, cooling halts until all of the liquid freezes. Solid B freezes out as well as solid A. During this eutectic halt, there are at first three phases: liquid with the eutectic composition, solid A, and solid B. As heat continues to be withdrawn from the system, the amount of liquid decreases and the amounts of the solids increase until finally only $0.60\mol$ of solid A and $0.40\mol$ of solid B are present. The temperature then begins to decrease again and the system point enters the two-phase area for solid A and solid B; tie lines in this area extend from $z\B{=}0$ to $z\B{=}1$.
Temperature–composition phase diagrams such as this are often mapped out experimentally by observing the cooling curve (temperature as a function of time) along isopleths of various compositions. This procedure is thermal analysis. A break in the slope of a cooling curve at a particular temperature indicates the system point has moved from a one-phase liquid area to a two-phase area of liquid and solid. A temperature halt indicates the temperature is either the freezing point of the liquid to form a solid of the same composition, or else a eutectic temperature.
Figure 13.2 Temperature–composition phase diagrams with single eutectics.
(a) Two pure solids and a liquid mixture (E. W. Washburn, International Critical Tables of Numerical Data, Physics, Chemistry and Technology, Vol. IV, McGraw-Hill, New York, 1928, p. 98).
(b) Two solid solutions and a liquid mixture.
Figure 13.2 shows two temperature–composition phase diagrams with single eutectic points. The left-hand diagram is for the binary system of chloroform and carbon tetrachloride, two liquids that form nearly ideal mixtures. The solid phases are pure crystals, as in Fig. 13.1. The right-hand diagram is for the silver–copper system and involves solid phases that are solid solutions (substitutional alloys of variable composition). The area labeled s$\aph$ is a solid solution that is mostly silver, and s$\bph$ is a solid solution that is mostly copper. Tie lines in the two-phase areas do not end at a vertical line for a pure solid component as they do in the system shown in the left-hand diagram. The three phases that can coexist at the eutectic temperature of $\tx{1,052}\K$ are the melt of the eutectic composition and the two solid solutions.
Figure 13.3 Temperature–composition phase diagram for the binary system of $\alpha$-naphthylamine (A) and phenol (B) at $1\br$ (J. C. Philip, J. Chem. Soc., 83, 814–834, 1903).
Section 12.5.4 discussed the possibility of the appearance of a solid compound when a binary liquid mixture is cooled. An example of this behavior is shown in Fig. 13.3, in which the solid compound contains equal amounts of the two components $\alpha$-naphthylamine and phenol. The possible solid phases are pure A, pure B, and the solid compound AB. Only one or two of these solids can be present simultaneously in an equilibrium state. The vertical line in the figure at $z\B=0.5$ represents the solid compound. The temperature at the upper end of this line is the melting point of the solid compound, $29\units{\(\degC$}\). The solid melts congruently to give a liquid of the same composition. A melting process with this behavior is called a dystectic reaction. The cooling curve for liquid of this composition would display a halt at the melting point.
The phase diagram in Fig. 13.3 has two eutectic points. It resembles two simple phase diagrams like Fig. 13.1 placed side by side. There is one important difference: the slope of the freezing-point curve (liquidus curve) is nonzero at the composition of a pure component, but is zero at the composition of a solid compound that is completely dissociated in the liquid (as derived theoretically in Sec. 12.5.4). Thus, the curve in Fig. 13.3 has a relative maximum at the composition of the solid compound ($z\B=0.5$) and is rounded there, instead of having a cusp—like a Romanesque arch rather than a Gothic arch.
Figure 13.4 Temperature–composition phase diagram for the binary system of H$_2$O and NaCl at $1\br$. (Data from Roger Cohen-Adad and John W. Lorimer, Alkali Metal and Ammonium Chlorides in Water and Heavy Water (Binary Systems), Solubility Data Series, Vol. 47, Pergamon Press, Oxford, 1991; and E. W. Washburn, International Critical Tables of Numerical Data, Physics, Chemistry and Technology, Vol. III, McGraw-Hill, New York, 1928.)
An example of a solid compound that does not melt congruently is shown in Fig. 13.4. The solid hydrate $\ce{NaCl*2H2O}$ is $61.9\%$ NaCl by mass. It decomposes at $0\units{\(\degC$}\) to form an aqueous solution of composition $26.3\%$ NaCl by mass and a solid phase of anhydrous NaCl. These three phases can coexist at equilibrium at $0\units{\(\degC$}\). A phase transition like this, in which a solid compound changes into a liquid and a different solid, is called incongruent or peritectic melting, and the point on the phase diagram at this temperature at the composition of the liquid is a peritectic point.
Figure 13.4 shows there are two other temperatures at which three phases can be present simultaneously: $-21\units{\(\degC$}\), where the phases are ice, the solution at its eutectic point, and the solid hydrate; and $109\units{\(\degC$}\), where the phases are gaseous H$_2$O, a solution of composition $28.3\%$ NaCl by mass, and solid NaCl. Note that both segments of the right-hand boundary of the one-phase solution area have positive slopes, meaning that the solubilities of the solid hydrate and the anhydrous salt both increase with increasing temperature.
13.2.3 Partially-miscible liquids
When two liquids that are partially miscible are combined in certain proportions, phase separation occurs (Sec. 11.1.6). Two liquid phases in equilibrium with one another are called conjugate phases. Obviously the two phases must have different compositions or they would be identical; the difference is called a miscibility gap. A binary system with two phases has two degrees of freedom, so that at a given temperature and pressure each conjugate phase has a fixed composition.
Figure 13.5 Temperature–composition phase diagram for the binary system of methyl acetate (A) and carbon disulfide (B) at $1\br$ (data from P. Ferloni and G. Spinolo, Int. DATA Ser., Sel. Data Mixtures, Ser. A, 70, 1974). All phases are liquids. The open circle indicates the critical point.
The typical dependence of a miscibility gap on temperature is shown in Fig. 13.5. The miscibility gap (the difference in compositions at the left and right boundaries of the two-phase area) decreases as the temperature increases until at the upper consolute temperature, also called the upper critical solution temperature, the gap vanishes. The point at the maximum of the boundary curve of the two-phase area, where the temperature is the upper consolute temperature, is the consolute point or critical point. At this point, the two liquid phases become identical, just as the liquid and gas phases become identical at the critical point of a pure substance. Critical opalescence (Sec. 8.2.3) is observed in the vicinity of this point, caused by large local composition fluctuations. At temperatures at and above the critical point, the system is a single binary liquid mixture.
Suppose we combine $6.0\mol$ of component A (methyl acetate) and $4.0\mol$ of component B (carbon disulfide) in a cylindrical vessel and adjust the temperature to $200\K$. The overall mole fraction of B is $z\B=0.40$. The system point is at point a in the two-phase region. From the positions of points b and c at the ends of the tie line through point a, we find the two liquid layers have compositions $x\B\aph=0.20$ and $x\B\bph=0.92$. Since carbon disulfide is the more dense of the two pure liquids, the bottom layer is phase $\phb$, the layer that is richer in carbon disulfide. According to the lever rule, the ratio of the amounts in the two phases is given by $\frac{n\bph}{n\aph} = \frac{z\B-x\B\aph}{x\B\bph-z\B} = \frac{0.40-0.20}{0.92-0.40} = 0.38 \tag{13.2.2}$ Combining this value with $n\aph+n\bph=10.0\mol$ gives us $n\aph=7.2\mol$ and $n\bph=2.8\mol$.
If we gradually add more carbon disulfide to the vessel while gently stirring and keeping the temperature constant, the system point moves to the right along the tie line. Since the ends of this tie line have fixed positions, neither phase changes its composition, but the amount of phase $\phb$ increases at the expense of phase $\pha$. The liquid–liquid interface moves up in the vessel toward the top of the liquid column until, at overall composition $z\B=0.92$ (point c), there is only one liquid phase.
Now suppose the system point is back at point a and we raise the temperature while keeping the overall composition constant at $z\B=0.40$. The system point moves up the isopleth a–d. The phase diagram shows that the ratio $(z\B-x\B\aph)/(x\B\bph-z\B)$ decreases during this change. As a result, the amount of phase $\pha$ increases, the amount of phase $\phb$ decreases, and the liquid–liquid interface moves down toward the bottom of the vessel until at $217\K$ (point d) there again is only one liquid phase.
13.2.4 Liquid–gas systems with ideal liquid mixtures
Figure 13.6 Phase diagrams for the binary system of toluene (A) and benzene (B). The curves are calculated from Eqs. 13.2.6 and 13.2.7 and the saturation vapor pressures of the pure liquids.
(a) Pressure–composition diagram at $T=340\K$.
(b) Temperature–composition diagram at $p=1\br$.
Toluene and benzene form liquid mixtures that are practically ideal and closely obey Raoult’s law for partial pressure. For the binary system of these components, we can use the vapor pressures of the pure liquids to generate the liquidus and vaporus curves of the pressure–composition and temperature–composition phase diagram. The results are shown in Fig. 13.6. The composition variable $z\A$ is the overall mole fraction of component A (toluene).
The equations needed to generate the curves can be derived as follows. Consider a binary liquid mixture of components A and B and mole fraction composition $x\A$ that obeys Raoult’s law for partial pressure (Eq. 9.4.2): $p\A = x\A p\A^* \qquad p\B = (1-x\A)p\B^* \tag{13.2.3}$ Strictly speaking, Raoult’s law applies to a liquid–gas system maintained at a constant pressure by means of a third gaseous component, and $p\A^*$ and $p\B^*$ are the vapor pressures of the pure liquid components at this pressure and the temperature of the system. However, when a liquid phase is equilibrated with a gas phase, the partial pressure of a constituent of the liquid is practically independent of the total pressure (Sec. 12.8.1), so that it is a good approximation to apply the equations to a binary liquid–gas system and treat $p\A^*$ and $p\B^*$ as functions only of $T$.
When the binary system contains a liquid phase and a gas phase in equilibrium, the pressure is the sum of $p\A$ and $p\B$, which from Eq. 13.2.3 is given by \begin{gather} \s {\begin{split} p & = x\A p\A^* + (1-x\A)p\B^* \cr & = p\B^* + (p\A^*-p\B^*)x\A \end{split} } \tag{13.2.4} \cond{($C{=}2$, ideal liquid mixture)} \end{gather} where $x\A$ is the mole fraction of A in the liquid phase. Equation 13.2.4 shows that in the two-phase system, $p$ has a value between $p\A^*$ and $p\B^*$, and that if $T$ is constant, $p$ is a linear function of $x\A$. The mole fraction composition of the gas in the two-phase system is given by $y\A = \frac{p\A}{p} = \frac{x\A p\A^*}{p\B^* + (p\A^*-p\B^*)x\A } \tag{13.2.5}$
A binary two-phase system has two degrees of freedom. At a given $T$ and $p$, each phase must have a fixed composition. We can calculate the liquid composition by rearranging Eq. 13.2.4: \begin{gather} \s {x\A = \frac{p-p\B^*}{p\A^*-p\B^*}} \tag{13.2.6} \cond{($C{=}2$, ideal liquid mixture)} \end{gather} The gas composition is then given by \begin{gather} \s {\begin{split} y\A & = \frac{p\A}{p} = \frac{x\A p\A^*}{p} \cr & = \left( \frac{p-p\B^*}{p\A^*-p\B^*}\right) \frac{p\A^*}{p} \end{split} } \tag{13.2.7} \cond{($C{=}2$, ideal liquid mixture)} \end{gather}
If we know $p\A^*$ and $p\B^*$ as functions of $T$, we can use Eqs. 13.2.6 and 13.2.7 to calculate the compositions for any combination of $T$ and $p$ at which the liquid and gas phases can coexist, and thus construct a pressure–composition or temperature–composition phase diagram.
In Fig. 13.6(a), the liquidus curve shows the relation between $p$ and $x\A$ for equilibrated liquid and gas phases at constant $T$, and the vaporus curve shows the relation between $p$ and $y\A$ under these conditions. We see that $p$ is a linear function of $x\A$ but not of $y\A$.
In a similar fashion, the liquidus curve in Fig. 13.6(b) shows the relation between $T$ and $x\A$, and the vaporus curve shows the relation between $T$ and $y\A$, for equilibrated liquid and gas phases at constant $p$. Neither curve is linear.
Figure 13.7 Liquidus and vaporus surfaces for the binary system of toluene (A) and benzene. Cross-sections through the two-phase region are drawn at constant temperatures of $340\K$ and $370\K$ and at constant pressures of $1\br$ and $2\br$. Two of the cross-sections intersect at a tie line at $T=370\K$ and $p=1\br$, and the other cross-sections are hatched in the direction of the tie lines.
A liquidus curve is also called a bubble-point curve or a boiling-point curve. Other names for a vaporus curve are dew-point curve and condensation curve. These curves are actually cross-sections of liquidus and vaporus surfaces in a three-dimensional $T$–$p$–$z\A$ phase diagram, as shown in Fig. 13.7. In this figure, the liquidus surface is in view at the front and the vaporus surface is hidden behind it.
13.2.5 Liquid–gas systems with nonideal liquid mixtures
Most binary liquid mixtures do not behave ideally. The most common situation is positive deviations from Raoult’s law. (In the molecular model of Sec. 11.1.5, positive deviations correspond to a less negative value of $k\subs{AB}$ than the average of $k\subs{AA}$ and $k\subs{BB}$.) Some mixtures, however, have specific A–B interactions, such as solvation or molecular association, that prevent random mixing of the molecules of A and B, and the result is then negative deviations from Raoult’s law. If the deviations from Raoult’s law, either positive or negative, are large enough, the constant-temperature liquidus curve exhibits a maximum or minimum and azeotropic behavior results.
Figure 13.8 Binary system of methanol (A) and benzene at $45\units{\(\degC$}\) (Hossein Toghiani, Rebecca K. Toghiani, and Dabir S. Viswanath, J. Chem. Eng. Data, 39, 63–67, 1994).
(a) Partial pressures and total pressure in the gas phase equilibrated with liquid mixtures. The dashed lines indicate Raoult’s law behavior.
(b) Pressure–composition phase diagram at $45\units{\(\degC$}\). Open circle: azeotropic point at $z\A=0.59$ and $p=60.5\units{kPa}$.
Figure 13.8 shows the azeotropic behavior of the binary methanol-benzene system at constant temperature. In Fig. 13.8(a), the experimental partial pressures in a gas phase equilibrated with the nonideal liquid mixture are plotted as a function of the liquid composition. The partial pressures of both components exhibit positive deviations from Raoult’s law, consistent with the statement in Sec. 12.8.2 that if one constituent of a binary liquid mixture exhibits positive deviations from Raoult’s law, with only one inflection point in the curve of fugacity versus mole fraction, the other constituent also has positive deviations from Raoult’s law. The total pressure (equal to the sum of the partial pressures) has a maximum value greater than the vapor pressure of either pure component. The curve of $p$ versus $x\A$ becomes the liquidus curve of the pressure–composition phase diagram shown in Fig. 13.8(b). Points on the vaporus curve are calculated from $p=p\A/y\A$.
In practice, the data needed to generate the liquidus and vaporus curves of a nonideal binary system are usually obtained by allowing liquid mixtures of various compositions to boil in an equilibrium still at a fixed temperature or pressure. When the liquid and gas phases have become equilibrated, samples of each are withdrawn for analysis. The partial pressures shown in Fig. 13.8(a) were calculated from the experimental gas-phase compositions with the relations $p\A=y\A p$ and $p\B=p-p\A$.
If the constant-temperature liquidus curve has a maximum pressure at a liquid composition not corresponding to one of the pure components, which is the case for the methanol–benzene system, then the liquid and gas phases are mixtures of identical compositions at this pressure. This behavior was deduced at the end of Sec. 12.8.3. On the pressure–composition phase diagram, the liquidus and vaporus curves both have maxima at this pressure, and the two curves coincide at an azeotropic point. A binary system with negative deviations from Raoult’s law can have an isothermal liquidus curve with a minimum pressure at a particular mixture composition, in which case the liquidus and vaporus curves coincide at an azeotropic point at this minimum. The general phenomenon in which equilibrated liquid and gas mixtures have identical compositions is called azeotropy, and the liquid with this composition is an azeotropic mixture or azeotrope (Greek: boils unchanged). An azeotropic mixture vaporizes as if it were a pure substance, undergoing an equilibrium phase transition to a gas of the same composition.
Figure 13.9 Liquidus and vaporus surfaces for the binary system of methanol (A) and benzene (Hossein Toghiani, Rebecca K. Toghiani, and Dabir S. Viswanath, J. Chem. Eng. Data, 39, 63–67, 1994). Cross-sections are hatched in the direction of the tie lines. The dashed curve is the azeotrope vapor-pressure curve.
If the liquidus and vaporus curves exhibit a maximum on a pressure–composition phase diagram, then they exhibit a minimum on a temperature–composition phase diagram. This relation is explained for the methanol–benzene system by the three-dimensional liquidus and vaporus surfaces drawn in Fig. 13.9. In this diagram, the vaporus surface is hidden behind the liquidus surface. The hatched cross-section at the front of the figure is the same as the pressure–composition diagram of Fig. 13.8(b), and the hatched cross-section at the top of the figure is a temperature–composition phase diagram in which the system exhibits a minimum-boiling azeotrope.
A binary system containing an azeotropic mixture in equilibrium with its vapor has two species, two phases, and one relation among intensive variables: $x\A =y\A$. The number of degrees of freedom is then $F = 2+s-r-P = 2+2-1-2 = 1$; the system is univariant. At a given temperature, the azeotrope can exist at only one pressure and have only one composition. As $T$ changes, so do $p$ and $z\A$ along an azeotrope vapor-pressure curve as illustrated by the dashed curve in Fig. 13.9.
Figure 13.10 Temperature–composition phase diagrams of binary systems exhibiting (a) no azeotropy, (b) a minimum-boiling azeotrope, and (c) a maximum-boiling azeotrope. Only the one-phase areas are labeled; two-phase areas are hatched in the direction of the tie lines.
Figure 13.10 summarizes the general appearance of some relatively simple temperature–composition phase diagrams of binary systems. If the system does not form an azeotrope (zeotropic behavior), the equilibrated gas phase is richer in one component than the liquid phase at all liquid compositions, and the liquid mixture can be separated into its two components by fractional distillation. The gas in equilibrium with an azeotropic mixture, however, is not enriched in either component. Fractional distillation of a system with an azeotrope leads to separation into one pure component and the azeotropic mixture.
Figure 13.11 Temperature–composition phase diagrams of binary systems with partially-miscible liquids exhibiting (a) the ability to be separated into pure components by fractional distillation, (b) a minimum-boiling azeotrope, and (c) boiling at a lower temperature than the boiling point of either pure component. Only the one-phase areas are labeled; two-phase areas are hatched in the direction of the tie lines.
More complicated behavior is shown in the phase diagrams of Fig. 13.11. These are binary systems with partially-miscible liquids in which the boiling point is reached before an upper consolute temperature can be observed.
13.2.6 Solid–gas systems
Figure 13.12 Pressure–composition phase diagram for the binary system of CuSO$_4$ (A) and H$_2$O (B) at $25\units{\(\degC$}\) (Thomas S. Logan, J. Chem. Educ., 35, 148–149, 1958; E. W. Washburn, International Critical Tables of Numerical Data, Physics, Chemistry and Technology, Vol. VII, McGraw-Hill, New York, 1930, p. 263).
As an example of a two-component system with equilibrated solid and gas phases, consider the components $\ce{CuSO4}$ and $\ce{H2O}$, denoted A and B respectively. In the pressure–composition phase diagram shown in Fig. 13.12, the composition variable $z\B$ is as usual the mole fraction of component B in the system as a whole.
The anhydrous salt and its hydrates (solid compounds) form the series of solids $\ce{CuSO4}$, $\ce{CuSO4*H2O}$, $\ce{CuSO4*3H2O}$, and $\ce{CuSO4*5H2O}$. In the phase diagram these formulas are abbreviated A, AB, AB$_3$, and AB$_5$. The following dissociation equilibria (dehydration equilibria) are possible: \begin{align*} \ce{CuSO4*H2O}\tx{(s)} & \arrows \ce{CuSO4}\tx{(s)} + \ce{H2O}\tx{(g)}\cr \ce{1/2CuSO4*3H2O}\tx{(s)} & \arrows \ce{1/2CuSO4*H2O}\tx{(s)} + \ce{H2O}\tx{(g)}\cr \ce{1/2CuSO4*5H2O}\tx{(s)} & \arrows \ce{1/2CuSO4*3H2O}\tx{(s)} + \ce{H2O}\tx{(g)} \end{align*} The equilibria are written above with coefficients that make the coefficient of H$_2$O(g) unity. When one of these equilibria is established in the system, there are two components and three phases; the phase rule then tells us the system is univariant and the pressure has only one possible value at a given temperature. This pressure is called the dissociation pressure of the higher hydrate.
The dissociation pressures of the three hydrates are indicated by horizontal lines in Fig. 13.12. For instance, the dissociation pressure of $\ce{CuSO4*5H2O}$ is $1.05\timesten{-2}\units{\(\br$}\). At the pressure of each horizontal line, the equilibrium system can have one, two, or three phases, with compositions given by the intersections of the line with vertical lines. A fourth three-phase equilibrium is shown at $p=3.09\timesten{-2}\units{\(\br$}\); this is the equilibrium between solid $\ce{CuSO4*5H2O}$, the saturated aqueous solution of this hydrate, and water vapor.
Consider the thermodynamic equilibrium constant of one of the dissociation reactions. At the low pressures shown in the phase diagram, the activities of the solids are practically unity and the fugacity of the water vapor is practically the same as the pressure, so the equilibrium constant is almost exactly equal to $p\subs{d}/p\st$, where $p\subs{d}$ is the dissociation pressure of the higher hydrate in the reaction. Thus, a hydrate cannot exist in equilibrium with water vapor at a pressure below the dissociation pressure of the hydrate because dissociation would be spontaneous under these conditions. Conversely, the salt formed by the dissociation of a hydrate cannot exist in equilibrium with water vapor at a pressure above the dissociation pressure because hydration would be spontaneous.
If the system contains dry air as an additional gaseous component and one of the dissociation equilibria is established, the partial pressure $p\subs{H\(_2$O}\) of H$_2$O is equal (approximately) to the dissociation pressure $p\subs{d}$ of the higher hydrate. The prior statements regarding dissociation and hydration now depend on the value of $p\subs{H\(_2$O}\). If a hydrate is placed in air in which $p\subs{H\(_2$O}\) is less than $p\subs{d}$, dehydration is spontaneous; this phenomenon is called efflorescence (Latin: blossoming). If $p\subs{H\(_2$O}\) is greater than the vapor pressure of the saturated solution of the highest hydrate that can form in the system, the anhydrous salt and any of its hydrates will spontaneously absorb water and form the saturated solution; this is deliquescence (Latin: becoming fluid).
If the two-component equilibrium system contains only two phases, it is bivariant corresponding to one of the areas in Fig. 13.12. Here both the temperature and the pressure can be varied. In the case of areas labeled with two solid phases, the pressure has to be applied to the solids by a fluid (other than H$_2$O) that is not considered part of the system.
13.2.7 Systems at high pressure
Figure 13.13 Pressure–temperature–composition behavior in the binary heptane–ethane system (W. B. Kay, Ind. Eng. Chem., 30, 459–465, 1938). The open circles are critical points; the dashed curve is the critical curve. The dashed line a–b illustrates retrograde condensation at $450\K$.
Binary phase diagrams begin to look different when the pressure is greater than the critical pressure of either of the pure components. Various types of behavior have been observed in this region. One common type, that found in the binary system of heptane and ethane, is shown in Fig. 13.13. This figure shows sections of a three-dimensional phase diagram at five temperatures. Each section is a pressure–composition phase diagram at constant $T$. The two-phase areas are hatched in the direction of the tie lines. At the left end of each tie line (at low $z\A$) is a vaporus curve, and at the right end is a liquidus curve. The vapor pressure curve of pure ethane ($z\A{=}0$) ends at the critical point of ethane at $305.4\K$; between this point and the critical point of heptane at $540.5\K$, there is a continuous critical curve, which is the locus of critical points at which gas and liquid mixtures become identical in composition and density.
Consider what happens when the system point is at point a in Fig. 13.13 and the pressure is then increased by isothermal compression along line a–b. The system point moves from the area for a gas phase into the two-phase gas–liquid area and then out into the gas-phase area again. This curious phenomenon, condensation followed by vaporization, is called retrograde condensation.
Under some conditions, an isobaric increase of $T$ can result in vaporization followed by condensation; this is retrograde vaporization.
Figure 13.14 Pressure–temperature–composition behavior in the binary xenon–helium system (J. de Swann Arons and G. A. M. Diepen, J. Chem. Phys., 44, 2322–2330, 1966). The open circles are critical points; the dashed curve is the critical curve.
A different type of high-pressure behavior, that found in the xenon–helium system, is shown in Fig. 13.14. Here, the critical curve begins at the critical point of the less volatile component (xenon) and continues to higher temperatures and pressures than the critical temperature and pressure of either pure component. The two-phase region at pressures above this critical curve is sometimes said to represent gas–gas equilibrium, or gas–gas immiscibility, because we would not usually consider a liquid to exist beyond the critical points of the pure components. Of course, the coexisting phases in this two-phase region are not gases in the ordinary sense of being tenuous fluids, but are instead high-pressure fluids of liquid-like densities. If we want to call both phases gases, then we have to say that pure gaseous substances at high pressure do not necessarily mix spontaneously in all proportions as they do at ordinary pressures.
If the pressure of a system is increased isothermally, eventually solid phases will appear; these are not shown in Figs. 13.13 and Fig. 13.14. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/13%3A_The_Phase_Rule_and_Phase_Diagrams/13.02%3A__Phase_Diagrams-_Binary_Systems.txt |
An underlined problem number or problem-part letter indicates that the numerical answer appears in Appendix I.
13.1 Consider a single-phase system that is a gaseous mixture of $\mathrm{N}_{2}, \mathrm{H}_{2}$, and $\mathrm{NH}_{3}$. For each of the following cases, find the number of degrees of freedom and give an example of the independent intensive variables that could be used to specify the equilibrium state, apart from the total amount of gas.
(a) There is no reaction.
(b) The reaction $\mathrm{N}_{2}(\mathrm{~g})+3 \mathrm{H}_{2}(\mathrm{~g}) \rightarrow 2 \mathrm{NH}_{3}(\mathrm{~g})$ is at equilibrium.
(c) The reaction is at equilibrium and the system is prepared from $\mathrm{NH}_{3}$ only.
13.2 How many components has a mixture of water and deuterium oxide in which the equilibrium $\mathrm{H}_{2} \mathrm{O}+\mathrm{D}_{2} \mathrm{O} \rightleftharpoons 2$ HDO exists?
13.3 Consider a system containing only $\mathrm{NH}_{4} \mathrm{Cl}(\mathrm{s}), \mathrm{NH}_{3}(\mathrm{~g})$, and $\mathrm{HCl}(\mathrm{g}) .$ Assume that the equilibrium $\mathrm{NH}_{4} \mathrm{Cl}(\mathrm{s}) \rightleftharpoons \mathrm{NH}_{3}(\mathrm{~g})+\mathrm{HCl}(\mathrm{g})$ exists.
(a) Suppose you prepare the system by placing solid $\mathrm{NH}_{4} \mathrm{Cl}$ in an evacuated flask and heating to $400 \mathrm{~K}$. Use the phase rule to decide whether you can vary the pressure while both phases remain in equilibrium at $400 \mathrm{~K}$.
(b) According to the phase rule, if the system is not prepared as described in part (a) could you vary the pressure while both phases remain in equilibrium at $400 \mathrm{~K}$ ? Explain.
(c) Rationalize your conclusions for these two cases on the basis of the thermodynamic equilibrium constant. Assume that the gas phase is an ideal gas mixture and use the approximate expression $K=p_{\mathrm{NH}_{3}} p_{\mathrm{HCl}} /\left(p^{\circ}\right)^{2}$.
13.4 Consider the lime-kiln process $\mathrm{CaCO}_{3}(\mathrm{~s}) \rightarrow \mathrm{CaO}(\mathrm{s})+\mathrm{CO}_{2}(\mathrm{~g})$. Find the number of intensive variables that can be varied independently in the equilibrium system under the following conditions:
(a) The system is prepared by placing calcium carbonate, calcium oxide, and carbon dioxide in a container.
(b) The system is prepared from calcium carbonate only.
(c) The temperature is fixed at $1000 \mathrm{~K}$.
13.5 What are the values of $C$ and $F$ in systems consisting of solid $\mathrm{AgCl}$ in equilibrium with an aqueous phase containing $\mathrm{H}_{2} \mathrm{O}, \mathrm{Ag}^{+}(\mathrm{aq}), \mathrm{Cl}^{-}(\mathrm{aq}), \mathrm{Na}^{+}(\mathrm{aq})$, and $\mathrm{NO}_{3}^{-}(\mathrm{aq})$ prepared in the following ways? Give examples of intensive variables that could be varied independently.
(a) The system is prepared by equilibrating excess solid $\mathrm{AgCl}$ with an aqueous solution of $\mathrm{NaNO}_{3}$.
(b) The system is prepared by mixing aqueous solutions of $\mathrm{AgNO}_{3}$ and $\mathrm{NaCl}$ in arbitrary proportions; some solid $\mathrm{AgCl}$ forms by precipitation.
13.6 How many degrees of freedom has a system consisting of solid $\mathrm{NaCl}$ in equilibrium with an aqueous phase containing $\mathrm{H}_{2} \mathrm{O}, \mathrm{Na}^{+}(\mathrm{aq}), \mathrm{Cl}^{-}(\mathrm{aq}), \mathrm{H}^{+}(\mathrm{aq})$, and $\mathrm{OH}^{-}(\mathrm{aq})$ ? Would it be possible to independently vary $T, p$, and $m_{\mathrm{OH}^{-}}$? If so, explain how you could do this.
13.7 Consult the phase diagram shown in Fig. $13.4$ on page 430. Suppose the system contains $36.0 \mathrm{~g}$ (2.00 mol) $\mathrm{H}_{2} \mathrm{O}$ and $58.4 \mathrm{~g}$ (1.00 mol) $\mathrm{NaCl}$ at $25^{\circ} \mathrm{C}$ and 1 bar.
(a) Describe the phases present in the equilibrium system and their masses.
(b) Describe the changes that occur at constant pressure if the system is placed in thermal contact with a heat reservoir at $-30^{\circ} \mathrm{C}$.
(c) Describe the changes that occur if the temperature is raised from $25^{\circ} \mathrm{C}$ to $120^{\circ} \mathrm{C}$ at constant pressure.
(d) Describe the system after $200 \mathrm{~g} \mathrm{H}_{2} \mathrm{O}$ is added at $25^{\circ} \mathrm{C}$.
Table 13.1 Aqueous solubilities of sodium sulfate decahydrate and anhydrous sodium sulfate ${ }^{a}$
\begin{tabular}{lccc}
\hline $\mathrm{Na}_{2} \mathrm{SO}_{4} \cdot 10 \mathrm{H}_{2} \mathrm{O}$ & & \multicolumn{2}{c}{$\mathrm{Na}_{2} \mathrm{SO}_{4}$} \
\cline { 1 - 2 } \cline { 5 }$t /{ }^{\circ} \mathrm{C}$ & $x_{\mathrm{B}}$ & $t /{ }^{\circ} \mathrm{C}$ & $x_{\mathrm{B}}$ \
\hline 10 & $0.011$ & 40 & $0.058$ \
15 & $0.016$ & 50 & $0.056$ \
20 & $0.024$ & & \
25 & $0.034$ & & \
30 & $0.048$ & & \
\hline${ }^{a}$ Ref. [59], p. 179-180. & & \
& & &
\end{tabular}
13.8 Use the following information to draw a temperature-composition phase diagram for the binary system of $\mathrm{H}_{2} \mathrm{O}(\mathrm{A})$ and $\mathrm{Na}_{2} \mathrm{SO}_{4}(\mathrm{~B})$ at $p=1$ bar, confining $t$ to the range $-20$ to $50^{\circ} \mathrm{C}$ and $z_{\mathrm{B}}$ to the range $0-0.2$. The solid decahydrate, $\mathrm{Na}_{2} \mathrm{SO}_{4} \cdot 10 \mathrm{H}_{2} \mathrm{O}$, is stable below $32.4^{\circ} \mathrm{C}$. The anhydrous salt, $\mathrm{Na}_{2} \mathrm{SO}_{4}$, is stable above this temperature. There is a peritectic point for these two solids and the solution at $x_{\mathrm{B}}=0.059$ and $t=32.4^{\circ} \mathrm{C}$. There is a eutectic point for ice, $\mathrm{Na}_{2} \mathrm{SO}_{4} \cdot 10 \mathrm{H}_{2} \mathrm{O}$, and the solution at $x_{\mathrm{B}}=0.006$ and $t=-1.3^{\circ} \mathrm{C}$. Table $13.1$ gives the temperature dependence of the solubilities of the ionic solids.
Table 13.2 Data for Problem 13.9. Temperatures of saturated solutions of aqueous iron(III) chloride at $p=$ 1 bar (\left(\mathrm{A}=\mathrm{FeCl}_{3}, \mathrm{~B}=\mathrm{H}_{2} \mathrm{O}\right)^{a}\)
\begin{tabular}{crcccr}
\hline$x_{\mathrm{A}}$ & $t /{ }^{\circ} \mathrm{C}$ & $x_{\mathrm{A}}$ & $t /{ }^{\circ} \mathrm{C}$ & $x_{\mathrm{A}}$ & $t /{ }^{\circ} \mathrm{C}$ \
\hline $0.000$ & $0.0$ & $0.119$ & $35.0$ & $0.286$ & $56.0$ \
$0.020$ & $-10.0$ & $0.143$ & $37.0$ & $0.289$ & $55.0$ \
$0.032$ & $-20.5$ & $0.157$ & $36.0$ & $0.293$ & $60.0$ \
$0.037$ & $-27.5$ & $0.173$ & $33.0$ & $0.301$ & $69.0$ \
$0.045$ & $-40.0$ & $0.183$ & $30.0$ & $0.318$ & $72.5$ \
$0.052$ & $-55.0$ & $0.195$ & $27.4$ & $0.333$ & $73.5$ \
$0.053$ & $-41.0$ & $0.213$ & $32.0$ & $0.343$ & $72.5$ \
$0.056$ & $-27.0$ & $0.222$ & $32.5$ & $0.358$ & $70.0$ \
$0.076$ & $0.0$ & $0.232$ & $30.0$ & $0.369$ & $66.0$ \
$0.083$ & $10.0$ & $0.238$ & $35.0$ & $0.369$ & $80.0$ \
$0.093$ & $20.0$ & $0.259$ & $50.0$ & $0.373$ & $100.0$ \
$0.106$ & $30.0$ & $0.277$ & $55.0$ & & \
\hline
\end{tabular}
${ }^{a}$ Data from Ref. [59], page $193 .$
13.9 Iron(III) chloride forms various solid hydrates, all of which melt congruently. Table $13.2$ on the preceding page lists the temperatures $t$ of aqueous solutions of various compositions that are saturated with respect to a solid phase.
(a) Use these data to construct a $t-z_{\mathrm{B}}$ phase diagram for the binary system of $\mathrm{FeCl}_{3}$ (A) and $\mathrm{H}_{2} \mathrm{O}$ (B). Identify the formula and melting point of each hydrate. Hint: derive a formula for the mole ratio $n_{\mathrm{B}} / n_{\mathrm{A}}$ as a function of $x_{\mathrm{A}}$ in a binary mixture.
(b) For the following conditions, determine the phase or phases present at equilibrium and the composition of each.
1. $t=-70.0^{\circ} \mathrm{C}$ and $z_{\mathrm{A}}=0.100$
2. $t=50.0^{\circ} \mathrm{C}$ and $z_{\mathrm{A}}=0.275$
$\overline $ Ref. [59], p. $95 .$
13.10 Figure $13.19$ is a temperature-composition phase diagram for the binary system of water (A) and phenol (B) at 1 bar. These liquids are partially miscible below $67^{\circ} \mathrm{C}$. Phenol is more dense than water, so the layer with the higher mole fraction of phenol is the bottom layer. Suppose you place $4.0 \mathrm{~mol}$ of $\mathrm{H}_{2} \mathrm{O}$ and $1.0 \mathrm{~mol}$ of phenol in a beaker at $30^{\circ} \mathrm{C}$ and gently stir to allow the layers to equilibrate.
(a) What are the compositions of the equilibrated top and bottom layers?
(b) Find the amount of each component in the bottom layer.
(c) As you gradually stir more phenol into the beaker, maintaining the temperature at $30^{\circ} \mathrm{C}$, what changes occur in the volumes and compositions of the two layers? Assuming that one layer eventually disappears, what additional amount of phenol is needed to cause this to happen?
13.11 The standard boiling point of propane is $-41.8{ }^{\circ} \mathrm{C}$ and that of $n$-butane is $-0.2{ }^{\circ} \mathrm{C}$. Table $13.3$ on the next page lists vapor pressure data for the pure liquids. Assume that the liquid mixtures obey Raoult's law.
(a) Calculate the compositions, $x_{\mathrm{A}}$, of the liquid mixtures with boiling points of $-10.0^{\circ} \mathrm{C}$, $-20.0^{\circ} \mathrm{C}$, and $-30.0^{\circ} \mathrm{C}$ at a pressure of $1 \mathrm{bar}$.
(b) Calculate the compositions, $y_{\mathrm{A}}$, of the equilibrium vapor at these three temperatures.
Table $13.3$ Saturation vapor pressures of propane (A) and $n$-butane (B)
\begin{tabular}{ccc}
\hline$t /{ }^{\circ} \mathrm{C}$ & $p_{\mathrm{A}}^{*} /$ bar & $p_{\mathrm{B}}^{*} / \mathrm{bar}$ \
\hline$-10.0$ & $3.360$ & $0.678$ \
$-20.0$ & $2.380$ & $0.441$ \
$-30.0$ & $1.633$ & $0.275$ \
\hline
\end{tabular}
(c) Plot the temperature-composition phase diagram at $p=1$ bar using these data, and label the areas appropriately.
(d) Suppose a system containing $10.0 \mathrm{~mol}$ propane and $10.0 \mathrm{~mol} n$-butane is brought to a pressure of 1 bar and a temperature of $-25^{\circ} \mathrm{C}$. From your phase diagram, estimate the compositions and amounts of both phases.
Table 13.4 Liquid and gas compositions in the two-phase system of 2-propanol (A) and benzene at $45^{\circ} \mathrm{C}^{a}$
\begin{tabular}{llcccc}
\hline$x_{\mathrm{A}}$ & $y_{\mathrm{A}}$ & $p / \mathrm{kPa}$ & $x_{\mathrm{A}}$ & $y_{\mathrm{A}}$ & $p / \mathrm{kPa}$ \
\hline 0 & 0 & $29.89$ & $0.5504$ & $0.3692$ & $35.32$ \
$0.0472$ & $0.1467$ & $33.66$ & $0.6198$ & $0.3951$ & $34.58$ \
$0.0980$ & $0.2066$ & $35.21$ & $0.7096$ & $0.4378$ & $33.02$ \
$0.2047$ & $0.2663$ & $36.27$ & $0.8073$ & $0.5107$ & $30.28$ \
$0.2960$ & $0.2953$ & $36.45$ & $0.9120$ & $0.6658$ & $25.24$ \
$0.3862$ & $0.3211$ & $36.29$ & $0.9655$ & $0.8252$ & $21.30$ \
$0.4753$ & $0.3463$ & $35.93$ & $1.0000$ & $1.0000$ & $18.14$ \
\hline
\end{tabular}
${ }^{a}$ Ref. [24].
13.12 Use the data in Table $13.4$ to draw a pressure-composition phase diagram for the 2-propanolbenzene system at $45^{\circ} \mathrm{C}$. Label the axes and each area.
Table 13.5 Liquid and gas compositions in the twophase system of acetone (A) and chloroform at $35.2{ }^{\circ} \mathrm{C}$
\begin{tabular}{lllccc}
\hline$x_{\mathrm{A}}$ & $y_{\mathrm{A}}$ & $p / \mathrm{kPa}$ & $x_{\mathrm{A}}$ & $y_{\mathrm{A}}$ & $p / \mathrm{kPa}$ \
\hline 0 & 0 & $39.08$ & $0.634$ & $0.727$ & $36.29$ \
$0.083$ & $0.046$ & $37.34$ & $0.703$ & $0.806$ & $38.09$ \
$0.200$ & $0.143$ & $34.92$ & $0.815$ & $0.896$ & $40.97$ \
$0.337$ & $0.317$ & $33.22$ & $0.877$ & $0.936$ & $42.62$ \
$0.413$ & $0.437$ & $33.12$ & $0.941$ & $0.972$ & $44.32$ \
$0.486$ & $0.534$ & $33.70$ & $1.000$ & $1.000$ & $45.93$ \
$0.577$ & $0.662$ & $35.09$ & & & \
\hline
\end{tabular}
${ }^{a}$ Ref. [179], p. $286 .$ | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/13%3A_The_Phase_Rule_and_Phase_Diagrams/13.04%3A_Chapter_13_Problems.txt |
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An electrochemical cell is a system in which passage of an electric current through an electrical circuit is linked to an internal cell reaction. A galvanic cell, or voltaic cell, is an electrochemical cell that, when isolated, has an electric potential difference between its terminals; the cell is said to be a seat of electromotive force.
The cell reaction in a galvanic cell differs in a fundamental way from the same reaction (i.e., one with the same reaction equation) taking place in a reaction vessel that is not part of an electrical circuit. In the reaction vessel, the reactants and products are in the same phase or in phases in contact with one another, and the reaction advances in the spontaneous direction until reaction equilibrium is reached. This reaction is the direct reaction.
The galvanic cell, in contrast, is arranged with the reactants physically separated from one another so that the cell reaction can advance only when an electric current passes through the cell. If there is no current, the cell reaction is constrained from taking place. When the electrical circuit is open and the cell is isolated from its surroundings, a state of thermal, mechanical, and transfer equilibrium is rapidly reached. In this state of cell equilibrium or electrochemical equilibrium, however, reaction equilibrium is not necessarily present—that is, if the reactants and products were moved to a reaction vessel at the same activities, there might be spontaneous advancement of the reaction.
As will be shown, measurements of the cell potential of a galvanic cell are capable of yielding precise values of molar reaction quantities of the cell reaction and thermodynamic equilibrium constants, and of mean ionic activity coefficients in electrolyte solutions.
14: Galvanic Cells
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$\newcommand{\cond}[1]{\$-2.5pt]{}\tag*{#1}}$ $\newcommand{\nextcond}[1]{\[-5pt]{}\tag*{#1}}$ $\newcommand{\R}{8.3145\units{J\,K\per\,mol\per}} % gas constant value$ $\newcommand{\Rsix}{8.31447\units{J\,K\per\,mol\per}} % gas constant value - 6 sig figs$ $\newcommand{\jn}{\hspace3pt\lower.3ex{\Rule{.6pt}{2ex}{0ex}}\hspace3pt}$ $\newcommand{\ljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}} \hspace3pt}$ $\newcommand{\lljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace1.4pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace3pt}$ 14.1.1 Elements of a galvanic cell We will treat a galvanic cell as a system. The cell has two metal wires called terminals that pass through the system boundary. Within the cell are phases that can conduct an electric current and are collectively called electrical conductors. Each terminal is attached to an electron conductor that is usually a metal, but might also be graphite or a semiconductor. Each electron conductor is in contact with an ionic conductor, usually an electrolyte solution, through which ions but not electrons can move. Both of the electron conductors can be in contact with the same ionic conductor; or they can be in contact with separate ionic conductors, in which case the ionic conductors contact one another at a liquid junction. The general arrangement of the physical elements of a galvanic cell is therefore \[ \tx{terminal – electron conductor – ionic conductor(s) – electron conductor – terminal}$ Both terminals must be the same metal (usually copper) in order for it to be possible to measure the electric potential difference between them.
The combination of an electron conductor and the ionic conductor in contact with it is called an electrode, or half-cell. (The term “electrode” is sometimes used to refer to just the electron conductor.) To describe a galvanic cell, it is conventional to distinguish the left and right electrodes. In this way, we establish a left–right association with the reactants and products of the reactions at the electrodes.
14.1.2 Cell diagrams
The cell of Fig. 14.1 has a single electrolyte phase with essentially the same composition at both electrodes, and is an example of a cell without liquid junction or cell without transference. As an example of a cell with transference, consider the cell diagram $\ce{Zn} \jn \ce{Zn2+}\tx{(aq)} \ljn \ce{Cu2+}\tx{(aq)} \jn \ce{Cu}$ This is the zinc–copper cell depicted in Fig. 14.2, sometimes called a Daniell cell. The two electrolyte phases are separated by a liquid junction represented in the cell diagram by the dashed vertical bar. If the liquid junction potential can be assumed to be negligible, the liquid junction is instead represented by a pair of dashed vertical bars: \begin{equation*} \ce{Zn} \jn \ce{Zn2+}\tx{(aq)} \lljn \ce{Cu2+}\tx{(aq)} \jn \ce{Cu} \end{equation*}
14.1.4 Advancement and charge
The electron number or charge number, $z$, of the cell reaction is defined as the amount of electrons entering at the right terminal per unit advancement of the cell reaction. $z$ is a positive dimensionless quantity equal to $|\nu\subs{e}|$, where $\nu\subs{e}$ is the stoichiometric number of the electrons in either of the electrode reactions whose sum is the cell reaction.
Because both electrode reactions are written with the same value of $|\nu\subs{e}|$, the advancements of these reactions and of the cell reaction are all described by the same advancement variable $\xi$. For an infinitesimal change $\dif\xi$, an amount of electrons equal to $z\dif\xi$ enters the system at the right terminal, an equal amount of electrons leaves at the left terminal, and there is no buildup of charge in any of the internal phases.
The Faraday constant $F$ is a physical constant defined as the charge per amount of protons, and is equal to the product of the elementary charge (the charge of a proton) and the Avogadro constant: $F=eN\subs{A}$. Its value to five significant figures is $F=96,485\units{C mol\(^{-1}$}\). The charge per amount of electrons is $-F$. Thus, the charge entering the right terminal during advancement $\dif\xi$ is $\dQ\sys = -zF\dif\xi \tag{14.1.1}$ | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/14%3A_Galvanic_Cells/14.01%3A_Cell_Diagrams_and_Cell_Reactions.txt |
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$\newcommand{\cond}[1]{\$-2.5pt]{}\tag*{#1}}$ $\newcommand{\nextcond}[1]{\[-5pt]{}\tag*{#1}}$ $\newcommand{\R}{8.3145\units{J\,K\per\,mol\per}} % gas constant value$ $\newcommand{\Rsix}{8.31447\units{J\,K\per\,mol\per}} % gas constant value - 6 sig figs$ $\newcommand{\jn}{\hspace3pt\lower.3ex{\Rule{.6pt}{2ex}{0ex}}\hspace3pt}$ $\newcommand{\ljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}} \hspace3pt}$ $\newcommand{\lljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace1.4pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace3pt}$ This e-book will denote the molar reaction Gibbs energy of a cell reaction by $\Delsub{r}G\cell$. This notation distinguishes it from the molar reaction Gibbs energy $\Delsub{r}G$ of the direct reaction, which may have a different value because in the cell the chemical potential of an ionic species is affected by the electric potential of its phase. $\Delsub{r}G\cell$ is defined by $\Delsub{r}G\cell \defn \sum_i\nu_i\mu_i \tag{14.3.1}$ where the sum is over the reactants and products of the cell reaction. $\Delsub{r}G\cell$ is also equal to the partial derivative $\pd{G\cell}{\xi}{T,p}$, where $\xi$ is the advancement of the cell reaction. 14.3.1 Relation between $\Delsub{r}G\cell$ and $\Eeq$ When a galvanic cell is in a zero-current equilibrium state, both electrode reactions are at reaction equilibrium. In the electrode reaction at the left electrode, electrons are a product with stoichiometric number equal to $z$. At the right electrode, electrons are a reactant with stoichiometric number equal to $-z$. We can write the conditions for electrode reaction equilibria as follows: $\tx{At the left electrode} \quad \sum_i\nu_i\mu_i + z\mue(\tx{LE}) = 0 \tag{14.3.2}$ $\tx{At the right electrode} \quad \sum_j\nu_j\mu_j - z\mue(\tx{RE}) = 0 \tag{14.3.3}$ In these equations, the sum over $i$ is for the chemical species (excluding electrons) of the electrode reaction at the left electrode, and the sum over $j$ is for the chemical species of the electrode reaction at the right electrode. $\mue(\tx{LE})$ is the chemical potential of electrons in the electron conductor of the left electrode, and $\mue(\tx{RE})$ is the chemical potential of electrons in the electron conductor of the right electrode. Adding Eqs. 14.3.2 and 14.3.3, we obtain $\sum_i\nu_i\mu_i + \sum_j\nu_j\mu_j + z[ \mue(\tx{LE})-\mue(\tx{RE}) ] = 0 \tag{14.3.4}$ The first two terms on the left side of Eq. 14.3.4 are sums over all the reactants and products of the cell reaction. From Eq. 14.3.1, we recognize the sum of these terms as the molar reaction Gibbs energy of the cell reaction: $\sum_i\nu_i\mu_i + \sum_j\nu_j\mu_j = \Delsub{r}G\cell \tag{14.3.5}$ Substituting from Eq. 14.3.5 into Eq. 14.3.4 and solving for $\Delsub{r}G\cell$, we obtain $\Delsub{r}G\cell = -z[ \mue(\tx{LE})-\mue(\tx{RE}) ] \tag{14.3.6}$ In a zero-current equilibrium state, there is electron transfer equilibrium between the left electron conductor and the left terminal, and between the right electron conductor and the right terminal: $\mue(\tx{LE})=\mue(\tx{LT})$ and $\mue(\tx{RE})=\mue(\tx{RT})$, where $\mue(\tx{LT})$ and $\mue(\tx{RT})$ are the chemical potentials of electrons in the left terminal and right terminal, respectively. Thus we can rewrite Eq. 14.3.6 as $\Delsub{r}G\cell = -z[ \mue(\tx{LT})-\mue(\tx{RT}) ] \tag{14.3.7}$ Making substitutions from Eq. 14.2.2 for $\mue(\tx{LT})$ and $\mue(\tx{RT})$, and recognizing that $\mue(0)$ is the same in both terminals because they have the same composition, we obtain $\begin{split} \Delsub{r}G\cell & = -zF(\phi\subs{R}-\phi\subs{L}) \cr & = -zF\Eeq \end{split} \tag{14.3.8}$ We can see from Eq. 14.3.1 that the value of $\Delsub{r}G\cell$ has nothing to do with the composition of the terminals. The relations of Eq. 14.3.8 were derived for a cell with both terminals made of the same metal. We can make the following deductions for such a cell: 1. Equation 14.3.8 can be derived by a different route. According to Eq. 5.8.6, reversible electrical work at constant $T$ and $p$ is equal to the Gibbs energy change: $\dw\subs{el, rev}=\dif G\cell$. Making the substitution $\dw\subs{el, rev}=\Eeq\dQ\sys$ (from Eq. 3.8.8), with $\dQ\sys$ set equal to $-zF\dif\xi$ (Eq. 14.1.1), followed by division by $\dif\xi$, gives $-zF\Eeq = \pd{G\cell}{\xi}{T,p}$, or $\Delsub{r}G\cell = -zF\Eeq$. Strictly speaking, this derivation applies only to a cell without a liquid junction. In a cell with a liquid junction, the electric current is carried across the junction by different ions depending on the direction of the current, and the cell is therefore not reversible. 14.3.2 Relation between $\Delsub{r}G\cell$ and $\Delsub{r}G$ Now imagine a reaction vessel that has the same temperature and pressure as the galvanic cell, and contains the same reactants and products at the same activities as in the cell. This reaction vessel, unlike the cell, is not part of an electrical circuit. In it, the reactants and products are in direct contact with one another, so there is no constraint preventing a spontaneous direct reaction. For example, the reaction vessel corresponding to the zinc–copper cell of Fig. 14.2 would have zinc and copper strips in contact with a solution of both ZnSO$_4$ and CuSO$_4$. Another example is the slow direct reaction in a cell without liquid junction described in Sec. 14.2.1. Let the reaction equation of the direct reaction be written with the same stoichiometric numbers $\nu_i$ as in the reaction equation for the cell reaction. The direct reaction in the reaction vessel is described by this equation or its reverse, depending on which direction is spontaneous for the given activities. The question now arises whether the molar reaction Gibbs energy $\Delsub{r}G\cell$ of the cell reaction is equal to the molar reaction Gibbs energy $\Delsub{r}G$ of the direct reaction. Both $\Delsub{r}G\cell$ and $\Delsub{r}G$ are defined by the sum $\sum_i\!\nu_i\mu_i$. Both reactions have the same values of $\nu_i$, but the values of $\mu_i$ for charged species are in general different in the two systems because the electric potentials are different. Consider first a cell without a liquid junction. This kind of cell has a single electrolyte solution, and all of the reactant and product ions of the cell reaction are in this solution phase. The same solution phase is present in the reaction vessel during the direct reaction. When all ions are in the same phase, the value of $\sum_i\!\nu_i\mu_i$ is independent of the electric potentials of any of the phases (see the comment following Eq. 11.8.4), so that the molar reaction Gibbs energies are the same for the cell reaction and the direct reaction: \begin{gather} \s{ \Delsub{r}G\cell = \Delsub{r}G } \tag{14.3.9} \cond{(no liquid junction)} \end{gather} Next, consider a cell with two electrolyte solutions separated by a liquid junction. For the molar reaction Gibbs energy of the cell reaction, we write $\Delsub{r}G\cell = \sum_i\nu_i\mu_i(\phi_i) + \sum_j\nu_j\mu_j(\phi_j) \tag{14.3.10}$ The sums here include all of the reactants and products appearing in the cell reaction, those with index $i$ being at the left electrode and those with index $j$ at the right electrode. Let the solution at the left electrode be phase $\pha$ and the solution at the right electrode be phase $\phb$. Then making the substitution $\mu_i(\phi)=\mu_i(0)+z_iF\phi$ (Eq. 10.1.6) gives us $\Delsub{r}G\cell = \sum_i\nu_i\mu_i(0) + \sum_j\nu_j\mu_j(0) + \sum_i\nu_i z_i F\phi\aph + \sum_j\nu_j z_j F\phi\bph \tag{14.3.11}$ The sum of the first two terms on the right side of Eq. 14.3.11 is the molar reaction Gibbs energy of a reaction in which the reactants and products are in phases of zero electric potential. According to the comment following Eq. 11.8.4, the molar reaction Gibbs energy would be the same if the ions were in a single phase of any electric potential. Consequently the sum $\sum_i\!\nu_i\mu_i(0){+}\sum_j\!\nu_j\mu_j(0)$ is equal to $\Delsub{r}G$ for the direct reaction. The conservation of charge during advancement of the electrode reactions at the left electrode and the right electrode is expressed by $\sum_i\!\nu_i z_i - z = 0$ and $\sum_j\!\nu_j z_j + z = 0$, respectively. Equation 14.3.11 becomes \begin{gather} \s{ \Delsub{r}G\cell = \Delsub{r}G - zF\Ej } \tag{14.3.12} \cond{(cell with liquid junction)} \end{gather} where $\Ej = \phi\bph-\phi\aph$ is the liquid junction potential. Finally, in Eqs. 14.3.9 and 14.3.12 we replace $\Delsub{r}G\cell$ by $-zF\Eeq$ (Eq. 14.3.8) and solve for $\Eeq$: \begin{gather} \s{ \Eeq = -\frac{\Delsub{r}G}{zF} } \tag{14.3.13} \cond{(cell without liquid junction)} \end{gather} \begin{gather} \s{ \Eeq = -\frac{\Delsub{r}G}{zF} + \Ej } \tag{14.3.14} \cond{(cell with liquid junction)} \end{gather} $\Eeq$ can be measured with great precision. If a reaction can be carried out in a galvanic cell without liquid junction, Eq. 14.3.13 provides a way to evaluate $\Delsub{r}G$ under given conditions. If the reaction can only be carried out in a cell with a liquid junction, Eq. 14.3.14 can be used for this purpose provided that the liquid junction potential $\Ej$ can be assumed to be negligible or can be estimated from theory. Note that the cell has reaction equilibrium only if $\Delsub{r}G$ is zero. The cell has thermal, mechanical, and transfer equilibrium when the electric current is zero and the cell potential is the zero-current cell potential $\Eeq$. Equations 14.3.13 and 14.3.14 show that in order for the cell to also have reaction equilibrium, $\Eeq$ must equal the liquid junction potential if there is a liquid junction, or be zero otherwise. These are the conditions of an exhausted, “dead” cell that can no longer do electrical work. 14.3.3 Standard molar reaction quantities Consider a hypothetical galvanic cell in which each reactant and product of the cell reaction is in its standard state at unit activity, and in which a liquid junction if present has a negligible liquid junction potential. The equilibrium cell potential of this cell is called the standard cell potential of the cell reaction, $\Eeq\st$. An experimental procedure for evaluating $\Eeq\st$ will be described in Sec. 14.5. In this hypothetical cell, $\Delsub{r}G\cell$ is equal to the standard molar reaction Gibbs energy $\Delsub{r}G\st$. From Eq. 14.3.13, or Eq. 14.3.14 with $\Ej$ assumed equal to zero, we have $\Delsub{r}G\st=-zF\Eeq\st \tag{14.3.15}$ $\Delsub{r}G\st$ is the molar reaction Gibbs energy when each reactant and product is at unit activity and, if it is an ion, is in a phase of zero electric potential. Since $\Delsub{r}G\st$ is equal to $-RT\ln K$ (Eq. 11.8.10), we can write $\ln K = \frac{zF}{RT}\Eeq\st \tag{14.3.16}$ Equation 14.3.16 allows us to evaluate the thermodynamic equilibrium constant $K$ of the cell reaction by a noncalorimetric method. Consider for example the cell \[ \ce{Ag} \jn \ce{Ag+}\tx{(aq)} \lljn \ce{Cl-}\tx{(aq)} \jn \ce{AgCl}\tx{(s)} \jn \ce{Ag}$ in which the pair of dashed vertical bars indicates a liquid junction of negligible liquid junction potential. The electrode reactions are \begin{align*} & \tx{Ag(s)} \arrow \tx{Ag$^+$(aq)} + \tx{e$^-$} \cr & \tx{AgCl(s)} + \tx{e$^-$} \arrow \tx{Ag(s)} + \tx{Cl$^-$(aq)} \end{align*} and the cell reaction is \begin{equation*} \ce{AgCl}\tx{(s)} \arrow \ce{Ag+}\tx{(aq)} + \ce{Cl-}\tx{(aq)} \end{equation*} The equilibrium constant of this reaction is the solubility product $K\subs{s}$ of silver chloride (Sec. 12.5.5). At $298.15\K$, the standard cell potential is found to be $\Eeq\st=-0.5770\V$. We can use this value in Eq. 14.3.16 to evaluate $K\subs{s}$ at $298.15\K$ (see Prob. 14.5).
Equation 14.3.16 also allows us to evaluate the standard molar reaction enthalpy by substitution in Eq. 12.1.13: \begin{gather} \s{ \begin{split} \Delsub{r}H\st & = RT^2\frac{\dif\ln K}{\dif T}\cr & = zF\left(T\frac{\dif \Eeq\st}{\dif T}-\Eeq\st \right) \end{split} } \tag{14.3.17} \cond{(no solute standard states} \nextcond{based on concentration)} \end{gather} Finally, by combining Eqs. 14.3.15 and 14.3.17 with $\Delsub{r}G\st=\Delsub{r}H\st-T\Delsub{r}S\st$, we obtain an expression for the standard molar reaction entropy: \begin{gather} \s{ \Delsub{r}S\st = zF\frac{\dif \Eeq\st}{\dif T} } \tag{14.3.18} \cond{(no solute standard states} \nextcond{based on concentration)} \end{gather}
Because $G$, $H$, and $S$ are state functions, the thermodynamic equilibrium constant and the molar reaction quantities evaluated from $\Eeq\st$ and $\dif\Eeq\st/\dif T$ are the same quantities as those for the reaction when it takes place in a reaction vessel instead of in a galvanic cell. However, the heats at constant $T$ and $p$ are not the same (Sec. 11.3.1). During a reversible cell reaction, $\dif S$ must equal $\dq/T$, and $\dq/\dif\xi$ is therefore equal to $T\Delsub{r}S\st$ during a cell reaction taking place reversibly under standard state conditions at constant $T$ and $p$. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/14%3A_Galvanic_Cells/14.03%3A_Molar_Reaction_Quantities_of_the_Cell_Reaction.txt |
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$\newcommand{\cond}[1]{\$-2.5pt]{}\tag*{#1}}$ $\newcommand{\nextcond}[1]{\[-5pt]{}\tag*{#1}}$ $\newcommand{\R}{8.3145\units{J\,K\per\,mol\per}} % gas constant value$ $\newcommand{\Rsix}{8.31447\units{J\,K\per\,mol\per}} % gas constant value - 6 sig figs$ $\newcommand{\jn}{\hspace3pt\lower.3ex{\Rule{.6pt}{2ex}{0ex}}\hspace3pt}$ $\newcommand{\ljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}} \hspace3pt}$ $\newcommand{\lljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace1.4pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace3pt}$ The standard cell potential $\Eeq\st$ of a cell reaction is the equilibrium cell potential of the hypothetical galvanic cell in which each reactant and product of the cell reaction is in its standard state and there is no liquid junction potential. The value of $\Eeq\st$ for a given cell reaction with given choices of standard states is a function only of temperature. The measured equilibrium cell potential $\Eeq$ of an actual cell, however, depends on the activities of the reactants and products as well as on temperature and the liquid junction potential, if present. To derive a relation between $\Eeq$ and activities for a cell without liquid junction, or with a liquid junction of negligible liquid junction potential, we substitute expressions for $\Delsub{r}G$ and for $\Delsub{r}G\st$ from Eqs. 14.3.13 and Eq. 14.3.15 into $\Delsub{r}G = \Delsub{r}G\st + RT\ln Q\subs{rxn}$ (Eq. 11.8.8) and solve for $\Eeq$: \begin{gather} \s {\Eeq = \Eeq\st - \frac{RT}{zF}\ln Q\subs{rxn} } \tag{14.4.1} \cond{(no liquid junction, or $\Ej{=}0$)} \end{gather} Equation 14.4.1 is the Nernst equation for the cell reaction. Here $Q\subs{rxn}$ is the reaction quotient for the cell reaction defined by Eq. 11.8.6: $Q\subs{rxn}=\prod_i a_i^{\nu_i}$. The rest of this section will assume that the cell reaction takes place in a cell without liquid junction, or in one in which $\Ej$ is negligible. If each reactant and product of the cell reaction is in its standard state, then each activity is unity and $\ln Q\subs{rxn}$ is zero. We can see from the Nernst equation that the equilibrium cell potential $\Eeq$ in this case has its standard value $\Eeq\st$, as expected. A decrease in product activities or an increase in reactant activities decreases the value of $\ln Q\subs{rxn}$ and increases $\Eeq$, as we would expect since $\Eeq$ should be greater when the forward cell reaction has a greater tendency for spontaneity. If the cell reaction comes to reaction equilibrium, as it will if we short-circuit the cell terminals with an external wire, the value of $Q\subs{rxn}$ becomes equal to the thermodynamic equilibrium constant $K$, and the Nernst equation becomes $\Eeq=\Eeq\st-(RT/zF)\ln K$. The term $(RT/zF)\ln K$ is equal to $\Eeq\st$ (Eq. 14.3.16), so $\Eeq$ becomes zero—the cell is “dead” and is incapable of performing electrical work on the surroundings. At $T{=}298.15\K$ ($25.00\units{\(\degC$}\)), the value of $RT/F$ is $0.02569\V$, and we can write the Nernst equation in the compact form \begin{gather} \s{ \Eeq = \Eeq\st - \frac{0.02569\V}{z}\ln Q\subs{rxn} } \tag{14.4.2} \cond{($T{=}298.15\K$)} \end{gather} As an illustration of an application of the Nernst equation, consider the reaction equation \[ \ce{H2}\tx{(g)} + \ce{2AgCl}\tx{(s)} \ra \ce{2H+}\tx{(aq)}+\ce{2Cl-}\tx{(aq)}+\ce{2Ag}\tx{(s)}$ This reaction takes place in a cell without liquid junction (Fig. 14.1), and the electrolyte solution can be aqueous HCl. The expression for the reaction quotient is $Q\subs{rxn} = \frac{a_+^2 a_-^2 a\subs{Ag}^2}{a\subs{H$_2$}a\subs{AgCl}^2} \tag{14.4.3}$ We may usually with negligible error approximate the pressure factors of the solids and solutes by unity. The activities of the solids are then 1, the solute activities are $a_+=\g_+ m_+/m\st$ and $a_-=\g_- m_-/m\st$, and the hydrogen activity is $a\subs{H\(_2$}=\fug\subs{H$_2$}/p\st\). The ion molalities $m_+$ and $m_-$ are equal to the HCl molality $m\B$. The expression for $Q\subs{rxn}$ becomes $Q\subs{rxn} = \frac{\g_+^2\g_-^2\left(m\B/m\st\right)^4} {\fug\subs{H$_2$}/p\st} = \frac{\g_{\pm}^4\left(m\B/m\st\right)^4}{\fug\subs{H$_2$}/p\st} \tag{14.4.4}$ and the Nernst equation for this cell is $\begin{split} \Eeq & = \Eeq\st - \frac{RT}{2F}\ln\frac{\g_{\pm}^4(m\B/m\st)^4} {\fug\subs{H$_2$}/p\st} \cr & = \Eeq\st - \frac{2RT}{F}\ln\g_{\pm} - \frac{2RT}{F}\ln\frac{m\B}{m\st} + \frac{RT}{2F}\ln\frac{\fug\subs{H$_2$}}{p\st} \end{split} \tag{14.4.5}$ By measuring $\Eeq$ for a cell with known values of $m\B$ and $\fug\subs{H\(_2$}\), and with a derived value of $\Eeq\st$, we can use this equation to find the mean ionic activity coefficient $\g_{\pm}$ of the HCl solute. This is how the experimental curve for aqueous HCl in Fig. 10.3 was obtained.
We can always multiply each of the stoichiometric coefficients of a reaction equation by the same positive constant without changing the meaning of the reaction. How does this affect the Nernst equation for the reaction equation above? Suppose we decide to multiply the stoichiometric coefficients by one-half: \begin{equation*} \ce{1/2H2}\tx{(g)}+\ce{AgCl}\tx{(s)} \arrow \ce{H+}\tx{(aq)}+\ce{Cl-}\tx{(aq)} + \tx{Ag(s)} \end{equation*} With this changed reaction equation, the value of $z$ is changed from 2 to 1 and the Nernst equation becomes $\Eeq = \Eeq\st - \frac{RT}{F}\ln\frac{\g_{\pm}^2(m\B/m\st)^2} {(\fug\subs{H$_2$}/p\st)^{1/2}} \tag{14.4.6}$ which yields the same value of $\Eeq$ for given cell conditions as Eq. 14.4.5. This value must of course be unchanged, because physically the cell is the same no matter how we write its cell reaction, and measurable physical quantities such as $\Eeq$ are unaffected. However, molar reaction quantities such as $\Delsub{r}G$ and $\Delsub{r}G\st$ do depend on how we write the cell reaction, because they are changes per extent of reaction. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/14%3A_Galvanic_Cells/14.04%3A_The_Nernst_Equation.txt |
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As we have seen, the value of the standard cell potential $\Eeq\st$ of a cell reaction has useful thermodynamic applications. The value of $\Eeq\st$ for a given cell reaction depends only on temperature. To evaluate it, we can extrapolate an appropriate function to infinite dilution where ionic activity coefficients are unity.
To see how this procedure works, consider again the cell reaction $\ce{H2}\tx{(g)} + \ce{2AgCl}\tx{(s)} \arrow \ce{2H+}\tx{(aq)}+\ce{2Cl-}\tx{(aq)}+\ce{2Ag}\tx{(s)}$. The cell potential depends on the molality $m\B$ of the HCl solute according to Eq. 14.4.5. We can rearrange the equation to $\Eeq\st = \Eeq + \frac{2RT}{F}\ln\g_{\pm} + \frac{2RT}{F}\ln\frac{m\B}{m\st} - \frac{RT}{2F}\ln\frac{\fug\subs{H$_2$}}{p\st} \tag{14.5.1}$ For given conditions of the cell, we can measure all quantities on the right side of Eq. 14.5.1 except the mean ionic activity coefficient $\g_\pm$ of the electrolyte. We cannot know the exact value of $\ln\g_{\pm}$ for any given molality until we have evaluated $\Eeq\st$. We do know that as $m\B$ approaches zero, $\g_{\pm}$ approaches unity and $\ln\g_{\pm}$ must approach zero. The Debye–Hückel formula of Eq. 10.4.7 is a theoretical expression for $\ln\g_{\pm}$ that more closely approximates the actual value the lower is the ionic strength. Accordingly, we define the quantity $E\cell' = \Eeq + \frac{2RT}{F}\left(-\frac{A\sqrt{m\B}}{1+Ba\sqrt{m\B}}\right) +\frac{2RT}{F}\ln\frac{m\B}{m\st} - \frac{RT}{2F}\ln\frac{\fug\subs{H$_2$}}{p\st} \tag{14.5.2}$ The expression in parentheses is the Debye–Hückel formula for $\ln\g_{\pm}$ with $I_m$ replaced by $m\B$. The constants $A$ and $B$ have known values at any temperature (Sec. 10.4), and $a$ is an ion-size parameter for which we can choose a reasonable value. At a given temperature, we can evaluate $E\cell'$ experimentally as a function of $m\B$.
The expression on the right side of Eq. 14.5.1 differs from that of Eq. 14.5.2 by contributions to $(2RT/F)\ln\g_{\pm}$ not accounted for by the Debye–Hückel formula. Since these contributions approach zero in the limit of infinite dilution, the extrapolation of measured values of $E\cell'$ to $m\B{=}0$ yields the value of $\Eeq\st$.
Figure 14.5 $E\cell'$ (defined by Eq. 14.5.2) as a function of HCl molality for the cell of Fig. 14.1 at $298.15\K$. Data from Herbert S. Harned and Russell W. Ehlers, J. Am. Chem. Soc., 54, 1350–1357, 1932, with $\fug\subs{H\(_2$}\) set equal to $p\subs{H\(_2$}\) and the parameter $a$ set equal to $4.3\timesten{-10}\units{m}$. The dashed line is a least-squares fit to a linear relation.
Figure 14.5 shows this extrapolation using data from the literature. The extrapolated value indicated by the filled circle is $\Eeq\st=0.2222\V$, and the uncertainty is on the order of only $0.1\units{mV}$. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/14%3A_Galvanic_Cells/14.05%3A_Evaluation_of_the_Standard_Cell_Potential.txt |
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Section 14.5 explained how, by measuring the equilibrium cell potential of a galvanic cell at different electrolyte molalities, we can evaluate the standard cell potential $\Eeq\st$ of the cell reaction. It is not necessary to carry out this involved experimental procedure for each individual cell reaction of interest. Instead, we can calculate $\Eeq\st$ from standard electrode potentials.
By convention, standard electrode potentials use a standard hydrogen electrode as a reference electrode. A standard hydrogen electrode is a hydrogen electrode, such as the electrode shown at the left in Fig. 14.1, in which the species H$_2$(g) and H$^+$(aq) are in their standard states. Since these are hypothetical gas and solute standard states, the standard hydrogen electrode is a hypothetical electrode—not one we can actually construct in the laboratory.
A standard electrode potential $E\st$ is defined as the standard cell potential of a cell with a hydrogen electrode at the left and the electrode of interest at the right. For example, the cell in Fig. 14.1 with cell diagram \begin{equation*} \ce{Pt} \jn \ce{H2}\tx{(g)} \jn \ce{HCl}\tx{(aq)} \jn \ce{AgCl}\tx{(s)} \jn \ce{Ag} \end{equation*} has a hydrogen electrode at the left and a silver–silver chloride electrode at the right. The standard electrode potential of the silver–silver chloride electrode, therefore, is equal to the standard cell potential of this cell.
Since a cell with hydrogen electrodes at both the left and right has a standard cell potential of zero, the standard electrode potential of the hydrogen electrode is zero at all temperatures. The standard electrode potential of any other electrode is nonzero and is a function only of temperature.
Consider the following three cells constructed from various combinations of three different electrodes: a hydrogen electrode, and two electrodes denoted L and R.
• We wish to calculate the standard cell potential $\Eeq\st$ of cell 1 from the standard electrode potentials $E\subs{L}\st$ and $E\subs{R}\st$.
If we write the cell reactions of cells 1 and 2 using the same value of the electron number $z$ for both, we find that their sum is the cell reaction for cell 3 with the same value of $z$. Call these reactions 1, 2, and 3, respectively: $\tx{(reaction 1)}+\tx{(reaction 2)}=\tx{(reaction 3)} \tag{14.6.1}$
Equation 14.6.3 is a general relation applicable to any galvanic cell. It should be apparent that we can use the relation to calculate the standard electrode potential of an electrode from the standard electrode potential of a different electrode and the standard cell potential of a cell that contains both electrodes. Neither electrode has to be a hydrogen electrode, which is difficult to work with experimentally.
Using Eq. 14.6.3 to calculate standard cell potentials from standard electrode potentials saves a lot of experimental work. For example, measurement of $\Eeq\st$ for ten different cells, only one of which needs to include a hydrogen electrode, provides values of $E\st$ for ten electrodes other than $E\st{=}0$ for the hydrogen electrode. From these ten values of $E\st$, values of $\Eeq\st$ can be calculated for 35 other cells without hydrogen electrodes. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/14%3A_Galvanic_Cells/14.06%3A_Standard_Electrode_Potentials.txt |
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$\newcommand{\rev}{\subs{rev}} % reversible$
$\newcommand{\irr}{\subs{irr}} % irreversible$
$\newcommand{\fric}{\subs{fric}} % friction$
$\newcommand{\diss}{\subs{diss}} % dissipation$
$\newcommand{\el}{\subs{el}} % electrical$
$\newcommand{\cell}{\subs{cell}} % cell$
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$\newcommand{\dotprod}{\small\bullet}$
$\newcommand{\fug}{f} % fugacity$
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$\newcommand{\G}{\varGamma} % activity coefficient of a reference state (pressure factor)$
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$\newcommand{\Ej}{E\subs{j}} % liquid junction potential$
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$\newcommand{\cond}[1]{\$-2.5pt]{}\tag*{#1}}$ $\newcommand{\nextcond}[1]{\[-5pt]{}\tag*{#1}}$ $\newcommand{\R}{8.3145\units{J\,K\per\,mol\per}} % gas constant value$ $\newcommand{\Rsix}{8.31447\units{J\,K\per\,mol\per}} % gas constant value - 6 sig figs$ $\newcommand{\jn}{\hspace3pt\lower.3ex{\Rule{.6pt}{2ex}{0ex}}\hspace3pt}$ $\newcommand{\ljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}} \hspace3pt}$ $\newcommand{\lljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace1.4pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace3pt}$ An underlined problem number or problem-part letter indicates that the numerical answer appears in Appendix I. 14.1 The state of a galvanic cell without liquid junction, when its temperature and pressure are uniform, can be fully described by values of the variables $T$, $p$, and $\xi$. Find an expression for $\dif G$ during a reversible advancement of the cell reaction, and use it to derive the relation $\Delsub{r}G\cell = -zF\Eeq$ (Eq. 14.3.8). (Hint: Eq. 3.8.8.) 14.2 Before 1982 the standard pressure was usually taken as $1\units{atm}$. For the cell shown in Fig. 14.1, what correction is needed, for a value of $\Eeq\st$ obtained at $25\units{\(\degC$}\) and using the older convention, to change the value to one corresponding to a standard pressure of $1\br$? Equation 14.3.15 can be used for this calculation. 14.3 Careful measurements (Roger G. Bates and Vincent E. Bower, J. Res. Natl. Bur. Stand. (U.S.), 53, 283–290, 1954) of the equilibrium cell potential of the cell \begin{equation*} \ce{Pt}\jn \ce{H2}\tx{(g)}\jn \ce{HCl}\tx{(aq)}\jn \ce{AgCl}\tx{(s)}\jn \ce{Ag} \end{equation*} yielded, at $298.15\K$ and using a standard pressure of $1\br$, the values $\Eeq\st=0.22217\V$ and $\dif \Eeq\st/\dif T=-6.462\timesten{-4}\units{V K\(^{-1}$}\). (The requested calculated values are close to, but not exactly the same as, the values listed in Appendix H, which are based on the same data combined with data of other workers.) (a) Evaluate $\Delsub{r}G\st$, $\Delsub{r}S\st$, and $\Delsub{r}H\st$ at $298.15\K$ for the reaction \[ \textstyle \frac{1}{2}\ce{H2}\tx{(g)}+\ce{AgCl}\tx{(s)} \arrow \ce{H+}\tx{(aq)}+\ce{Cl-}\tx{(aq)}+\ce{Ag}\tx{(s)}$
(b) Problem 12.18 showed how the standard molar enthalpy of formation of the aqueous chloride ion may be evaluated based on the convention $\Delsub{f}H\st(\ce{H+}, \tx{aq})=0$. If this value is combined with the value of $\Delsub{r}H\st$ obtained in part (a) of the present problem, the standard molar enthalpy of formation of crystalline silver chloride can be evaluated. Carry out this calculation for $T=298.15\K$ using the value $\Delsub{f}H\st(\ce{Cl-}, \tx{aq})=-167.08\units{kJ mol\(^{-1}$}\) (Appendix H).
(c) By a similar procedure, evaluate the standard molar entropy, the standard molar entropy of formation, and the standard molar Gibbs energy of formation of crystalline silver chloride at $298.15\K$. You need the following standard molar entropies evaluated from spectroscopic and calorimetric data: \begin{array}{lll} S\m\st(\ce{H2}, \tx{g})=130.68\units{J K$^{-1}$ mol$^{-1}$} & \qquad & S\m\st(\ce{Cl2}, \tx{g})=223.08\units{J K$^{-1}$ mol$^{-1}$} \cr S\m\st(\ce{Cl-}, \tx{aq})=56.60\units{J K$^{-1}$ mol$^{-1}$} & & S\m\st(\ce{Ag}, \tx{s})=42.55\units{J K$^{-1}$ mol$^{-1}$} \end{array}
14.4
The standard cell potential of the cell $\ce{Ag} \jn \ce{AgCl}\tx{(s)} \jn \ce{HCl}\tx{(aq)}\jn \ce{Cl2}\tx{(g)}\jn \ce{Pt}$ has been determined over a range of temperature (G. Faita, P. Longhi, and T. Mussini, J. Electrochem. Soc., 114, 340–343, 1967). At $T{=}298.15\K$, the standard cell potential was found to be $\Eeq\st=1.13579\V$, and its temperature derivative was found to be $\dif \Eeq\st/\dif T=-5.9863\timesten{-4}\units{V K\(^{-1}$}\).
(a) Write the cell reaction for this cell.
(b) Use the data to evaluate the standard molar enthalpy of formation and the standard molar Gibbs energy of formation of crystalline silver chloride at $298.15\K$. (Note that this calculation provides values of quantities also calculated in Prob. 14.3 using independent data.)
14.5
Use data in Sec. 14.3.3 to evaluate the solubility product of silver chloride at $298.15\K$.
14.6
The equilibrium cell potential of the galvanic cell $\ce{Pt} \jn \ce{H2}(\tx{g}, \fug{=}1\br) \jn \ce{HCl}(\tx{aq}, 0.500\units{mol kg$^{-1}$}) \jn \tx{Cl$_2$(g, $\fug{=}1\br$)} \jn \tx{Pt}$ is found to be $\Eeq=1.410\V$ at $298.15\K$. The standard cell potential is $\Eeq\st=1.360\V$.
(a) Write the cell reaction and calculate its thermodynamic equilibrium constant at $298.15\K$.
(b) Use the cell measurement to calculate the mean ionic activity coefficient of aqueous HCl at $298.15\K$ and a molality of $0.500\units{mol kg\(^{-1}$}\).
14.7
Consider the following galvanic cell, which combines a hydrogen electrode and a calomel electrode: $\ce{Pt} \jn \ce{H2}\tx{(g)} \jn \ce{HCl}\tx{(aq)} \jn \ce{Hg2Cl2}\tx{(s)} \jn \ce{Hg}\tx{(l)} \jn \ce{Pt}$
(a) Write the cell reaction.
(b) At $298.15\K$, the standard cell potential of this cell is $\Eeq\st=0.2680\V$. Using the value of $\Delsub{f}G\st$ for the aqueous chloride ion in Appendix H, calculate the standard molar Gibbs energy of formation of crystalline mercury(I) chloride (calomel) at $298.15\K$.
(c) Calculate the solubility product of mercury(I) chloride at $298.15\K$. The dissolution equilibrium is $\ce{Hg2Cl2}\tx{(s)}\arrows \ce{Hg2^2+}\tx{(aq)}+\ce{2Cl-}\tx{(aq)}$. Take values for the standard molar Gibbs energies of formation of the aqueous ions from Appendix H.
14.8
Table 14.1 lists equilibrium cell potentials obtained with the following cell at $298.15\K$ (Albert S. Keston, J. Am. Chem. Soc., 57, 1671–1673, 1935): \begin{equation*} \tx{Pt} \jn \tx{H}_2\tx{(g, $1.01\br$)} \jn \tx{HBr(aq, $m\B$)} \jn \tx{AgBr(s)} \jn \tx{Ag} \end{equation*} Use these data to evaluate the standard electrode potential of the silver-silver bromide electrode at this temperature to the nearest millivolt. (Since the electrolyte solutions are quite dilute, you may ignore the term $Ba\sqrt{m\B}$ in Eq. 14.5.2.)
14.9
The cell diagram of a mercury cell can be written \begin{equation*} \ce{Zn}\tx{(s)} \jn \ce{ZnO}\tx{(s)} \jn \ce{NaOH}\tx{(aq)} \jn \ce{HgO}\tx{(s)} \jn \ce{Hg}\tx{(l)} \end{equation*}
(a) Write the electrode reactions and cell reaction with electron number $z=2$.
(b) Use data in Appendix H to calculate the standard molar reaction quantities $\Delsub{r}H\st$, $\Delsub{r}G\st$, and $\Delsub{r}S\st$ for the cell reaction at $298.15\K$.
(c) Calculate the standard cell potential of the mercury cell at $298.15\K$ to the nearest $0.01\V$.
(d) Evaluate the ratio of heat to advancement, $\dq/\dif\xi$, at a constant temperature of $298.15\K$ and a constant pressure of $1\br$, for the cell reaction taking place in two different ways: reversibly in the cell, and spontaneously in a reaction vessel that is not part of an electrical circuit.
(e) Evaluate $\dif\Eeq\st/\dif T$, the temperature coefficient of the standard cell potential. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/14%3A_Galvanic_Cells/14.07%3A_Chapter_14_Problems.txt |
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The official definitions of the base units given in the IUPAC Green Book (E. Richard Cohen et al, Quantities, Units and Symbols in Physical Chemistry, 3rd edition, RSC Publishing, Cambridge, 2007, Sec. 3.3) are as follows.
• The metre is the length of path traveled by light in vacuum during a time interval of 1/299,792,458 of a second.
[This e-book uses the alternative spelling meter.]
• The kilogram is the unit of mass; it is equal to the mass of the international prototype of the kilogram.
[The international prototype is a platinum-iridium cylinder stored in a vault of the International Bureau of Weights and Measures in Sèvres near Paris, France.]
• The second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom. This definition refers to a caesium atom at rest at a temperature of $0\K$.
• The kelvin, unit of thermodynamic temperature, is the fraction 1/$273.16$ of the thermodynamic temperature of the triple point of water. This definition refers to water having the isotopic composition defined exactly by the following amount-of-substance ratios: 0.000 155 76 mole of ${}^2$H per mole of ${}^1$H, 0.000 379 9 mole of ${}^{17}$O per mole of ${}^{16}$O, and 0.002 005 2 mole of ${}^{18}$O per mole of ${}^{16}$O.
• The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in $0.012$ kilogram of carbon 12; its symbol is “mol”. When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles. In this definition, it is understood that unbound atoms of carbon 12, at rest and in their ground state, are referred to.
• The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 metre apart in vacuum, would produce between these conductors a force equal to $2\timesten{-7}$ newton per metre of length.
• The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency $540\timesten{12}$ hertz and that has a radiant intensity in that direction of 1/683 watt per steradian. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/15%3A_Appendices/15.01%3A_Appendix_A-_Definitions_of_the_SI_Base_Units.txt |
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The following table lists values from The Nist Reference on Constants, Units, and Uncertainty of fundamental physical constants used in thermodynamic calculations. Except for those marked “exact,” they are the 2010 CODATA (Committee on Data for Science and Technology) recommended values. The number in parentheses at the end of a value is the standard deviation uncertainty in the right-most digits of the value. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/15%3A_Appendices/15.02%3A_Appendix_B-_Physical_Constants.txt |
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This appendix lists the symbols for most of the variable physical quantities used in this e-book. The symbols are those recommended in the IUPAC Green Book (Ian Mills et al, Quantities, Units and Symbols in Physical Chemistry, 2nd edition, Blackwell, Oxford, 1993) except for quantities followed by an asterisk ($^*$). The first table lists Roman letter symbols, and the second lists Greek letter symbols.
\begin{array}{lll} \hline \textbf{Symbol} & \textbf{Physical quantity} & \textbf{SI unit} \ \hline A & \tx{Helmholtz energy} & \tx{J} \ \As & \tx{surface area} & \tx{m}^2 \ a & \tx{activity} & \tx{(dimensionless)} \ B & \tx{second virial coefficient} & \tx{m}^3 \tx{ mol}^{-1} \ C & \tx{number of components}^* & \tx{(dimensionless)} \ C_p & \tx{heat capacity at constant pressure} & \tx{J K}^{-1} \ C_V & \tx{heat capacity at constant volume} & \tx{J K}^{-1} \ c & \tx{concentration} & \tx{mol m}^{-3} \ E & \tx{energy} & \tx{J} \ & \tx{electrode potential} & \tx{V} \ \boldsymbol{E} & \tx{electric field strength} & \tx{V m}^{-1} \ E\cell & \tx{cell potential} & \tx{V} \ \Ej & \tx{liquid junction potential} & \tx{V} \ E\sys & \tx{system energy in a lab frame} & \tx{J} \ F & \tx{force} & \tx{N} \ & \tx{number of degrees of freedom}^* & \tx{(dimensionless)} \ \fug & \tx{fugacity} & \tx{Pa} \ g & \tx{acceleration of free fall} & \tx{m s}^{-2} \ G & \tx{Gibbs energy} & \tx{J} \ h & \tx{height, elevation} & \tx{m} \ H & \tx{enthalpy} & \tx{J} \ \boldsymbol{H} & \tx{magnetic field strength} & \tx{A m}^{-1} \ I & \tx{electric current} & \tx{A} \ I_m & \tx{ionic strength, molality basis} & \tx{mol kg}^{-1} \ I_c & \tx{ionic strength, concentration basis} & \tx{mol m}^{-3} \ K & \tx{thermodynamic equilibrium constant} & \tx{(dimensionless)} \ K\subs{a} & \tx{acid dissociation constant} & \tx{(dimensionless)} \ K_p & \tx{equilibrium constant, pressure basis} & \tx{Pa}^{\sum\nu} \ K\subs{s} & \tx{solubility product} & \tx{(dimensionless)} \ \kHi & \tx{Henry’s law constant of species }i, \ & \quad \tx{mole fraction basis} & \tx{Pa} \ k_{c,i} & \tx{Henry’s law constant of species }i, \ & \quad \tx{concentration basis}^* & \tx{Pa m}^3\tx{ mol}^{-1} \ k_{m,i} & \tx{Henry’s law constant of species }i, \ & \quad \tx{molality basis}^* & \tx{Pa kg mol}^{-1} \ l & \tx{length, distance} & \tx{m} \ L & \tx{relative partial molar enthalpy}^* & \tx{J mol}^{-1} \ M & \tx{molar mass} & \tx{kg mol}^{-1} \ \boldsymbol{M} & \tx{magnetization} & \tx{A m}^{-1} \ M\subs{r} & \tx{relative molecular mass (molecular weight)} & \tx{(dimensionless)} \ m & \tx{mass} & \tx{kg} \ m_i & \tx{molality of species }i & \tx{mol kg}^{-1} \ N & \tx{number of entities (molecules, atoms, ions,} \ & \quad \tx{formula units, etc.)} & \tx{(dimensionless)} \ n & \tx{amount of substance} & \tx{mol} \ P & \tx{number of phases}^* & \tx{(dimensionless)} \ p & \tx{pressure} & \tx{Pa} \ & \tx{partial pressure} & \tx{Pa} \ \boldsymbol{P} & \tx{dielectric polarization} & \tx{C m}^{-2} \ Q & \tx{electric charge} & \tx{C} \ Q\sys & \tx{charge entering system at right conductor}^* & \tx{C} \ Q\subs{rxn} & \tx{reaction quotient}^* & \tx{(dimensionless)} \ q & \tx{heat} & \tx{J} \ R\el & \tx{electric resistance}^* & \Omega \ S & \tx{entropy} & \tx{J K}^{-1} \ s & \tx{solubility} & \tx{mol m}^{-3} \ & \tx{number of species}^* & \tx{(dimensionless)} \ T & \tx{thermodynamic temperature} & \tx{K} \ t & \tx{time} & \tx{s} \ & \tx{Celsius temperature} & \degC \ U & \tx{internal energy} & \tx{J} \ V & \tx{volume} & \tx{m}^3 \ v & \tx{specific volume} & \tx{m}^3\tx{ kg}^{-1} \ & \tx{velocity, speed} & \tx{m s}^{-1} \ w & \tx{work} & \tx{J} \ & \tx{mass fraction (weight fraction)} & \tx{(dimensionless)} \ w\el & \tx{electrical work}^* & \tx{J} \ w' & \tx{nonexpansion work}^* & \tx{J} \ x & \tx{mole fraction in a phase} & \tx{(dimensionless)} \ & \tx{Cartesian space coordinate} & \tx{m} \ y & \tx{mole fraction in gas phase} & \tx{(dimensionless)} \ & \tx{Cartesian space coordinate} & \tx{m} \ Z & \tx{compression factor (compressibility factor)} & \tx{(dimensionless)} \ z & \tx{mole fraction in multiphase system}^* & \tx{(dimensionless)} \ & \tx{charge number of an ion} & \tx{(dimensionless)}\ & \tx{electron number of cell reaction} & \tx{(dimensionless)} \ & \tx{Cartesian space coordinate} & \tx{m} \ \hline \end{array}
\begin{array}{lll} \hline \textbf{Symbol} & \textbf{Physical quantity} & \textbf{SI unit} \ \hline \alpha & \tx{degree of reaction, dissociation, etc.} & \tx{(dimensionless)} \ & \tx{cubic expansion coefficient} & \tx{K}^{-1} \ \g & \tx{surface tension} & \tx{N m}^{-1}, \tx{J m}^{-2} \ \g_i & \tx{activity coefficient of species i,} \ & \quad \tx{pure liquid or solid standard state}^* & \tx{(dimensionless)} \ \g_{m,i} & \tx{activity coefficient of species i,} \ & \quad \tx{molality basis} & \tx{(dimensionless)} \ \g_{c,i} & \tx{activity coefficient of species i,} \ & \quad \tx{concentration basis} & \tx{(dimensionless)} \ \g_{x,i} & \tx{activity coefficient of species i,} \ & \quad \tx{mole fraction basis} & \tx{(dimensionless)} \ \g_{\pm} & \tx{mean ionic activity coefficient} & \tx{(dimensionless)} \ \G & \tx{pressure factor (activity of a reference state)}^* & \tx{(dimensionless)} \ \epsilon & \tx{efficiency of a heat engine} & \tx{(dimensionless)} \ & \tx{energy equivalent of a calorimeter}^* & \tx{J K}^{-1} \ \vartheta & \tx{angle of rotation} & \tx{(dimensionless)} \ \kappa & \tx{reciprocal radius of ionic atmosphere} & \tx{m}^{-1} \ \kappa _T & \tx{isothermal compressibility} & \tx{Pa}^{-1} \ \mu & \tx{chemical potential} & \tx{J mol}^{-1} \ \mu\subs{JT} & \tx{Joule–Thomson coefficient} & \tx{K Pa}^{-1} \ \nu & \tx{number of ions per formula unit} & \tx{(dimensionless)} \ & \tx{stoichiometric number} & \tx{(dimensionless)} \ \nu_+ & \tx{number of cations per formula unit} & \tx{(dimensionless)} \ \nu_- & \tx{number of anions per formula unit} & \tx{(dimensionless)} \ \xi & \tx{advancement (extent of reaction)} & \tx{mol} \ \varPi & \tx{osmotic pressure} & \tx{Pa} \ \rho & \tx{density} & \tx{kg m}^{-3} \ \tau & \tx{torque}^* & \tx{J} \ \phi & \tx{fugacity coefficient} & \tx{(dimensionless)} \ & \tx{electric potential} & \tx{V} \ \Del\phi & \tx{electric potential difference} & \tx{V} \ \phi_m & \tx{osmotic coefficient, molality basis} & \tx{(dimensionless)} \ \varPhi_L & \tx{relative apparent molar enthalpy of solute}^* & \tx{J} mol^{-1} \ \omega & \tx{angular velocity} & \tx{s}^{-1} \ \hline \end{array} | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/15%3A_Appendices/15.03%3A_Appendix_C-_Symbols_for_Physical_Quantities.txt |
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$\newcommand{\Rsix}{8.31447\units{J\,K\per\,mol\per}} % gas constant value - 6 sig figs$
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$\newcommand{\ljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}} \hspace3pt}$
$\newcommand{\lljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace1.4pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace3pt}$
D.1 Physical States
These abbreviations for physical states (states of aggregation) may be appended in parentheses to chemical formulas or used as superscripts to symbols for physical quantities. All but “mixt” are listed in the IUPAC Green Book (E. Richard Cohen et al, Quantities, Units and Symbols in Physical Chemistry, 3rd edition, RSC Publishing, Cambridge, 2007, p. 54).
\begin{array}{@{}ll@{}} \tx{g} & \tx{gas or vapor} \cr \tx{l} & \tx{liquid} \cr \tx{f} & \tx{fluid (gas or liquid)} \cr \tx{s} & \tx{solid} \cr \tx{cd} & \tx{condensed phase (liquid or solid)} \cr \tx{cr} & \tx{crystalline} \cr \tx{mixt} & \tx{mixture} \cr \tx{sln} & \tx{solution} \cr \tx{aq} & \tx{aqueous solution} \cr \tx{aq$,\infty$} & \tx{aqueous solution at infinite dilution} \end{array}
D.2 Subscripts for Chemical Processes
These abbreviations are used as subscripts to the $\Del$ symbol. They are listed in the IUPAC Green Book (E. Richard Cohen et al, Quantities, Units and Symbols in Physical Chemistry, 3rd edition, RSC Publishing, Cambridge, 2007, p. 59–60).
The combination $\Delsub{p}$, where “p” is any one of the abbreviations below, can be interpreted as an operator: $\Delsub{p} \defn \partial/\partial\xi\subs{p}$ where $\xi\subs{p}$ is the advancement of the given process at constant temperature and pressure. For example, $\Delsub{c}H = \pd{H}{\xi\subs{c}}{T,p}$ is the molar differential enthalpy of combustion.
\begin{array}{@{}ll@{}} \tx{vap} & \tx{vaporization, evaporation (l $\ra$ g)}\cr \tx{sub} & \tx{sublimation (s $\ra$ g)}\cr \tx{fus} & \tx{melting, fusion (s $\ra$ l)}\cr \tx{trs} & \tx{transition between two phases}\cr \tx{mix} & \tx{mixing of fluids}\cr \tx{sol} & \tx{solution of a solute in solvent}\cr \tx{dil} & \tx{dilution of a solution}\cr \tx{ads} & \tx{adsorption} \cr \tx{dpl} & \tx{displacement} \cr \tx{imm} & \tx{immersion} \cr \tx{r} & \tx{reaction in general} \cr \tx{at} & \tx{atomization} \cr \tx{c} & \tx{combustion reaction} \cr \tx{f} & \tx{formation reaction}\cr \end{array}
D.3 Superscripts
These abbreviations and symbols are used as superscripts to symbols for physical quantities. All but $'$, int, and ref are listed as recommended superscripts in the IUPAC Green Book (E. Richard Cohen et al, Quantities, Units and Symbols in Physical Chemistry, 3rd edition, RSC Publishing, Cambridge, 2007, p. 60).
\begin{array}{@{}ll@{}} \st & \tx{standard} \cr ^* & \tx{pure substance} \cr ' & \tx{Legendre transform of a thermodynamic potential} \cr \infty & \tx{infinite dilution} \cr \tx{id} & \tx{ideal} \cr \tx{int} & \tx{integral} \cr \textsf{E} & \tx{excess quantity} \cr \tx{ref} & \tx{reference state} \end{array} | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/15%3A_Appendices/15.04%3A_Appendix_D-_Miscellaneous_Abbreviations_and_Symbols.txt |
$\newcommand{\tx}[1]{\text{#1}} % text in math mode$
$\newcommand{\subs}[1]{_{\text{#1}}} % subscript text$
$\newcommand{\sups}[1]{^{\text{#1}}} % superscript text$
$\newcommand{\st}{^\circ} % standard state symbol$
$\newcommand{\id}{^{\text{id}}} % ideal$
$\newcommand{\rf}{^{\text{ref}}} % reference state$
$\newcommand{\units}[1]{\mbox{\thinspace#1}}$
$\newcommand{\K}{\units{K}} % kelvins$
$\newcommand{\degC}{^\circ\text{C}} % degrees Celsius$
$\newcommand{\br}{\units{bar}} % bar (\bar is already defined)$
$\newcommand{\Pa}{\units{Pa}}$
$\newcommand{\mol}{\units{mol}} % mole$
$\newcommand{\V}{\units{V}} % volts$
$\newcommand{\timesten}[1]{\mbox{\,\times\,10^{#1}}}$
$\newcommand{\per}{^{-1}} % minus one power$
$\newcommand{\m}{_{\text{m}}} % subscript m for molar quantity$
$\newcommand{\CVm}{C_{V,\text{m}}} % molar heat capacity at const.V$
$\newcommand{\Cpm}{C_{p,\text{m}}} % molar heat capacity at const.p$
$\newcommand{\kT}{\kappa_T} % isothermal compressibility$
$\newcommand{\A}{_{\text{A}}} % subscript A for solvent or state A$
$\newcommand{\B}{_{\text{B}}} % subscript B for solute or state B$
$\newcommand{\bd}{_{\text{b}}} % subscript b for boundary or boiling point$
$\newcommand{\C}{_{\text{C}}} % subscript C$
$\newcommand{\f}{_{\text{f}}} % subscript f for freezing point$
$\newcommand{\mA}{_{\text{m},\text{A}}} % subscript m,A (m=molar)$
$\newcommand{\mB}{_{\text{m},\text{B}}} % subscript m,B (m=molar)$
$\newcommand{\mi}{_{\text{m},i}} % subscript m,i (m=molar)$
$\newcommand{\fA}{_{\text{f},\text{A}}} % subscript f,A (for fr. pt.)$
$\newcommand{\fB}{_{\text{f},\text{B}}} % subscript f,B (for fr. pt.)$
$\newcommand{\xbB}{_{x,\text{B}}} % x basis, B$
$\newcommand{\xbC}{_{x,\text{C}}} % x basis, C$
$\newcommand{\cbB}{_{c,\text{B}}} % c basis, B$
$\newcommand{\mbB}{_{m,\text{B}}} % m basis, B$
$\newcommand{\kHi}{k_{\text{H},i}} % Henry's law constant, x basis, i$
$\newcommand{\kHB}{k_{\text{H,B}}} % Henry's law constant, x basis, B$
$\newcommand{\arrow}{\,\rightarrow\,} % right arrow with extra spaces$
$\newcommand{\arrows}{\,\rightleftharpoons\,} % double arrows with extra spaces$
$\newcommand{\ra}{\rightarrow} % right arrow (can be used in text mode)$
$\newcommand{\eq}{\subs{eq}} % equilibrium state$
$\newcommand{\onehalf}{\textstyle\frac{1}{2}\D} % small 1/2 for display equation$
$\newcommand{\sys}{\subs{sys}} % system property$
$\newcommand{\sur}{\sups{sur}} % surroundings$
$\renewcommand{\in}{\sups{int}} % internal$
$\newcommand{\lab}{\subs{lab}} % lab frame$
$\newcommand{\cm}{\subs{cm}} % center of mass$
$\newcommand{\rev}{\subs{rev}} % reversible$
$\newcommand{\irr}{\subs{irr}} % irreversible$
$\newcommand{\fric}{\subs{fric}} % friction$
$\newcommand{\diss}{\subs{diss}} % dissipation$
$\newcommand{\el}{\subs{el}} % electrical$
$\newcommand{\cell}{\subs{cell}} % cell$
$\newcommand{\As}{A\subs{s}} % surface area$
$\newcommand{\E}{^\mathsf{E}} % excess quantity (superscript)$
$\newcommand{\allni}{\{n_i \}} % set of all n_i$
$\newcommand{\sol}{\hspace{-.1em}\tx{(sol)}}$
$\newcommand{\solmB}{\tx{(sol,\,m\B)}}$
$\newcommand{\dil}{\tx{(dil)}}$
$\newcommand{\sln}{\tx{(sln)}}$
$\newcommand{\mix}{\tx{(mix)}}$
$\newcommand{\rxn}{\tx{(rxn)}}$
$\newcommand{\expt}{\tx{(expt)}}$
$\newcommand{\solid}{\tx{(s)}}$
$\newcommand{\liquid}{\tx{(l)}}$
$\newcommand{\gas}{\tx{(g)}}$
$\newcommand{\pha}{\alpha} % phase alpha$
$\newcommand{\phb}{\beta} % phase beta$
$\newcommand{\phg}{\gamma} % phase gamma$
$\newcommand{\aph}{^{\alpha}} % alpha phase superscript$
$\newcommand{\bph}{^{\beta}} % beta phase superscript$
$\newcommand{\gph}{^{\gamma}} % gamma phase superscript$
$\newcommand{\aphp}{^{\alpha'}} % alpha prime phase superscript$
$\newcommand{\bphp}{^{\beta'}} % beta prime phase superscript$
$\newcommand{\gphp}{^{\gamma'}} % gamma prime phase superscript$
$\newcommand{\apht}{\small\aph} % alpha phase tiny superscript$
$\newcommand{\bpht}{\small\bph} % beta phase tiny superscript$
$\newcommand{\gpht}{\small\gph} % gamma phase tiny superscript$
$\newcommand{\upOmega}{\Omega}$
$\newcommand{\dif}{\mathop{}\!\mathrm{d}} % roman d in math mode, preceded by space$
$\newcommand{\Dif}{\mathop{}\!\mathrm{D}} % roman D in math mode, preceded by space$
$\newcommand{\df}{\dif\hspace{0.05em} f} % df$
$\newcommand{\dBar}{\mathop{}\!\mathrm{d}\hspace-.3em\raise1.05ex{\Rule{.8ex}{.125ex}{0ex}}} % inexact differential$
$\newcommand{\dq}{\dBar q} % heat differential$
$\newcommand{\dw}{\dBar w} % work differential$
$\newcommand{\dQ}{\dBar Q} % infinitesimal charge$
$\newcommand{\dx}{\dif\hspace{0.05em} x} % dx$
$\newcommand{\dt}{\dif\hspace{0.05em} t} % dt$
$\newcommand{\difp}{\dif\hspace{0.05em} p} % dp$
$\newcommand{\Del}{\Delta}$
$\newcommand{\Delsub}[1]{\Delta_{\text{#1}}}$
$\newcommand{\pd}[3]{(\partial #1 / \partial #2 )_{#3}} % \pd{}{}{} - partial derivative, one line$
$\newcommand{\Pd}[3]{\left( \dfrac {\partial #1} {\partial #2}\right)_{#3}} % Pd{}{}{} - Partial derivative, built-up$
$\newcommand{\bpd}[3]{[ \partial #1 / \partial #2 ]_{#3}}$
$\newcommand{\bPd}[3]{\left[ \dfrac {\partial #1} {\partial #2}\right]_{#3}}$
$\newcommand{\dotprod}{\small\bullet}$
$\newcommand{\fug}{f} % fugacity$
$\newcommand{\g}{\gamma} % solute activity coefficient, or gamma in general$
$\newcommand{\G}{\varGamma} % activity coefficient of a reference state (pressure factor)$
$\newcommand{\ecp}{\widetilde{\mu}} % electrochemical or total potential$
$\newcommand{\Eeq}{E\subs{cell, eq}} % equilibrium cell potential$
$\newcommand{\Ej}{E\subs{j}} % liquid junction potential$
$\newcommand{\mue}{\mu\subs{e}} % electron chemical potential$
$\newcommand{\defn}{\,\stackrel{\mathrm{def}}{=}\,} % "equal by definition" symbol$
$\newcommand{\D}{\displaystyle} % for a line in built-up$
$\newcommand{\s}{\smash[b]} % use in equations with conditions of validity$
$\newcommand{\cond}[1]{\-2.5pt]{}\tag*{#1}}$ $\newcommand{\nextcond}[1]{\[-5pt]{}\tag*{#1}}$ $\newcommand{\R}{8.3145\units{J\,K\per\,mol\per}} % gas constant value$ $\newcommand{\Rsix}{8.31447\units{J\,K\per\,mol\per}} % gas constant value - 6 sig figs$ $\newcommand{\jn}{\hspace3pt\lower.3ex{\Rule{.6pt}{2ex}{0ex}}\hspace3pt}$ $\newcommand{\ljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}} \hspace3pt}$ $\newcommand{\lljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace1.4pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace3pt}$ E.1 Derivatives Let $f$ be a function of the variable $x$, and let $\Del f$ be the change in $f$ when $x$ changes by $\Del x$. Then the derivative $\df/\dx$ is the ratio $\Del f/\Del x$ in the limit as $\Del x$ approaches zero. The derivative $\df/\dx$ can also be described as the rate at which $f$ changes with $x$, and as the slope of a curve of $f$ plotted as a function of $x$. The following is a short list of formulas likely to be needed. In these formulas, $u$ and $v$ are arbitrary functions of $x$, and $a$ is a constant. \begin{align*} & \frac{\dif(u^a)}{\dx} = au^{a-1}\frac{\dif u}{\dx} \cr & \frac{\dif (uv)}{\dx} = u\frac{\dif v}{\dx}+v\frac{\dif u}{\dx} \cr & \frac{\dif(u/v)}{\dx} = \left( \frac{1}{v^2} \right) \left( v\frac{\dif u}{\dx}-u\frac{\dif v}{\dx} \right) \cr & \frac{\dif\ln(ax)}{\dx} = \frac{1}{x} \cr & \frac{\dif(e^{ax})}{\dx} = ae^{ax} \cr & \frac{\df(u)}{\dx} = \frac{\df(u)}{\dif u}\cdot\frac{\dif u}{\dx} \end{align*} E.2 Partial Derivatives If $f$ is a function of the independent variables $x$, $y$, and $z$, the partial derivative $\pd{f}{x}{y,z}$ is the derivative $\df/\dx$ with $y$ and $z$ held constant. It is important in thermodynamics to indicate the variables that are held constant, as $\pd{f}{x}{y,z}$ is not necessarily equal to $\pd{f}{x}{a,b}$ where $a$ and $b$ are variables different from $y$ and $z$. The variables shown at the bottom of a partial derivative should tell you which variables are being used as the independent variables. For example, if the partial derivative is $\D\Pd{f}{y}{a,b}$ then $f$ is being treated as a function of $y$, $a$, and $b$. E.3 Integrals Let $f$ be a function of the variable $x$. Imagine the range of $x$ between the limits $x'$ and $x''$ to be divided into many small increments of size $\Del x_i (i = 1, 2, \ldots)$. Let $f_i$ be the value of $f$ when $x$ is in the middle of the range of the $i$th increment. Then the integral \[ \int_{x'}^{x''} \!\! f \dx is the sum $\sum_i f_i \Del x_i$ in the limit as each $\Del x_i$ approaches zero and the number of terms in the sum approaches infinity. The integral is also the area under a curve of $f$ plotted as a function of $x$, measured from $x = x'$ to $x = x''$. The function $f$ is the integrand, which is integrated over the integration variable $x$.
This e-book uses the following integrals: \begin{align*} & \int_{x'}^{x''} \!\! \dx = x''-x' \cr & \int_{x'}^{x''}\frac{\dx}{x} = \ln \left| \frac{x''}{x'} \right| \cr & \int_{x'}^{x''} \!\! x^a \dx = \frac{1}{a+1} \left[ (x'')^{a+1} - (x')^{a+1} \right] \qquad \tx{($a$ is a constant other than $-1$)} \cr & \int_{x'}^{x''}\!\!\frac{\dx}{ax+b} = \frac{1}{a}\ln\left|\frac{ax''+b}{ax'+b}\right| \qquad \tx{($a$ is a constant)} \end{align*} Here are examples of the use of the expression for the third integral with $a$ set equal to $1$ and to $-2$: \begin{align*} & \int_{x'}^{x''} \!\! x \dx = \frac{1}{2}\left[(x'')^2-(x')^2\right] \cr & \int_{x'}^{x''} \! \frac{\dx}{x^2} = -\left( \frac{1}{x''} - \frac{1}{x'} \right) \end{align*}
E.4 Line Integrals
A line integral is an integral with an implicit single integration variable that constraints the integration to a path.
The most frequently-seen line integral in this e-book, $\int\!p\dif V$, will serve as an example. The integral can be evaluated in three different ways:
1. The integrand $p$ can be expressed as a function of the integration variable $V$, so that there is only one variable. For example, if $p$ equals $c/V$ where $c$ is a constant, the line integral is given by $\int\!p\dif V=c\int_{V_1}^{V_2}(1/V)\dif V = c\ln(V_2/V_1)$.
2. If $p$ and $V$ can be written as functions of another variable, such as time, that coordinates their values so that they follow the desired path, this new variable becomes the integration variable.
3. The desired path can be drawn as a curve on a plot of $p$ versus $V$; then $\int\!p\dif V$ is equal in value to the area under the curve. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/15%3A_Appendices/15.05%3A_Appendix_E-_Calculus_Review.txt |
$\newcommand{\tx}[1]{\text{#1}} % text in math mode$
$\newcommand{\subs}[1]{_{\text{#1}}} % subscript text$
$\newcommand{\sups}[1]{^{\text{#1}}} % superscript text$
$\newcommand{\st}{^\circ} % standard state symbol$
$\newcommand{\id}{^{\text{id}}} % ideal$
$\newcommand{\rf}{^{\text{ref}}} % reference state$
$\newcommand{\units}[1]{\mbox{\thinspace#1}}$
$\newcommand{\K}{\units{K}} % kelvins$
$\newcommand{\degC}{^\circ\text{C}} % degrees Celsius$
$\newcommand{\br}{\units{bar}} % bar (\bar is already defined)$
$\newcommand{\Pa}{\units{Pa}}$
$\newcommand{\mol}{\units{mol}} % mole$
$\newcommand{\V}{\units{V}} % volts$
$\newcommand{\timesten}[1]{\mbox{\,\times\,10^{#1}}}$
$\newcommand{\per}{^{-1}} % minus one power$
$\newcommand{\m}{_{\text{m}}} % subscript m for molar quantity$
$\newcommand{\CVm}{C_{V,\text{m}}} % molar heat capacity at const.V$
$\newcommand{\Cpm}{C_{p,\text{m}}} % molar heat capacity at const.p$
$\newcommand{\kT}{\kappa_T} % isothermal compressibility$
$\newcommand{\A}{_{\text{A}}} % subscript A for solvent or state A$
$\newcommand{\B}{_{\text{B}}} % subscript B for solute or state B$
$\newcommand{\bd}{_{\text{b}}} % subscript b for boundary or boiling point$
$\newcommand{\C}{_{\text{C}}} % subscript C$
$\newcommand{\f}{_{\text{f}}} % subscript f for freezing point$
$\newcommand{\mA}{_{\text{m},\text{A}}} % subscript m,A (m=molar)$
$\newcommand{\mB}{_{\text{m},\text{B}}} % subscript m,B (m=molar)$
$\newcommand{\mi}{_{\text{m},i}} % subscript m,i (m=molar)$
$\newcommand{\fA}{_{\text{f},\text{A}}} % subscript f,A (for fr. pt.)$
$\newcommand{\fB}{_{\text{f},\text{B}}} % subscript f,B (for fr. pt.)$
$\newcommand{\xbB}{_{x,\text{B}}} % x basis, B$
$\newcommand{\xbC}{_{x,\text{C}}} % x basis, C$
$\newcommand{\cbB}{_{c,\text{B}}} % c basis, B$
$\newcommand{\mbB}{_{m,\text{B}}} % m basis, B$
$\newcommand{\kHi}{k_{\text{H},i}} % Henry's law constant, x basis, i$
$\newcommand{\kHB}{k_{\text{H,B}}} % Henry's law constant, x basis, B$
$\newcommand{\arrow}{\,\rightarrow\,} % right arrow with extra spaces$
$\newcommand{\arrows}{\,\rightleftharpoons\,} % double arrows with extra spaces$
$\newcommand{\ra}{\rightarrow} % right arrow (can be used in text mode)$
$\newcommand{\eq}{\subs{eq}} % equilibrium state$
$\newcommand{\onehalf}{\textstyle\frac{1}{2}\D} % small 1/2 for display equation$
$\newcommand{\sys}{\subs{sys}} % system property$
$\newcommand{\sur}{\sups{sur}} % surroundings$
$\renewcommand{\in}{\sups{int}} % internal$
$\newcommand{\lab}{\subs{lab}} % lab frame$
$\newcommand{\cm}{\subs{cm}} % center of mass$
$\newcommand{\rev}{\subs{rev}} % reversible$
$\newcommand{\irr}{\subs{irr}} % irreversible$
$\newcommand{\fric}{\subs{fric}} % friction$
$\newcommand{\diss}{\subs{diss}} % dissipation$
$\newcommand{\el}{\subs{el}} % electrical$
$\newcommand{\cell}{\subs{cell}} % cell$
$\newcommand{\As}{A\subs{s}} % surface area$
$\newcommand{\E}{^\mathsf{E}} % excess quantity (superscript)$
$\newcommand{\allni}{\{n_i \}} % set of all n_i$
$\newcommand{\sol}{\hspace{-.1em}\tx{(sol)}}$
$\newcommand{\solmB}{\tx{(sol,\,m\B)}}$
$\newcommand{\dil}{\tx{(dil)}}$
$\newcommand{\sln}{\tx{(sln)}}$
$\newcommand{\mix}{\tx{(mix)}}$
$\newcommand{\rxn}{\tx{(rxn)}}$
$\newcommand{\expt}{\tx{(expt)}}$
$\newcommand{\solid}{\tx{(s)}}$
$\newcommand{\liquid}{\tx{(l)}}$
$\newcommand{\gas}{\tx{(g)}}$
$\newcommand{\pha}{\alpha} % phase alpha$
$\newcommand{\phb}{\beta} % phase beta$
$\newcommand{\phg}{\gamma} % phase gamma$
$\newcommand{\aph}{^{\alpha}} % alpha phase superscript$
$\newcommand{\bph}{^{\beta}} % beta phase superscript$
$\newcommand{\gph}{^{\gamma}} % gamma phase superscript$
$\newcommand{\aphp}{^{\alpha'}} % alpha prime phase superscript$
$\newcommand{\bphp}{^{\beta'}} % beta prime phase superscript$
$\newcommand{\gphp}{^{\gamma'}} % gamma prime phase superscript$
$\newcommand{\apht}{\small\aph} % alpha phase tiny superscript$
$\newcommand{\bpht}{\small\bph} % beta phase tiny superscript$
$\newcommand{\gpht}{\small\gph} % gamma phase tiny superscript$
$\newcommand{\upOmega}{\Omega}$
$\newcommand{\dif}{\mathop{}\!\mathrm{d}} % roman d in math mode, preceded by space$
$\newcommand{\Dif}{\mathop{}\!\mathrm{D}} % roman D in math mode, preceded by space$
$\newcommand{\df}{\dif\hspace{0.05em} f} % df$
$\newcommand{\dBar}{\mathop{}\!\mathrm{d}\hspace-.3em\raise1.05ex{\Rule{.8ex}{.125ex}{0ex}}} % inexact differential$
$\newcommand{\dq}{\dBar q} % heat differential$
$\newcommand{\dw}{\dBar w} % work differential$
$\newcommand{\dQ}{\dBar Q} % infinitesimal charge$
$\newcommand{\dx}{\dif\hspace{0.05em} x} % dx$
$\newcommand{\dt}{\dif\hspace{0.05em} t} % dt$
$\newcommand{\difp}{\dif\hspace{0.05em} p} % dp$
$\newcommand{\Del}{\Delta}$
$\newcommand{\Delsub}[1]{\Delta_{\text{#1}}}$
$\newcommand{\pd}[3]{(\partial #1 / \partial #2 )_{#3}} % \pd{}{}{} - partial derivative, one line$
$\newcommand{\Pd}[3]{\left( \dfrac {\partial #1} {\partial #2}\right)_{#3}} % Pd{}{}{} - Partial derivative, built-up$
$\newcommand{\bpd}[3]{[ \partial #1 / \partial #2 ]_{#3}}$
$\newcommand{\bPd}[3]{\left[ \dfrac {\partial #1} {\partial #2}\right]_{#3}}$
$\newcommand{\dotprod}{\small\bullet}$
$\newcommand{\fug}{f} % fugacity$
$\newcommand{\g}{\gamma} % solute activity coefficient, or gamma in general$
$\newcommand{\G}{\varGamma} % activity coefficient of a reference state (pressure factor)$
$\newcommand{\ecp}{\widetilde{\mu}} % electrochemical or total potential$
$\newcommand{\Eeq}{E\subs{cell, eq}} % equilibrium cell potential$
$\newcommand{\Ej}{E\subs{j}} % liquid junction potential$
$\newcommand{\mue}{\mu\subs{e}} % electron chemical potential$
$\newcommand{\defn}{\,\stackrel{\mathrm{def}}{=}\,} % "equal by definition" symbol$
$\newcommand{\D}{\displaystyle} % for a line in built-up$
$\newcommand{\s}{\smash[b]} % use in equations with conditions of validity$
$\newcommand{\cond}[1]{\[-2.5pt]{}\tag*{#1}}$
$\newcommand{\nextcond}[1]{\[-5pt]{}\tag*{#1}}$
$\newcommand{\R}{8.3145\units{J\,K\per\,mol\per}} % gas constant value$
$\newcommand{\Rsix}{8.31447\units{J\,K\per\,mol\per}} % gas constant value - 6 sig figs$
$\newcommand{\jn}{\hspace3pt\lower.3ex{\Rule{.6pt}{2ex}{0ex}}\hspace3pt}$
$\newcommand{\ljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}} \hspace3pt}$
$\newcommand{\lljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace1.4pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace3pt}$
A state function is a property of a thermodynamic system whose value at any given instant depends only on the state of the system at that instant (Sec. 2.4).
F.1 Differentials
The differential $\df$ of a state function $f$ is an infinitesimal change of $f$. Since the value of a state function by definition depends only on the state of the system, integrating $\df$ between an initial state $1$ and a final state $2$ yields the change in $f$, and this change is independent of the path: $\int_{f_1}^{f_2}\!\df=f_2-f_1=\Del f \tag{F.1.1}$ A differential with this property is called an exact differential. The differential of a state function is always exact.
F.2 Total Differential
A state function $f$ treated as a dependent variable is a function of a certain number of independent variables that are also state functions. The total differential of $f$ is $\df$ expressed in terms of the differentials of the independent variables and has the form $\df = \Pd{f}{x}{}\dx + \Pd{f}{y}{}\dif y + \Pd{f}{z}{}\dif z + \ldots \tag{F.2.1}$ There are as many terms in the expression on the right side as there are independent variables. Each partial derivative in the expression has all independent variables held constant except the variable shown in the denominator.
Figure F.1 interprets this expression for a function $f$ of the two independent variables $x$ and $y$. The shaded plane represents a small element of the surface $f = f(x,y)$.
Consider a system with three independent variables. If we choose these independent variables to be $x$, $y$, and $z$, the total differential of the dependent state function $f$ takes the form $\df = a\dx + b\dif y + c\dif z \tag{F.2.2}$ where we can identify the coefficients as $a=\Pd{f}{x}{y,z} \qquad b=\Pd{f}{y}{x,z} \qquad c=\Pd{f}{z}{x,y} \tag{F.2.3}$ These coefficients are themselves, in general, functions of the independent variables and may be differentiated to give mixed second partial derivatives; for example: $\Pd{a}{y}{x,z} = \frac{\partial^2 f}{\partial y \partial x} \qquad \Pd{b}{x}{y,z} = \frac{\partial^2 f}{\partial x \partial y} \tag{F.2.4}$ The second partial derivative $\partial^2 f/\partial y\partial x$, for instance, is the partial derivative with respect to $y$ of the partial derivative of $f$ with respect to $x$. It is a theorem of calculus that if a function $f$ is single valued and has continuous derivatives, the order of differentiation in a mixed derivative is immaterial. Therefore the mixed derivatives $\partial^2 f/\partial y\partial x$ and $\partial^2 f/\partial x\partial y$, evaluated for the system in any given state, are equal: $\Pd{a}{y}{x,z} = \Pd{b}{x}{y,z} \tag{F.2.5}$ The general relation that applies to a function of any number of independent variables is $\Pd{X}{y}{} = \Pd{Y}{x}{} \tag{F.2.6}$ where $x$ and $y$ are any two of the independent variables, $X$ is $\partial f/\partial x$, $Y$ is $\partial \f/\partial y$, and each partial derivative has all independent variables held constant except the variable shown in the denominator. This general relation is the Euler reciprocity relation, or reciprocity relation for short. A necessary and sufficient condition for $\df$ to be an exact differential is that the reciprocity relation is satisfied for each pair of independent variables.
F.3 Integration of a Total Differential
If the coefficients of the total differential of a dependent variable are known as functions of the independent variables, the expression for the total differential may be integrated to obtain an expression for the dependent variable as a function of the independent variables.
For example, suppose the total differential of the state function $f(x,y,z)$ is given by Eq. F.2.2 and the coefficients are known functions $a(x,y,z)$, $b(x,y,z)$, and $c(x,y,z)$. Because $f$ is a state function, its change between $f(0,0,0)$ and $f(x',y',z')$ is independent of the integration path taken between these two states. A convenient path would be one with the following three segments:
1. The expression for $f(x,y,z)$ is then the sum of the three integrals and a constant of integration.
Here is an example of this procedure applied to the total differential $\df = (2xy)\dx + (x^2+z)\dif y + (y-9z^2)\dif z \tag{F.3.1}$ An expression for the function $f$ in this example is given by the sum $\begin{split} f & = \int_0^{x'}\!(2x\cdot 0)\dx + \int_0^{y'}\![(x')^2+0]\dif y + \int_0^{z'}\!(y'-9z^2)\dif z + C\cr & = 0 + x^2y + ( yz-9z^3/3 ) + C\cr & = x^2y + yz - 3z^3 + C \end{split} \tag{F.3.2}$ where primes are omitted on the second and third lines because the expressions are supposed to apply to any values of $x$, $y$, and $z$. $C$ is an integration constant. You can verify that the third line of Eq. F.3.2 gives the correct expression for $f$ by taking partial derivatives with respect to $x$, $y$, and $z$ and comparing with Eq. F.3.1.
A different kind of integration can be used to express a dependent extensive property in terms of independent extensive properties. An extensive property of a thermodynamic system is one that is additive, and an intensive property is one that is not additive and has the same value everywhere in a homogeneous region (Sec. 2.1.1). Suppose we have a state function $f$ that is an extensive property with the total differential $\df = a\dx + b\dif y + c\dif z + \ldots \tag{F.3.3}$ where the independent variables $x,y,z,\ldots$ are extensive and the coefficients $a,b,c,\ldots$ are intensive. If the independent variables include those needed to describe an open system (for example, the amounts of the substances), then it is possible to integrate both sides of the equation from a lower limit of zero for each of the extensive functions while holding the intensive functions constant: $\int_0^{f'}\!\!\df = a\int_0^{x'}\!\!\dx + b\int_0^{y'}\!\!\dif y + c\int_0^{z'}\!\!\dif z + \ldots \tag{F.3.4}$ $f' = ax'+by'+cz'+\ldots \tag{F.3.5}$ Note that a term of the form $c\dif u$ where $u$ is intensive becomes zero when integrated with intensive functions held constant, because $\dif u$ is this case is zero.
F.4 Legendre Transforms
A Legendre transform of a state function is a linear change of one or more of the independent variables made by subtracting products of conjugate variables.
To understand how this works, consider a state function $f$ whose total differential is given by $\df=a\dx+b\dif y+c\dif z \tag{F.4.1}$ In the expression on the right side, $x$, $y$, and $z$ are being treated as the independent variables. The pairs $a$ and $x$, $b$ and $y$, and $c$ and $z$ are conjugate pairs. That is, $a$ and $x$ are conjugates, $b$ and $y$ are conjugates, and $c$ and $z$ are conjugates.
For the first example of a Legendre transform, we define a new state function $f_1$ by subtracting the product of the conjugate variables $a$ and $x$: $f_1 \defn f-ax \tag{F.4.2}$ The function $f_1$ is a Legendre transform of $f$. We take the differential of Eq. F.4.2 $\df_1 = \df - a\dx - x\dif a \tag{F.4.3}$ and substitute for $\df$ from Eq. F.4.1: $\begin{split} \df_1 & = (a\dx+b\dif y+c\dif z) - a\dx - x\dif a \cr & = -x\dif a + b\dif y + c\dif z \end{split} \tag{F.4.4}$ Equation F.4.4 gives the total differential of $f_1$ with $a$, $y$, and $z$ as the independent variables. The functions $x$ and $a$ have switched places as independent variables. What we did in order to let $a$ replace $x$ as an independent variable was to subtract from $f$ the product of the conjugate variables $a$ and $x$.
Because the right side of Eq. F.4.4 is an expression for the total differential of the state function $f_1$, we can use the expression to identify the coefficients as partial derivatives of $f_1$ with respect to the new set of independent variables: $-x = \Pd{f_1}{a}{y,z} \qquad b = \Pd{f_1}{y}{a,z} \qquad c = \Pd{f_1}{z}{a,y} \tag{F.4.5}$ We can also use Eq. F.4.4 to write new reciprocity relations, such as $-\Pd{x}{y}{a,z} = \Pd{b}{a}{y,z} \tag{F.4.6}$
We can make other Legendre transforms of $f$ by subtracting one or more products of conjugate variables. A second example of a Legendre transform is $f_2 \defn f - by - cz \tag{F.4.7}$ whose total differential is $\begin{split} \df_2 & = \df - b\dif y - y\dif b - c\dif z - z\dif c \cr & = a\dx - y\dif b - z\dif c \end{split} \tag{F.4.8}$ Here $b$ has replaced $y$ and $c$ has replaced $z$ as independent variables. Again, we can identify the coefficients as partial derivatives and write new reciprocity relations.
If we have an algebraic expression for a state function as a function of independent variables, then a Legendre transform preserves all the information contained in that expression. To illustrate this, we can use the state function $f$ and its Legendre transform $f_2$ described above. Suppose we have an expression for $f(x,y,z)$—this is $f$ expressed as a function of the independent variables $x$, $y$, and $z$. Then by taking partial derivatives of this expression, we can find according to Eq. F.2.3 expressions for the functions $a(x,y,z)$, $b(x,y,z)$, and $c(x,y,z)$.
Now we perform the Legendre transform of Eq. F.4.7: $f_2=f-by-cz$ with total differential $\df_2=a\dx-y\dif b-z\dif c$ (Eq. F.4.8). The independent variables have been changed from $x$, $y$, and $z$ to $x$, $b$, and $c$.
We want to find an expression for $f_2$ as a function of these new variables, using the information available from the original function $f(x,y,z)$. To do this, we eliminate $z$ from the known functions $b(x,y,z)$ and $c(x,y,z)$ and solve for $y$ as a function of $x$, $b$, and $c$. We also eliminate $y$ from $b(x,y,z)$ and $c(x,y,z)$ and solve for $z$ as a function of $x$, $b$, and $c$. This gives us expressions for $y(x,b,c)$ and $z(x,b,c)$ which we substitute into the expression for $f(x,y,z)$, turning it into the function $f(x,b,c)$. Finally, we use the functions of the new variables to obtain an expression for $f_2(x,b,c)=f(x,b,c)-by(x,b,c)-cz(x,b,c)$.
The original expression for $f(x,y,z)$ and the new expression for $f_2(x,b,c)$ contain the same information. We could take the expression for $f_2(x,b,c)$ and, by following the same procedure with the Legendre transform $f=f_2+by+cz$, retrieve the expression for $f(x,y,z)$. Thus no information is lost during a Legendre transform. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/15%3A_Appendices/15.06%3A_Appendix_F-_Mathematical_Properties_of_State_Functions.txt |
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$\newcommand{\phg}{\gamma} % phase gamma$
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$\newcommand{\dq}{\dBar q} % heat differential$
$\newcommand{\dw}{\dBar w} % work differential$
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$\newcommand{\dx}{\dif\hspace{0.05em} x} % dx$
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$\newcommand{\R}{8.3145\units{J\,K\per\,mol\per}} % gas constant value$
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The values in these tables are for a temperature of 298.15 K (25.00 $\degC$) and the standard pressure $p\st = 1\units{bar}$. Solute standard states are based on molality. A crystalline solid is denoted by cr.
Most of the values in this table come from a project of the Committee on Data for Science and Technology (CODATA) to establish a set of recommended, internally consistent values of thermodynamic properties. The values of $\Delsub{f}H\st$ and $S\m\st$ shown with uncertainties are values recommended by CODATA (J. D. Cox, D. D. Wagman, and V. A. Medvedev, CODATA Key Values for Thermodynamics, Hemisphere Publishing Corp., New York, 1989).
\begin{array}{lccc} \hline \textbf{Inorganic substance} & \boldsymbol{\D{\frac{\Delsub{f}H\st}{\textbf{kJ mol}^{-1}}}} & \boldsymbol{\D{\frac{S\m\st}{\textbf{J K}^{-1}\textbf{ mol}^{-1}}}} & \boldsymbol{\D{\frac{\Delsub{f}G\st}{\textbf{kJ mol}^{-1}}}} \ \hline \tx{Ag(cr)} & 0 & 42.55\pm0.20 & 0 \ \tx{AgCl(cr)} & -127.01\pm0.05 & 96.25\pm0.20 & -109.77 \ \tx{C(cr, graphite)} & 0 & 5.74\pm0.10 & 0 \ \tx{CO(g)} & -110.53\pm0.17 & 197.660\pm0.004 & -137.17 \ \tx{CO$_2$(g)} & -393.51\pm0.13 & 213.785\pm0.010 & -394.41 \ \tx{Ca(cr)} & 0 & 41.59\pm0.40 & 0 \ \tx{CaCO$_3$(cr, calcite)} & -1206.9 & 92.9 & -1128.8 \ \tx{CaO(cr)} & -634.92\pm0.90 & 38.1\pm0.4 & -603.31 \ \tx{Cl$_2$(g)} & 0 & 223.081\pm0.010 & 0 \ \tx{F$_2$(g)} & 0 & 202.791\pm0.005 & 0 \ \tx{H$_2$(g)} & 0 & 130.680\pm0.003 & 0 \ \tx{HCl(g)} & -92.31\pm0.10 & 186.902\pm0.005 & -95.30 \ \tx{HF(g)} & -273.30\pm0.70 & 173.779\pm0.003 & -275.40 \ \tx{HI(g)} & 26.50\pm0.10 & 206.590\pm0.004 & 1.70 \ \tx{H$_2$O(l)} & -285.830\pm0.040 & 69.95\pm0.03 & -237.16 \ \tx{H$_2$O(g)} & -241.826\pm0.040 & 188.835\pm0.010 & -228.58 \ \tx{H$_2$S(g)} & -20.6\pm0.5 & 205.81\pm0.05 & -33.44 \ \tx{Hg(l)} & 0 & 75.90\pm0.12 & 0 \ \tx{Hg(g)} & 61.38\pm0.04 & 174.971\pm0.005 & 31.84 \ \tx{HgO(cr, red)} & -90.79\pm0.12 & 70.25\pm0.30 & -58.54 \ \tx{Hg$_2$Cl$_2$(cr)} & -265.37\pm0.40 & 191.6\pm0.8 & -210.72 \ \tx{I$_2$(cr)} & 0 & 116.14\pm0.30 & 0 \ \tx{K(cr)} & 0 & 64.68\pm0.20 & 0 \ \tx{KI(cr)} & -327.90 & 106.37 & -323.03 \ \tx{KOH(cr)} & -424.72 & 78.90 & -378.93 \ \tx{N$_2$(g)} & 0 & 191.609\pm0.004 & 0 \ \tx{NH$_3$(g)} & -45.94\pm0.35 & 192.77\pm0.05 & -16.41 \ \tx{NO$_2$(g)} & 33.10 & 240.04 & 51.22 \ \tx{N$_2$O$_4$(g)} & 9.08 & 304.38 & 97.72 \ \tx{Na(cr)} & 0 & 51.30\pm0.20 & 0 \ \tx{NaCl(cr)} & -411.12 & 72.11 & -384.02 \ \tx{O$_2$(g)} & 0 & 205.152\pm0.005 & 0 \ \tx{O$_3$(g)} & 142.67 & 238.92 & 163.14 \ \tx{P(cr, white)} & 0 & 41.09\pm0.25 & 0 \ \tx{S(cr, rhombic)} & 0 & 32.054\pm0.050 & 0 \ \tx{SO$_2$(g)} & -296.81\pm0.20 & 248.223\pm0.050 & -300.09 \ \tx{Si(cr)} & 0 & 18.81\pm0.08 & 0 \ \tx{SiF$_4$(g)} & -1615.0\pm0.8 & 282.76\pm0.50 & -1572.8 \ \tx{SiO$_2$(cr, $\alpha$-quartz)} & -910.7\pm1.0 & 41.46\pm0.20 & -856.3 \ \tx{Zn(cr)} & 0 & 41.63\pm0.15 & 0 \[1mm] \tx{ZnO(cr)} & -350.46\pm0.27 & 43.65\pm0.40 & -320.48 \ \hline \end{array}
\begin{array}{lccc} \hline \textbf{Organic compound} & \boldsymbol{\D{\frac{\Delsub{f}H\st}{\textbf{kJ mol}^{-1}}}} & \boldsymbol{\D{\frac{S\m\st}{\textbf{J K}^{-1}\textbf{ mol}^{-1}}}} & \boldsymbol{\D{\frac{\Delsub{f}G\st}{\textbf{kJ mol}^{-1}}}} \ \hline \tx{CH$_4$(g)} & -74.87 & 186.25 & -50.77 \ \tx{CH$_3$OH(l)} & -238.9 & 127.2 & -166.6 \ \tx{CH$_3$CH$_2$OH(l)} & -277.0 & 159.9 & -173.8 \ \tx{C$_2$H$_2$(g)} & 226.73 & 200.93 & 209.21 \ \tx{C$_2$H$_4$(g)} & 52.47 & 219.32 & 68.43 \ \tx{C$_2$H$_6$(g)} & -83.85 & 229.6 & -32.00 \ \tx{C$_3$H$_8$(g)} & -104.7 & 270.31 & -24.3 \ \tx{C$_6$H$_6$(l, benzene)} & 49.04 & 173.26 & 124.54 \ \hline \end{array}
\begin{array}{lccc} \hline \textbf{Ionic solute} & \boldsymbol{\D{\frac{\Delsub{f}H\st}{\textbf{kJ mol}^{-1}}}} & \boldsymbol{\D{\frac{S\m\st}{\textbf{J K}^{-1}\textbf{ mol}^{-1}}}} & \boldsymbol{\D{\frac{\Delsub{f}G\st}{\textbf{kJ mol}^{-1}}}} \ \hline \tx{Ag$^+$(aq)} & 105.79\pm0.08 & 73.45\pm0.40 & 77.10 \ \tx{CO$_3^{2-}$(aq)} & -675.23\pm0.25 & -50.0\pm1.0 & -527.90 \ \tx{Ca$^{2+}$(aq)} & -543.0\pm1.0 & -56.2\pm1.0 & -552.8 \ \tx{Cl$^-$(aq)} & -167.08\pm0.10 & 56.60\pm0.20 & -131.22 \ \tx{F$^-$(aq)} & -335.35\pm0.65 & -13.8\pm0.8 & -281.52 \ \tx{H$^+$(aq)} & 0 & 0 & 0 \ \tx{HCO$_3^-$(aq)} & -689.93\pm2.0 & 98.4\pm0.5 & -586.90 \ \tx{HS$^-$(aq)} & -16.3\pm1.5 & 67\pm 5 & 12.2 \ \tx{HSO$_4^-$(aq)} & -886.9\pm1.0 & 131.7\pm3.0 & -755.4 \ \tx{Hg$_2^{2+}$(aq)} & 166.87\pm0.50 & 65.74\pm0.80 & 153.57 \ \tx{I$^-$(aq)} & -56.78\pm0.05 & 106.45\pm0.30 & -51.72 \ \tx{K$^+$(aq)} & -252.14\pm0.08 & 101.20\pm0.20 & -282.52 \ \tx{NH$_4^+$(aq)} & -133.26\pm0.25 & 111.17\pm0.40 & -79.40 \ \tx{NO$_3^-$(aq)} & -206.85\pm0.40 & 146.70\pm0.40 & -110.84 \ \tx{Na$^+$(aq)} & -240.34\pm0.06 & 58.45\pm0.15 & -261.90 \ \tx{OH$^-$(aq)} & -230.015\pm0.040 & -10.90\pm0.20 & -157.24 \ \tx{S$^{2-}$(aq)} & 33.1 & -14.6 & 86.0 \ \tx{SO$_4^{2-}$(aq)} & -909.34\pm0.40 & 18.50\pm0.40 & -744.00 \ \hline \end{array} | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/15%3A_Appendices/15.08%3A_Appendix_H-_Standard_Molar_Thermodynamic_Properties.txt |
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$\newcommand{\CVm}{C_{V,\text{m}}} % molar heat capacity at const.V$
$\newcommand{\Cpm}{C_{p,\text{m}}} % molar heat capacity at const.p$
$\newcommand{\kT}{\kappa_T} % isothermal compressibility$
$\newcommand{\A}{_{\text{A}}} % subscript A for solvent or state A$
$\newcommand{\B}{_{\text{B}}} % subscript B for solute or state B$
$\newcommand{\bd}{_{\text{b}}} % subscript b for boundary or boiling point$
$\newcommand{\C}{_{\text{C}}} % subscript C$
$\newcommand{\f}{_{\text{f}}} % subscript f for freezing point$
$\newcommand{\mA}{_{\text{m},\text{A}}} % subscript m,A (m=molar)$
$\newcommand{\mB}{_{\text{m},\text{B}}} % subscript m,B (m=molar)$
$\newcommand{\mi}{_{\text{m},i}} % subscript m,i (m=molar)$
$\newcommand{\fA}{_{\text{f},\text{A}}} % subscript f,A (for fr. pt.)$
$\newcommand{\fB}{_{\text{f},\text{B}}} % subscript f,B (for fr. pt.)$
$\newcommand{\xbB}{_{x,\text{B}}} % x basis, B$
$\newcommand{\xbC}{_{x,\text{C}}} % x basis, C$
$\newcommand{\cbB}{_{c,\text{B}}} % c basis, B$
$\newcommand{\mbB}{_{m,\text{B}}} % m basis, B$
$\newcommand{\kHi}{k_{\text{H},i}} % Henry's law constant, x basis, i$
$\newcommand{\kHB}{k_{\text{H,B}}} % Henry's law constant, x basis, B$
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$\newcommand{\sur}{\sups{sur}} % surroundings$
$\renewcommand{\in}{\sups{int}} % internal$
$\newcommand{\lab}{\subs{lab}} % lab frame$
$\newcommand{\cm}{\subs{cm}} % center of mass$
$\newcommand{\rev}{\subs{rev}} % reversible$
$\newcommand{\irr}{\subs{irr}} % irreversible$
$\newcommand{\fric}{\subs{fric}} % friction$
$\newcommand{\diss}{\subs{diss}} % dissipation$
$\newcommand{\el}{\subs{el}} % electrical$
$\newcommand{\cell}{\subs{cell}} % cell$
$\newcommand{\As}{A\subs{s}} % surface area$
$\newcommand{\E}{^\mathsf{E}} % excess quantity (superscript)$
$\newcommand{\allni}{\{n_i \}} % set of all n_i$
$\newcommand{\sol}{\hspace{-.1em}\tx{(sol)}}$
$\newcommand{\solmB}{\tx{(sol,\,m\B)}}$
$\newcommand{\dil}{\tx{(dil)}}$
$\newcommand{\sln}{\tx{(sln)}}$
$\newcommand{\mix}{\tx{(mix)}}$
$\newcommand{\rxn}{\tx{(rxn)}}$
$\newcommand{\expt}{\tx{(expt)}}$
$\newcommand{\solid}{\tx{(s)}}$
$\newcommand{\liquid}{\tx{(l)}}$
$\newcommand{\gas}{\tx{(g)}}$
$\newcommand{\pha}{\alpha} % phase alpha$
$\newcommand{\phb}{\beta} % phase beta$
$\newcommand{\phg}{\gamma} % phase gamma$
$\newcommand{\aph}{^{\alpha}} % alpha phase superscript$
$\newcommand{\bph}{^{\beta}} % beta phase superscript$
$\newcommand{\gph}{^{\gamma}} % gamma phase superscript$
$\newcommand{\aphp}{^{\alpha'}} % alpha prime phase superscript$
$\newcommand{\bphp}{^{\beta'}} % beta prime phase superscript$
$\newcommand{\gphp}{^{\gamma'}} % gamma prime phase superscript$
$\newcommand{\apht}{\small\aph} % alpha phase tiny superscript$
$\newcommand{\bpht}{\small\bph} % beta phase tiny superscript$
$\newcommand{\gpht}{\small\gph} % gamma phase tiny superscript$
$\newcommand{\upOmega}{\Omega}$
$\newcommand{\dif}{\mathop{}\!\mathrm{d}} % roman d in math mode, preceded by space$
$\newcommand{\Dif}{\mathop{}\!\mathrm{D}} % roman D in math mode, preceded by space$
$\newcommand{\df}{\dif\hspace{0.05em} f} % df$
$\newcommand{\dBar}{\mathop{}\!\mathrm{d}\hspace-.3em\raise1.05ex{\Rule{.8ex}{.125ex}{0ex}}} % inexact differential$
$\newcommand{\dq}{\dBar q} % heat differential$
$\newcommand{\dw}{\dBar w} % work differential$
$\newcommand{\dQ}{\dBar Q} % infinitesimal charge$
$\newcommand{\dx}{\dif\hspace{0.05em} x} % dx$
$\newcommand{\dt}{\dif\hspace{0.05em} t} % dt$
$\newcommand{\difp}{\dif\hspace{0.05em} p} % dp$
$\newcommand{\Del}{\Delta}$
$\newcommand{\Delsub}[1]{\Delta_{\text{#1}}}$
$\newcommand{\pd}[3]{(\partial #1 / \partial #2 )_{#3}} % \pd{}{}{} - partial derivative, one line$
$\newcommand{\Pd}[3]{\left( \dfrac {\partial #1} {\partial #2}\right)_{#3}} % Pd{}{}{} - Partial derivative, built-up$
$\newcommand{\bpd}[3]{[ \partial #1 / \partial #2 ]_{#3}}$
$\newcommand{\bPd}[3]{\left[ \dfrac {\partial #1} {\partial #2}\right]_{#3}}$
$\newcommand{\dotprod}{\small\bullet}$
$\newcommand{\fug}{f} % fugacity$
$\newcommand{\g}{\gamma} % solute activity coefficient, or gamma in general$
$\newcommand{\G}{\varGamma} % activity coefficient of a reference state (pressure factor)$
$\newcommand{\ecp}{\widetilde{\mu}} % electrochemical or total potential$
$\newcommand{\Eeq}{E\subs{cell, eq}} % equilibrium cell potential$
$\newcommand{\Ej}{E\subs{j}} % liquid junction potential$
$\newcommand{\mue}{\mu\subs{e}} % electron chemical potential$
$\newcommand{\defn}{\,\stackrel{\mathrm{def}}{=}\,} % "equal by definition" symbol$
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$\newcommand{\cond}[1]{\[-2.5pt]{}\tag*{#1}}$
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$\newcommand{\R}{8.3145\units{J\,K\per\,mol\per}} % gas constant value$
$\newcommand{\Rsix}{8.31447\units{J\,K\per\,mol\per}} % gas constant value - 6 sig figs$
$\newcommand{\jn}{\hspace3pt\lower.3ex{\Rule{.6pt}{2ex}{0ex}}\hspace3pt}$
$\newcommand{\ljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}} \hspace3pt}$
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3.3(b)
$q = -w = 1.00\timesten{5}\units{J}$
3.4(c)
$w=1.99\timesten{3}\units{J}$, $q=-1.99\timesten{3}\units{J}$.
3.5
$0.0079\%$
3.6(c)
$V_2 \ra nRV_1/(C_V+nR)$, $T_2 \ra \infty$. For $C_V=(3/2)nR$, $V_2/V_1 \ra 0.4$.
3.11
$9.58\timesten{3}\units{s}$ ($2\units{hr}$ $40\units{min}$)
4.4
$\Del S=0.054\units{J K\(^{-1}$}\)
4.5
$\Del S = 549\units{J K\(^{-1}$}\) for both processes; $\int\!\dq/T\subs{ext} = 333\units{J K\(^{-1}$}\) and $0$.
5.4(a)
$\D S = nR\ln\left[cT^{3/2}\left(\frac{V}{n}-b\right)\right] + \left(\frac{5}{2}\right)nR$
5.5(a)
$q=0$, $w=1.50\timesten{4}\units{J}$, $\Del U=1.50\timesten{4}\units{J}$, $\Del H=2.00\timesten{4}\units{J}$
5.5(c)
$\Del S=66.7\units{J K\(^{-1}$}\)
6.1
$S\m \approx 151.6\units{J K\(^{-1}$ mol$^{-1}$}\)
7.6(a)
$\alpha=8.519\timesten{-4}\units{K\(^{-1}$}\)
$\kappa_t=4.671\timesten{-5}\units{bar\(^{-1}$}\)
$\pd{p}{T}{V}=18.24\units{bar K\(^{-1}$}\)
$\pd{U}{V}{T}=5437\br$
7.6(b)
$\Del p \approx 1.8\br$
7.7(b)
$\pd{\Cpm}{p}{T}=-4.210\timesten{-8}\units{J K\(^{-1}$ Pa$^{-1}$ mol$^{-1}$}\)
7.8(b)
$8\timesten{-4}\units{K\(^{-1}$}\)
7.11
$5.001\timesten{3}\units{J}$
7.12
$\Del H = 2.27\timesten{4}\units{J}$, $\Del S = 43.6\units{J K\(^{-1}$}\)
7.13(a)
$\Cpm\st=42.3\units{J K\(^{-1}$ mol$^{-1}$}\)
7.13(b)
$\Cpm \approx 52.0\units{J K\(^{-1}$ mol$^{-1}$}\)
7.14(a)
$2.56\units{J K\(^{-1}$ g$^{-1}$}\)
7.15(b)
$\fug = 17.4\br$
7.16(a)
$\phi=0.739$, $\fug=148\br$
7.16(b)
$B = -7.28\timesten{-5}\units{m\(^3$ mol$^{-1}$}\)
8.2(a)
$S\m\st\liquid =253.6\units{J K\(^{-1}$ mol$^{-1}$}\)
8.2(b)
$\Delsub{vap}S\st=88.6\units{J K\(^{-1}$ mol$^{-1}$}\), $\Delsub{vap}H\st=2.748\timesten{4}\units{J mol\(^{-1}$}\)
8.4
$4.5\timesten{-3}\br$
8.5
$19\units{J mol\(^{-1}$}\)
8.6(a)
$352.82\K$
8.6(b)
$3.4154\timesten{4}\units{J mol\(^{-1}$}\)
8.7(a)
$3.62\timesten{3}\units{Pa K\(^{-1}$}\)
8.7(b)
$3.56\timesten{3}\units{Pa K\(^{-1}$}\)
8.7(c)
$99.60\units{\(\degC$}\)
8.8(b)
$\Delsub{vap}H\st = 4.084\timesten{4}\units{J mol\(^{-1}$}\)
8.9
$0.93\units{mol}$
9.2(b)
$V\A(x\B=0.5) \approx 125.13\units{cm\(^3$ mol$^{-1}$}\)
$V\B(x\B=0.5)\approx 158.01\units{cm\(^3$ mol$^{-1}$}\)
$V\B^{\infty} \approx 157.15\units{cm\(^3$ mol$^{-1}$}\)
9.4
real gas: $p=1.9743\br$
ideal gas: $p=1.9832\br$
9.5(a)
$x\subs{N\(_2$} = 8.83\timesten{-6}\)
$x\subs{O\(_2$} = 4.65\timesten{-6}\)
$y\subs{N\(_2$} = 0.763\)
$y\subs{O\(_2$} = 0.205\)
9.5(b)
$x\subs{N\(_2$} = 9.85\timesten{-6}\)
$x\subs{O\(_2$} = 2.65\timesten{-6}\)
$y\subs{N\(_2$} = 0.851\)
$y\subs{O\(_2$} = 0.117\)
9.7(b)
$\fug\A=0.03167\br$, $\fug\A=0.03040\br$
9.8(a)
In the mixture of composition $x\A=0.9782$, the activity coefficient is $\g\B \approx 11.5$.
9.9(d)
$k\subs{H,A} \approx 680\units{kPa}$
9.11
Values for $m\B/m\st=20$: $\g\A=1.026$, $\g\mbB=0.526$; the limiting slopes are $\dif\g\A/\dif(m\B/m\st)=0$, $\dif\g\mbB/\dif(m\B/m\st)=-0.09$
9.13
$p\subs{N\(_2$} = 0.235\br\)
$y\subs{N\(_2$} = 0.815\)
$p\subs{O\(_2$} = 0.0532\br\)
$y\subs{O\(_2$} = 0.185\)
$p=0.288\br$
9.14(b)
$h=1.2\units{m}$
9.15(a)
$p(7.20\units{cm})-p(6.95\units{cm})=1.2\br$
9.15(b)
$M\B=187\units{kg mol\(^{-1}$}\)
mass binding ratio ${} = 1.37$
10.2
$\g{\pm} = 0.392$
11.1
$\Delsub{r}H\st = -63.94\units{kJ mol\(^{-1}$}\)
$K=4.41\timesten{-2}$
11.2(b)
$\Delsub{f}H\st$: no change
$\Delsub{f}S\st$: subtract $0.219\units{J K\(^{-1}$ mol$^{-1}$}\)
$\Delsub{f}G\st$: add $65\units{J mol\(^{-1}$}\)
11.3
$p(298.15\K)=2.6\timesten{-6}\br$
$p(273.15\K) = 2.7\timesten{-7}\br$
11.4(a)
$-240.34\units{kJ mol\(^{-1}$}\), $-470.36\units{kJ mol\(^{-1}$}\), $-230.02\units{kJ mol\(^{-1}$}\)
11.4(b)
$-465.43\units{kJ mol\(^{-1}$}\)
11.4(c)
$-39.82\units{kJ mol\(^{-1}$}\)
11.5
$\Del H = 0.92\units{kJ}$
11.6
$L\A=-0.405\units{J mol\(^{-1}$}\)
$L\B=0.810\units{kJ mol\(^{-1}$}\)
11.7(a)
State 1:
$n\subs{C\(_6$H$_{14}$}=7.822\timesten{-3}\mol\)
$n\subs{H\(_2$O}=0.05560\mol\)
amount of O$_2$ consumed: $0.07431\mol$
State 2:
$n\subs{H\(_2$O}=0.11035\mol\)
$n\subs{CO\(_2$}=0.04693\mol\)
\tx{mass of H$_2$O}=$1.9880\units{g}$
11.7(b)
$V\m\tx{(C\(_6$H$_{14}$)} = 131.61\units{cm$^3$ mol$^{-1}$}\)
$V\m\tx{(H\(_2$O)} = 18.070\units{cm$^3$ mol$^{-1}$}\)
11.7(c)
State 1: $V\tx{(C\(_6$H$_{14}$)} = 1.029\units{cm$^3$}\)
$V\tx{(H\(_2$O)} = 1.005\units{cm$^3$}\)
$V\sups{g} = 348.0\units{cm\(^3$}\)
State 2:
$V\tx{(H\(_2$O)} = 1.994\units{cm$^3$}\)
$V\sups{g} = 348.0\units{cm\(^3$}\)
11.7(d)
State 1:
$n\subs{O\(_2$}=0.429\mol\)
State 2:
$n\subs{O\(_2$}=0.355\mol\)
$y\subs{O\(_2$}=0.883\)
$y\subs{CO\(_2$}=0.117\)
11.7(e)
State 2:
$p_2 = 27.9\br$
$p\subs{O\(_2$} = 24.6\br\)
$p\subs{CO\(_2$} = 3.26\br\)
11.7(f)
$\fug\subs{H\(_2$O}(0.03169\br ) = 0.03164\br\)
State 1: $\fug\subs{H\(_2$O} = 0.03234\br\)
State 2: $\fug\subs{H\(_2$O} = 0.03229\br\)
11.7(g)
State 1:
$\phi\subs{H\(_2$O}=0.925\)
$\phi\subs{O\(_2$}=0.981\)
$\fug\subs{O\(_2$}= 29.4\br\)
State 2:
$\phi\subs{H\(_2$O}=0.896\)
$\phi\subs{O\(_2$}=0.983\)
$\phi\subs{CO\(_2$}=0.910\)
$\fug\subs{O\(_2$}=24.2\br\)
$\fug\subs{CO\(_2$}=2.97\br\)
11.7(h)
State 1:
$n\subs{H\(_2$O}\sups{g}=5.00\timesten{-4}\mol\)
$n\subs{H\(_2$O}\sups{l} =0.05510\mol\)
State 2:
$n\subs{H\(_2$O}\sups{g}=5.19\timesten{-4}\mol\)
$n\subs{H\(_2$O}\sups{l} =0.10983\mol\)
11.7(i)
State 1:
$k_{m,\tx{O\(_2$}}= 825\units{bar kg mol$^{-1}$}\)
$n\subs{O\(_2$} = 3.57\timesten{-5}\mol\)
State 2:
$k_{m,\tx{O\(_2$}}= 823\units{bar kg mol$^{-1}$}\)
$k_{m,\tx{CO\(_2$}}= 30.8\units{bar kg mol$^{-1}$}\)
$n\subs{O\(_2$} = 5.85\timesten{-5}\mol\)
$n\subs{CO\(_2$} = 1.92\timesten{-4}\mol\)
11.7(j)
H$_2$O vaporization: $\Del U = +20.8\units{J}$
H$_2$O condensation: $\Del U = -21.6\units{J}$
11.7(k)
O$_2$ dissolution: $\Del U = -0.35\units{J}$
O$_2$ desolution: $\Del U = 0.57\units{J}$
CO$_2$ desolution: $\Del U = 3.32\units{J}$
11.7(l)
C$_6$H$_{14}$(l) compression: $\Del U=-1.226\units{J}$
solution compression: $\Del U=-0.225\units{J}$
solution decompression: $\Del U=0.414\units{J}$
11.7(m)
O$_2$ compression: $\Del U=-81\units{J}$
gas mixture: $\dif B/\dif T = 0.26\timesten{-6}\units{m\(^3$K$^{-1}$ mol$^{-1}$}\)
gas mixture expansion: $\Del U=87\units{J}$
11.7(n)
$\Del U = 8\units{J}$
11.7(o)
$\Delsub{c}U\st = -4154.4\units{kJ mol\(^{-1}$}\)
11.7(p)
$\Delsub{c}H\st = -4163.1\units{kJ mol\(^{-1}$}\)
11.8
$\Delsub{f}H\st = -198.8\units{kJ mol\(^{-1}$}\)
11.9
$T_2=2272\K$
11.10
$p\tx{(O\(_2$)} =2.55\timesten{-5}\br\)
11.11(a)
$K=3.5\timesten{41}$
11.11(b)
$p\subs{H\(_2$}=2.8\timesten{-42}\br\)
$N\subs{H\(_2$}=6.9\timesten{-17}\)
11.11(c)
$t=22\units{s}$
11.12(b)
$p \approx 1.5\timesten{4}\br$
11.13(c)
$K=0.15$
12.1(b)
$T=1168\K$
$\Delsub{r}H\st=1.64\timesten{5}\units{J mol\(^{-1}$}\)
12.4
$K\subs{f}=1.860\units{K kg mol\(^{-1}$}\)
$K\bd=0.5118\units{K kg mol\(^{-1}$}\)
12.5
$M\B \approx 5.6\timesten{4}\units{g mol\(^{-1}$}\)
12.6
$\Delsub{sol,B}H\st/\tx{kJ mol\(^{-1}$}=-3.06, 0, 6.35\)
$\Delsub{sol,B}S\st/\tx{J K\(^{-1}$ mol$^{-1}$} = -121.0, -110.2, -88.4\)
12.7(a)
$m_+\aph = m_-\aph = 1.20\timesten{-3}\units{mol kg\(^{-1}$} \)
$m_+\bph = 1.80\timesten{-3}\units{mol kg\(^{-1}$} \)
$m_-\bph = 0.80\timesten{-3}\units{mol kg\(^{-1}$} \)
$m\subs{P} = 2.00\timesten{-6}\units{mol kg\(^{-1}$} \)
12.8(a)
$p\sups{l} =2.44\br$
12.8(b)
$\fug(2.44\br)-\fug(1.00\br)=3.4\timesten{-5}\br$
12.10(a)
$x\B=1.8\timesten{-7}$
$m\B = 1.0\timesten{-5}\units{mol kg\(^{-1}$}\)
12.10(b)
$\Delsub{sol,B}H\st = -1.99\timesten{4}\units{J mol\(^{-1}$}\)
12.10(c)
$K=4.4\timesten{-7}$
$\Delsub{r}H\st=9.3\units{kJ mol\(^{-1}$}\)
12.13(a)
$p=92399.6\Pa$, $y\B=0.965724$
12.13(b)
$\phi\A=0.995801$
12.13(c)
$\fug\A = 3164.47\Pa$
12.13(d)
$y\B = 0.965608$
12.13(e)
$Z = 0.999319$
12.13(f)
$p = 92347.7\Pa$
12.13(g)
$\kHB = 4.40890\timesten{9}\Pa$
12.15(a)
$\g\xbB=0.9826$
12.15(b)
$x\B=4.19\timesten{-4}$
12.16
$K=1.2\timesten{-6}$
12.17(a)
$\alpha=0.129$
$m_+=1.29\timesten{-3}\units{mol kg\(^{-1}$}\)
12.17(b)
$\alpha=0.140$
12.18
$\Delsub{f}H\st(\tx{Cl\(^-$, aq}) = -167.15\units{kJ mol$^{-1}$}\)
$S\m\st(\tx{Cl\(^-$, aq}) = 56.46\units{J K$^{-1}$ mol$^{-1}$}\)
12.19(a)
$K\subs{s} = 1.783\timesten{-10}$
12.20(a)
$\Delsub{r}H\st=-65.769\units{kJ mol\(^{-1}$}\)
12.20(b)
$\Delsub{f}H\st\tx{(Ag\(^+$, aq)} =105.84\units{kJ mol$^{-1}$}\)
13.1(a)
$F=4$
13.1(b)
$F=3$
13.1(c)
$F=2$
13.10(a)
$x\B\tx{(top)} =0.02$, $x\B\tx{(bottom)} =0.31$
13.10(b)
$n\A = 2.1\mol$, $n\B = 1.0 \mol$
14.3(a)
$\Delsub{r}G\st=-21.436\units{kJ mol\(^{-1}$}\)
$\Delsub{r}S\st=-62.35\units{J K\(^{-1}$ mol$^{-1}$}\)
$\Delsub{r}H\st=-40.03\units{kJ mol\(^{-1}$}\)
14.3(b)
$\Delsub{f}H\st(\tx{AgCl, s})=-127.05\units{kJ mol\(^{-1}$}\)
14.3(c)
$S\m\st(\tx{AgCl, s})=96.16\units{J K\(^{-1}$ mol$^{-1}$}\)
$\Delsub{f}S\st(\tx{AgCl, s})=-57.93$\units{J K$^{-1}$ mol$^{-1}$}
$\Delsub{f}G\st(\tx{AgCl, s})=-109.78\units{kJ mol\(^{-1}$}\)
14.4(b)
$\Delsub{f}H\st(\tx{AgCl, s})=-126.81\units{kJ mol\(^{-1}$}\)
$\Delsub{f}G\st(\tx{AgCl, s})=-109.59\units{kJ mol\(^{-1}$}\)
14.5
$K\subs{s} = 1.76\timesten{-10}$
14.6(b)
$\g_{\pm}=0.756$
14.7(b)
$\Delsub{f}G\st=-210.72\units{kJ mol\(^{-1}$}\)
14.7(c)
$K\subs{s}=1.4\timesten{-18}$
14.8
$E\st = 0.071\V$
14.9(c)
$\Eeq\st=1.36\V$
14.9(d)
In the cell:
$\dq/\dif\xi = 2.27\units{kJ mol\(^{-1}$}\)
In a reaction vessel:
$\dq/\dif\xi = -259.67\units{kJ mol\(^{-1}$}\)
14.9(e)
$\dif\Eeq\st/\dif T = 3.9\timesten{-5}\units{V K\(^{-1}$}\) | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/DeVoes_Thermodynamics_and_Chemistry/15%3A_Appendices/15.09%3A_Appendix_I-_Answers_to_Selected_Problems.txt |
1 - Introduction
General Remarks
Electron Paramagnetic Resonance (EPR) spectroscopy is less well known and less widely applied than NMR spectroscopy. The reason is that EPR spectroscopy requires unpaired electrons and electron pairing is usually energetically favorable. Hence, only a small fraction of pure substances exhibit EPR signals, whereas NMR spectroscopy is applicable to almost any compound one can think of. On the other hand, as electron pairing underlies the chemical bond, unpaired electrons are associated with reactivity. Accordingly, EPR spectroscopy is a very important technique for understanding radical reactions, electron transfer processes, and transition metal catalysis, which are all related to the ’reactivity of the unpaired electron’. Some species with unpaired electrons are chemically stable and can be used as spin probes to study systems where NMR spectroscopy runs into resolution limits or cannot provide sufficient information for complete characterization of structure and dynamics. This lecture course introduces the basics for applying EPR spectroscopy on reactive or catalytically active species as well as on spin probes.
Many concepts in EPR spectroscopy are related to similar concepts in NMR spectroscopy. Hence, the lectures on EPR spectroscopy build on material that has been introduced before in the lectures on NMR spectroscopy. This material is briefly repeated and enhanced in this script and similarities as well as differences are pointed out. Such a linked treatment of the two techniques is not found in introductory textbooks. By emphasizing this link, the course emphasizes understanding of the physics that underlies NMR and EPR spectroscopy instead of focusing on individual application fields. We aim for understanding of spectra at a fundamental level and for understanding how parameters of the spin Hamiltonian can be measured with the best possible sensitivity and resolution.
Chapter 2 of the script introduces electron spin, relates it to nuclear spin, and discusses, which interactions contribute to the spin Hamiltonian of a paramagnetic system. Chapter 3 treats the electron Zeeman interaction, the deviation of the $g$ value of a bound electron from the $g$ value of a free electron, and the manifestation of $g$ anisotropy in solid-state EPR spectra. Chapter 4 introduces the hyperfine interaction between electron and nuclear spins, which provides most information on electronic and spatial structure of paramagnetic centers. Spectral manifestation in the liquid and solid state is considered for spectra of the electron spin and of the nuclear spins. Chapter 5 discusses phenomena that occur when the hyperfine interaction is so large that the high-field approximation is violated for the nuclear spin. In this situation, formally forbidden transitions become partially allowed and mixing of energy levels leads to changes in resonance frequencies. Chapter 6 discusses how the coupling between electron spins is described in the spin Hamiltonian, depending on its size. Throughout Chapters $3-6$, the introduced interactions of the electron spin are related to electronic and spatial structure.
Chapter 7-9 are devoted to experimental techniques. In Chapter 7 , continuous-wave $(\mathrm{CW})$ EPR is introduced as the most versatile and sensitive technique for measuring EPR spectra. The requirements for obtaining well resolved spectra with high signal-to-noise ratio are derived from first physical principles. Chapter 8 discusses two techniques for measuring hyperfine couplings in nuclear frequency spectra, where they are better resolved than in EPR spectra. Electron nuclear double resonance (ENDOR) experiments use electron spin polarization and detection of electron spins in order to enhance sensitivity of such measurements, but still rely on direct excitation of the nuclear spins. Electron spin echo envelope modulation (ESEEM) experiments rely on the forbidden electron-nuclear spin transitions discussed in Chapter 5. Chapter 9 treats the measurement of distance distributions in the nanometer range by separating the dipole-dipole coupling between electron spins from other interactions.
The final Chapter 10 introduces spin probing and spin trapping and, at the same time, demonstrates the application of concepts that were introduced in earlier Chapters.
At some points (dipole-dipole coupling, explanation of CW EPR spectroscopy in terms of the Bloch equations) this lecture script significantly overlaps with the NMR part of the lecture script. This is intended in order to make the EPR script reasonably self-contained. Note also that this lecture script serves two purposes. First, it should serve as a help in studying the subject and preparing for the examination. Second, it is reference material when you later encounter paramagnetic species in your own research and need to obtain information on them by EPR spectroscopy.
Suggested Reading & Electronic Resources
There is no textbook on EPR spectroscopy that treats all material of this course on a basic level. However, many of the concepts are covered by a title from the Oxford Chemistry Primer series by Chechik, Carter, and Murphy [CCM16]. Physically minded students may also appreciate the older standard textbook by Weil, Bolton, and Wertz [WBW94].
For some of the simulated spectra and worked examples in these lecture notes, Matlab scripts or Mathematica notebooks are provided on the lecture homepage. Part of the numerical simulations is based on EasySpin by Stefan Stoll (http://wWW. easyspin.org/) and another part on SPIDYAN by Stephan Pribitzer (http://www. epr . ethz. ch/software. html). Computations with product operator formalism require the Mathematica package SpinOp.m by Serge Boentges, which is available on the course homepage. An alternative larger package for such analytical computations is SpinDynamica by Malcolm Levitt (http://Www. spindynamica. soton. ac. uk/). Last, but not least the most extensive package for numerical simulations of magnetic resonance experiments is SPINACH by Ilya Kuprov et al. (http://spindynamics. org/ Spinach.php). For quantum-chemical computations of spin Hamiltonian parameters, the probably most versatile program is the freely available package ORCA (https://orcaforum . cec.mpg.de/).
Magnetic resonance of the free electron
The magnetic moment of the free electron Differences between EPR and NMR spectroscopy
Interactions in electron-nuclear spin systems General consideration on spin interactions The electron-nuclear spin Hamiltonian
$2.1$ Magnetic resonance of the free electron
The magnetic moment of the free electron
As an elementary particle, the electron has an intrinsic angular momentum called spin. The spin quantum number is $S=1 / 2$, so that in an external magnetic field along $z$, only two possible values can be observed for the $z$ component of this angular momentum, $+\hbar / 2$, corresponding to magnetic quantum number $m_{S}=+1 / 2(\alpha$ state $)$ and $-\hbar / 2$, corresponding to magnetic quantum number $m_{S}=-1 / 2$ ( $\beta$ state $)$. The energy difference between the corresponding two states of the electron results from the magnetic moment associated with spin. For a classical rotating particle with elementary charge $e$, angular momentum $J=\hbar S$ and mass $m_{e}$, this magnetic moment computes to
$\vec{\mu}_{\text {classical }}=\frac{e}{2 m_{e}} \vec{J}$
The charge-to-mass ratio $e / m_{e}$ is much larger for the electron than the corresponding ratio for a nucleus, where it is of the order of $-e / m_{p}$, where $m_{p}$ is the proton mass. By introducing the Bohr magneton $\mu_{\mathrm{B}}=\hbar e /\left(2 m_{e}\right)=9.27400915(23) \times 10^{-24} \mathrm{~J} \mathrm{~T}^{-1}$ and the quantum-mechanical correction factor $g$, we can rewrite Eq. (2.1) as
$\vec{\mu}_{\mathrm{e}}=g \mu_{\mathrm{B}} \vec{S}$
Dirac-relativistic quantum mechanics provides $g=2$, a correction that can also be found in a non-relativistic derivation. Exact measurements have shown that the $g$ value of a free electron deviates slightly from $g=2$. The necessary correction can be derived by quantum electrodynamics, leading to $g_{e}=2.00231930437378(2)$. The energy difference between the two spin states of a free electron in an external magnetic field $B_{0}$ is given by
$\hbar \omega_{S}=g_{e} \mu_{\mathrm{B}} B_{0}$
so that the gyromagnetic ratio of the free electron is $\gamma_{e}=-g_{e} \mu_{\mathrm{B}} / \hbar$. This gyromagnetic ratio corresponds to a resonance frequency of $28.025 \mathrm{GHz}$ at a field of $1 \mathrm{~T}$, which is by a factor of about 658 larger than the nuclear Zeeman frequency of a proton.
Differences between EPR and NMR spectroscopy
Most of the differences between NMR and EPR spectroscopy result from this much larger magnetic moment of the electron. Boltzmann polarization is larger by this factor and at the same magnetic field the detected photons have an energy larger by this factor. Relaxation times are roughly by a factor $658^{2}$ shorter, allowing for much faster repetition of EPR experiments compared to NMR experiments. As a result, EPR spectroscopy is much more sensitive. Standard instrumentation with an electromagnet working at a field of about $0.35 \mathrm{~T}$ and at microwave frequencies of about $9.5 \mathrm{GHz}$ (X band) can detect about $10^{10}$ spins, if the sample has negligible dielectric microwave losses. In aqueous solution, organic radicals can be detected at concentrations down to $10 \mathrm{nM}$ in a measurement time of a few minutes.
Due to the large magnetic moment of the electron spin the high-temperature approximation may be violated without using exotic equipment. The spin transition energy of a free electron matches thermal energy $k_{\mathrm{B}} T$ at a temperature of $4.5 \mathrm{~K}$ and a field of about $3.35 \mathrm{~T}$ corresponding to a frequency of about $94 \mathrm{GHz}$ (W band). Likewise, the high-field approximation may break down. The dipole-dipole interaction between two electron spins is by a factor of $658^{2}$ larger than between two protons and two unpaired electrons can come closer to each other than two protons. The zero-field splitting that results from such coupling can amount to a significant fraction of the electron Zeeman interaction or can even exceed it at the magnetic fields, where EPR experiments are usually performed $(0.1-10 \mathrm{~T})$. The hyperfine coupling between an electron and a nucleus can easily exceed the nuclear Zeeman frequency, which leads to breakdown of the high-field approximation for the nuclear spin.
$2.2$ Interactions in electron-nuclear spin systems
General consideration on spin interactions
Spins interact with magnetic fields. The interaction with a static external magnetic field $B_{0}$ is the Zeeman interaction, which is usually the largest spin interaction. At sufficiently large fields, where the high-field approximation holds, the Zeeman interaction determines the quantization direction of the spin. In this situation, $m_{S}$ is a good quantum number and, if the high-field approximation also holds for a nuclear spin $I_{i}$, the magnetic quantum number $m_{I, i}$ is also a good quantum number. The energies of all spin levels can then be expressed by parameters that quantify spin interactions and by the magnetic quantum numbers. The vector of all magnetic quantum numbers defines the state of the spin system.
Spins also interact with the local magnetic fields induced by other spins. Usually, unpaired electrons are rare, so that each electron spin interacts with several nuclear spins in its vicinity, whereas each nuclear spin interacts with only one electron spin (Fig. 2.1). The hyperfine interaction between the electron and nuclear spin is usually much smaller than the electron Zeeman interaction, with exceptions for transition metal ions. In contrast, for nuclei in the close vicinity of the electron spin, the hyperfine interaction may be larger than the nuclear Zeeman interaction at the fields where EPR spectra are usually measured. In this case, which is discussed in Chapter 6, the high-field approximation breaks down and $m_{I, i}$ is not a good quantum number. Hyperfine couplings to nuclei are relevant as long as they are at least as large as the transverse relaxation rate $1 / T_{2 n}$ of the coupled nuclear spin. Smaller couplings are unresolved.
In some systems, two or more unpaired electrons are so close to each other that their coupling exceeds their transverse relaxation rates $1 / T_{2 \mathrm{e}}$. In fact, the isotropic part of this coupling can by far exceed the electron Zeeman interaction and often even thermal energy $k_{\mathrm{B}} T$ if two unpaired electrons reside in different molecular orbitals of the same organic molecule (triplet state molecule) or if several unpaired electrons belong to a high-spin state of a transition metal or rare earth metal ion. In this situation, the system is best described in a coupled representation with an
Figure 2.1: Scheme of interactions in electron-nuclear spin systems. All spins have a Zeeman interaction with the external magnetic field $B_{0}$. Electron spins (red) interact with each other by the dipole-dipole interaction through space and by exchange due to overlap of the singly occupied molecular orbitals (green). Each electron spin interacts with nuclear spins (blue) in its vicinity by hyperfine couplings (purple). Couplings between nuclear spins are usually negligible in paramagnetic systems, as are chemical shifts. These two interactions are too small compared to the relaxation rate in the vicinity of an electron spin.
electron group spin $S>1 / 2$. The isotropic coupling between the individual electron spins does not influence the sublevel splitting for a given group spin quantum number $S$. The anisotropic coupling, which does lead to sublevel splitting, is called the zero-field or fine interaction. If the electron Zeeman interaction by far exceeds the spin-spin coupling, it is more convenient to describe the system in terms of the individual electron spins $S_{i}=1 / 2$. The isotropic exchange coupling $J$, which stems from overlap of two singly occupied molecular orbitals (SOMOs), then does contribute to level splitting. In addition, the dipole-dipole coupling through space between two electron spins also contributes.
Concept $2.2 .1$ - Singly occupied molecular orbital (SOMO). Each molecular orbital can be occupied by two electrons with opposite magnetic spin quantum number $m_{S}$. If a molecular orbital is singly occupied, the electron is unpaired and its magnetic spin quantum number can be changed by absorption or emission of photons. The orbital occupied by the unpaired electron is called a singly occupied molecular orbital (SOMO). Several unpaired electrons can exist in the same molecule or metal complex, i.e., there may be several SOMOs.
Nuclear spins in the vicinity of an electron spin relax much faster than nuclear spins in diamagnetic substances. ${ }^{1}$ Their transverse relaxation rates $1 / T_{2 n, i}$ thus exceed couplings between nuclear spins and chemical shifts. These interactions, which are very important in NMR spectroscopy, are negligible in EPR spectroscopy. For nuclear spins $1 / 2$ no information on the chemical identity of a nucleus can be obtained, unless its hyperfine coupling is understood. The element can be identified via the nuclear Zeeman interaction. For nuclear spins $I_{i}>1 / 2$, information on the chemical identity is encoded in the nuclear quadrupole interaction, whose magnitude usually exceeds $1 / T_{2 n, i}$.
An overview of all interactions and their typical magnitude in frequency units is given in Figure 2.2. This Figure also illustrates another difference between EPR and NMR spectroscopy. Several interactions, such as the zero-field interaction, the hyperfine interaction, larger dipole-dipole and exchange couplings between electron spins and also the anisotropy of the electron Zeeman interaction usually exceed the excitation bandwidth of the strongest and shortest microwave pulses
${ }^{1}$ There is an exception. If the electron spin longitudinal relaxation rate exceeds the nuclear Zeeman interaction by far, nuclear spin relaxation is hardly affected by the presence of the electron spin. In this situation, EPR spectroscopy is impossible, however. that are available. NMR pulses sequences that rely on the ability to excite the full spectrum of a certain type of spins thus cannot easily be adapted to EPR spectroscopy.
The electron-nuclear spin Hamiltonian
Considering all interactions discussed in Section 2.2.1, the static spin Hamiltonian of an electron-nuclear spin system in angular frequency units can be written as
where index $i$ runs over all nuclear spins, indices $k$ and $l$ run over electron spins and the symbol $\mathrm{T}$ denotes the transpose of a vector or vector operator. Often, only one electron spin and one nuclear spin have to be considered at once, so that the spin Hamiltonian simplifies drastically. For electron group spins $S>1$, terms with higher powers of spin operators can be significant. We do not consider this complication here.
The electron Zeeman interaction $\hat{\mathcal{H}}_{\mathrm{EZ}}$ is, in general, anisotropic and therefore parametrized by $g$ tensors $\mathbf{g}_{k}$. It is discussed in detail in Chapter 3 . In the nuclear Zeeman interaction $\hat{\mathcal{H}}_{\mathrm{NZ}}$, the nuclear Zeeman frequencies $\omega_{I, i}$ depend only on the element and isotope and thus can be specified without knowing electronic and spatial structure of the molecule. The hyperfine interaction is again anisotropic and thus characterized by tensors $\mathbf{A}_{k i}$. It is discussed in detail in Chapter 4. All electron-electron interactions are explained in Chapter 5 . The zero-field interaction $\hat{\mathcal{H}}_{\mathrm{ZFI}}$ is purely anisotropic and thus characterized by traceless tensors $\mathbf{D}_{k}$. The exchange interaction is often purely isotropic $\hat{\mathcal{H}}_{\mathrm{EX}}$ and any anisotropic contribution cannot be experimentally distinguished from the purely anisotropic dipole-dipole interaction $\hat{\mathcal{H}}_{\mathrm{DD}}$. Hence, the former interaction is characterized by scalars $J_{k l}$ and the latter interaction by tensors $\mathbf{D}_{k l}$. Finally, the nuclear quadrupole interaction $\hat{\mathcal{H}}_{\mathrm{NQI}}$ is characterized by traceless tensors $\mathbf{P}_{i}$.
\begin{aligned} & \hat{\mathcal{H}}_{0}=\hat{\mathcal{H}}_{\mathrm{EZ}}+\hat{\mathcal{H}}_{\mathrm{NZ}}+\hat{\mathcal{H}}_{\mathrm{HFI}}+\hat{\mathcal{H}}_{\mathrm{ZFI}}+\hat{\mathcal{H}}_{\mathrm{EX}}+\hat{\mathcal{H}}_{\mathrm{DD}}+\hat{\mathcal{H}}_{\mathrm{NQI}} \ & =\frac{\mu_{\mathrm{B}}}{\hbar} \sum_{k} \vec{B}_{0}^{\mathrm{T}} \mathbf{g}_{k} \overrightarrow{\hat{S}}_{k}+\sum_{i} \omega_{I, i} \hat{I}_{z, i}+\sum_{k} \sum_{i} \overrightarrow{\hat{S}}_{k}^{\mathrm{T}} \mathbf{A}_{k i} \overrightarrow{\hat{I}}_{i}+\sum_{S_{k}>1 / 2} \overrightarrow{\hat{S}}_{k}^{\mathrm{T}} \mathbf{D}_{k} \overrightarrow{\hat{S}}_{k} \ & +\sum_{k} \sum_{l \neq k} J_{k l} \hat{S}_{z, k} \hat{S}_{z, l}+\sum_{k} \sum_{l \neq k} \overrightarrow{\hat{S}}_{k}^{\mathrm{T}} \mathbf{D}_{k l} \overrightarrow{\hat{S}}_{l}+\sum_{I_{i}>1 / 2} \overrightarrow{\hat{I}}_{i}^{\mathrm{T}} \mathbf{P}_{i} \overrightarrow{\hat{I}}_{i} \end{aligned}
Electron Zeeman interaction
Zero field interaction
Dipole-dipole interaction
between weakly coupled electron spins
Homogeneous EPR linewidths
Figure 2.2: Relative magnitude of interactions that contribute to the Hamiltonian of electron-nuclear spin systems.
Physical origin of the $g$ shift Electron Zeeman Hamiltonian Spectral manifestation of the electron Zeeman interaction
Liquid solution Solid state
$3.1$ Physical origin of the $g$ shift
Bound electrons are found to have $g$ values that differ from the value $g_{e}$ for the free electron. They depend on the orientation of the paramagnetic center with respect to the magnetic field vector $\vec{B}_{0}$. The main reason for this $g$ value shift is coupling of spin to orbital angular momentum of the electron. Spin-orbit coupling $(\mathrm{SOC})$ is a purely relativistic effect and is thus larger if orbitals of heavy atoms contribute to the SOMO. In most molecules, orbital angular momentum is quenched in the ground state. For this reason, SOC leads only to small or moderate $g$ shifts and can be treated as a perturbation. Such a perturbation treatment is not valid if the ground state is degenerate or near degenerate.
The perturbation treatment considers excited states where the unpaired electron is not in the SOMO of the ground state. Such excited states are slightly admixed to the ground state and the mixing arises from the orbital angular momentum operator. For simplicity, we consider a case where the main contribution to the $g$ shift arises from orbitals localized at a single, dominating atom and by single-electron SOC. To second order in perturbation theory, the matrix elements of the $g$ tensor can then be expressed as
$g_{i j}=g_{e} \delta_{i j}+2 \lambda \Lambda_{i j}$
where $\delta_{i j}$ is a Kronecker delta, the factor $\lambda$ in the shift term is the spin-orbit coupling constant for the dominating atom, and the matrix elements $\Lambda_{i j}$ are computed as
$\Lambda_{i j}=\sum_{n \neq 0} \frac{\left\langle 0\left|\hat{l}_{i}\right| n\right\rangle\left\langle n\left|\hat{l}_{j}\right| 0\right\rangle}{\epsilon_{0}-\epsilon_{n}}$
where indices $i$ and $j$ run over the Cartesian directions $x, y$, and $z$. The operators $\hat{l}_{x}, \hat{l}_{y}$, and $\hat{l}_{z}$ are Cartesian components of the angular momentum operator, $|n\rangle$ designates the orbital where the unpaired electron resides in an excited-state electron configuration, counted from $n=0$ for the SOMO of the ground state configuration. The energy of that orbital is $\epsilon_{n}$.
Since the product of the overlap integrals in the numerator on the right-hand side of Eq. (3.2) is usually positive, the sign of the $g$ shift is determined by the denominator. The denominator is positive if a paired electron from a fully occupied orbital is promoted to the ground-state SOMO and negative if the unpaired electron is promoted to a previously unoccupied orbital (Figure Ground state
Excitation of a paired electron
$\varepsilon_{2}=$
$\varepsilon_{1}=$
$\begin{array}{ll}\varepsilon_{-1} & \mid \ & \ \varepsilon_{-2} & \end{array}$ $\varepsilon_{0}-\varepsilon_{n}>0$
$\varepsilon_{0}-\varepsilon_{n}<0$
Figure 3.1: Admixture of excited states by orbital angular momentum operators leads to a $g$ shift by spin-orbit coupling. The energy difference in the perturbation expression is positive for excitation of a paired electron to the ground-state SOMO and negative for excitation of the paired electron to a higher energy orbital.
3.1). Because the energy gap between the SOMO and the lowest unoccupied orbital (LUMO) is usually larger than the one between occupied orbitals, terms with positive numerator dominate in the sum on the right-hand side of Eq. (3.2). Therefore, positive $g$ shifts are more frequently encountered than negative ones.
The relevant spin-orbit coupling constant $\lambda$ depends on the element and type of orbital. It scales roughly with $Z^{4}$, where $Z$ is the nuclear charge. Unless there is a very low lying excited state (near degeneracy of the ground state), contributions from heavy nuclei dominate. If their are none, as in organic radicals consisting of only hydrogen and second-row elements, $g$ shifts of only $\Delta g<10^{-2}$ are observed, typical shifts are $1 \ldots 3 \times 10^{-3}$. Note that this still exceeds typical chemical shifts in NMR by one to two orders magnitude. For first-row transition metals, $g$ shifts are of the order of $10^{-1}$.
For rare-earth ions, the perturbation treatment breaks down. The Landé factor $g_{J}$ can then be computed from the term symbol for a doublet of levels
$g_{J}=1+\frac{J(J+1)+S(S+1)-L(L+1)}{2 J(J+1)}$
where $J$ is the quantum number for total angular momentum and $L$ the quantum number for orbital angular momentum. The principal values of the $g$ tensor are $\epsilon_{x} g_{J}, \epsilon_{y} g_{J}$, and $\epsilon_{z} g_{J}$, where the $\epsilon_{i}$ with $i=x, y, z$ are differences between the eigenvalues of $\hat{L}_{i}$ for the two levels.
If the structure of a paramagnetic center is known, the $g$ tensor can be computed by quantum chemistry. This works quite well for organic radicals and reasonably well for most first-row transition metal ions. Details are explained in [KBE04].
The $g$ tensor is a global property of the SOMO and is easily interpretable only if it is dominated by the contribution at a single atom, which is often, but not always, the case for transition metal and rare earth ion complexes. If the paramagnetic center has a $C_{n}$ symmetry axis with $n \geq 3$, the $g$ tensor has axial symmetry with principal values $g_{x}=g_{y}=g_{\perp}, g_{z}=g_{\|}$. For cubic or tetrahedral symmetry the $g$ value is isotropic, but not necessarily equal to $g_{e}$. Isotropic $g$ values are also encountered to a very good approximation for transition metal and rare earth metal ions with half-filled shells, such as in Mn(II) complexes ( $3 d^{5}$ electron configuration) and Gd(III) complexes $\left(4 f^{7}\right)$.
$3.2$ Electron Zeeman Hamiltonian
We consider a single electron spin $S$ and thus drop the sum and index $k$ in $\hat{\mathcal{H}}_{\mathrm{EZ}}$ in Eq. (2.4). In the principal axes system (PAS) of the $g$ tensor, we can then express the electron Zeeman Hamiltonian as
\begin{aligned} \hat{\mathcal{H}}_{\mathrm{EZ}} &=\frac{\mu_{\mathrm{B}}}{\hbar} B_{0}(\cos \phi \sin \theta \quad \sin \phi \sin \theta \quad \cos \theta)\left(\begin{array}{ccc} g_{x} & 0 & 0 \ 0 & g_{y} & 0 \ 0 & 0 & g_{z} \end{array}\right)\left(\begin{array}{c} \hat{S}_{x} \ \hat{S}_{y} \ \hat{S}_{z} \end{array}\right) \ &=\frac{\mu_{\mathrm{B}}}{\hbar} B_{0}\left(g_{x} \cos \phi \sin \theta \hat{S}_{x}+g_{y} \sin \phi \sin \theta \hat{S}_{y}+g_{z} \cos \theta \hat{S}_{z}\right) \end{aligned}
where $B_{0}$ is the magnetic field, $g_{x}, g_{y}$, and $g_{z}$ are the principal values of the $g$ tensor and the polar angles $\phi$ and $\theta$ determine the orientation of the magnetic field in the PAS.
This Hamiltonian is diagonalized by the Bleaney transformation, providing
$\hat{\mathcal{H}}_{\mathrm{EZ}}^{B T}=\frac{\mu_{\mathrm{B}}}{\hbar} g_{\mathrm{eff}} B_{0} \hat{S}_{z}$
with the effective $g$ value at orientation $(\phi, \theta)$
$g_{\mathrm{eff}}(\phi, \theta)=\sqrt{g_{x}^{2} \sin ^{2} \theta \cos ^{2} \phi+g_{y}^{2} \sin ^{2} \theta \sin ^{2} \phi+g_{z}^{2} \cos ^{2} \theta}$
If anisotropy of the $g$ tensor is significant, the $z$ axis in Eq. (3.5) is tilted from the direction of the magnetic field. This effect is negligible for most organic radicals, but not for transition metal ions or rare earth ions. Eq. (3.6) for the effective $g$ values describes an ellipsoid (Figure $3.2$ ).
Figure 3.2: Ellipsoid describing the orientation dependence of the effective $g$ value in the PAS of the $g$ tensor. At a given direction of the magnetic field vector $\vec{B}_{0}$ (red), $g_{\text {eff }}$ corresponds to distance between the origin and the point where $\vec{B}_{0}$ intersects the ellipsoid surface.
Concept 3.2.1 - Energy levels in the high-field approximation. In the high-field approximation the energy contribution of a Hamiltonian term to the level with magnetic quantum numbers $m_{S, k}$ and $m_{I, i}$ can be computed by replacing the $\hat{J}_{z, j}$ operators $(J=S, I, j=k, i)$ by the corresponding magnetic quantum numbers. This is because the magnetic quantum numbers are the eigenvalues of the $\hat{J}_{z, j}$ operators, all $\hat{J}_{z, j}$ operators commute with each other, and contributions with all other Cartesian spin operators are negligible in this approximation. For the electron Zeeman energy contribution is $m_{S} g_{\mathrm{eff}} \mu_{\mathrm{B}} B_{0} / \hbar$. If the high-field approximation is slightly violated, this expression corresponds to a first-order perturbation treatment. The selection rule for transitions in EPR spectroscopy is $\left|\Delta m_{S}\right|=1,\left|\Delta m_{I}\right|=0$ and it applies strictly as long as the high-field approximation applies strictly to all spins. This selection rule results from conservation of angular momentum on absorption of a microwave photon and from the fact that the microwave photon interacts with electron spin transitions. It follows that the first-order contribution of the electron Zeeman interaction to the frequencies of all electron spin transitions is the same, namely $g_{\mathrm{eff}} \mu_{\mathrm{B}} B_{0} / \hbar$. As we shall see in Chapter 7 , EPR spectra are usually measured at constant microwave frequency $\nu_{\mathrm{mw}}$ by sweeping the magnetic field $B_{0}$. The resonance field is then given by
$B_{0, \mathrm{res}}=\frac{h \nu_{\mathrm{mw}}}{g_{\mathrm{eff}} \mu_{\mathrm{B}}}$
For nuclear spin transitions, $\left|\Delta m_{S}\right|=0,\left|\Delta m_{I}\right|=1$, the electron Zeeman interaction does not contribute to the transition frequency.
Spectral manifestation of the electron Zeeman interaction
Liquid solution
In liquid solution, molecules tumble due to Brownian rotational diffusion. The time scale of this motion can be characterized by a rotational correlation time $\tau_{\text {rot }}$ that in non-viscous solvents is of the order of $10 \mathrm{ps}$ for small molecules, and of the order of $1 \mathrm{~ns}$ to 100 ns for proteins and other macromolecules. For a globular molecule with radius $r$ in a solvent with viscosity $\eta$, the rotational correlation time can be roughly estimated by the Stokes-Einstein law
$\tau_{\mathrm{r}}=\frac{4 \pi \eta r^{3}}{3 k_{\mathrm{B}} T}$
If this correlation time and the maximum difference $\Delta \omega$ between the transition frequencies of any two orientations of the molecule in the magnetic field fulfill the relation $\tau_{\mathrm{r}} \Delta \omega \ll 1$, anisotropy is fully averaged and only the isotropic average of the transition frequencies is observed. For somewhat slower rotation, modulation of the transition frequency by molecular tumbling leads to line broadening as it shortens the transverse relaxation time $T_{2}$. In the slow-tumbling regime, where $\tau_{\mathrm{r}} \Delta \omega \approx 1$, anisotropy is incompletely averaged and line width attains a maximum. For $\tau_{\mathrm{r}} \Delta \omega \gg 1$, the solid-state spectrum is observed. The phenomena can be described as a multi-site exchange between the various orientations of the molecule (see Section 10.1.4), which is analogous to the chemical exchange discussed in the NMR part of the lecture course.
For the electron Zeeman interaction, fast tumbling leads to an average resonance field
$B_{0, \mathrm{res}}=\frac{h \nu_{\mathrm{mw}}}{g_{\mathrm{iso}} \mu_{\mathrm{B}}}$
with the isotropic $g$ value $g_{\text {iso }}=\left(g_{x}+g_{y}+g_{z}\right) / 3$. For small organic radicals in non-viscous solvents at X-band frequencies around $9.5 \mathrm{GHz}$, line broadening from $g$ anisotropy is negligible. At W-band frequencies of $94 \mathrm{GHz}$ for organic radicals and already at X-band frequencies for small transition metal complexes, such broadening can be substantial. For large macromolecules or in viscous solvents, solid-state like EPR spectra can be observed in liquid solution.
Solid state
For a single-crystal sample, the resonance field at any given orientation can be computed by Eq. (3.7). Often, only microcrystalline powders are available or the sample is measured in glassy frozen solution. Under such conditions, all orientations contribute equally. With respect to the
Figure 3.3: Powder line shape for a $g$ tensor with axial symmetry. (a) The probability density to find an orientation with polar angle $\theta$ is proportional to the circumference of a circle a angle $\theta$ on a unit sphere. (b) Probability density $P(\theta)$. The effective $g$ value at angle $\theta$ is $\sqrt{g_{\perp}^{2}+g_{\|}^{2}+\cos (2 \theta)\left(g_{\|}^{2}-g_{\perp}^{2}\right) / 2}$. (c) Schematic powder line shape. The pattern corresponds to $g_{\perp}>g_{\|}$for a field sweep and to $g_{\perp}<g_{\|}$for a frequency sweep. Because of the frame tilting, the isotropic value $g_{\text {iso }}=\left(2 g_{\perp}+g_{\|}\right) / 3$ is not encountered at the magic angle, although the shift is small if $\Delta g=2\left(g_{\|}-g_{\perp}\right) / 3 \ll g_{\text {iso }}$.
polar angles, this implies that $\phi$ is uniformly distributed, whereas the probability to encounter a certain angle $\theta$ is proportional to $\sin \theta$ (Figure 3.3). The line shape of the absorption spectrum is most easily understood for axial symmetry of the $g$ tensor. Transitions are observed only in the range between the limiting resonance fields at $g_{\|}$and $g_{\perp}$. The spectrum has a global maximum at $g_{\perp}$ and a minimum at $g_{\|}$.
In CW EPR spectroscopy we do not observe the absorption line shape, but rather its first derivative (see Chapter 7). This derivative line shape has sharp features at the line shape singularities of the absorption spectrum and very weak amplitude in between (Figure $3.4$ ).
Concept 3.3.1 - Orientation selection. The spread of the spectrum of a powder sample or glassy frozen solution allows for selecting molecules with a certain orientation with respect to the magnetic field. For an axial $g$ tensor only orientations near the $z$ axis of the $g$ tensor PAS are selected when observing near the resonance field of $g_{\|}$. In contrast, when observing near the resonance field for $g_{\perp}$, orientations withing the whole $x y$ plane of the PAS contribute. For the case of orthorhombic symmetry with three distinct principal values $g_{x}, g_{y}$, and $g_{z}$, narrow sets of orientations can be observed at the resonance fields corresponding to the extreme $g$ values $g_{x}$ and $g_{z}$ (see right top panel in Figure 3.4). At the intermediate principal value $g_{y}$ a broad range of orientations contributes, because the same resonance field can be realized by orientations other than $\phi=90^{\circ}$ and $\theta=90^{\circ}$. Such orientation selection can enhance the resolution of ENDOR and ESEEM spectra (Chapter 8) and simplify their interpretation. axial symmetry
orthorhombic
Figure 3.4: Simulated X-band EPR spectra for systems with only $g$ anisotropy. The upper panels show absorption spectra as they can be measured by echo-detected field-swept EPR spectroscopy. The lower panels show the first derivative of the absorption spectra as they are detected by continuous-wave EPR. The unit-sphere pictures in the right upper panel visualize the orientations that are selected at the resonance fields corresponding to the principal values of the $g$ tensor.
Physical origin of the hyperfine interaction
The magnetic moments of an electron and a nuclear spin couple by the magnetic dipole-dipole interaction; similar to the dipole-dipole interaction between nuclear spins discussed in the NMR part of the lecture course. The main difference to the NMR case is that, in many cases, a point-dipole description is not a good approximation for the electron spin, as the electron is distributed over the SOMO. The nucleus under consideration can be considered as well localized in space. We now picture the SOMO as a linear combination of atomic orbitals. Contributions from spin density in an atomic orbital of another nucleus (population of the unpaired electron in such an atomic orbital) can be approximated by assuming that the unpaired electron is a point-dipole localized at this other nucleus.
For spin density in atomic orbitals on the same nucleus, we have to distinguish between types of atomic orbitals. In $s$ orbitals, the unpaired electron has finite probability density for residing at the nucleus, at zero distance $r_{S I}$ to the nuclear spin. This leads to a singularity of the dipole-dipole interaction, since this interaction scales with $r_{S I}^{-3}$. The singularity has been treated by Fermi. The contribution to the hyperfine coupling from spin density in $s$ orbitals on the nucleus under consideration is therefor called Fermi contact interaction. Because of the spherical symmetry of $s$ orbitals, the Fermi contact interaction is purely isotropic.
For spin density in other orbitals ( $p, d, f$ orbitals) on the nucleus under consideration, the dipole-dipole interaction must be averaged over the spatial distribution of the electron spin in these orbitals. This average has no isotropic contribution. Therefore, spin density in $p, d, f$ orbitals does not influence spectra of fast tumbling radicals or metal complexes in liquid solution and neither does spin density in $s$ orbitals of other nuclei. The isotropic couplings detected in solution result only from the Fermi contact interaction.
Since the isotropic and purely anisotropic contributions to the hyperfine coupling have different physical origin, we separate these contributions in the hyperfine tensor $\mathbf{A}_{k i}$ that describes the interaction between electron spin $S_{k}$ and nuclear spin $I_{i}$ :
$\mathbf{A}_{k i}=A_{\mathrm{iso}, k i}\left(\begin{array}{ccc} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{array}\right)+\mathbf{T}_{k i}$
where $A_{\text {iso, } k i}$ is the isotropic hyperfine coupling and $\mathbf{T}_{k i}$ the purely anisotropic coupling. In the following, we drop the electron and nuclear spin indices $k$ and $i$.
Dipole-dipole hyperfine interaction
The anisotropic hyperfine coupling tensor $\mathbf{T}$ of a given nucleus can be computed from the ground state wavefunction $\psi_{0}$ by applying the correspondence principle to the classical interaction between two point dipoles
$T_{i j}=\frac{\mu_{0}}{4 \pi \hbar} g_{e} \mu_{\mathrm{B}} g_{\mathrm{n}} \mu_{\mathrm{n}}\left\langle\psi_{0}\left|\frac{3 r_{i} r_{j}-\delta_{i j} r^{2}}{r^{5}}\right| \psi_{0}\right\rangle$
Such computations are implemented in quantum chemistry programs such as ORCA, ADF, or Gaussian. If the SOMO is considered as a linear combination of atomic orbitals, the contributions from an individual orbital can be expressed as the product of spin density in this orbital with a spatial factor that can be computed once for all. The spatial factors have been tabulated [KM85]. In general, nuclei of elements with larger electronegativity have larger spatial factors. At the same spatial factor, such as for isotopes of the same element, the hyperfine coupling is proportional to the nuclear $g$ value $g_{\mathrm{n}}$ and thus proportional to the gyromagnetic ratio of the nucleus. Hence, a deuterium coupling can be computed from a known proton coupling or vice versa.
A special situation applies to protons, alkali metals and earth alkaline metals, which have no significant spin densities in $p-, d-$, or $f$-orbitals. In this case, the anisotropic contribution can only arise from through-space dipole-dipole coupling to centers of spin density at other nuclei. In a point-dipole approximation the hyperfine tensor is then given by
$\mathbf{T}=\frac{\mu_{0}}{4 \pi \hbar} g_{e} \mu_{\mathrm{B}} g_{\mathrm{n}} \mu_{\mathrm{n}} \sum_{j \neq i} \rho_{j} \frac{3 \vec{n}_{i j} \vec{n}_{i j}^{\mathrm{T}}-\overrightarrow{1}}{R_{i j}^{3}}$
where the sum runs over all nuclei $j$ with significant spin density $\rho_{j}$ (summed over all orbitals at this nucleus) other than nucleus $i$ under consideration. The $R_{i j}$ are distances between the nucleus under consideration and the centers of spin density, and the $\vec{n}_{i j}$ are unit vectors along the direction from the considered nucleus to the center of spin density. For protons in transition metal complexes it is often a good approximation to consider spin density only at the central metal ion. The distance $R$ from the proton to the central ion can then be directly inferred from the anisotropic part of the hyperfine coupling.
Hyperfine tensor contributions $\mathbf{T}$ computed by any of these ways must be corrected for the influence of $\mathrm{SOC}$ if the $g$ tensor is strongly anisotropic. If the dominant contribution to $\mathrm{SOC}$ arises at a single nucleus, the hyperfine tensor at this nucleus ${ }^{1}$ can be corrected by
$\mathbf{T}^{(\mathrm{g})}=\frac{\mathbf{g} \mathbf{T}}{g_{e}}$
The product g $\mathbf{T}$ may have an isotropic part, although $\mathbf{T}$ is purely anisotropic. This isotropic pseudocontact contribution depends on the relative orientation of the $g$ tensor and the spin-only dipole-dipole hyperfine tensor $\mathbf{T}$. The correction is negligible for most organic radicals, but not for paramagnetic metal ions. If contributions to $\mathrm{SOC}$ arise from several centers, the necessary correction cannot be written as a function of the $g$ tensor.
Fermi contact interaction
The Fermi contact contribution takes the form
$A_{\text {iso }}=\rho_{s} \cdot \frac{2}{3} \frac{\mu_{0}}{\hbar} g_{e} \mu_{\mathrm{B}} g_{\mathrm{n}} \mu_{\mathrm{n}}\left|\psi_{0}(0)\right|^{2}$
${ }^{1}$ Most literature holds that the correction should be done for all nuclei. As pointed out by Frank Neese, this is not true. An earlier discussion of this point is found in [Lef67] where $\rho_{s}$ is the spin density in the $s$ orbital under consideration, $g_{\mathrm{n}}$ the nuclear $g$ value and $\mu_{\mathrm{n}}=\beta_{\mathrm{n}}=5.05078317(20) \cdot 10^{-27} \mathrm{~J} \mathrm{~T}^{-1}$ the nuclear magneton $\left(g_{\mathrm{n}} \mu_{\mathrm{n}}=\gamma_{\mathrm{n}} \hbar\right)$. The factor $\left|\psi_{0}(0)\right|^{2}$ denotes the probability to find the electron at this nucleus in the ground state with wave function $\psi_{0}$ and has been tabulated [KM85].
Figure 4.1: Transfer of spin density by the spin polarization mechanism. According to the Pauli principle, the two electrons in the C-H bond orbital must have opposite spin state. If the unpaired electron resides in a $p_{z}$ orbital on the $\mathrm{C}$ atom, for other electrons on the same $\mathrm{C}$ atom the same spin state is slightly favored, as this minimizes electrostatic repulsion. Hence, for the electron at the $\mathrm{H}$ atom, the opposite spin state (left panel) is slightly favored over the same spin state (right panel). Positive spin density in the $p_{z}$ orbital on the $\mathrm{C}$ atom induces some negative spin density in the $s$ orbital on the $\mathrm{H}$ atom.
Spin polarization
The contributions to the hyperfine coupling discussed up to this point can be understood and computed in a single-electron picture. Further contributions arise from correlation of electrons in a molecule. Assume that the $p_{z}$ orbital on a carbon atom contributes to the SOMO, so that the $\alpha$ spin state of the electron is preferred in that orbital (Fig. 4.1). Electrons in other orbitals on the same atom will then also have a slight preference for the $\alpha$ state (left panel), as electrons with the same spin tend to avoid each other and thus have less electrostatic repulsion. ${ }^{2}$ In particular, this means that the spin configuration in the left panel of Fig. $4.1$ is slightly more preferable than the one in the right panel. According to the Pauli principle, the two electrons that share the $s$ bond orbital of the $\mathrm{C}-\mathrm{H}$ bond must have antiparallel spin. Thus, the electron in the $s$ orbital of the hydrogen atom that is bound to the spin-carrying carbon atom has a slight preference for the $\beta$ state. This corresponds to a negative isotropic hyperfine coupling of the directly bound $\alpha$ proton, which is induced by the positive hyperfine coupling of the adjacent carbon atom. The effect is termed "spin polarization", although it has no physical relation to the polarization of electron spin transitions in an external magnetic field.
Spin polarization is important, as it transfers spin density from $p$ orbitals, where it is invisible in liquid solution and from carbon atoms with low natural abundance of the magnetic isotope ${ }^{13} \mathrm{C}$ to $s$ orbitals on protons, where it can be easily observed in liquid solution. This transfer occurs, both, in $\sigma$ radicals, where the unpaired electron is localized on a single atom, and in $\pi$ radicals, where it is distributed over the $\pi$ system. The latter case is of larger interest, as the distribution of the $\pi$ orbital over the nuclei can be mapped by measuring and assigning the isotropic proton hyperfine couplings. This coupling can be predicted by the McConnell equation
$A_{\mathrm{iso}, \mathrm{H}}=Q_{\mathrm{H}} \rho_{\pi}$
where $\rho_{\pi}$ is the spin density at the adjacent carbon atom and $Q_{\mathrm{H}}$ is a parameter of the order of $-2.5 \mathrm{mT}$, which slightly depends on structure of the $\pi$ system.
${ }^{2}$ This preference for electrons on the same atom to have parallel spin is also the basis of Hund’s rule.
Figure 4.2: Mapping of the LUMO and HOMO of an aromatic molecule via measurements of hyperfine couplings after one-electron reduction or oxidation. Reduction leads to an anion radical, whose SOMO is a good approximation to the lowest unoccupied molecular orbital (LUMO) of the neutral parent molecule. Oxidation leads to an cation radical, whose SOMO is a good approximation to the highest occupied molecular orbital (HOMO) of the neutral parent molecule.
The McConnell equation is mainly applied for mapping the LUMO and HOMO of aromatic molecules (Figure 4.2). An unpaired electron can be put into these orbitals by one-electron reduction or oxidation, respectively, without perturbing the orbitals too strongly. The isotropic hyperfine couplings of the hydrogen atom directly bound to a carbon atom report on the contribution of the $p_{z}$ orbital of this carbon atom to the $\pi$ orbital. The challenges in this mapping are twofold. First, it is hard to assign the observed couplings to the hydrogen atoms unless a model for the distribution of the $\pi$ orbital is already available. Second, the method is blind to carbon atoms without a directly bonded hydrogen atom.
Hyperfine Hamiltonian
We consider the interaction of a single electron spin $S$ with a single nuclear spin $I$ and thus drop the sums and indices $k$ and $i$ in $\hat{\mathcal{H}}_{\mathrm{HFI}}$ in Eq. (2.4). In general, all matrix elements of the hyperfine tensor $\mathbf{A}$ will be non-zero after the Bleaney transformation to the frame where the electron Zeeman interaction is along the $z$ axis (see Eq. 3.5). The hyperfine Hamiltonian is then given by
\begin{aligned} \hat{\mathcal{H}}_{\mathrm{HFI}}=&\left(\begin{array}{lll} \hat{S}_{x} & \hat{S}_{y} & \hat{S}_{z} \end{array}\right)\left(\begin{array}{ccc} A_{x x} & A_{x y} & A_{x z} \ A_{y x} & A_{y y} & A_{y z} \ A_{z x} & A_{z y} & A_{z z} \end{array}\right)\left(\begin{array}{c} \hat{I}_{x} \ \hat{I}_{y} \ \hat{I}_{z} \end{array}\right) \ =& A_{x x} \hat{S}_{x} \hat{I}_{x}+A_{x y} \hat{S}_{x} \hat{I}_{y}+A_{x y} \hat{S}_{x} \hat{I}_{z} \ &+A_{y x} \hat{S}_{y} \hat{I}_{x}+A_{y y} \hat{S}_{y} \hat{I}_{y}+A_{y z} \hat{S}_{y} \hat{I}_{z} \ &+A_{z x} \hat{S}_{z} \hat{I}_{x}+A_{z y} \hat{S}_{z} \hat{I}_{y}+A_{z z} \hat{S}_{z} \hat{I}_{z} \end{aligned}
Note that the $z$ axis of the nuclear spin coordinate system is parallel to the magnetic field vector $\vec{B}_{0}$ whereas the one of the electron spin system is tilted, if $g$ anisotropy is significant. Hence, the hyperfine tensor is not a tensor in the strict mathematical sense, but rather an interaction matrix.
In Eq. (4.7), the term $A_{z z} \hat{S}_{z} \hat{I}_{z}$ is secular and must always be kept. Usually, the high-field approximation does hold for the electron spin, so that all terms containing $\hat{S}_{x}$ or $\hat{S}_{y}$ operators are non-secular and can be dropped. The truncated hyperfine Hamiltonian thus reads
$\hat{\mathcal{H}}_{\mathrm{HFI}, \text { trunc }}=A_{z x} \hat{S}_{z} \hat{I}_{x}+A_{z y} \hat{S}_{z} \hat{I}_{y}+A_{z z} \hat{S}_{z} \hat{I}_{z}$
The first two terms on the right-hand side can be considered as defining an effective transverse coupling that is the sum of a vector with length $A_{z x}$ along $x$ and a vector of length $A_{z y}$ along $y$. The length of the sum vector is $B=\sqrt{A_{z x}^{2}+A_{z y}^{2}}$. The truncated hyperfine Hamiltonian simplifies if we take the laboratory frame $x$ axis for the nuclear spin along the direction of this effective transverse hyperfine coupling. In this frame we have
$\hat{\mathcal{H}}_{\text {HFI,trunc }}=A \hat{S}_{z} \hat{I}_{z}+B \hat{S}_{z} \hat{I}_{x}$
where $A=A_{z z}$ quantifies the secular hyperfine coupling and $B$ the pseudo-secular hyperfine coupling. The latter coupling must be considered if and only if the hyperfine coupling violates the high-field approximation for the nuclear spin (see Chapter 6).
If $g$ anisotropy is very small, as is the case for organic radicals, the $z$ axes of the two spin coordinate systems are parallel. In this situation and for a hyperfine tensor with axial symmetry, $A$ and $B$ can be expressed as
\begin{aligned} &A=A_{\mathrm{iso}}+T\left(3 \cos ^{2} \theta_{\mathrm{HFI}}-1\right) \ &B=3 T \sin \theta_{\mathrm{HFI}} \cos \theta_{\mathrm{HFI}} \end{aligned}
where $\theta_{\text {HFI }}$ is the angle between the static magnetic field $\vec{B}_{0}$ and the symmetry axis of the hyperfine tensor and $T$ is the anisotropy of the hyperfine coupling. The principal values of the hyperfine tensor are $A_{x}=A_{y}=A_{\perp}=A_{\text {iso }}-T$ and $A_{z}=A_{\|}=A_{\text {iso }}+2 T$. The pseudo-secular contribution $B$ vanishes along the principal axes of the hyperfine tensor, where $\theta_{\mathrm{HFI}}$ is either $0^{\circ}$ or $90^{\circ}$ or for a purely isotropic hyperfine coupling. Hence, the pseudo-secular contribution can also be dropped when considering fast tumbling radicals in the liquid state. We now consider the point-dipole approximation, where the electron spin is well localized on the length scale of the electron-nuclear distance $r$ and assume that $T$ arises solely from through-space interactions. This applies to hydrogen, alkali and earth alkali ions. We then find
$T=\frac{1}{r^{3}} \frac{\mu_{0}}{4 \pi \hbar} g_{e} \mu_{\mathrm{B}} g_{\mathrm{n}} \mu_{\mathrm{n}}$
For the moment we assume that the pseudo-secular contribution is either negligible or can be considered as a small perturbation. The other case is treated in Chapter 6 . To first order, the contribution of the hyperfine interaction to the energy levels is then given by $m_{S} m_{I} A$. In the EPR spectrum, each nucleus with spin $I$ generates $2 I+1$ electron spin transitions with $\left|\Delta m_{S}\right|=1$ that can be labeled by the values of $m_{I}=-I,-I+1, \ldots I$. In the nuclear frequency spectrum, each nucleus exhibits $2 S+1$ transitions with $\left|\Delta m_{I}\right|=1$. For nuclear spins $I>1 / 2$ in the solid state, each transition is further split into $2 I$ transitions by the nuclear quadrupole interaction. The contribution of the secular hyperfine coupling to the electron transition frequencies is $m_{I} A$, whereas it is $m_{S} A$ for nuclear transition frequencies. In both cases, the splitting between adjacent lines of a hyperfine multiplet is given by $A$.
Spectral manifestation of the hyperfine interaction
Liquid-solution EPR spectra
Since each nucleus splits each electron spin transition into $2 I+1$ transitions with different frequencies, the number of EPR transitions is $\prod_{i}\left(2 I_{i}+1\right)$. Some of these transitions may coincide if hyperfine couplings are the same or integer multiples of each other. An important case, where hyperfine couplings are exactly the same are chemically equivalent nuclei. For instance, two nuclei $I_{1}=I_{2}=1 / 2$ can have spin state combinations $\alpha_{1} \alpha_{2}, \alpha_{1} \beta_{2}, \beta_{1} \alpha_{2}$, and $\beta_{1} \beta_{2}$. The contributions to the transition frequencies are $\left(A_{1}+A_{2}\right) / 2,\left(A_{1}-A_{2}\right) / 2,\left(-A_{1}+A_{2}\right) / 2$, and
Figure 4.3: Hyperfine splitting in the EPR spectrum of the phenyl radical. The largest hyperfine coupling for the two equivalent ortho protons generates a triplet of lines with relative intensities $1: 2: 1$. The medium coupling to the two equivalent meta proton splits each line again into a $1: 2: 1$ pattern, leading to 9 lines with an intensity ratio of $1: 2: 1: 2: 4: 2: 1: 2: 1$. Finally, each line is split into a doublet by the small hyperfine coupling of the para proton, leading to 18 lines with intensity ratio $1: 1: 2: 2: 1: 1: 2: 2: 4: 4: 2: 2: 1: 1: 2: 2: 1$.
$\left(-A_{1}-A 2\right) / 2$. For equivalent nuclei with $A_{1}=A_{2}=A$ only three lines are observed with hyperfine shifts of $A, 0$, and $-A$ with respect to the electron Zeeman frequency. The unshifted center line has twice the amplitude than the shifted lines, leading to a $1: 2: 1$ pattern with splitting $A$. For $k$ equivalent nuclei with $I_{i}=1 / 2$ the number of lines is $k+1$ and the relative intensities can be inferred from Pascal’s triangle. For a group of $k_{i}$ equivalent nuclei with arbitrary spin quantum number $I_{i}$ the number of lines is $2 k_{i} I_{i}+1$. The multiplicities of groups of equivalent nuclei multiply. Hence, the total number of EPR lines is
$n_{\mathrm{EPR}}=\prod_{i}\left(2 k_{i} I_{i}+1\right)$
where index $i$ runs over the groups of equivalent nuclei.
Figure $4.3$ illustrates on the example of the phenyl radical how the multiplet pattern arises. For radicals with more extended $\pi$ systems, the number of lines can be very large and it may become impossible to fully resolve the spectrum. Even if the spectrum is fully resolved, analysis of the multiplet pattern may be a formidable task. An algorithm that works well for analysis of patterns with a moderate number of lines is given in [CCM16].
Liquid-solution nuclear frequency spectra
As mentioned in Section $4.2$ the secular hyperfine coupling $A$ can be inferred from nuclear frequency spectra as well as from EPR spectra. Line widths are smaller in the nuclear frequency spectra, since nuclear spins have longer transverse relaxation times $T_{2, i}$. Another advantage of nuclear frequency spectra arises from the fact that the electron spin interacts with all nuclear spins whereas each nuclear spin interacts with only one electron spin (Figure 4.4). The number of lines in nuclear frequency spectra thus grows only linearly with the number of nuclei, whereas
Figure 4.4: Topologies of an electron-nuclear spin system for EPR spectroscopy (a) and of a nuclear spin system typical for NMR spectroscopy (b). Because of the much larger magnetic moment of the electron spin, the electron spin "sees" all nuclei, while each nuclear spin in the EPR case sees only the electron spin. In the NMR case, each nuclear spin sees each other nuclear spin, giving rise to very rich, but harder to analyze information.
it grows exponentially in EPR spectra. In liquid solution, each group of equivalent nuclear spins adds $2 S+1$ lines, so that the number of lines for $N_{\text {eq }}$ such groups is
$n_{\mathrm{NMR}}=(2 S+1) N_{\mathrm{eq}}$
The nuclear frequency spectra in liquid solution can be measured by CW ENDOR, a technique that is briefly discussed in Section 8.1.2.
$d$
Figure 4.5: Energy level schemes (a,c) and nuclear frequency spectra (b,d) in the weak hyperfine coupling $(\mathrm{a}, \mathrm{b})$ and strong hyperfine coupling (c,d) cases for an electron-nuclear spin system $S=1 / 2$, $I=1 / 2$. Here, $\omega_{I}$ is assumed to be negative and $A$ is assumed to be positive. (a) In the weak-coupling case, $|A| / 2<\left|\omega_{I}\right|$, the two nuclear spin transitions (green) have frequencies $\left|\omega_{I}\right| \pm|A| / 2$. (b) In the weak-coupling case, the doublet is centered at frequency $\left|\omega_{I}\right|$ and split by $|A|$. (c) In the strong-coupling case, $|A| / 2>\left|\omega_{I}\right|$, levels cross for one of the electron spin states. The two nuclear spin transitions (green) have frequencies $|A| / 2 \pm\left|\omega_{I}\right|$. (d) In the strong-coupling case, the doublet is centered at frequency $|A| / 2$ and split by $2\left|\omega_{I}\right|$.
A complication in interpretation of nuclear frequency spectra can arise from the fact that the hyperfine interaction may be larger than the nuclear Zeeman interaction. This is illustrated in Figure 4.5. Only in the weak-coupling case with $|A| / 2<\left|\omega_{I}\right|$ the hyperfine doublet in nuclear frequency spectra is centered at $\left|\omega_{I}\right|$ and split by $|A|$. In the strong-coupling case, hyperfine sublevels cross for one of the electron spin states and the nuclear frequency $\left|\omega_{I}\right|-|A| / 2$ becomes negative. As the sign of the frequency is not detected, the line is found at frequency $|A| / 2-\left|\omega_{I}\right|$ instead, i.e., it is "mirrored" at the zero frequency. This results in a doublet centered at frequency $|A| / 2$ and split by $2\left|\omega_{I}\right|$. Recognition of such cases in well resolved liquid-state spectra is simplified by the fact that the nuclear Zeeman frequency $\left|\omega_{I}\right|$ can only assume a few values that are known if the nuclear isotopes in the molecule and the magnetic field are known. Figure $4.6$ illustrates how the nuclear frequency spectrum of the phenyl radical is constructed based on such considerations. The spectrum has only 6 lines, compared to the 18 lines that arise in the EPR spectrum in Figure 4.3.
Figure 4.6: Schematic ENDOR (nuclear frequency) spectrum of the phenyl radical at an X-band frequency where $\omega_{I} /(2 \pi) \approx 14 \mathrm{MHz}$. (a) Subspectrum of the two equivalent ortho protons. The strong-coupling case applies. (b) Subspectrum of the two equivalent meta protons. The weak-coupling case applies. (c) Subspectrum of the para proton. The weak-coupling case applies. (d) Complete spectrum.
Solid-state EPR spectra
In the solid state, construction of the EPR spectra is complicated by the fact that the electron Zeeman interaction is anisotropic. At each individual orientation of the molecule, the spectrum looks like the pattern in liquid state, but both the central frequency of the multiplet and the hyperfine splittings depend on orientation. As these frequency distributions are continuous, resolved splittings are usually observed only at the singularities of the line shape pattern of the interaction with the largest anisotropy. For organic radicals at X-band frequencies, often hyperfine anisotropy dominates. At high frequencies or for transition metal ions, often electron Zeeman anisotropy dominates. The exact line shape depends not only on the principal values of the $g$ tensor and the hyperfine tensors, but also on relative orientation of their PASs. The general case is complicated and requires numerical simulations, for instance, by EasySpin.
However, simple cases, where the hyperfine interaction of only one nucleus dominates and the PASs of the $g$ and hyperfine tensor coincide, are quite often encountered. For instance, Cu(II) complexes are often square planar and, if all four ligands are the same, have a $C_{4}$ symmetry axis. The $g$ tensor than has axial symmetry with the $C_{4}$ axis being the unique axis. The hyperfine tensors of ${ }^{63} \mathrm{Cu}$ and ${ }^{65} \mathrm{Cu}$ have the same symmetry and the same unique axis. The two isotopes both have spin $I=3 / 2$ and very similar gyromagnetic ratios. The spectra can thus be understood by considering one electron spin $S=1 / 2$ and one nuclear spin $I=3 / 2$ with axial $g$ and hyperfine tensors with a coinciding unique axis.
In this situation, the subspectra for each of the nuclear spin states $m_{I}=-3 / 2,-1 / 2,+1 / 2$, and $+3 / 2$ take on a similar form as shown in Figure $3.3$. The resonance field can be computed by solving
$\hbar \omega_{\mathrm{mw}}=\frac{\mu_{\mathrm{B}}}{B_{0, \mathrm{res}}}\left(2 g_{\perp}^{2} \sin ^{2} \theta+g_{\|}^{2} \cos ^{2} \theta\right)+m_{I}\left[A_{\text {iso }}+T\left(3 \cos ^{2} \theta-1\right)\right]$
where $\theta$ is the angle between the $C_{4}$ symmetry axis and the magnetic field vector $\vec{B}_{0}$. The singularities are encountered at $\theta=0^{\circ}$ and $\theta=90^{\circ}$ and correspond to angular frequencies $\mu_{\mathrm{B}} B_{0} g_{\|}+m_{I} A_{\|}$and $\mu_{\mathrm{B}} B_{0} g_{\perp}+m_{I} A_{\perp}$.
Figure 4.7: Construction of a solid-state EPR spectrum for a copper(II) complex with four equivalent ligands and square planar coordination. The $g_{\|}$and $A_{\|}$principal axes directions coincide with the $C_{4}$ symmetry axis of the complex (inset). (a) Subspectra for the four nuclear spin states with different magnetic spin quantum number $m_{I}$. (b) Absorption spectrum. (c) Derivative of the absorption spectrum.
The construction of a Cu(II) EPR spectrum according to these considerations is shown in Figure 4.7. The values of $g_{\|}$and $A_{\|}$can be inferred by analyzing the singularities near the low-field edge of the spectrum. Near the high-field edge, the hyperfine splitting $A_{\perp}$ is usually not resolved. Here, $g_{\perp}$ corresponds to the maximum of the absorption spectrum and to the zero crossing of its derivative.
Solid-state nuclear frequency spectra
Again, a simpler situation is encountered in nuclear frequency spectra, as the nuclear Zeeman frequency is isotropic and chemical shift anisotropy is negligibly small compared to hyperfine anisotropy. Furthermore, resolution is much better for the reasons discussed above, so that smaller hyperfine couplings and anisotropies can be detected. If anisotropy of the hyperfine coupling is dominated by through-space dipole-dipole coupling to a single center of spin density, as is often the case for protons, or by contribution from spin density in a single $p$ or $d$ orbital, as is often the case for other nuclei, the hyperfine tensor has nearly axial symmetry. In this case, one can infer from the line shapes whether the weak-or strong-coupling case applies and whether the isotropic hyperfine coupling is positive or negative (Figure 4.8). The case with $A_{\text {iso }}=0$ corresponds to the Pake pattern discussed in the NMR part of the lecture course.
Figure 4.8: Solid-state nuclear frequency spectra for cases with negative nuclear Zeeman frequency $\omega_{I}$. (a) Weak-coupling case with $A_{\text {iso }}>0$ and $A_{\text {iso }}>T$. (b) Weak-coupling case with $A_{\text {iso }}<0$ and $\left|A_{\text {iso }}\right|>T$. (a) Strong-coupling case with $A_{\text {iso }}>0$ and $A_{\text {iso }}>T$. (b) Strong-coupling case with $A_{\text {iso }}<0$ and $\left|A_{\text {iso }}\right|>T$.
Exchange interaction
Physical origin and consequences of the exchange interaction
If two unpaired electrons occupy SOMOs in the same molecule or in spatially close molecules, the wave functions $\psi_{1}$ and $\psi_{2}$ of the two SOMOs may overlap. The two unpaired electrons can couple either to a singlet state or to a triplet state. The energy difference between the singlet and triplet state is the exchange integral
$J=-2 e^{2} \iint \frac{\psi_{1}^{*}\left(r_{1}\right) \psi_{2}^{*}\left(r_{2}\right) \psi_{1}\left(r_{2}\right) \psi_{2}\left(r_{1}\right)}{\left|\vec{r}_{1} \vec{r}_{2}\right|} \mathrm{d} \vec{r}_{1} \mathrm{~d} \vec{r}_{2}$
There exist different conventions for the sign of $J$ and the factor 2 may be missing in parts of the literature. With the sign convention used here, the singlet state is lower in energy for positive $J$. Since the singlet state $S$ with spin wave function $(|\alpha \beta\rangle-|\beta \alpha\rangle) / \sqrt{2}$ is antisymmetric with respect to exchange of the two electrons and electrons are Fermions, it corresponds to the situation where the two electrons could also occupy the same orbital. This is a bonding orbital overlap, corresponding to an antiferromagnetic spin ordering. Negative $J$ correspond to a lower-lying triplet state, i.e., antibonding orbital overlap and ferromagnetic spin ordering. The triplet state has three substates with wave functions $|\alpha \alpha\rangle$ for the $\mathrm{T}_{+}$state, $(|\alpha \beta\rangle+|\beta \alpha\rangle) / \sqrt{2}$ for the $\mathrm{T}_{0}$ state, and $|\beta \beta\rangle$ for the $\mathrm{T}_{-}$state. The $\mathrm{T}_{+}$and $\mathrm{T}_{-}$state are eigenstates both in the absence and presence of the $J$ coupling. The states $\mathrm{S}$ and $\mathrm{T}_{0}$ are eigenstates for $J \gg \Delta \omega$, where $\Delta \omega$ is the difference between the electron Zeeman frequencies of the two spins. For the opposite case of $\Delta \omega \gg J$, the eigenstates are $|\alpha \beta\rangle$ and $|\beta \alpha\rangle$. The latter case corresponds to the high-field approximation with respect to the exchange interaction.
For strong exchange, $J \gg \Delta \omega$, the energies are approximately $-(3 / 4) J$ for the singlet state and $J / 4-\omega_{S}, J / 4$ and $J / 4+\omega_{S}$ for the triplet substates $\mathrm{T}_{-}, \mathrm{T}_{0}$, and $\mathrm{T}_{+}$, respectively, where $\omega_{S}$ is the electron Zeeman interaction, which is the same for both spins within this approximation. If $J \gg 2 \pi \nu_{\mathrm{mw}}$, microwave photons with energy $h \nu_{\mathrm{mw}}$ cannot excite transitions between the singlet and triplet subspace of spin Hilbert space. It is then convenient to use a coupled representation and consider the two subspaces separately from each other. The singlet subspace corresponds to a diamagnetic molecule and does not contribute to EPR spectra. The triplet subspace can be described by a group spin $S=1$ of the two unpaired electrons. In the coupled representation, $J$ does not enter the spin Hamiltonian, as it shifts all subspace levels by the same energy. For $J<0$, the triplet state is the ground state and is always observable by EPR spectroscopy. However, usually one has $J>0$ and the singlet state is the ground state. As long as $\hbar J$ does not exceed thermal energy $k_{\mathrm{B}} T$ by a large factor, the triplet state is thermally excited and observable. In this case, EPR signal amplitude may increase rather than decrease with increasing temperature. For organic molecules, this case is also rare. If $\hbar J \gg k_{\mathrm{B}} T$, the compound does not give an EPR signal. It may still be possible to observe the triplet state transiently after photoexcitation to an excited singlet state and intersystem crossing to the triplet state.
Weak exchange coupling is observed in biradicals with well localized SOMOs that are separated on length scales between $0.5$ and $1.5 \mathrm{~nm}$. In such cases, exchange coupling $J$ decreases exponentially with the distance between the two electrons or with the number of conjugated bonds that separate the two centers of spin density. If the two centers are not linked by a continuous chain of conjugated bonds, exchange coupling is rarely resolved at distances larger than $1.5 \mathrm{~nm}$. In any case, at such long distances exchange coupling is much smaller than the dipole-dipole coupling between the two unpaired electrons if the system is not conjugated. For weak exchange coupling, the system is more conveniently described in an uncoupled representation with two spins $S_{1}=1 / 2$ and $S_{2}=1 / 2$.
Exchange coupling is also significant during diffusional encounters of two paramagnetic molecules in liquid solution. Such dynamic Heisenberg spin exchange can be pictured as physical exchange of unpaired electrons between the colliding molecules. This causes a sudden change of the spin Hamiltonian, which leads to spin relaxation. A typical example is line broadening in EPR spectra of radicals by oxygen, which has a paramagnetic triplet ground state. If radicals of the same type collide, line broadening is also observed, but the effects on the spectra can be more subtle, since the spin Hamiltonians of the colliding radicals are the same. In this case, exchange of unpaired electrons between the radicals changes only spin state, but not the spin Hamiltonian.
Exchange Hamiltonian
The spin Hamiltonian contribution by weak exchange coupling is
$\hat{\mathcal{H}}_{\mathrm{EX}}=J\left(\hat{S}_{1 x} \hat{S}_{2 x}+\hat{S}_{1 y} \hat{S}_{2 y}+\hat{S}_{1 z} \hat{S}_{2 z}\right)$
This Hamiltonian is analogous to the $J$ coupling Hamiltonian in NMR spectroscopy. If the two spins have different $g$ values and the field is sufficiently high $\left(g \mu_{\mathrm{B}} B_{0} / \hbar \gg J\right)$, the exchange Hamiltonian can be truncated in the same way as the $J$ coupling Hamiltonian in heteronuclear NMR:
$\hat{\mathcal{H}}_{\text {EX,trunc }}=J \hat{S}_{1 z} \hat{S}_{2 z}$
Spectral manifestation of the exchange interaction
In the absence of hyperfine coupling, the situation is the same as for $J$ coupling in NMR spectroscopy. Exchange coupling between like spins (same electron Zeeman frequency) does not influence the spectra. For radicals in liquid solution, hyperfine coupling is usually observable. In this case, exchange coupling does influence the spectra even for like spins, as illustrated in Figure $5.1$ for two exchange-coupled electron spins $S_{1}=1 / 2$ and $S_{2}=1 / 2$ with each of them coupled exclusively to only one nuclear spin $\left(I_{1}=1\right.$ and $I_{2}=1$, respectively) with the same hyperfine coupling $A_{\mathrm{iso}}$. If the exchange coupling is much smaller than the isotropic hyperfine coupling, each of the individual lines of the hyperfine triplet further splits into three lines. If the splitting is very small, it may be noticeable only as a line broadening. At very large exchange coupling, the electron spins are uniformly distributed over the two exchange-coupled moieties. Hence, each of them has the same hyperfine coupling to both nuclei. This coupling is half the original hyperfine coupling, since, on average, the electron spin has only half the spin density in the orbitals of a given nucleus as compared to the case without exchange coupling. For intermediate exchange couplings, complex splitting patterns arise that are characteristic for the ratio between the exchange and hyperfine coupling.
Figure 5.1: Influence of the exchange coupling $J$ on EPR spectra with hyperfine coupling in liquid solution (simulation). Spectra are shown for two electron spins $S_{1}=1 / 2$ and $S_{2}=1 / 2$ with the same isotropic $g$ value and the same isotropic hyperfine coupling to a nuclear spin $I_{1}=1$ or $I_{2}=1$, respectively. In the absence of exchange coupling, a triplet with amplitude ratio $1: 1: 1$ is observed. For small exchange couplings, each line splits into a triplet. At intermediate exchange couplings, complicated patterns with many lines result. For very strong exchange coupling, each electron spin couples to both nitrogen nuclei with half the isotropic exchange coupling. A quintuplet with amplitude ratio $1: 2: 3: 2: 1$ is observed.
$5.2$ Dipole-dipole interaction
Physical picture
The magnetic dipole-dipole interaction between two localized electron spins with magnetic moments $\mu_{1}$ and $\mu_{2}$ takes the same form as the classical interaction between two magnetic point dipoles. The interaction energy
$E=-\frac{\mu_{0}}{4 \pi} \cdot \mu_{1} \mu_{2} \cdot \frac{1}{r^{3}} \cdot\left(2 \cos \theta_{1} \cos \theta_{2}-\sin \theta_{1} \sin \theta_{2} \cos \phi\right)$
generally depends on the two angles $\theta_{1}$ and $\theta_{2}$ that the point dipoles include with the vector between them and on the dihedral angle $\phi$ (Figure 5.2). The dipole-dipole interaction scales with the inverse cube of the distance between the two point dipoles.
In general, the two electron spins are spatially distributed in their respective SOMOs. The point-dipole approximation is still a good approximation if the distance $r$ is much larger than the spatial distribution of each electron spin. Further simplification is possible if $g$ anisotropy is much smaller than the isotropic $g$ value. In that case, the two spins are aligned parallel to the magnetic field and thus also parallel to each other, so that $\theta_{1}=\theta_{2}=\theta$ and $\phi=0$. Eq. (5.4) then simplifies to
$E=-\frac{\mu_{0}}{4 \pi} \cdot \mu_{1} \mu_{2} \cdot \frac{1}{r^{3}} \cdot\left(3 \cos ^{2} \theta-1\right)$
which is the form known from NMR spectroscopy.
Figure 5.2: Geometry of two magnetic point dipoles in general orientation. Angles $\theta_{1}$ and $\theta_{2}$ are included between the respective magnetic moment vectors $\vec{\mu}_{1}$ or $\vec{\mu}_{2}$ and the distance vector $\vec{r}$ between the point dipoles. Angle $\phi$ is the dihedral angle.
Dipole-dipole Hamiltonian
For two electron spins that are not necessarily aligned parallel to the external magnetic field, the dipole-dipole coupling term of the spin Hamiltonian assumes the form
$\widehat{H}_{\mathrm{dd}}=\widehat{S}_{1}^{\mathrm{T}} \underline{D} \widehat{S}_{2}=\frac{1}{r^{3}} \cdot \frac{\mu_{0}}{4 \pi \hbar} \cdot g_{1} g_{2} \mu_{\mathrm{B}}^{2}\left[\widehat{S}_{1} \widehat{S}_{2}-\frac{3}{r^{2}}\left(\widehat{S}_{1} \vec{r}\right)\left(\widehat{S}_{2} \vec{r}\right)\right]$
If the electrons are distributed in space, the Hamiltonian has to be averaged (integrated) over the two spatial distributions, since electron motion proceeds on a much faster time scale than an EPR experiment.
If the two unpaired electrons are well localized on the length scale of their distances and their spins are aligned parallel to the external magnetic field, the dipole-dipole Hamiltonian takes the form
$\hat{H}_{\mathrm{dd}}=\frac{1}{r^{3}} \cdot \frac{\mu_{0}}{4 \pi \hbar} \cdot g_{1} g_{2} \mu_{\mathrm{B}}^{2}[\hat{A}+\hat{B}+\hat{C}+\hat{D}+\hat{E}+\hat{F}]$
with the terms of the dipolar alphabet
\begin{aligned} \hat{A} &=\hat{S}_{z} \hat{I}_{z}\left(1-3 \cos ^{2} \theta\right) \ \hat{B} &=-\frac{1}{4}\left[\hat{S}^{+} \hat{I}^{-}+\hat{S}^{-} \hat{I}^{+}\right]\left(1-3 \cos ^{2} \theta\right) \ \hat{C} &=-\frac{3}{2}\left[\hat{S}^{+} \hat{I}_{z}+\hat{S}_{z} \hat{I}^{+}\right] \sin \theta \cos \theta e^{-i \phi} \ \hat{D} &=-\frac{3}{2}\left[\hat{S}^{-} \hat{I}_{z}+\hat{S}_{z} \hat{I}^{-}\right] \sin \theta \cos \theta e^{i \phi} \ \hat{E} &=-\frac{3}{4} \hat{S}^{+} \hat{I}^{+} \sin ^{2} \theta e^{-2 i \phi} \ \hat{F} &=-\frac{3}{4} \hat{S}^{-} \hat{I}^{-} \sin ^{2} \theta e^{2 i \phi} \end{aligned}
Usually, EPR spectroscopy is performed at fields where the electron Zeeman interaction is much larger than the dipole-dipole coupling, which has a magnitude of about $50 \mathrm{MHz}$ at a distance of $1 \mathrm{~nm}$ and of $50 \mathrm{kHz}$ at a distance of $10 \mathrm{~nm}$. In this situation, the terms $\hat{C}, \hat{D}, \hat{E}$, and $\hat{F}$ are non-secular and can be dropped. The $\hat{B}$ term is pseudo-secular and can be dropped only if
Figure 5.3: Explanation of dipole-dipole coupling between two spins in a local field picture. At the observer spin (blue) a local magnetic field is induced by the magnetic moment of the coupling partner spin (red). In the secular approximation only the $z$ component of this field is relevant, which is parallel or antiparallel to the external magnetic field $\vec{B}_{0}$. The magnitude of this $z$ component depends on angle $\theta$ between the external magnetic field and the spin-spin vector $\vec{r}$. For the $\alpha$ (left) and $\beta$ (right) states of the partner spin, the local field at the observer spin has the same magnitude, but opposite direction. In the high-temperature approximation, both these states are equally populated. The shift of the resonance frequency of the observer spin thus leads to a splitting of the observer spin transition, which is twice the product of the local field with the gyromagnetic ratio of the observer spin.
the difference between the electron Zeeman frequencies is much larger than the dipole-dipole coupling 1 . In electron electron double resonance (ELDOR) experiments, the difference of the Larmor frequencies of the two coupled spins can be selected via the difference of the two microwave frequencies. It is thus possible to excite spin pairs for which only the secular part of the spin Hamiltonian needs to be considered,
$\widehat{H}_{\mathrm{dd}}=\omega_{\perp}\left(1-3 \cos ^{2} \theta\right) \hat{S}_{z} \hat{I}_{z}$
with
$\omega_{\perp}=\frac{1}{r^{3}} \cdot \frac{\mu_{0}}{4 \pi \hbar} \cdot g_{1} g_{2} \mu_{\mathrm{B}}^{2}$
The dipole-dipole coupling then has a simple dependence on the angle $\theta$ between the external magnetic field $\vec{B}_{0}$ and the spin-spin vector $\vec{r}$ and the coupling can be interpreted as the interaction of the spin with the $z$ component of the local magnetic field that is induced by the magnetic dipole moment of the coupling partner (Figure 5.3). Since the average of the second Legendre polynomial $\left(1-3 \cos ^{2} \theta\right) / 2$ over all angles $\theta$ vanishes, the dipole-dipole interaction vanishes under fast isotropic motion. Measurements of this interaction are therefore performed in the solid state.
The dipole-dipole tensor in the secular approximation has the eigenvalues $\left(\omega_{\perp}, \omega_{\perp},-2 \omega_{\perp}\right)$. The dipole-dipole coupling $d$ at any orientation $\theta$ is given by
$d=\omega_{\perp}\left(1-3 \cos ^{2} \theta\right)$
Spectral manifestation of the dipole-dipole interaction
The energy level scheme and a schematic spectrum for a spin pair with fixed angle $\theta$ are shown in Figure $5.4 \mathrm{a}$ and b, respectively. The dipole-dipole couplings splits the transition of either coupled spin by $d$. If the sample is macroscopically isotropic, for instance a microcrystalline powder or a glassy frozen solution, all angles $\theta$ occur with probability $\sin \theta$. Each line of the dipolar doublet
${ }^{1}$ Hyperfine coupling of the electron spins can modify this condition.
Figure 5.4: Energy level scheme (a) and schematic spectrum (b) for a dipole-dipole coupled spin pair at fixed orientation $\theta$ with respect to the magnetic field. The electron Zeeman frequencies of the two spins are $\omega_{\mathrm{A}}$ and $\omega_{\mathrm{B}}$, respectively. Weak coupling $d \ll\left|\omega_{\mathrm{A}}-\omega_{\mathrm{B}}\right|$ is assumed. The dipolar splitting $d$ is the same for both spins. Depending on homogeneous linewidth $1 / T_{2}$, the splitting may or may not be resolved. If $\omega_{\mathrm{A}}$ and $\omega_{\mathrm{B}}$ are distributed, for instance by $g$ anisotropy, resolution is lost even for $d>1 / T_{2}$.
is then broadened to a powder pattern as illustrated in Figure 3.3. The powder pattern for the $\beta$ state of the partner spin is a mirror image of the one for the $\alpha$ state, since the frequency shifts by the local magnetic field have opposite sign for the two states. The superposition of the two axial powder patterns is called Pake pattern (Figure 5.5). The center of the Pake pattern corresponds to the magic angle $\theta_{\text {magic }}=\arccos \sqrt{1 / 3} \approx 54.7^{\circ}$. The dipole-dipole coupling vanishes at this angle.
Figure 5.5: Pake pattern observed for a dipole-dipole coupled spin pair. (a) The splitting of the dipolar doublet varies with angle $\theta$ between the spin-spin vector and the static magnetic field. Orientations have a probability $\sin \theta$. (b) The sum of all doublets for a uniform distribution of directions of the spin-spin vector is the Pake pattern. The "horns" are split by $\omega_{\perp}$ and the "shoulders" are split by $\omega_{\|}=2 \omega_{\perp}$. The center of the pattern corresponds to the magic angle.
The Pake pattern is very rarely observed in an EPR spectrum, since usually other anisotropic interactions are larger than the dipole-dipole interaction between electron spins. If the weakcoupling condition $d \ll\left|\omega_{\mathrm{A}}-\omega_{\mathrm{B}}\right|$ is fulfilled for the vast majority of all orientations, the EPR lineshape is well approximated by a convolution of the Pake pattern with the lineshape in the absence of dipole-dipole interaction. If the latter lineshape is known, for instance from measuring analogous samples that carry only one of the two electron spins, the Pake pattern can be extracted by deconvolution and the distance between the two electron spins can be inferred from the splitting $\omega_{\perp}$ by inverting Eq. (5.15).
Zero-field interaction
Physical picture
If several unpaired spins are very strongly exchange coupled, then they are best described by a group spin $S$. The concept is most easily grasped for the case of two electron spins that we have already discussed in Section 5.1.1. In this case, the singlet state with group spin $S=0$ is diamagnetic and thus not observable by EPR. The three sublevels of the observable triplet state with group spin $S=1$ correspond to magnetic quantum numbers $m_{S}=-1,0$, and $+1$ at high field. These levels are split by the electron Zeeman interaction. The transitions $m_{S}=-1 \leftrightarrow 0$ and $m_{S}=0 \leftrightarrow+1$ are allowed electron spin transitions, whereas the transition $m_{S}=-1 \leftrightarrow+1$ is a forbidden double-quantum transition.
At zero magnetic field, the electron Zeeman interaction vanishes, yet the three triplet sublevels are not degenerate, they exhibit zero-field splitting. This is because the unpaired electrons are also dipole-dipole coupled. Integration of Eq. (5.6) over the spatial distribution of the two electron spins in their respective SOMOs provides a zero-field interaction tensor $\mathbf{D}$ that can be cast in a form where it describes coupling of the group spin $S=1$ with itself [Rie07]. At zero field, the triplet sublevels are not described by the magnetic quantum number $m_{S}$, which is a good quantum number only if the electron Zeeman interaction is much larger than the zero-field interaction. Rather, the triplet sublevels at zero field are related to the principal axes directions of the zero-field interaction tensor and are therefore labeled $\mathrm{T}_{x}, \mathrm{~T}_{y}$, and $\mathrm{T}_{z}$, whereas the sublevels in the high-field approximation are labeled $\mathrm{T}_{-1}, \mathrm{~T}_{0}$, and $\mathrm{T}_{+1}$.
This concept can be extended to an arbitrary number of strongly coupled electron spins. Cases with up to 5 strongly coupled unpaired electrons occur for transition metal ions (d shell) and cases with up to 7 strongly coupled unpaired electrons occur for rare earth ions (f shell). According to Hund’s rule, in the absence of a ligand field the state with largest group spin $S$ is the ground state. Kramers ions with an odd number of unpaired electrons have a half-integer group spin $S$. They behave differently from non-Kramers ions with an even number of electrons and integer group spin $S$. This classification relates to Kramers’ theorem, which states that for a time-reversal symmetric system with half-integer total spin, all eigenstates occur as pairs (Kramers pairs) that are degenerate at zero magnetic field. As a consequence, for Kramers ions the ground state at zero field will split when a magnetic field is applied. For any microwave frequency there exists a magnetic field where the transition within the ground Kramers doublet is observable in an EPR spectrum. The same does not apply for integer group spin, where the ground state may not be degenerate at zero field. If the zero-field interaction is larger than the maximum available microwave frequency, non-Kramers ions may be unobservable by EPR spectroscopy although they exist in a paramagnetic high-spin state. Typical examples of such EPR silent non-Kramers ions are high-spin $\mathrm{Ni}(\mathrm{II})\left(3 \mathrm{~d}^{8}, S=1\right)$ and high-spin $\mathrm{Fe}(\mathrm{II})\left(3 \mathrm{~d}^{6}, S=2\right)$. In rare cases, non-Kramers ions are EPR observable, since the ground state can be degenerate at zero magnetic field if the ligand field features axial symmetry. Note also that "EPR silent" non-Kramers ions can become observable at sufficiently high microwave frequency and magnetic field.
For transition metal and rare earth ions, zero-field interaction is not solely due to the dipole-dipole interaction between the electron spins. Spin-orbit coupling also contributes, in many cases even stronger than the dipole-dipole interaction. Quantum-chemical prediction of the zero-field interaction is an active field of research. Quite reasonable predictions can be obtained for transition metal ions, whereas only order-of-magnitude estimates are usually possible for rare earth ions.
Zero-field interaction Hamiltonian
The zero-field interaction Hamiltonian is often given as
$\hat{\mathcal{H}}_{\mathrm{ZFI}}=\overrightarrow{\hat{S}}^{\mathrm{T}} \mathbf{D} \overrightarrow{\hat{S}}^{\overrightarrow{\mathrm{S}}}$
where ${ }^{T}$ denotes the transpose of the spin vector operator. In the principal axes system of the zero-field splitting (ZFS) tensor, the Hamiltonian simplifies to
\begin{aligned} \hat{\mathcal{H}}_{\mathrm{ZFI}} &=D_{x} \hat{S}_{x}^{2}+D_{y} \hat{S}_{y}^{2}+D_{z} \hat{S}_{z}^{2} \ &=D\left[S_{z}^{2}-\frac{1}{3} S(S+1)\right]+E\left(S_{x}^{2}-S_{y}^{2}\right) \end{aligned}
where $D=3 D_{z} / 2$ and $E=\left(D_{x}-D_{y}\right) / 2$. The reduction to two parameters is possible, since $\mathbf{D}$ is a traceless tensor. In other words, the zero-field interaction is purely anisotropic. The $D, E$ notation presumes that $D_{z}$ is the principal value with the largest absolute value $(D$ can be negative). Together with the absence of an isotropic component, this means that $D_{y}$, which is always the intermediate value, is either closer to $D_{x}$ than to $D_{z}$ or exactly in the middle between these two values. Accordingly, $|E| \leq|D / 3|$. At axial symmetry $E=0$. Axial symmetry applies if the system has a $C_{n}$ symmetry axis with $n \geq 3$. At cubic symmetry, both $D$ and $E$ are zero. For group spin $S \geq 2$, the leading term of the $\mathrm{ZFS}$ is then a hexadecapolar contribution that scales with the fourth power of the spin operators $\left(\hat{S}_{x}^{4}, \hat{S}_{y}^{4}, \hat{S}_{z}^{4}\right)$.
In the high-field approximation the ZFS contribution to the Hamiltonian is a $\omega_{D} S_{z}^{2}$ term. In other words, to first order in perturbation theory the contribution of the ZFS to the energy of a spin level with magnetic quantum number $m_{S}$ scales with $m_{S}^{2}$. For an allowed transition $m_{S} \leftrightarrow m_{S}+1$, this contribution is $\omega_{D}\left(2 m_{S}+1\right)$. This contribution vanishes for the central transition $m_{S}=-1 / 2 \leftrightarrow 1 / 2$ of Kramers ions. More generally, because of the scaling of the level energies with $m_{S}^{2}$ to first-order, the contribution of ZFS to transition frequencies vanishes for all $-m_{S} \leftrightarrow+m_{S}$ transitions.
Figure 5.6: Schematic CW EPR spectra for triplet states $(S=1)$ at high field. Simulations were performed at an X-band frequency of $9.6 \mathrm{GHz}$. (a) Axial symmetry $(D=1 \mathrm{GHz}, E=0)$. The spectrum is the derivative of a Pake pattern. (b) Orthorhombic symmetry $(D=1 \mathrm{GHz}, E=0.1 \mathrm{GHz})$.
Spectral manifestation of zero-field splitting
Spectra are most easily understood in the high-field approximation. Quite often, deviations from this approximation are significant for the ZFS (see Fig. 2.2), and such deviations are discussed later. The other limiting case, where the ZFS is much larger than the electron Zeeman interaction (Fe(III) and most rare earth ions), is discussed in Section 5.3.4.
For triplet states $(S=1)$ with axial symmetry of the ZFS tensor, the absorption spectrum is a Pake pattern (see Section 5.2.3). With continuous-wave EPR, the derivative of the absorption spectrum is detected, which has the appearance shown in Fig. 5.6(a). A deviation from axial symmetry leads to a splitting of the "horns" of the Pake pattern by $3 E$, whereas the "shoulders" of the pattern are not affected (Fig. 5.6(b)). Triplet states of organic molecules are often observed after optical excitation of a singlet state and intersystem crossing. Such intersystem crossing generally leads to different population of the zero-field triplet sublevels $\mathrm{T}_{x}, \mathrm{~T}_{y}$, and $\mathrm{T}_{z}$. In this situation the spin system is not at thermal equilibrium, but spin polarized. Such spin polarization affects relative intensity of the lineshape singularities in the spectra and even the sign of the signal may change. However, the singularities are still observed at the same resonance fields, i.e., the parameters $D$ and $E$ can still be read off the spectra as indicated in Fig. $5.6$.
Even if the populations of the triple sublevels have relaxed to thermal equilibrium, the spectrum may still differ from the high-field approximation spectrum, as is illustrated in Fig. $5.7$ for the excited naphtalene triplet (simulation performed with an example script of the software package EasySpin http://WWW . easyspin.org/). For $D=3 \mathrm{GHz}$ at a field of about $160 \mathrm{mT}$ (electron Zeeman frequency of about $4.8 \mathrm{GHz}$ ) the high-field approximation is violated and $m_{S}$ is no longer a good quantum number. Hence, the formally forbidden double-quantum transition $m_{S}=-1 \leftrightarrow+1$ becomes partially allowed. To first order in perturbation theory, this transition is not broadened by the ZFS. Therefore it is very narrow compared to the allowed transitions and appears with higher amplitude.
Figure 5.7: CW EPR spectrum of the excited naphtalene triplet at thermal equilibrium (simulation at an $\mathrm{X}$-band frequency of $9.6 \mathrm{GHz}) . D \approx 3 \mathrm{GHz}, E \approx 0.41 \mathrm{GHz}$ ). The red arrow marks the half-field signal, which corresponds to the formally forbidden double-quantum transition $m_{S}=-1 \leftrightarrow+1$.
For Kramers ions, the spectra are usually dominated by the central $m_{S}=-1 / 2 \leftrightarrow 1 / 2$ transition, which is not ZFS-broadened to first order. To second order in perturbation theory, the ZFS-broadening of this line scales inversely with magnetic field. Hence, whereas systems with $g$ anisotropy exhibit broadening proportional to the magnetic field $B_{0}$, central transitions of Kramers ions exhibit narrowing with $1 / B_{0}$. The latter systems can be detected with exceedingly high sensitivity at high fields if they do not feature significant $g$ anisotropy. This applies to systems with half-filled shells (e.g. $\mathrm{Mn}(\mathrm{II}), 3 \mathrm{~d}^{5} ; \mathrm{Gd}(\mathrm{III}), 4 \mathrm{f}^{7}$ ). In the case of Mn(II) (Figure 5.8) the narrow central transition is split into six lines by hyperfine coupling to the nuclear spin of ${ }^{55} \mathrm{Mn}$ (nuclear spin $I=5 / 2,100 \%$ natural abundance). Because of the $\left|2 m_{S}+1\right|$ scaling of anisotropic ZFS broadening of $m_{S} \leftrightarrow m_{S}+1$ transitions, satellite transitions become the broader the larger $\left|m_{S}\right|$ is for the involved levels. In the high-temperature approximation, the integral intensity in the absorption spectrum is the same for all transitions. Hence, broader transitions make a smaller contribution to the amplitude in the absorption spectrum and in its first derivative that is acquired by CW EPR.
Figure 5.8: CW EPR spectrum of a Mn(II) complex (simulation at a W-band frequency of $94 \mathrm{GHz}$ ). $D=0.6 \mathrm{GHz}, E=0.05 \mathrm{GHz}, A\left({ }^{55} \mathrm{Mn}\right)=253 \mathrm{MHz}$. The six intense narrow lines are the hyperfine multiplet of the central transition $m_{S}=-1 / 2 \leftrightarrow+1 / 2$.
The situation can be further complicated by $D$ and $E$ strain, which is a distribution of the $D$ and $E$ parameters due to a distribution in the ligand field. Such a case is demonstrated in Fig. $5.9$ for Gd(III) at a microwave frequency of $34 \mathrm{GHz}$ where second-order broadening of the central transition is still rather strong. In such a case, lineshape singularities are washed out and ZFS parameters cannot be directly read off the spectra. In CW EPR, the satellite transitions may remain unobserved as the derivative of the absorption lineshape is very small except for the central transition.
Effective spin $1 / 2$ in Kramers doublets
For some systems, such as Fe(III), ZFS is much larger than the electron Zeeman interaction at any experimentally attainable magnetic field. In this case, the zero-field interaction determines the quantization direction and the electron Zeeman interaction can be treated as a perturbation [Cas+60]. The treatment is simplest for axial symmetry $(E=0)$, where the quantization axis is the $z$ axis of the ZFS tensor. The energies in the absence of the magnetic field are
$\omega\left(m_{S}\right)=D m_{S}^{2}$
which for high-spin Fe(III) with $S=5 / 2$ gives three degenerate Kramers doublets corresponding to $m_{S}=\pm 5 / 2, \pm 3 / 2$, and $\pm 1 / 2$. If the magnetic field is applied along the $z$ axis of the ZFS tensor, $m_{S}$ is a good quantum number and there is simply an additional energy term $m_{S} g \mu_{\mathrm{B}} B_{0}$ with $g$ being the $g$ value for the half-filled shell, which can be approximated as $g=2$. Furthermore, in this situation only the $m_{S}=-1 / 2 \leftrightarrow 1 / 2$ transition is allowed. The Zeeman term leads to a splitting of the $m_{S}=\pm 1 / 2$ Kramers doublet that is proportional to $B_{0}$ and
Figure 5.9: Echo-detected EPR spectrum (absorption spectrum) of a Gd(III) complex with $D \approx 1.2$ GHz, a Gaussian distribution of $D$ with standard deviation of $0.24 \mathrm{GHz}$ and a correlated distribution of $E$ (simulation at a Q-band frequency of $34 \mathrm{GHz}$ courtesy of Dr. Maxim Yulikov). (a) Total spectrum (black) and contributions of the individual transitions (see legend). The signal from the central transition (blue) dominates. (b) Contributions of the satellite transitions scaled vertically for clarity.
corresponds to $g=2$. This Kramers doublet can thus be described as an effective spin $S^{\prime}=1 / 2$ with $g_{\mathrm{eff}}=2$.
If the magnetic field is perpendicular to the $\mathrm{ZFS}$ tensor $z$ axis, the $m_{S}=\pm 5 / 2$ and $\pm 3 / 2$ Kramers doublets are not split, since the $S_{x}$ and $S_{y}$ operator does not connect these levels. The $S_{x}$ operator has an off-diagonal element connecting the $m_{S}=\pm 1 / 2$ levels that is $\sqrt{S(S+1)+1 / 4} / 2=3 / 2$. Since the levels are degenerate in the absence of the electron Zeeman interaction, they become quantized along the magnetic field and $m_{S}$ is again a good quantum number of this Kramers doublet. The energies are $m_{S} 3 g \mu_{\mathrm{B}} B_{0}+D / 4$, so that the transition frequency is again proportional to $B_{0}$, but now with an effective $g$ value $g_{\text {eff }}=6$. Intermediate orientations can be described by assuming an effective $g$ tensor with axial symmetry and $g_{\perp}=6, g_{\|}=2$. This situation is encountered to a good approximation for high-spin Fe(III) in hemoglobins $\left(g_{\perp} \approx 5.88, g_{\|}=2.01\right)$.
For the non-axial case $(E \neq 0)$, the magnetic field $B_{0}$ will split all three Kramers doublets. To first order in perturbation theory the splitting is proportional to $B_{0}$, meaning that each Kramers doublet can be described by an effective spin $S^{\prime}=1 / 2$ with an effective $g$ tensor. Another simple case is encountered for extreme rhombicity, $E=D / 3$. By reordering principal axes (exchanging $z$ with either $x$ or $y$ ) one can the get rid of the $S_{z}^{2}$ term in Eq. (5.18), so that the ZFS Hamiltonian reduces to $E^{\prime}=\left(S_{x}^{2}-S_{y}^{2}\right)$ with $E^{\prime}=2 E$. The level pair corresponding to the new $z$ direction of the $\mathrm{ZFS}$ tensor has zero energy at zero magnetic field and it can be shown that it has an isotropic effective $g$ value $g_{\mathrm{eff}}=30 / 7 \approx 4.286$. Indeed, signals near $g=4.3$ are very often observed for high-spin Fe(III).
Physical picture
The $S=1 / 2, I=1 / 2$ spin system
The basic phenomena can be well understood in the simplest possible electron-nuclear spin system consisting of a single electron spin $S=1 / 2$ with isotropic $g$ value that is hyperfine coupled to a nuclear spin $I=1 / 2$ with a magnitude of the hyperfine coupling that is much smaller than the electron Zeeman interaction. In this situation the high-field approximation is valid for the electron spin, so that the hyperfine Hamiltonian can be truncated to the form given by Eq. (4.9). Because of the occurrence of an $\hat{S}_{z} \hat{I}_{x}$ operator in this Hamiltonian, we cannot simply transform the Hamiltonian to the rotating frame for the nuclear spin $I$. However, we don’t need to, as we shall consider only microwave irradiation. For the electron spin $S$, we transform to the rotating frame where this spin has a resonance offset $\Omega_{S}$. Hence, the total Hamiltonian takes the form
$\hat{H}_{0}=\Omega_{S} \hat{S}_{z}+\omega_{I} \hat{I}_{z}+A \hat{S}_{z} \hat{I}_{z}+B \hat{S}_{z} \hat{I}_{x}$
in the rotating frame for the electron spin and the laboratory frame for the nuclear spin. Such a Hamiltonian is a good approximation, for instance, for protons in organic radicals.
The Hamiltonian deviates from the Hamiltonian that would apply if the high-field approximation were also fulfilled for the nuclear spin. The difference is the pseudo-secular hyperfine coupling term $B \hat{S}_{z} \hat{I}_{x}$. As can be seen from Eq. (4.10), this term vanishes if the hyperfine interaction is purely isotropic, i.e. for sufficiently fast tumbling in liquid solution, ${ }^{1}$ and along the principal axes of the hyperfine tensor. Otherwise, the $B$ term can only be neglected if $\omega_{I} \gg A, B$, corresponding to the high-field approximation of the nuclear spin. Within the approximate range $2\left|\omega_{I}\right| / 5<|A|<10\left|\omega_{I}\right|$ the pseudo-secular interaction may affect transition frequencies and makes formally forbidden transitions with $\Delta m_{S}=1, \Delta m_{I}=1$ partially allowed, as $m_{I}$ is no longer a good quantum number.
Local fields at the nuclear spin
The occurrence of forbidden transitions can be understood in a semi-classical magnetization vector picture by considering local fields at the nuclear spin for the two possible states $\alpha_{S}$ and
${ }^{1}$ The product of rotational correlation time $\tau_{\mathrm{r}}$ and hyperfine anisotropy must be much smaller than unity
Figure 6.1: Local fields (multiplied by the gyromagnetic ratio $\gamma_{I}$ of the nuclear spin) at the nuclear spin in the two states $\alpha_{S}$ and $\beta_{S}$ of an electron spin $S=1 / 2$. The quantization axes are along the effective fields $\vec{\omega}_{\alpha} / \gamma_{I}$ and $\vec{\omega}_{\beta} / \gamma_{I}$ and are, thus, not parallel.
$\beta_{S}$ of the electron spin. These local fields are obtained from the parameters $\omega_{I}, A$, and $B$ of the Hamilton operator terms that act on the nuclear spin. When divided by the gyromagnetic ratio of the nuclear spin these terms have the dimension of a local magnetic field. The local field corresponding to the nuclear Zeeman interaction equals the static magnetic field $B_{0}$ and is the same for both electron spin states, since the expectation value of $\hat{I}_{z}$ does not depend on the electron spin state. It is aligned with the $z$ direction of the laboratory frame (blue arrow in Figure 6.1). Both hyperfine fields arise from Hamiltonian terms that contain an $\hat{S}_{z}$ factor, which has the expectation value $m_{S}=+1 / 2$ for the $\alpha_{S}$ state and $m_{S}=-1 / 2$ for the $\beta_{S}$ state. The $A$ term is aligned with the $z$ axis and directed towards $+z$ in the $\alpha_{S}$ state and towards $-z$ in the $\beta_{S}$ state, assuming $A>0$ (violet arrows). The $B$ term is aligned with the $x$ axis and directed towards $+x$ in the $\alpha_{S}$ state and towards $-x$ in the $\beta_{S}$ state, assuming $B>0$ (green arrows).
The effective fields at the nuclear spin in the two electron spins states are vector sums of the three local fields. Because of the $B$ component along $x$, they are tilted from the $z$ direction by angle $\eta_{\alpha}$ in the $\alpha_{S}$ state and by angle $\eta_{\beta}$ in the $\beta_{S}$ state. The length of the sum vectors are the nuclear transition frequencies in these two states and are given by
\begin{aligned} &\omega_{\alpha}=\sqrt{\left(\omega_{I}+A / 2\right)^{2}+B^{2} / 4} \ &\omega_{\beta}=\sqrt{\left(\omega_{I}-A / 2\right)^{2}+B^{2} / 4} \end{aligned}
For $\left|\omega_{I}\right|>2|A|$, the hyperfine splitting is given by
$\omega_{\mathrm{hfs}}=\left|\omega_{\alpha}-\omega_{\beta}\right|$
and the sum frequency is given by
$\omega_{\text {sum }}=\omega_{\alpha}+\omega_{\beta}$
For $\left|\omega_{I}\right|>2|A|$, the nuclear frequency doublet is centered at $\omega_{\text {sum }} / 2($ Fig. $6.2(\mathrm{c})$ ). The sum frequency is always larger than twice the nuclear Zeeman frequency. None of the nuclear frequencies can become zero, the minimum possible value $B / 2$ is attained in one of the electron spin states for matching of the nuclear Zeeman and hyperfine interaction at $2\left|\omega_{I}\right|=|A|$. For $\left|\omega_{I}\right|<2|A|$ the nuclear frequency doublet is split by $\omega_{\text {sum }}$ and centered at $\omega_{\text {hfs }} / 2($ Fig. $\left.6.2(\mathrm{~d}))\right)$. The tilt angles $\eta_{\alpha}$ and $\eta_{\beta}$ (Figure 6.1) can be inferred from trigonometric relations and are given by
\begin{aligned} &\eta_{\alpha}=\arctan \left(\frac{-B}{2 \omega_{I}+A}\right) \ &\eta_{\beta}=\arctan \left(\frac{-B}{2 \omega_{I}-A}\right) \end{aligned}
Consider now a situation where the electron spin is in its $\alpha_{S}$ state. The nuclear magnetization from all radicals in this state at thermal equilibrium is aligned with $\vec{\omega}_{\alpha}$. Microwave excitation causes transitions to the $\beta_{S}$ state. In this state, the local field at the nuclear spin is directed along $\vec{\omega}_{\beta}$. Hence, the nuclear magnetization vector from the radicals under consideration is tilted by angle $2 \eta$ (Figure 6.1) with respect to the current local field. It will start to precess around this local field vector. This corresponds to excitation of the nuclear spin by flipping the electron spin, which is a formally forbidden transition. Obviously, such excitation will occur only if angle $2 \eta$ differs from $0^{\circ}$ and from $180^{\circ}$. The case of $0^{\circ}$ corresponds to the absence of pseudo-secular hyperfine coupling $(B=0)$ and is also attained in the limit $|A| \ll\left|\omega_{I}\right|$. The situation $2 \eta \rightarrow 180^{\circ}$ is attained in the limit of very strong secular hyperfine coupling, $|A| \gg\left|\omega_{I}\right|$. Forbidden transitions are observed for intermediate hyperfine coupling. Maximum excitation of nuclear spins is expected when the two quantization axes are orthogonal with respect to each other, $2 \eta=90^{\circ}$.
Figure 6.2: Electron-nuclear spin system $S=1 / 2, I=1 / 2$ in the presence of pseudo-secular hyperfine coupling. (a) Level scheme. In EPR, $\Delta m_{S}=1, \Delta m_{I}=0$ transitions are allowed (red), in NMR $\Delta m_{S}=0, \Delta m_{I}=1$ transitions are allowed (blue), and the zero- and double-quantum transitions with $\Delta m_{S}=1, \Delta m_{I}=1$ are formally forbidden. (b) EPR stick spectrum. Allowed transitions have transition probability $\cos ^{2} \eta$ and forbidden transitions probability $\sin ^{2} \eta$. The spectrum is shown for $\left|\omega_{I}\right|>2|A|$. For $\left|\omega_{I}\right|<2|A|$, the forbidden transitions lie inside the allowed transition doublet. (c) NMR spectrum for $\left|\omega_{I}\right|>2|A|$. (d) NMR spectrum for $\left|\omega_{I}\right|<2|A|$.
Product operator formalism with pseudo-secular interactions
Transformation of $\hat{S}_{x}$ to the eigenbasis
Excitation and detection in EPR experiments are described by the $\hat{S}_{x}$ and $\hat{S}_{y}$ operators in the rotating frame. These operators act only on electron spin transitions and thus formalize the spectroscopic selection rules. If the spin Hamiltonian contains off-diagonal terms, such as the pseudo-secular $B \hat{S}_{z} \hat{I}_{x}$ term in Eq. (6.1), the eigenbasis deviates from the basis of the electron spin rotating frame/nuclear spin laboratory frame in which the Hamiltonian is written and in which the excitation and detection operators are linear combinations of $\hat{S}_{x}$ and $\hat{S}_{y}$. In order to understand which transitions are driven and detected with what transition moment, we need to transform $\hat{S}_{x}$ to the eigenbasis (the transformation of $\hat{S}_{y}$ is analogous). This can be done by product operator formalism and can be understood in the local field picture.
The Hamiltonian in the eigenbasis has no off-diagonal elements, meaning that all quantization axes are along $z$. Thus, we can directly infer from Fig. $6.1$ that, in the $\alpha_{S}$ state, we need a counterclockwise (mathematically positive) rotation by tilt angle $\eta_{\alpha}$ about the $y$ axis, which is pointing into the paper plane. In the $\beta_{S}$ state, we need a clockwise (mathematically negative) rotation by tilt angle $\eta_{\beta}$ about the $y$ axis. The electron spin states can be selected by the projection operators $\hat{S}^{\alpha}$ and $\hat{S}^{\beta}$, respectively. Hence, we have to apply rotations $\eta_{\alpha} \hat{S}^{\alpha} \hat{I}_{y}$ and $-\eta_{\beta} \hat{S}^{\beta} \hat{I}_{y}$. These two rotations commute, as the $\alpha_{S}$ and $\beta_{S}$ subspaces are distinct when $m_{S}$ is a good quantum number. For the rotation into the eigenbasis, we can write a unitary matrix
\begin{aligned} \hat{U}_{\mathrm{EB}} &=\exp \left\{-i\left(\eta_{\alpha} \hat{S}^{\alpha} \hat{I}_{y}-\eta_{\beta} \hat{S}^{\beta} \hat{I}_{y}\right)\right\} \ &=\exp \left\{-i\left(\xi \hat{I}_{y}+\eta 2 \hat{S}_{z} \hat{I}_{y}\right)\right\} \end{aligned}
where $\xi=\left(\eta_{\alpha}-\eta_{\beta}\right) / 2$ and $\eta=\left(\eta_{\alpha}+\eta_{\beta}\right) / 2$. Note that the definition of angle $\eta$ corresponds to the one given graphically in Fig. 6.1.2 The two new rotations about $\hat{I}_{y}$ and $\hat{S}_{z} \hat{I}_{y}$ also commute. Furthermore, $\hat{I}_{y}$ commutes with $\hat{S}_{x}\left(\right.$ and $\hat{S}_{y}$ ), so that the transformation of $\hat{S}_{x}$ to the eigenbasis reduces to
$\hat{S}_{x} \stackrel{\eta 2 \hat{S}_{z} \hat{I}_{y}}{\longrightarrow} \cos \eta \hat{S}_{x}+\sin \eta 2 \hat{S}_{y} \hat{I}_{y}$
The transition moment for the allowed transitions that are driven by $\hat{S}_{x}$ is multiplied by a factor $\cos \eta \leq 1$, i.e. it becomes smaller when $\eta \neq 0$. In order to interpret the second term, it is best rewritten in terms of ladder operators $\hat{S}^{+}=\hat{S}_{x}+i \hat{S}_{y}$ and $\hat{S}^{-}=\hat{S}_{x}-i \hat{S}_{y}$. We find
$2 \hat{S}_{y} \hat{I}_{y}=\frac{1}{2}\left(\hat{S}^{+} \hat{I}^{-}+\hat{S}^{-} \hat{I}^{+}-\hat{S}^{+} \hat{I}^{+}-\hat{S}^{-} \hat{I}^{-}\right)$
In other words, this term drives the forbidden electron-nuclear zero- and double-quantum transitions (Fig. 6.2(a)) with a transition proportional to $\sin \eta$.
In a CW EPR experiment, each transition must be both excited and detected. In other words, the amplitude is proportional to the square of the transition moment, which is the transition probability. Allowed transitions thus have an intensity proportional to $\cos ^{2} \eta$ and forbidden transitions a transition probability proportional to $\sin ^{2} \eta$ (Fig. 6.2(b)).
General product operator computations for a non-diagonal Hamiltonian
In a product operator computation, terms of the Hamiltonian can be applied one after the other if and only if they pairwise commute. This is not the case for the Hamiltonian in Eq. (6.1). However, application of $\hat{U}_{\mathrm{EB}}$ diagonalizes the Hamiltonian:
$\hat{H}_{0} \stackrel{\eta \hat{S}_{z} \hat{I}_{y}}{\longrightarrow} \Omega_{S} \hat{S}_{z}+\omega_{\mathrm{sum}} / 2 \hat{I}_{z}+\omega_{\mathrm{hfi}} \hat{S}_{z} \hat{I}_{z}$
This provides a simple recipe for product operator computations in the presence of the pseudosecular hyperfine coupling. Free evolution and transition-selective pulses are computed in the
${ }^{2}$ We have used $\hat{S}^{\alpha}=\hat{\mathbb{1}} / 2+\hat{S}_{z}$ and $\hat{S}^{\beta}=\hat{\mathbb{1}} / 2-\hat{S}_{z}$. eigenbasis, using the Hamiltonian on the right-hand side of relation (6.9). For application of non-selective pulses, the density operator needs to be transformed to the electron spin rotating frame/nuclear spin laboratory frame basis by applying $\hat{U}_{\mathrm{EB}}^{\dagger}$. In product operator formalism this corresponds to a product operator transformation $\stackrel{-\eta \hat{S}_{z} \hat{I}_{y}}{\longrightarrow}$. After application of non-selective pulses, the density operator needs to be backtransformed to the eigenbasis. Detection also needs to be performed in the electron spin rotating frame/nuclear spin laboratory frame basis.
This concept can be extended to any non-diagonal Hamiltonian as long as one can find a unitary transformation that transforms the Hamiltonian to its eigenbasis and can be expressed by a single product operator term or a sum of pairwise commuting product operator terms.
Generation and detection of nuclear coherence by electron spin excitation
Nuclear coherence generator $(\pi / 2)-\tau-(\pi / 2)$
We have seen that a single microwave pulse can excite coherence on forbidden electron-nuclear zero- and double-quantum transitions. In principle, this provides access to the nuclear frequencies $\omega_{\alpha}$ and $\omega_{\beta}$, which are differences of frequencies of allowed and forbidden electron spin transitions, as can be inferred from Fig. 6.2(a,b). Indeed, the decay of an electron spin Hahn echo $(\pi / 2)-\tau-(\pi)-\tau-$ echo as a function of $\tau$ is modulated with frequencies $\omega_{\alpha}$ and $\omega_{\beta}($ as well as with $\omega_{\text {hfi }}$ and $\omega_{\text {sum }}$ ). Such modulation arises from forbidden transitions during the refocusing pulse, which redistribute coherence among the four transitions. The coherence transfer echoes are modulated by the difference of the resonance frequencies before and after the transfer by the $\pi$ pulse, in which the resonance offset $\Omega_{S}$ cancels, while the nuclear spin contributions do not cancel. This two-pulse ESEEM experiment is not usually applied for measuring hyperfine couplings, as the appearance of the combination frequencies $\omega_{\text {hfi }}$ and $\omega_{\text {sum }}$ complicates the spectra and linewidth is determined by electron spin transverse relaxation, which is much faster the nuclear spin transverse relaxation.
Better resolution and simpler spectra can be obtained by indirect observation of the evolution of nuclear coherence. Such coherence can be generated by first applying a $\pi / 2$ pulse to the electron spins, which will generate electron spin coherence on allowed transitions with amplitude proportional to $\cos \eta$ and on forbidden transitions with amplitude proportional to sin $\eta$. After a delay $\tau$ a second $\pi / 2$ pulse is applied. Note that the block $(\pi / 2)-\tau-(\pi / 2)$ is part of the EXSY and NOESY experiments in NMR. The second $\pi / 2$ pulse will generate an electron spin magnetization component along $z$ for half of the existing electron spin coherence, i.e., it will "switch off" half the electron spin coherence and convert it to polarization. However, for the coherence on forbidden transitions, there is a chance $\cos \eta$ that the nuclear spin is not flipped, i.e. that the coherent superposition of the nuclear spin states survives. For electron spin coherence on allowed transitions, there is a chance $\sin \eta$ that the "switching off" of the electron coherence will lead to a "switching on" of nuclear coherences. Hence, in both these pathways there is a probability proportional to $\sin \eta \cos \eta=\sin (2 \eta) / 2$ that nuclear coherence is generated. The delay $\tau$ is required, since at $\tau=0$ the different nuclear coherence components have opposite phase and cancel.
The nuclear coherence generated by the block $(\pi / 2)-\tau-(\pi / 2)$ can be computed by product operator formalism as outlined in Section 6.2.2. We find
\begin{aligned} &\left\langle\hat{S}^{\alpha} \hat{I}_{x}\right\rangle=-\sin \left(\Omega_{S} \tau\right) \sin (2 \eta) \sin \left(\frac{\omega_{\beta}}{2} \tau\right) \cos \left(\omega_{\alpha} \tau\right) \ &\left\langle\hat{S}^{\alpha} \hat{I}_{y}\right\rangle=-\sin \left(\Omega_{S} \tau\right) \sin (2 \eta) \sin \left(\frac{\omega_{\beta}}{2} \tau\right) \sin \left(\omega_{\alpha} \tau\right) \ &\left\langle\hat{S}^{\beta} \hat{I}_{x}\right\rangle=-\sin \left(\Omega_{S} \tau\right) \sin (2 \eta) \sin \left(\frac{\omega_{\alpha}}{2} \tau\right) \cos \left(\omega_{\beta} \tau\right) \ &\left\langle\hat{S}^{\beta} \hat{I}_{y}\right\rangle=-\sin \left(\Omega_{S} \tau\right) \sin (2 \eta) \sin \left(\frac{\omega_{\alpha}}{2} \tau\right) \sin \left(\omega_{\beta} \tau\right) \end{aligned}
This expression can be interpreted in the following way. Nuclear coherence is created with a phase as if it had started to evolve as $\hat{I}_{x}$ at time $\tau=0$ (last cosine factors on the right-hand side of each line). It is modulated as a function of the electron spin resonance offset $\Omega_{S}$ and zero exactly on resonance (first factor on each line). The integral over an inhomogeneously broadened, symmetric EPR line is also zero, since $\int_{-\infty}^{\infty} \sin \left(\Omega_{S} \tau\right) \mathrm{D} \Omega_{S}=0$. However, this can be compensated later by applying another $\pi / 2$ pulse. The amplitude of the nuclear coherence generally scales with $\sin 2 \eta$, since one allowed and one forbidden transfer are required to excite it and $\sin (\eta) \cos (\eta)=\sin (2 \eta) / 2$ (second factor). The third factor on the right-hand side of lines 1 and 2 tells that the amplitude of the coherence with frequency $\omega_{\alpha}$ is modulated as a function of $\tau$ with frequency $\omega_{\beta}$. Likewise, the amplitude of the coherence with frequency $\omega_{\beta}$ is modulated as a function of $\tau$ with frequency $\omega_{\alpha}$ (lines 3 and 4 ). At certain values of $\tau$ no coherence is created at the transition with frequency $\omega_{\alpha}$, at other times maximum coherence is generated. Such behavior is called blind-spot behavior. In order to detect all nuclear frequencies, an experiment based on the $(\pi / 2)-\tau-(\pi / 2)$ nuclear coherence generator has to be repeated for different values of $\tau$. Why and how CW EPR spectroscopy is done Sensitivity advantages of CW EPR spectroscopy The CW EPR experiment
Considerations on sample preparation Theoretical description of CW EPR Spin packet lineshape Saturation
7 - CW EPR Spectroscopy
Why and how CW EPR spectroscopy is done
Sensitivity advantages of CW EPR spectroscopy
In NMR spectroscopy, CW techniques have been almost completely displaced by Fourier transform (FT) techniques, except for a few niche applications. FT techniques have a sensitivity advantage if the spectrum contains large sections of baseline and the whole spectrum can be excited simultaneously by the pulses. Neither condition is usually fulfilled in EPR spectroscopy. For two reasons, FT techniques lose sensitivity in EPR spectroscopy compared to the CW experiment. First, while typical NMR spectra comfortably fit into the bandwidth of a welldesigned critically coupled radiofrequency resonance circuit, EPR spectra are much broader than the bandwidth of a microwave resonator with high quality factor. Broadening detection bandwidth and proportionally lowering the quality factor $Q$ of the resonator reduces signal-to-noise ratio unless the absorption lineshape is infinitely broad. A quality factor of the order of 10 ’ 000 , which can be achieved with cavity resonators, corresponds to a bandwidth of roughly $1 \mathrm{MHz}$ at X-band frequencies around $9.6 \mathrm{GHz}$. The intrinsic high sensitivity of detection in such a narrow band can be used only in a CW experiment. Second, even if the resonator is overcoupled to a much lower quality factor or resonators with intrinsically lower $Q$ are used (the sensitivity loss can partially be compensated by a higher filling factor of such resonators), residual power from a high-power microwave pulse requires about $100 \mathrm{~ns}$ in order to decay below the level of an EPR signal. This dead time is often a significant fraction of the transverse relaxation time of electron spins, which entails signal loss by relaxation. In contrast, in NMR spectroscopy the dead time is usually negligibly short compared to relaxation times. In many cases, the dead time in pulsed EPR spectroscopy even strongly exceeds $T_{2}$. In this situation FT EPR is impossible, even with echo refocusing, while CW EPR spectra can still be measured. This case usually applies to transition metal complexes at room temperature and to many rare earth metal complexes and high-spin Fe(III) complexes even down to the boiling point of liquid helium at normal pressure (4.2 K). For these reasons, any unknown potentially paramagnetic sample should first be characterized by CW EPR spectroscopy. Pulsed EPR techniques are required if the resolution of CW EPR spectroscopy provides insufficient information to assign a structure. This applies mainly to small hyperfine couplings in organic radicals and of ligand nuclei in transition metal complexes (see Chapter 8) and to the measurement of distances between electron spins in the nanometer range (see Chapter 9). At temperatures where pulse EPR signals can be obtained, measurement of relaxation times is also easier and more precise with pulsed EPR techniques.
Figure 7.1: Scheme of a CW EPR spectrometer. Microwave from a fixed-frequency source is passed through an attenuator for adjusting its power and then through a circulator to the sample. Microwave that comes back from the sample passes on a different way through the same circulator and is combined with reference microwave of adjustable power (bias) and phase before it is detected by a microwave diode. The output signal of this diode enters a phase-sensitive detector (PSD) where it is demodulated with respect to the field modulation frequency (typically $100 \mathrm{kHz}$ ) and at the same time amplified. The output signal of the PSD is digitized and further processed in a computer. The spectrum is obtained by sweeping the static magnetic field $B_{0}$ at constant microwave frequency.
The CW EPR experiment
Since the bandwidth of an optimized microwave resonator is much smaller than the typical width of EPR spectra, it is impractical to sweep the frequency at constant magnetic field in order to obtain a spectrum. Instead, the microwave frequency is kept constant and coincides with the resonator frequency at all times. The resonance condition for the spins is established by sweeping the magnetic field $B_{0}$. Another difficulty arises from the weak magnetic coupling of the spins to the exciting electromagnetic field. Only a very small fraction of the excitation power is therefore observed. This problem is solved as follows. First, direct transmission of excitation power to the detector is prevented by a circulator (Figure 7.1). Power that enters port 1 can only leave to the sample through port 2 . Power that comes from the sample through port 2 can only leave to the detector diode through port 3. Second, the resonator is critically coupled. This means that all microwave power coming from the source that is incident to the resonator enters the resonator and is converted to heat by the impedance (complex resistance) of the resonator. If the sample is off resonant and thus does not absorb microwave, no microwave power leaves the resonator through port 3. If now the magnetic field $B_{0}$ is set to the resonance condition and the sample resonantly absorbs microwave, this means that the impedance of resonator + sample has changed. The resonator is no longer critically coupled and some of the incoming microwave is reflected. This microwave leaves the circulator through port 3 and is incident on the detector diode.
This reflected power at resonance absorption can be very weak at low sample concentration. It is therefore important to detect it sensitively. A microwave diode is only weakly sensitive to a change in incident power at low power (Fig. 7.2, input voltage is proportional to the square root of power). The diode is most sensitive to amplitude changes near its operating point, marked green in Fig. 7.2. Hence, the diode must be biased to its operating point by adding constant power from a reference arm. The phase of the reference arm must be adjusted so that microwave coming from the resonator and microwave coming from the reference arm interfere constructively.
Figure 7.2: Characteristic curve of a microwave detection diode. At small input voltage, the diode is rather insensitive to changes in input voltage. At the operating point (green), dependence of output current on input voltage is linear and has maximum slope. This corresponds to $200 \mu \mathrm{A}$ output current. If input voltage is too large, the diode is destroyed (red point).
A further problem arises from the fact that microwave diodes are broadband detectors. On the one hand, this is useful, since samples can significantly shift resonator frequency. On the other hand, broadband detectors also collect noise from a broad frequency band. This decreases signal-to-noise ratio and must be countered by limiting the detection bandwidth to the signal bandwidth or even below. Such bandwidth limitation can be realized most easily by effect modulation and phase sensitive detection. By applying a small sinusoidal magnetic field modulation with typical frequency of $100 \mathrm{kHz}$ and typical amplitude of $0.01-1 \mathrm{mT}$, the signal component at detector diode output becomes modulated with the same frequency, whereas noise is uncorrelated to the modulation. Demodulation with a reference signal from the field modulation generator (Figure 7.1) by a phase-sensitive detector amplifies the signal and limits bandwidth to the modulation frequency.
Effect modulation with phase-sensitive detection measures the derivative of the absorption lineshape, as long as the modulation amplitude $\Delta B_{0}$ is much smaller than the width of the EPR line (Fig. 7.3). Since signal-to-noise ratio is proportional to $\Delta B_{0}$, one usually measures at $\Delta B_{0} \approx \Delta B_{\mathrm{pp}} / 3$, where lineshape distortion is tolerable for almost all applications. Precise lineshape analysis may require $\Delta B_{0} \leq \Delta B_{\mathrm{pp}} / 5$, whereas maximum sensitivity at the expense of significant artificial line broadening is obtained at $\Delta B_{0}=\Delta B_{\mathrm{pp}}$. The modulation frequency should not be broader than the linewidth in frequency units. However, with the standard modulation frequency of $100 \mathrm{kHz}$ that corresponds on a magnetic field scale to only $3.6 \mu \mathrm{T}$ at $g=g_{e}$, this is rarely a problem.
Considerations on sample preparation
Since electron spins have a much larger magnetic moment than nuclear spins, electron-electron couplings lead to significant line broadening in concentrated solutions. Concentrations of paramagnetic centers should not usually exceed $1 \mathrm{mM}$ in order to avoid such broadening. For organic radicals in liquid solution it may be necessary to dilute the sample to $100 \mu \mathrm{M}$ in order to achieve ultimate resolution. For paramagnetic metal dopants in diamagnetic host compounds, at most $1 \%$ of the diamagnetic sites should be substituted by paramagnetic centers. Such concentrations can be detected easily and with good signal-to-noise ratio. For most samples, good spectra can be obtained down to the $1 \mu \mathrm{M}$ range in solution and down to the 100 ppm dopant range in solids.
Line broadening in liquid solution can also arise from diffusional collision of paramagnetic
Figure 7.3: Detection of the derivative lineshape by field modulation. The situation is considered at the instantaneous field during a field sweep (vertical dashed line) that is slow compared to the field modulation frequency of $100 \mathrm{kHz}$. Modulation of the magnetic field with amplitude $\Delta B_{0}$ (blue) causes a modulation of the output signal $V$ (red) with the same frequency and an amplitude $\Delta V$. Phase-sensitive detection measures this amplitude $\Delta V$, which is proportional to the derivative of the grey absorption lineshape and to $\Delta B_{0}$, as long as $\Delta B_{0}$ is much smaller than the peak-to-peak linewidth $\Delta B_{\mathrm{pp}}$ of the line. In practice, $\Delta B_{0}<\Delta B_{\mathrm{pp}} / 3$ is usually acceptable. For precise lineshape analysis, $\Delta B_{0}<\Delta B_{\mathrm{pp}} / 5$ is recommended.
Figure 7.4: Relaxation enhancement by collisional exchange with oxygen in solution. (a) Situation before diffusional encounter. As an example, triplet oxygen is assumed to be in a $\mathrm{T}_{-}$state (red), whereas the spin of a nitroxide radical is assumed to be $\alpha$ (green). (b) The oxygen molecule and nitroxide radical have collided during diffusional encounter. Their wavefunctions overlap and the three unpaired electrons cannot be distinguished from each other (grey). (c) After separation, the three unpaired electrons have been redistributed arbitrarily to the two molecules. For example, oxygen may now be in the $\mathrm{T}_{0}$ state (red) and the nitroxide in the $\beta$ state (green). The electron spin of the nitroxide radical has flipped.
species with paramagnetic triplet oxygen (Figure 7.4). During such a collision, wavefunctions of the two molecules overlap and, since electrons are undistinguishable particles, spin states of all unpaired electrons in both molecules are arbitrarily redistributed when the two molecules separate again. The stochastic diffusional encounters thus lead to additional flips of the observed electron spins, which corresponds to relaxation and shortens longitudinal relaxation time $T_{1}$. Since the linewidth is proportional to $T_{2}$ and $T_{2}$ cannot be longer than $2 T_{1}$, frequent collisional encounters of paramagnetic species lead to line broadening. Such line broadening increases with decreasing viscosity (faster diffusion) and increasing oxygen concentration. The effect is stronger in apolar solvents, where oxygen solubility is higher than in polar solvents, but it is often significant even in aqueous solution. Best resolution is obtained if the sample is free of oxygen. The same mechanism leads to line broadening at high concentration of a paramagnetic species in liquid solution. In the solid state, line broadening at high concentration is mainly due to dipole-dipole coupling. Often, the anisotropically broadened EPR spectrum in the solid state is of interest, as it provides information on $g$ anisotropy and anisotropic hyperfine couplings. This may require freezing of a solution of the species of interest. Usually, the species will precipitate if the solvent crystallizes, which may cause line broadening and, in extreme cases, even collapse of the hyperfine structure and averaging of $g$ anisotropy by exchange between neighboring paramagnetic species. These problems are prevented if the solvent forms a glass, as is often the case for solvents that have methyl groups or can form hydrogen bonds in very different geometries. Typical glass-forming solvents are toluene, 2-methyltetrahydrofuran, ethanol, ethylene glycol, and glycerol. Aqueous solutions require addition of at least $25 \%$ glycerol as a cryoprotectant. In most cases, crystallization will still occur on slow cooling. Samples are therefore shock frozen by immersion of the sample tube into liquid nitrogen. Glass tubes would break on direct immersion into liquid nitrogen, but EPR spectra have to be measured in fused silica sample tubes anyway, since glass invariably contains a detectable amount of paramagnetic iron impurities.
Theoretical description of CW EPR
This section overlaps with Section $2.7$ of the NMR lecture notes.
Spin packet lineshape
All spins in a sample that have the same resonance frequency form a spin packet. In the following we also assume that all spins of a spin packet have the same longitudinal and transverse relaxation times $T_{1}$ and $T_{2}$, respectively. If the number of spins in the spin packet is sufficiently large, we can assign a magnetization vector to the spin packet. Dynamics of this magnetization vector with equilibrium magnetization $M_{0}$ during microwave irradiation is described by the Bloch equations in the rotating frame. In EPR spectroscopy, it is unusual to use the gyromagnetic ratio. Hence, we shall denote the resonance offset by
$\Omega_{S}=\frac{g \mu_{\mathrm{B}}}{\hbar} B_{0}-2 \pi \nu_{\mathrm{mw}}$
where $\nu_{\mathrm{mw}}$ is the microwave frequency in frequency units. The rotating-frame Bloch equations for the three components of the magnetization vector can then be written as
\begin{aligned} \frac{\mathrm{d} M_{x}}{\mathrm{~d} t} &=-\Omega_{S} M_{y}-\frac{M_{x}}{T_{2}} \ \frac{\mathrm{d} M_{y}}{\mathrm{~d} t} &=\Omega_{S} M_{x}-\omega_{1} M_{z}-\frac{M_{y}}{T_{2}} \ \frac{\mathrm{d} M_{z}}{\mathrm{~d} t} &=\omega_{1} M_{y}-\frac{M_{z}-M_{0}}{T_{1}} \end{aligned}
where $\omega_{1}=g_{\perp} \mu_{\mathrm{B}} B_{1} / \hbar$ is the microwave field amplitude in angular frequency units and $g_{\perp}$ is the mean $g$ value in the plane perpendicular to the static magnetic field. The apparent sign difference for the $\Omega_{S}$ and $\omega_{1}$ terms arises from the different sense of spin precession for electron spins compared to nuclear spins with a positive gyromagnetic ratio.
If the spin packet is irradiated at constant microwave frequency, constant microwave power, and constant static magnetic field $B_{0}$ for a sufficiently long time (roughly $5 T_{1}$ ), the magnetization vector attains a steady state. Although the static field is swept in a CW EPR experiment, assuming a steady state is a good approximation, since the field sweep is usually slow compared to $T_{2}$ and $T_{1}$. Faster sweeps correspond to the rapid scan regime that is not treated in this lecture course. In the steady state, the left-hand sides of the differential equations (7.2) for the magnetization vector components must all be zero,
\begin{aligned} 0 &=-\Omega_{S} M_{y}-\frac{M_{x}}{T_{2}} \ 0 &=\Omega_{S} M_{x}-\omega_{1} M_{z}-\frac{M_{y}}{T_{2}} \ 0 &=\omega_{1} M_{y}-\frac{M_{z}-M_{0}}{T_{1}} \cdot\langle\text { stationary state }\rangle \end{aligned}
This linear system of equations has the solution
\begin{aligned} M_{x} &=M_{0} \omega_{1} \frac{\Omega T_{2}^{2}}{1+\Omega^{2} T_{2}^{2}+\omega_{1}^{2} T_{1} T_{2}} \ M_{y} &=-M_{0} \omega_{1} \frac{T_{2}}{1+\Omega^{2} T_{2}^{2}+\omega_{1}^{2} T_{1} T_{2}} \ M_{z} &=M_{0} \frac{1+\Omega^{2} T_{2}^{2}}{1+\Omega^{2} T_{2}^{2}+\omega_{1}^{2} T_{1} T_{2}},\langle\text { stationary state }\rangle \end{aligned}
where $M_{z}$ is not usually detected, $M_{x}$ is in phase with the exciting microwave irradiation and corresponds to the dispersion signal, and $M_{y}$ is out of phase with the exciting irradiation and corresponds to the absorption line. Unperturbed lineshapes are obtained in the linear regime, where the saturation parameter
$S=\omega_{1}^{2} T_{1} T_{2}$
fulfills $S \ll 1$. One can easily ascertain from Eq. (7.4) that in the linear regime $M_{y}$ increases linearly with increasing $\omega_{1}$, which corresponds to proportionality of the signal to the square root of microwave power. $M_{z}$ is very close to the equilibrium magnetization. Within this regime, a decrease of $6 \mathrm{~dB}$ in microwave attenuation, i.e., a power increase by $6 \mathrm{~dB}$, increases signal amplitude by a factor of 2 . Lineshape does not depend on $\omega_{1}$ in the linear regime. Therefore, it is good practice to measure at the highest microwave power that is still well within the linear regime, as this corresponds to maximum signal-to-noise ratio. For higher power the line is broadened.
Within the linear regime, $M_{y}$ takes the form of a Lorentzian absorption line
$M_{y}(\Omega)=M_{0} \omega_{1} T_{2} \frac{1}{1+\Omega^{2} T_{2}^{2}},\langle\text { linear regime }\rangle$
with linewidth $1 / T_{2}$ in angular frequency units. The peak-to-peak linewidth of the first derivative of the absorption line is $\Gamma_{\mathrm{pp}}=2 / \sqrt{3} T_{2}$. Since $\mathrm{CW}$ EPR spectra are measured by sweeping magnetic field, we need to convert to magnetic field units,
$\Gamma_{\text {pp,field sweep }}=\frac{2}{\sqrt{3} T_{2}} \cdot \frac{\hbar}{g \mu_{\mathrm{B}}}$
The linewidth of a spin packet is called homogeneous linewidth. If $T_{2}$ is the same for all spin packets, this homogeneous linewidth is proportional to $1 / \mathrm{g}$, a fact that needs to be taken into account in lineshape simulations for systems with large $g$ anisotropy. For most samples, additional line broadening arises from unresolved hyperfine couplings and, in the solid state, $g$ anisotropy. Therefore, $T_{2}$ cannot usually be obtained by applying Eq. (7.7) to the experimentally observed peak-to-peak linewidth.
Saturation
For microwave power larger than in the linear regime, the peak-to-peak linewidth increases by a factor $1+S$. If a weak signal needs to be detected with maximum signal-to-noise ratio it is advantageous to increase power beyond the linear regime, but not necessarily to the maximum available level. For very strong irradiation, $S \gg 1$, the term 1 can be neglected in the denominator of Eqs. (7.4) for the magnetization vector components. The on-resonance amplitude of the absorption line is then given by
$M_{y}(\Omega=0)=M_{0} / \omega_{1} T_{1},\left\langle\omega_{1}^{2} T_{1} T_{2} \gg 1\right\rangle$
i.e., it is inversely proportional to $\omega_{1}$. In this regime, the amplitude decreases with increasing power of the microwave irradiation.
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Figure 7.5: Progressive saturation measurement on the membrane protein LHCII solubilized in detergent micelles in nitrogen atmosphere. Residue V229 was mutated to cysteine and spin-labelled by iodoacetamidoPROXYL. Experimental data points (red) were obtained at microwave attenuations of $23,20,17,11$, and $8 \mathrm{~dB}$ with a full power $(0 \mathrm{~dB})$ of $200 \mathrm{~mW}$. The fit by Eq. (7.9) (black line) provides $P_{1 / 2}=3 \mathrm{~mW}$ and $\epsilon=1.24$.
Semi-quantitative information on spin relaxation can be obtained by the progressive power saturation experiment, where the EPR spectrum is measured as a function of microwave power $P_{\text {mw }}$. Usually, the peak-to-peak amplitude of the larges signal in the spectrum is plotted as a function of $\sqrt{P_{\mathrm{mw}}}$. Such saturation curves can be fitted by the equation
$A\left(P_{\mathrm{mw}}\right)=\frac{I \sqrt{P_{\mathrm{mw}}}}{\left[1+\left(2^{1 / \epsilon}-1\right) P_{\mathrm{mw}} / P_{1 / 2}\right]^{\epsilon}}$
where the inhomogeneity parameter $\epsilon$ takes the value $1.5$ in the homogeneous limit and $0.5$ in the inhomogeneous limit. Usually, $\epsilon$ is not known beforehand and is treated as a fit parameter. The other fit parameters are $I$, which is the slope of the amplitude increase with the square root of microwave power in the linear regime, and $P_{1 / 2}$, which is the half-saturation power. More precisely, $P_{1 / 2}$ is the incident mw power where $A$ is reduced to half of its unsaturated value. Figure $7.5$ shows experimental data from a progressive saturation measurement on spin-labelled mutant V229C of major plant light harvesting complex LHCII solubilized in detergent micelles in a nitrogen atmosphere and a fit of this data by Eq. (7.9).
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ENDOR
Advantages of electron-spin based detection of nuclear frequency spectra
Nuclear frequency spectra in the liquid (Section 4.3.2) and solid states (4.3.4) exhibit much better hyperfine resolution than EPR spectra, because the former spectra feature fewer and narrower lines. In fact, small hyperfine couplings to ligand nuclei in metal complexes are not usually resolved in EPR spectra and only the largest hyperfine couplings may be resolved in solid-state EPR spectra. The nuclear frequency spectra cannot be measured by a dedicated NMR spectrometer because they extend over several Megahertz to several tens of Megahertz, whereas NMR spectrometers are designed for excitation and detection bandwidths of a few tens of kilohertz. Furthermore, electron spin transitions have 660 times more polarization than proton transitions and more than that for other nuclei. Their larger magnetic moment also leads to higher detection sensitivity. It is thus advantageous to transfer polarization from electron spins to nuclear spins and to backtransfer the response of the nuclear spins to the electron spins for detection. Two classes of experiments can achieve this, electron nuclear double resonance (ENDOR) experiments, discussed in Section $8.1$ and electron spin echo envelope modulation (ESEEM) experiments discussed in Section 8.2.
Types of ENDOR experiments
An ENDOR experiment can be performed with strong CW irradiation of both electron and nuclear spins. In this CW ENDOR experiment, an electron spin transition is partially saturated, $S \gg 1$ in Eq. (7.5). By driving a nuclear spin transition that shares an energy level with the saturated transition, additional relaxation pathways are opened up. The electron spin transition under observation is thus partially desaturated, and an increase in the EPR signal is observed. The experiment is performed at constant magnetic field with strong microwave irradiation at a maximum of the first-derivative absorption spectrum (i.e. the CW EPR spectrum) and the EPR signal is recorded as a function of the frequency of additional radiofrequency irradiation, which must fulfill the saturation condition $S \gg 1$ for the nuclear spins. Usually, the radiofrequency irradiation is frequency modulated and the response is detected with another phase-sensitive detector, which leads to observation of the first derivative of the nuclear frequency spectrum. The CW ENDOR experiment depends critically on a balance of relaxation times, so that in the solid state sufficient sensitivity may only be achieved in a certain temperature range. Furthermore, simultaneous strong continuous irradiation by both microwave and radiofrequency, while keeping resonator frequency and temperature constant, is experimentally challenging. Therefore, CW ENDOR has been largely replaced by pulsed ENDOR techniques. However, for liquid solution samples CW ENDOR is usually the only applicable ENDOR technique.
The conceptually simplest pulsed ENDOR experiment is Davies ENDOR (Section 8.1.3), where saturation of the EPR transition is replaced by inversion by a $\pi$ pulse (Fig. 8.1(a)). A subsequent radiofrequency $\pi$ pulse, which is on-resonant with a transition that shares a level with the inverted EPR transition, changes population of this level and thus polarization of the EPR observer transition. This polarization change as a function of the radiofrequency is observed by a Hahn echo experiment on the observer transition. The approach works well for moderately large hyperfine couplings $(>3 \mathrm{MHz})$, in particular for ${ }^{14} \mathrm{~N}$ nuclei directly coordinated to a transition metal ion or for protons at hydrogen-bonding distance or distances up to about $4 \AA$. As we shall see in Section 8.1.3, the experiment is rather insensitive for very small hyperfine couplings. <figure>$1<figcaption>$2</figcaption></figure>
Figure 8.1: Pulsed ENDOR sequences. (a) Davies ENDOR. A selective inversion $\pi$ pulse on the electron spins is followed by a delay $T$ and Hahn echo detection (red). During microwave interpulse delay $T$, a frequency-variable radiofrequency $\pi$ pulse is applied (blue). If this pulse is on resonant with a nuclear transition, the inverted echo recovers (pale blue). (b) Mims ENDOR. An non-selective stimulated echo sequence with interpulse delays $\tau$ and $T$ is applied to the electron spins (red). During microwave interpulse delay $T$, a frequency-variable radiofrequency $\pi$ pulse is applied (blue). If this pulse is on resonant with a nuclear transition, the stimulated echo is attenuated (pale blue).
The smallest hyperfine couplings can be detected with the Mims ENDOR experiment that is based on the stimulated echo sequence $(\pi / 2)-\tau-(\pi / 2)-T-(\pi / 2)-\tau-e c h o($ Fig. $8.1($ b) $)$ ). The preparation block $(\pi / 2)-\tau-(\pi / 2)$ creates a polarization grating of the functional form $A\left(\Omega_{S}\right) \cos \left(\Omega_{S} \tau\right)$, where $A\left(\Omega_{S}\right)$ is the EPR absorption spectrum as a function of the resonance offset $\Omega_{S}$ and $\tau$ is the delay between the two $\pi / 2$ microwave pulses. A radiofrequency $\pi$ pulse with variable frequency is applied during time $T$ when the electron spin magnetization is aligned with the $z$ axis. If this pulse is on resonant with a nuclear transition that shares a level with the observer EPR transition, half of the polarization grating is shifted by the hyperfine splitting $A_{\text {eff }}$, as will also become apparent in Section 8.1.3. For $A_{\text {eff }} \tau=2(k+1) \pi$ with integer $k$ the polarization grating is destroyed by destructive interference. Since the stimulated echo is the free induction decay (FID) of this polarization grating, it is canceled by a radiofrequency pulse that is on resonant with a nuclear transition. It is apparent that the radiofrequency $\pi$ pulse has no effect for $A_{\text {eff }} \tau=2 k \pi$ with integer $k$, where the original and frequency-shifted gratings interfere constructively. Hence, the Mims ENDOR experiment features blind spots as a function of interpulse delay $\tau$. These blind spots are not a serious problem for very small hyperfine couplings $A_{\mathrm{eff}} \ll \pi / \tau$. Note however that the first blind spot corresponds to $A_{\mathrm{eff}}=0$. Hence, long interpulse delays $\tau$ are required in order to detect very small hyperfine couplings, and this leads to strong echo attenuation by a factor $\exp \left(-2 \tau / T_{2}\right)$ due to electron spin transverse relaxation. It can be shown that maximum sensitivity for very small couplings is attained approximately at $\tau=T_{2}$.
<figure>$1<figcaption>$2</figcaption></figure>
Figure 8.2: Polarization transfer in Davies ENDOR. (a) Level populations at thermal equilibrium, corresponding to green label 0 in Fig. 8.1(a). The electron transitions (red, pale red) are much more strongly polarized than the nuclear transitions (blue, pale blue). (b) Level populations after a selective mw inversion pulse on resonance with the $|\beta \alpha\rangle \leftrightarrow|\alpha \alpha\rangle$ transition (dark red), corresponding to green label 1 in Fig. 8.1(a). A state of two-spin order is generated, where the two electron spin transitions are polarized with opposite sign and the same is true for the two nuclear spin transitions. (c) Level populations after a selective rf inversion pulse on resonance with the $|\alpha \alpha\rangle \leftrightarrow|\alpha \beta\rangle$ transition (dark blue), corresponding to green label 2 in Fig. 8.1(a). The electron spin observer transition $|\beta \alpha\rangle \leftrightarrow|\alpha \alpha\rangle$ is no longer inverted, but only saturated.
Davies ENDOR
The Davies ENDOR experiment is most easily understood by looking at the polarization transfers. At thermal equilibrium the electron spin transitions (red and pale red) are much more strongly polarized than the nuclear spin transitions (Fig. 8.2(a)). Their frequencies differ by an effective hyperfine splitting $A_{\text {eff }}$ to a nuclear spin $I=1 / 2$ that is color-coded blue. The first microwave $\pi$ pulse is transition-selective, i.e., it has an excitation bandwidth that is smaller than $A_{\text {eff }}$. Accordingly, it inverts only one of the two electron spin transitions. We assume that the $|\beta \alpha\rangle \leftrightarrow|\alpha \alpha\rangle$ transition (red) is inverted and the $|\beta \beta\rangle \leftrightarrow|\alpha \beta\rangle$ transition (pale red) is not inverted; the other case is analogous. Such transition-selective inversion leads to a state of two-spin order, where all individual transitions in the two-spin system are polarized. However, the two electron spin transitions are polarized with opposite sign and the two nuclear transitions are also polarized with opposite sign (Fig. 8.2(b)). Now a radiofrequency $\pi$ pulse is applied. If this pulse is not resonant with a nuclear transition, the state of two-spin order persists and the observer electron spin transition (red) is still inverted. The radiofrequency pulse is also transition-selective. We now assume that this pulse inverts the $|\alpha \alpha\rangle \leftrightarrow|\alpha \beta\rangle$ transition (blue); the other case is again analogous. After such a resonant radiofrequency pulse, the two nuclear transitions are polarized with equal sign and the two electron spin transitions are saturated with no polarization existing on them (Fig. 8.2(c)). After the radiofrequency $\pi$ pulse a microwave Hahn echo sequence is applied resonant with the observer transition (Fig. 8.1(a)). If the radiofrequency pulse was off resonant (situation as in Fig. 8.2(b)), an inverted echo is observed. If, on the other hand, the radiofrequency pulse was on resonant (situation as in Fig. 8.2(c)) no echo is observed. In practice, polarization transfers are not complete and a weak echo is still observed. However, an on-resonant radiofrequency pulse causes some recovery of the inverted echo. If the radiofrequency is varied, recovery of the inverted echo is observed at all frequencies where the radiofrequency pulse is resonant with a nuclear transition.
$a$
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C
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$b$
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$d$
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Figure 8.3: Spectral hole burning explanation of Davies ENDOR. (a) An inhomogeneously broadened EPR line with width $\Gamma_{\text {inhom }}$ (red) consists of many narrower homogeneously broadened lines with linewidth $\Gamma_{\text {hom }}$. (b) Long weak microwave irradiation saturates the on-resonant spin packet and does not significantly affect off-resonant spin packets. A spectral hole is burnt into the inhomogeneously broadened line, which can be as narrow as $\Gamma_{\text {hom. }}$. (c) A selective microwave $\pi$ pulse burns an inversion hole into the EPR line whose width is approximately the inverse width of the pulse. (d) Situation after applying an on-resonant radiofrequency pulse. For the spin packet, where the microwave pulse was on-resonant with the $|\beta \alpha\rangle \leftrightarrow|\alpha \alpha\rangle$ transition, half of the spectral hole is shifted by $A_{\text {eff }}$ to lower EPR frequencies. For the spin packet where the microwave pulse was on-resonant with the $|\beta \beta\rangle \leftrightarrow|\alpha \beta\rangle$ transition, half of the spectral hole is shifted by $A_{\text {eff }}$ to higher EPR frequencies. Considering both cases, half of the hole persists, corresponding to saturation. Two side holes with a quarter of the depth of the inversion hole are created at $\omega_{\mathrm{mw}} \pm A_{\mathrm{eff}}$. These side holes do not contribute to the echo signal, as long as they are outside the detection window (pale red) whose width is determined by the excitation bandwidth of the Hahn echo detection sequence.
Further understanding of Davies ENDOR is gained by considering an inhomogeneously broadened EPR line (Fig. 8.3). In such a line with width $\Gamma_{\text {inhom }}$, each individual spin packet with much narrower width $\Gamma_{\text {hom }}$ can, in principle, be selectively excited. A long rectangular $\pi$ pulse inverts the on-resonant spin packet and partially inverts spin packets roughly over a bandwidth corresponding to the inverse length of the pulse. In Davies ENDOR, pulse lengths between 50 and $400 \mathrm{~ns}$, corresponding to excitation bandwidths between 20 and $2.5 \mathrm{MHz}$ are typical. Such a pulse creates an inversion hole centered at the microwave frequency $\omega_{\text {mw }}$. In an $S=1 / 2, I=1 / 2$ electron-nuclear spin system, two on-resonant spin packets exist, those where $\omega_{\mathrm{mw}}$ is the frequency of the $|\beta \alpha\rangle \leftrightarrow|\alpha \alpha\rangle$ transition and those where it is the frequency of the $|\beta \beta\rangle \leftrightarrow|\alpha \beta\rangle$ transition. For the former spin packet, inversion of the nuclear spin from the $|\beta\rangle$ to the $|\alpha\rangle$ state increases the EPR frequency by the effective hyperfine splitting $A_{\text {eff }}$, whereas for the latter packet, inversion from the $|\alpha\rangle$ to the $|\beta\rangle$ state decreases it by $A_{\text {eff }}$. In both cases half of the inversion hole is shifted to a side hole, leaving a saturation hole at $\omega_{m w}$ and creating a saturation side hole. The saturation center holes of the two spin packets coincide in frequency and combine to a saturation hole in the inhomogeneously broadened line. At the side hole frequencies $\omega_{\mathrm{mw}} \pm A_{\mathrm{eff}}$, only one of the two spin packets contributes to the hole, so that the side holes are only half as deep. The Hahn echo subsequence in the Davies ENDOR sequence must have a detection bandwidth that covers only the central hole (pale red in Fig. $8.3(\mathrm{~d})$ ), since no ENDOR effect would be observed if the side hole would also be covered. For this purpose, the detection bandwidth of the Hahn echo sequence can be limited either by using sufficiently long microwave pulses or by using a sufficiently long integration gate for the inverted echo.
In any case, a Davies ENDOR effect is only be observed if $A_{\text {eff }}$ exceeds the width of the original inversion hole. The smaller $A_{\mathrm{eff}}$, the longer the first inversion pulse needs to be and the fewer spin packets contribute to the signal. In general, hyperfine splittings much smaller than the homogeneous linewidth $\Gamma_{\text {hom }}=1 / T_{2}$ in the EPR spectrum cannot be detected. In practice, Davies ENDOR becomes very insensitive for $\pi$ pulse lengths exceeding 400 ns. If broadening of the inversion hole by electron spin relaxation is negligible, the suppression of signals with small hyperfine couplings in Davies ENDOR can be described by a selectivity parameter
$\eta_{\mathrm{S}}=\frac{A_{\mathrm{eff}} t_{\pi}^{(1)}}{2 \pi}$
where $t_{\pi}^{(1)}$ the length of the first mw $\pi$ pulse. Maximum absolute ENDOR intensity $V_{\max }$ is obtained for $\eta_{\mathrm{S}}=\sqrt{2} / 2$. As a function of $\eta_{\mathrm{S}}$, the absolute ENDOR intensity is given by
$V\left(\eta_{\mathrm{S}}\right)=V_{\max }\left(\frac{\sqrt{2} \eta_{\mathrm{S}}}{\eta_{\mathrm{S}}^{2}+1 / 2}\right)$
The hyperfine contrast selectivity described by Eq. (8.2) can be used for spectral editing. For instance, ${ }^{14} \mathrm{~N}$ ENDOR signals of directly coordinated ligand nitrogen atoms in transition metal complexes with $A_{\text {eff }}$ of the order of $20-40 \mathrm{MHz}$ overlap with ${ }^{1} \mathrm{H}$ ENDOR signals of weakly coupled ligand protons at X-band frequencies. At an inversion pulse length of about $50 \mathrm{~ns}^{1} \mathrm{H}$ ENDOR signals are largely suppressed.
The sensitivity advantage of Mims ENDOR for very small hyperfine couplings can also be understood in the hole burning picture. Instead of a single center hole, a preparation block $(\pi / 2)-\tau-(\pi / 2)$ with nonselective microwave pulses creates a polarization grating that can be imagined as a comb of many holes that are spaced by frequency difference $1 / \tau$. The width of each hole is approximately $1 / 2 \tau$. The width of the comb of holes is determined by the inverse length of the non-selective $\pi / 2$ pulses, which are typically $10 \mathrm{~ns}$ long. For small couplings, where $t_{\pi}^{(1)}$ in Davies ENDOR needs to be very long, more than an order of magnitude more spin packets take part in a Mims ENDOR experiment than in a Davies ENDOR experiment. The Mims ENDOR effect arises from the shift of one quarter of the polarization grating by frequency difference $+A_{\text {eff }}$ and one quarter of the grating by $-A_{\text {eff }}$. The shifted gratings interfer with the grating at the center frequency. Depending on $A_{\text {eff }}$ and on the periodicity $1 / \tau$ of the grating, this interference is destructive (ENDOR effect) or constructive (no ENDOR effect).
ESEEM and HYSCORE
ENDOR or ESEEM?
In ESEEM experiments, polarization transfer from electron spins to nuclear spins and detection of nuclear frequencies on electron spin transitions are based on the forbidden electron-nuclear transitions discussed in Chapter 6. Much of the higher polarization of the electron spin transitions is lost in such experiments, since the angle $2 \eta$ between the quantization axes of the nuclear spin in the two electron spin states is usually small and the depth of nuclear echo modulations is sin $2 \eta$. Furthermore, modulations vanish along the principal axes of the hyperfine tensor, where $B=0$ and thus $\eta=0$. Therefore, lineshape singularities are missing in one-dimensional ESEEM spectra, which significantly complicates lineshape analysis. For this reason, one-dimensional ESEEM experiments are not usually competitive with ENDOR experiments, at least if the ENDOR experiments can be performed at $\mathrm{Q}$-band frequencies $(\approx 34 \mathrm{GHz})$ or even higher frequencies. An exception arises for weakly coupled "remote" $14 \mathrm{~N}$ nuclei in transition metal complexes where exact cancellation between the nuclear Zeeman and the hyperfine interactions can be achieved for one of the electron spin states at X-band frequencies or slightly below. In this situation, pure nuclear quadrupole frequencies are observed, which leads to narrow lines and easily interpretable spectra. One-dimensional ESEEM data are also useful for determining local proton or deuterium concentrations around a spin label, which can be used as a proxy for water accessibility (Section 10.1.6).
The main advantage of ESEEM compared to ENDOR spectroscopy is the easier extension of ESEEM to a two-dimensional correlation experiment. Hyperfine sublevel correlation (HYSCORE) spectroscopy 8.2.3 resolves overlapping signal from different elements, simplifies peak assignment, and allows for direct determination of hyperfine tensor anisotropy even if the lineshape singularities are not observed.
Three-pulse ESEEM
The HYSCORE experiment is a two-dimensional extension of the three-pulse ESEEM experiment that we will treat first. In this experiment, the amplitude of a stimulated echo after is observed with the pulse sequence $(\pi / 2)-\tau-(\pi / 2)-t-(\pi / 2)-\tau-e c h o$ as a function of the variable interpulse delay $t$ at fixed interpulse delay $\tau$ (Fig. 8.4). The block $(\pi / 2)-\tau-(\pi / 2)$ serves as a nuclear coherence generator, as discussed in Section 6.3.1 and, simultaneously, creates the polarization grating discussed in the context of the Mims ENDOR experiment (Section 8.1.2). In fact, most of the thermal equilibrium magnetization is converted to the polarization grating whose FID after the final $\pi / 2$ pulse is the stimulated echo, while only a small fraction is transferred to nuclear coherence. The phase of the nuclear coherence determines how much of it contributes to the stimulated echo after back transfer to electron spin coherence by the last $\pi / 2$ pulse. For an electron-nuclear spin system $S=1 / 2, I=1 / 2$ this phase evolves with frequencies $\omega_{\alpha}$ or $\omega_{\beta}$ if during interpulse delay $t$ the electron spin is in its $\alpha$ or $\beta$ state, respectively. Hence, the part of the stimulated echo that arises from back transferred nuclear coherence is modulated as a function of $t$ with frequencies $\omega_{\alpha}$ and $\omega_{\beta}$.
An expression for the echo envelope modulation can be derived by product operator formalism using the concepts explained in Section 6.2. Disregarding relaxation, the somewhat lengthy derivation provides
$V_{3 \mathrm{p}}(\tau, t)=\frac{1}{2}\left[V_{\alpha}(\tau, t)+V_{\beta}(\tau, t)\right],$
where the terms $V_{\alpha}(\tau, t)$ and $V_{\beta}(\tau, t)$ correspond to contributions with the electron spin in its $\alpha$ or $\beta$ state, respectively, during interpulse delay $t$. These terms are given by
\begin{aligned} &V_{\alpha}(\tau, t)=1-\frac{k}{2}\left\{1-\cos \left[\omega_{\beta} \tau\right]\right\}\left\{1-\cos \left[\omega_{\alpha}(t+\tau)\right]\right\} \ &V_{\beta}(\tau, t)=1-\frac{k}{2}\left\{1-\cos \left[\omega_{\alpha} \tau\right]\right\}\left\{1-\cos \left[\omega_{\beta}(t+\tau)\right]\right\} \end{aligned}
The factors $\cos \left[\omega_{\beta} \tau\right]$ for the $V_{\alpha}$ term and $\cos \left[\omega_{\alpha} \tau\right]$ for the $V_{\beta}$ term describe the blind spot behavior of three-pulse ESEEM. The modulation depth $k$ is given by
$k=\sin ^{2} 2 \eta=\left(\frac{B \omega_{I}}{\omega_{\alpha} \omega_{\beta}}\right)^{2}$
For small hyperfine couplings, $A, B \ll \omega_{I}$, we have $\omega_{\alpha} \approx \omega_{\beta} \approx \omega_{I}$, so that Eq. (8.5) reduces to
$k=\frac{B^{2}}{\omega_{I}^{2}}$
i.e., the modulation depth is inversely proportional to the square of the magnetic field. Using Eqs. (4.10) and (4.11) we find for protons not too close to a well localized unpaired electron
$k=\frac{9}{4}\left(\frac{\mu_{0}}{4 \pi}\right)^{2}\left(\frac{g \mu_{B}}{B_{0}}\right)^{2} \frac{\sin ^{2}\left(2 \theta_{\mathrm{HFI}}\right)}{r^{6}}$
where $\theta_{\mathrm{HFI}}$ is the angle between the electron-proton axis and the static magnetic field $B_{0}$.
Because of the star topology of electron-nuclear spin systems (Fig. 4.4(a)), Eq. (8.3) can be easily extended by a product rule to multiple nuclei with spins $I_{l}=1 / 2$, where $l$ is an index that runs over all nuclei. One finds
$V_{3 \mathrm{p}}(\tau, t)=\frac{1}{2}\left[\prod_{l} V_{\alpha, l}(\tau, t)+\prod_{l} V_{\beta, l}(\tau, t)\right]$
In the weak modulation limit, where all modulation depths $k_{l}$ fulfill the condition $k_{l} \ll 1$, the ESEEM spectrum due to several coupled nuclei is the sum of the spectra of the individual nuclei.
a <figure>$1<figcaption>$2</figcaption></figure>
$b$
Figure 8.4: Pulse sequences for three-pulse ESEEM (a) and HYSCORE (b). In three-pulse ESEEM, time $t$ is varied and time $\tau$ is fixed. In HYSCORE, times $t_{1}$ and $t_{2}$ are varied independently in order to obtain a two-dimensional data set.
HYSCORE
The HYSCORE experiment is derived from the three-pulse ESEEM experiment by inserting a microwave $\pi$ pulse midway through the evolution of nuclear coherence. This splits the interpulse delay $t$ into two interpulse delays $t_{1}$ and $t_{2}$ (Fig. 8.4(b)), which are varied independently to provide a two-dimensional data set $V\left(t_{1}, t_{2}\right)$ that depends parametrically on fixed interpulse delay $\tau$. The inserted $\pi$ pulse inverts the electron spin state. Hence, coherence that has evolved with frequency $\omega_{\alpha}$ during interpulse delay $t_{1}$ evolves with frequency $\omega_{\beta}$ during interpulse delay $t_{2}$ and vice versa. In the weak modulation limit, the HYSCORE experiment correlates only frequencies $\omega_{\alpha}$ and $\omega_{\beta}$ of the same nuclear spin. The full modulation expression for the HYSCORE experiment contains a constant contribution and contributions that vary only with respect to either $t_{1}$ or $t_{2}$. These contributions can be removed by background correction with low-order polynomial functions along both dimensions. The remaining modulation corresponds to only cross peaks and can be expressed as
$V_{4 \mathrm{p}}\left(t_{1}, t_{2} ; \tau\right)=\frac{k}{2} \sin \left(\frac{\omega_{\alpha} \tau}{2}\right) \sin \left(\frac{\omega_{\beta} \tau}{2}\right)\left[V^{(\alpha \beta)}\left(t_{1}, t_{2} ; \tau\right)+V^{(\beta \alpha)}\left(t_{1}, t_{2} ; \tau\right)\right]$
with
\begin{aligned} &V^{(\alpha \beta)}\left(t_{1}, t_{2} ; \tau\right)=\cos ^{2} \eta \cos \left(\omega_{\alpha} t_{1}+\omega_{\beta} t_{2}+\omega_{\operatorname{sum}} \frac{\tau}{2}\right)-\sin ^{2} \eta \cos \left(\omega_{\alpha} t_{1}-\omega_{\beta} t_{2}+\omega_{\mathrm{hfi}} \frac{\tau}{2}\right) \ &V^{(\beta \alpha)}\left(t_{1}, t_{2} ; \tau\right)=\cos ^{2} \eta \cos \left(\omega_{\beta} t_{1}+\omega_{\alpha} t_{2}+\omega_{\mathrm{sum}} \frac{\tau}{2}\right)-\sin ^{2} \eta \cos \left(\omega_{\beta} t_{1}-\omega_{\alpha} t_{2}+\omega_{\mathrm{hfi}} \frac{\tau}{2}\right) \end{aligned}
In this representation with unsigned nuclear frequencies, one has $\eta<45^{\circ}$ for the weak coupling case $\left(|A|<2\left|\omega_{I}\right|\right)$ and $\eta>45^{\circ}$ for the strong coupling case $\left(|A|>2\left|\omega_{I}\right|\right)$, as can be inferred from Fig. 6.1. Hence, $\cos ^{2} \eta>\sin ^{2} \eta$ in the weak coupling case and $\sin ^{2} \eta>\cos ^{2} \eta$ in the strong coupling case. In the weak coupling case, the cross peaks that correlate nuclear frequencies with the same sign ( $\cos ^{2} \eta$ terms) are much stronger than those that correlate frequencies with opposite $\operatorname{sign}\left(\sin ^{2} \eta\right.$ terms) whereas it is the other way around in the strong coupling case. Therefore, the two cases can be easily distinguished in HYSCORE spectra, since the cross peaks appear in different quadrants (Fig. 8.5). Furthermore, disregarding a small shift that arises from the pseudo-secular part $B$ of the hyperfine coupling (see below), the cross peaks of a given isotope with spin $I=1 / 2$ are situated on parallels to the anti-diagonal that corresponds to the nuclear Zeeman frequency $\nu_{I}$. This frequency in turn can be computed from the nuclear $g$ value (or gyromagnetic ratio $\gamma$ ) and the static magnetic field $B_{0}$. Peak assignment for $I=1 / 2$ nuclei is thus straightforward. For nuclei with $I>1 / 2$ the peaks are further split by the nuclear quadrupole interaction. Unless this splitting is much smaller than both the hyperfine interaction and the nuclear Zeeman interaction $\left({ }^{2} \mathrm{H},{ }^{6} \mathrm{Li}\right)$, numerical simulations are required to assign the peaks and extract the hyperfine and nuclear quadrupole coupling.
<figure>$1<figcaption>$2</figcaption></figure>
Figure 8.5: Schematic HYSCORE spectrum for the phenyl radical (compare Fig. 4.6). Note that hyperfine couplings are given here in frequency units, not angular frequency units. Signals from weakly coupled nuclei appear in the right $(+,+)$ quadrant. To first order, these peaks are situated on a line parallel to the anti-diagonal that intersects the $\nu_{2}$ axis at $2 \nu_{I}$. The doublets are centered at $\nu_{I}$ and split by the respective hyperfine couplings. Signals from strongly coupled nuclei appear in the (-,+) quadrant. To first order, these peaks are situated on two lines parallel to the anti-diagonal that intersect the $\nu_{2}$ axis at $-2 \nu_{I}$ and $2 \nu_{I}$. The doublets are centered at half the hyperfine coupling and split by $2 \nu_{I}$.
The small pseudo-secular shift of the correlation peaks with respect to the anti-diagonal contains information on the anisotropy $T$ of the hyperfine interaction (Fig. 8.5). In the solid state, the cross peaks from different orientations $\theta_{\mathrm{HFI}}$ form curved ridges. For a hyperfine tensor with axial symmetry, as it is encountered for protons not too close to a well-localized unpaired electron, the maximum shift in the diagonal direction corresponds to $\theta_{\mathrm{HFI}}=45^{\circ}$ and is given by $9 T^{2} / 32\left|\omega_{I}\right|$. Since $\omega_{I}$ is known, $T$, and thus the electron-proton distance $r$ can be computed from this maximum shift. If $A_{\text {iso }} \ll \omega_{I}$, which is usually the case, the orientation with maximum shift is at the same time the orientation with maximum modulation depth.
The curved ridges end at their intersection with the parallel to the anti-diagonal. These points correspond to the principal values of the hyperfine tensor and modulation depth is zero at these points. However, it is usually possible to fit the theoretical ridge to the experimentally observed ridge, as the curvature near $\theta_{\mathrm{HFI}}=45^{\circ}$ together with the position of the $\theta_{\mathrm{HFI}}=45^{\circ}$ point fully determines the problem.
<figure>$1<figcaption>$2</figcaption></figure>
Figure 8.6: Schematic HYSCORE spectrum for a proton with an axial hyperfine tensor with anisotropy $T$ and isotropic component $A_{\text {iso }}$. The correlation peaks from different orientations form curved ridges (red). Curvature is the stronger the larger the anisotropy is and the ratio of squared anisotropy to the nuclear Zeeman frequency determines the maximum shift with respect to the $2 \omega_{I}$ anti-diagonal.
Analysis of HYSCORE spectra requires some precaution due to the blind-spot behavior (factor $\sin \left(\frac{\omega_{\alpha} \tau}{2}\right) \sin \left(\frac{\omega_{\beta} \tau}{2}\right)$ in Eq. (8.9)) and due to orientation selection by the limited bandwidth of the microwave pulses that is much smaller than spectral width for transition metal complexes. It is therefore prudent to measure HYSCORE spectra at several values of $\tau$ and at several observer positions within the EPR spectrum.
<figure>$1<figcaption>$2</figcaption></figure>
<figure>$1<figcaption>$2</figcaption></figure>
At a distance of $1 \mathrm{~nm}$ between two localized unpaired electrons, splitting $\omega_{\perp}$ between the "horns" of the Pake pattern is about $52 \mathrm{MHz}$ for two electron spins. Even strongly inhomogeneously broadened EPR spectra usually have features narrower than that (about $2 \mathrm{mT}$ in a magnetic field sweep). Depending on the width of the narrowest features in the EPR spectrum and on availability of an experimental spectrum or realistic simulated spectrum in the absence of dipole-dipole coupling, distances up to $1.5 \ldots 2.5 \mathrm{~nm}$ can be estimated from dipolar broadening by lineshape analysis. At distances below $1.2 \mathrm{~nm}$, such analysis becomes uncertain due to the contribution from exchange coupling between the two electron spins, which cannot be computed by first principles and cannot be predicted with sufficient accuracy by quantum-chemical computations. If the two unpaired electrons are linked by a continuous chain of conjugated bonds, exchange coupling can be significant up to much longer distances.
Distance measurements are most valuable for spin labels or native paramagnetic centers in biomolecules or synthetic macromolecules and supramolecular assemblies. In such systems, if the two unpaired electrons are not linked by a $\pi$-electron system, exchange coupling is negligible with respect to dipole-dipole coupling for distances longer than $1.5 \mathrm{~nm}$. Such systems can often assume different molecular conformations, i.e. their structure is not perfectly defined. Structural characterization thus profits strongly from the possibility to measure distance distributions on length scales that are comparable to the dimension of these systems. This dimension is of the order of 2 to $20 \mathrm{~nm}$, corresponding to $\omega_{\perp}$ between $7 \mathrm{MHz}$ and $7 \mathrm{kHz}$. In order to infer the distance distribution, this small dipole-dipole coupling needs to be separated from larger anisotropic interactions.
This separation of interactions is possible by observing the resonance frequency change for one spin in a pair (blue in Fig. 5.3) that is induced by flipping the spin of its coupling partner (red). In Fig. $9.1$ the resonance frequency of the observer spin before the flip of its coupling partner is indicated by a dashed line. If the coupling partner is in its $|\alpha\rangle$ state before the flip (left panel in Fig. 5.3), the local field at the observer spin will increase by $\Delta B$ upon flipping the coupling partner. This causes an increase of the resonance frequency of the observer spin by the dipole-dipole coupling $d$ (see Eq. (5.16)). If the coupling partner is in its $|\beta\rangle$ state before the flip (right panel in Fig. 5.3), the local field at the observer spin will decrease by $\Delta B$ upon flipping the coupling partner. This causes an decrease of the resonance frequency of the observer spin by the dipole-dipole coupling $d$. In the high-temperature approximation, both these cases have the
<figure>$1<figcaption>$2</figcaption></figure>
Figure 9.1: Resonance frequency shift of an observer spin (blue transitions) by the change $\pm \Delta B$ in local magnetic field that arises upon a flip of a second spin that is dipole-dipole coupled to the observer spin. Compare Fig. $5.3$ for the local field picture.
same probability. Hence, half of the observer spins will experience a frequency change $+d$ and the other half will experience a frequency change $-d$. If the observer spin evolves with changed frequency for some time $t$, phases $\pm d t$ will be gained compared to the situation without flipping the coupling partner. The additional phase can be observed as a cosine modulation $\cos (d t)$ for both cases, as the cosine is an even function.
DEER
The four-pulse DEER experiment
The most commonly used experiment for distance distribution measurements in the nanometer range is the four-pulse double electron electron resonance (DEER) experiment (Figure 9.2), which is sometimes also referred to as pulsed electron electron double resonance (PELDOR) experiment. All interactions of the observer spin are refocused twice by two $\pi$ pulses at times $2 \tau_{1}$ and $2 \tau_{1}+2 \tau_{2}$ after the initial $\pi / 2$ pulse. Repeated refocusing is necessary since all spin packets must be in phase at $t=0$ and overlap of the pump $\pi$ pulse with the $\pi / 2$ observer pulse would lead to signal distortion. The first refocusing with interpulse delay $\tau_{1}$ restores the situation (1) immediately after the $\pi / 2$ pulse with phase $x$, where the magnetization vectors of all spin packets are aligned with the $-y$ axis. ${ }^{1}$ In practice, coherence is excited on both observer spin transitions (blue in the energy level panels), but for clarity we consider only observer spin coherence that is on the upper transition and is symbolized by a wavy line in panel (1).
During time $t$ after the first refocusing, magnetization vectors of spin packets with different resonance offset dephase (panel (2)). Only the on-resonant spin packet, marked dark blue, is still aligned with the $-y$ direction. The pump pulse flips the coupling partner and thus transfers the coherence to the lower observer spin transition. The resonance frequency of this transition is shifted by the dipole-dipole coupling $d$ in all spin packets. Observer spin magnetization further dephases until the time just before application of the second observer $\pi$ pulse (3)) and, in addition, the whole bundle of spin packet magnetization vectors precesses counterclockwise with the frequency shift $d$. The originally on-resonant spin packet thus gains phase $d\left(\tau_{2}-t\right)$ before the second observer $\pi$ pulse is applied. The second observer $\pi$ pulse with phase $x$ corresponds to a $180^{\circ}$ rotation about the $x$ axis. This mirrors the bundle of magnetization vectors with respect to the $y$ axis, inverting phase of the observer spin coherence (panel (4)). The bundle, which still
<figure>$1<figcaption>$2</figcaption></figure> the first refocusing and all following magnetization panels are mirrored with respect to the $x$ axis. <figure>$1<figcaption>$2</figcaption></figure>
Figure 9.2: Four-pulse DEER sequence, coherence transfers, and evolution of the observer spin magnetization. Pulses shown in blue are applied to the observer spin, the pump pulse shown in red is applied to its coupling partner. The echo at time $2 \tau_{1}$ (dashed blue line) is not observed. Interpulse delays $\tau_{1}$ and $\tau_{2}$ are fixed, time $t$ is varied, and the echo amplitude is observed as a function of $t$.
precesses counterclockwise with angular frequency $d$ now lags the $+y$ axis by phase $d\left(\tau_{2}-t\right)$. During the final interpulse delay of length $\tau_{2}$ the bundle as a whole gains phase $d \tau_{2}$ (grey arrow in panel (4) and simultaneously realigns along its center due to echo refocusing. However, the center corresponding to the originally on-resonant spin packet does not end up along $+y$, as it would have in the absence of the pump pulse. Rather, this spin packet has gained phase $d t$ with respect to the $+y$ direction (panel (5)). The magnetization vector component along $+y$, which corresponds to the echo signal, is given by $\cos (d t)$.
The distance range of the DEER experiment is limited towards short distances by the requirement that, for echo refocusing, the observer pulses must excite both observer transitions, which are split by $d$ and, for coherence transfer, the pump pulse must excite both transitions of the coupling partner, which are also split by $d$. In other words, both the observer refocused echo subsequence and the pump pulse must have an excitation bandwidth that exceeds $d$. This requirement sets a lower distance bound of about $1.8 \mathrm{~nm}$ at X-band frequencies and of about $1.5 \mathrm{~nm}$ at $Q$-band frequencies. A limit towards long distances arises, since several dipolar oscillations need to be observed for inferring the width or even shape of a distance distribution and at least one oscillation needs to be observed for determining the mean distance. This requires $t>2 \pi / d$. On the other hand, we have $t<\tau_{2}$ and the fixed interpulse delay $\tau_{2}$ cannot be much longer than the transverse relaxation time $T_{2}$, since otherwise coherence has completely relaxed and no echos is observed. Electron spin transverse relaxation times are of the order of a few microseconds. Depending on sample type (see Section $9.1 .2$ ), $\tau_{2}$ can be chosen between $1.5$ and $20 \mu \mathrm{s}$, corresponding to maximum observable distances between 5 and $12 \mathrm{~nm}$.
Sample requirements
In the wanted coherence transfer pathway of the DEER experiment, observer pulses exclusively excite observer spins and the pump pulse exclusively excites the coupling partner. The excitation bandwidth must be sufficiently large to cover the dipole-dipole coupling $d$ at all orientations, i.e., larger than $\omega_{\|}-2 \omega_{\perp}$. If the two coupled spins have the same EPR spectrum, this spectrum must be broader than twice this minimum excitation bandwidth. This condition is fulfilled for nitroxide spin labels (Chapter 10) and transition metal ions at all EPR frequencies, whereas some organic radicals, such as trityl radicals, have spectra that are too narrow at X-band or even Q-band frequencies. Furthermore, $T_{2}$ must be sufficiently long for at least the observer spins. This condition can be fulfilled for almost all $S=1 / 2$ species at temperatures of $10 \mathrm{~K}$ (transition metal complexes) or $50 \cdots 80 \mathrm{~K}$ (organic radicals), but may require cooling below $4.2 \mathrm{~K}$ for some high-spin species. For high-spin species with a half-filled valence shell, such as Mn(II) $(S=5 / 2)$ or $\mathrm{Gd}(\mathrm{III})(S=7 / 2)$ measurement temperatures of $10 \mathrm{~K}$ are also sufficient.
Sample concentration should be sufficiently low for intermolecular distances to be much longer than intramolecular distances. For short distances, concentrations up to $200 \mu \mathrm{M}$ are possible, but concentrations of $10 \cdots 50 \mu \mathrm{M}$ provide better results, if a spectrometer with sufficient sensitivity is available. Depending on distance and $T_{2}$, measurements can be performed down to concentrations of $10 \cdots 1 \mu \mathrm{M}$. For membrane proteins reconstituted into liposomes, data quality is not only a function of bulk spin concentration, but also of lipid-to-protein ratio. This parameter needs to be optimized for each new protein. Required sample volume varies between a few microliters (W-band frequencies) and $150 \mu \mathrm{L}$ with $50 \mu \mathrm{L}$ at Q-band frequencies usually being optimal.
If concentration is not too high and the low-temperature limit of transverse relaxation can be attained, $T_{2}$ depends on the concentration and type of protons around the observer spin. Deuteration of the solvent and cryoprotectant (usually glycerol) usually dramatically improve data quality. If the matrix can be perdeuterated, deuteration of the protein or nuclei acid may further prolong $T_{2}$ and extend distance range or improve signal-to-noise ratio.
Complications arise if more than two unpaired electrons are found in the same molecule, but these complications can usually be solved. However, none of the spin pairs should have a distance shorter than the lower limit of the accessible distance range.
$9.2$ Conversion of dipolar evolution data to distance distributions
Expression for the DEER signal
In Section 9.1.1 we have seen that the echo is modulated with $\cos (d t)$. Usually, this applies only to a fraction $\lambda$ of the echo, because the pump pulse excites only a fraction $\lambda$ of all spin packets of the coupling partner of the observer spin. Therefore, the echo signal for an isolated pair of electron spins in a fixed orientation $\theta$ with respect to the magnetic field is described by
$F(t, r, \theta)=F(0)\{1-\lambda(\theta)[1-\cos (2 d(r, \theta) t)]\}$
where the dependence $d(\theta)$ is given by Eqs. (5.16) and (5.15). The dependence $\lambda(\theta)$ cannot be expressed in closed form, but often $\lambda$ is so weakly correlated with $\theta$ that it can be assumed as a constant, empirical parameter. In this situation, Eq. (9.1) can be integrated over all orientations
$F(t, r)=\int_{0}^{\pi / 2} F(t, r, \theta) \sin \theta \mathrm{d} \theta$
The pump pulse inverts not only the coupling partner of the observer spin in the same molecule, but also electron spins in remote other molecules. If these neighboring spins are homogeneously distributed in space, the background factor $B(t)$ that arises from them assumes the form
$B(t)=\exp \left(-\frac{2 \pi g^{2} \mu_{\mathrm{B}}^{2} \mu_{0} N_{\mathrm{A}}}{9 \sqrt{3} \hbar} \lambda^{\prime} c t\right)$
where the orientation-averaged inversion efficiency $\lambda^{\prime}$ is the fraction of spins excited by the pump pulse, $g$ is an average $g$ value, and $c$ is the total concentration of spins. For subtle reasons, $\lambda^{\prime}$ differs significantly from the empirical two-spin modulation depth $\lambda$. Homogeneous distributions of neighboring spins that are nearly confined to a plane or a line give rise to a stretched exponential background function $B(t)=\exp \left[-(k t)^{D / 3}\right]$, where $D$ is a fractional dimension of the distribution that is usually somewhat larger than 2 or 1 for nearly planar or linear distributions, respectively. The total DEER signal is given by
$V(t, r)=F(t, r) B(t)$
If distance $r$ is distributed with normalized probability density $P(r)\left(\int_{0}^{\infty} P(r) \mathrm{d} r=1\right)$, the form factor $F(t)$ needs to be replaced by $F_{P}(t)=\int_{0}^{\infty} P(r) F(t, r) \mathrm{d} r$. <figure>$1<figcaption>$2</figcaption></figure>
Figure 9.3: Background correction in DEER spectroscopy. (a) Primary data $V(t)$ (simulation) normalized to $V(0)$. Dipolar modulation decays until a time $t_{\mathrm{dec}}$. An exponentially decay function (red) is fitted to the data in the range $t_{\mathrm{dec}} \leq t \leq t_{\max }$, where $t_{\max }<\tau_{2}$ is the maximum dipolar evolution time. This background function $b(t)$ is extrapolated to the range $0 \leq t<t_{\mathrm{dec}}$ (ochre). (b) The form factor $F(t)$ is obtained by normalizing the background function, $B(t)=b(t) / b(0)$ and dividing the normalized primary data $V(t) / V(0)$ by $B(t)$. It decays to a constant level $1-\lambda$, where $\lambda$ is the modulation depth. The red curve is a simulation corresponding to the distance distribution extracted by Tikhonov regularization with optimum regularization parameter $\alpha$.
Background correction
The information on the distance distribution $P(r)$ is contained in $F(t)$, which must thus be separated from $B(t)$. Often, the distribution is sufficiently broad for dipolar oscillations to decay within a time $t_{\mathrm{dec}}$ shorter than the maximum dipolar evolution time $t_{\max }$ (Fig. 9.3(a)). For $t_{\mathrm{dec}} \leq t \leq t_{\max }$, the primary signal is then given by $b(t)=(1-\lambda) \exp \left[-(k t)^{D / 3}\right]$ plus noise. The expression for $b(t)$ is fitted to the primary data in this range (red line in Fig. 9.3(a)). In some cases, for instance for soluble proteins, a homogeneous distribution of molecules in three dimensions can be assumed, so that $D=3$ can be fixed. Otherwise, $D$ is treated as a fit parameter, as are $k$ and $\lambda$. The background function $B(t)$ is obtained by extrapolating $b(t)$ to the range $0 \leq t \leq t_{\mathrm{dec}}$ (ochre line) and dividing it by $b(0)=1-\lambda$. According to Eq. (9.4), the form factor $F(t) / F(0)$ results by dividing $V(t) / V(0)$ by $B(t)$. For narrow distance distributions, oscillations in $V(t) / V(0)$ may endure until the longest attainable $t_{\max }$. This does not create a problem if at least the first oscillation is completed well before $t_{\max }$. All the following oscillations have very similar amplitude and do not bias the background fit. As a rule of thumb, a good estimate for $B(t)$ can be obtained by fitting data at $t \geq t_{\max } / 2$ if $d t_{\max } \geq 4 \pi$, i.e., if two full oscillations can be observed. If the data trace is shorter than that, background fitting is fraught with uncertainty. Wrong background correction may suppress long distances or create artificial peaks at long distances.
Tikhonov regularization with non-negativity constraint
In order to extract the distance distribution $P(r)$ from the experimental form factor $F(t) / F(0)$, we need to remove the constant contribution and renormalize to the dipolar evolution function
$D(t)=\frac{F(t) / F(0)-(1-\lambda)}{\lambda}$
and invert the integral equation $D(t)=\int_{0}^{\infty} P(r) K(t, r) \mathrm{d} r$, where the kernel $K(t, r)$ is given by
$K(t, r)=\int_{0}^{1} \cos \left[\left(3 z^{2}-1\right) \omega_{\perp}(r) t\right] \mathrm{d} z$
Here, we have substituted $\cos \theta$ by $z, \sin \theta \mathrm{d} \theta$ by $-\mathrm{d} \cos \theta$ and reversed direction of the integration, which compensated for the negative $\operatorname{sign}$ in $-\mathrm{d} \cos \theta$.
In practice, $D(t)$ is digitized and given as a vector at sampling times $t_{i}$. Likewise, it is sufficient to compute $P(r)$ as a vector at sampling distances $r_{k}$. The integral equation is thus transformed to a matrix equation
$\vec{D}=\mathbf{K} \vec{P}$
Unfortunately, this matrix equation cannot easily be inverted, since the rows of kernel $\mathbf{K}$ are not orthogonal, i.e., the scalar product of dipolar evolution function vectors at different $r_{k}$ is not zero. The weak linear dependence of the rows makes the problem ill-posed. Small deviations of the experimental $\vec{D}$ from the "true" $\vec{D}_{\text {ideal }}$, for instance due to noise, cause large deviations of $\vec{P}$ from the true distance distribution. This problem can be solved only by taking into account additional information.
First, we know that, as a probability density, $P(r) \geq 0$ at all $r$. Hence, we can impose a non-negativity constraint on $\vec{P}$. It turns out that this is not sufficient for stabilizing the solution. Noise can be fitted by ragged distance distributions with many narrow peaks, although we know that the distance distribution must be smooth, as it arises from a continuous distribution of molecular conformations. Tikhonov regularization imposes a smoothness restraint by minimizing
$G_{\alpha}=\rho+\alpha \eta$
where
$\rho=\|\mathbf{K} \vec{P}-\vec{D}\| \|^{2}$
is the mean square deviation between experimental and simulated data and
$\eta=\left\|\hat{L}^{(2)} \vec{P}\right\|^{2}$
is the square norm of the second derivative, which can be computed from $\vec{P}$ by multiplication with the second derivative operator $\hat{L}^{(2)}$. The regularization parameter $\alpha$ determines the relative <figure>$1<figcaption>$2</figcaption></figure>
Figure 9.4: Tikhonov regularization of the data shown in Fig. 9.3. (a) L curve. The optimum regularization parameter corresponds to the corner (green circle) and provides the simulation shown in Fig. 9.3(b) as well as the extracted distance distribution shown as a black line in panel (c) of the current Figure. The red circle marks a too large regularization parameter that leads to oversmoothing. (b) Input form factor (black) and simulation for the too large regularization parameter corresponding to the red circle in the $L$ curve. (c) Theoretical distance distribution used for simulating a noiseless form factor (green) and distance distribution extracted from the noisy form factor with optimum regularization parameter corresponding to the green circle in the $\mathrm{L}$ curve (black). (d) (c) Theoretical distance distribution used for simulating a noiseless form factor (green) and distance distribution extracted from the noisy form factor with a too large regularization parameter corresponding to the red circle in the $\mathrm{L}$ curve (black).
weight of the smoothing restraint with respect to mean square deviation between experimental and simulated data. A parametric plot of $\log \eta$ versus $\log \rho$ as a function of $\alpha$ is approximately L-shaped (Fig. 9.4). For very small $\alpha$, roughness $\eta$ of the distance distribution can be decreased strongly without increasing mean square deviation $\rho$ very much. For large $\alpha, \vec{P}$ is already smooth and a further increase of $\alpha$ will lead only to a small decrease in roughness $\eta$, but to a large increase in $\rho$, since the overly broadened distance distribution no longer fits the dipolar oscillations. Hence, in a mathematical sense, the optimum regularization parameter corresponds to the corner of the $L$ curve. At this regularization parameter the extracted distance distribution (black line in Fig. 9.4(c)) is only slightly broader than the true distance distribution (green line) and the simulated form factor (red line in Fig. 9.3(b)) agrees with the experimental form factor (black line), except for the white noise contribution. If the regularization parameter is too large (red circle in Fig. 9.4(a)), the simulated form factor is overdamped (red line in Fig. 9.4(b)) and the distance distribution unrealistically broad (black line in Fig. 9.4(d)). For a too small regularization parameter the distance distribution unrealistically splits into several narrow peaks and the simulated form factor fits part of the noise (not shown). This error cannot be as clearly discerned in the simulated form factor as overdamping can be discerned. Undersmoothing is apparent only in the L curve.
<figure>$1<figcaption>$2</figcaption></figure>
Nitroxide spin probes and labels Spin probes and labels
Nitroxide radicals
The nitroxide EPR spectrum Influence of dynamics on the nitroxide spectrum Polarity and proticity
Water accessibility
Oxygen accessibility Local pH measurements
Spin traps
0 - Spin Probes and Spin Traps
Nitroxide spin probes and labels
Spin probes and labels
Spin probes are stable paramagnetic species that are admixed to a sample in order to obtain structural or dynamical information on their environment and, thus, indirectly on the sample. Spin labels are spin probes that are covalently attached to a molecule of interest, often at a specific site. As compared to more direct characterization of structure and dynamics by other techniques, EPR spectroscopy on spin probes may be able to access other length and time scales or may be applicable in aggregation states or environments where these other techniques exhibit low resolution or do not yield any signal. Site-directed spin labeling (SDSL) has the advantage that assignment of the signal to primary molecular structure is already known and that a specific site in a complex system can be studied without disturbance from signals of other parts of the system. This approach profits from the rarity of paramagnetic centers. For instance, many proteins and most nucleic acids and lipids are diamagnetic. If a spin label is introduced at a selected site, EPR information is specific to this particular site.
In principle, any stable paramagnetic species can serve as a spin probe. Some paramagnetic metal ions can substitute for diamagnetic ions native to the system under study, as they have similar charge and ionic radius or with similar complexation properties as the native ions. This applies to $\mathrm{Mn}(\mathrm{II})$, which can often substitute for $\mathrm{Mg}(\mathrm{II})$ without affecting function of proteins or nucleic acids, or Ln(III) lanthanide ions, which bind to Ca(II) sites. Paramagnetic metal ions can also be attached to proteins by engineering binding sites with coordinating amino acids, such as histidine, or by site-directed attachment of a metal ligand to the biomolecule. Such approaches are used for lanthanide ions, in particular Gd(III), and Cu(II).
For many spin probe approaches, organic radicals are better suited than metal ions, since in radicals the unpaired electron has closer contact to its environment (ligands screen environmental access of metal ions, in particular for lanthanide ions) and the EPR spectra are narrower, which allows for some experiments that cannot be performed on species with very broad spectra. Among organic radicals, nitroxides are the most versatile class of spin probes, mainly because of their relatively small size, comparable to an amino acid side group or nucleobase, and because of hyperfine and $g$ tensor anisotropy of a magnitude that is convenient for studying dynamics (Section 10.1.4). Triarylmethyl (TAM) radicals are chemically even more inert than nitroxide radicals and have slower relaxation times in liquid solution. Currently they are much less in use than nitroxide radicals, mainly because they are not commercially available and much harder to synthesize than nitroxide radicals. <figure>$1<figcaption>$2</figcaption></figure>
Figure 10.1: Structures of nitroxide probes. $\mathbf{1}$ TEMPO derivatives. $\mathbf{2}$ PROXYL derivatives. $\mathbf{3}$ pH-sensitive imidazolidine nitroxide. 4 DOXYL derivatives. 5 Methanethiosulfonate spin label (MTSL)
Nitroxide radicals
The nitroxide radical is defined by the $\mathrm{N}-\mathrm{O}^{\bullet}$ group, which is isoelectronic with the carbonyl group $(\mathrm{C}=\mathrm{O})$ and can thus be replaced in approximate force field and molecular dynamics computations by a $\mathrm{C}=\mathrm{O}$ group. The unpaired electron is distributed over both atoms, which contributes to radical stability, with a slight preference for the oxygen atom. Nitroxide radicals become stable on the time scale of months or years if both $\alpha$ positions are sterically protected, for instance by attaching two methyl groups to each of the $\alpha \mathrm{C}$ atoms (Fig. 10.1). Nitroxides of this type are thermally stable up to temperatures of about $140^{\circ} \mathrm{C}$, but they are easily reduced to the corresponding hydroxylamines, for instance by ascorbic acid, and are unstable at very low and very high pH. Nitroxides with five-membered rings (structures $\mathbf{2}, \boldsymbol{3}$, and $\mathbf{5}$ ) tend to be chemically more stable than those with six-membered rings (6). The five-membered rings also have less conformational freedom than the six-membered rings.
Spin probes can be addressed to certain environments in heterogeneous systems by choice of appropriate substituents $\mathrm{R}$ (Fig. 10.1). The unsubstituted species $(\mathrm{R}=\mathrm{H})$ are hydrophobic and partition preferably to nonpolar environments. Preference for hydrogen bond acceptors is achieved by hydroxyl derivatives $(\mathrm{R}=\mathrm{OH})$, whereas ionic environments can be addressed by a carboxylate group at sufficiently high $\mathrm{pH}\left(\mathrm{R}=\mathrm{COO}^{-}\right)$or by a trimethyl ammonium group $(\mathrm{R}=$ $\left.\mathrm{N}\left(\mathrm{CH}_{3}\right)_{3}^{+}\right)$. Reactive groups $\mathrm{R}$ are used for SDSL, such as the methanethiosulfonate group in the dehydro-PROXYL derivative MTSL 5 , which selectively reacts with thiol groups under mild conditions. Thiol groups can be introduced into proteins by site-directed point mutation of an amino acid to cysteine and to RNA by replacement of a nucleobase by thiouridine. In DOXYL derivatives $\mathbf{4}$, a six-membered ring is spiro-linked to an alkyl chain, which can be part of stearic acid or of lipid molecules. The $\mathrm{N}^{\circ} \mathrm{O}^{\bullet}$ group in DOXYL derivatives is rigidly attached to the alkyl chain and nearly parallel to the axis of a hypothetical all-trans chain.
The nitroxide EPR spectrum
To a good approximation, the spin system of a nitroxide radical can be considered as an electron spin $S=1 / 2$ coupled to the nuclear spin $I=1$ of the ${ }^{14} \mathrm{~N}$ atom of the $\mathrm{N}-\mathrm{O}^{\bullet}$ group. Hyperfine
<figure>$1<figcaption>$2</figcaption></figure>
Figure 10.2: The EPR spectrum and molecular frame of nitroxide radicals. (a) The hyperfine sublevels corresponding to the three possible ${ }^{14} \mathrm{~N}$ magnetic quantum numbers $m_{I}=-1,0,1$ are shifted by $m_{S} m_{I} A\left({ }^{14} \mathrm{~N}\right)$. Allowed transitions are those with $\Delta m_{S}=1$ and $\Delta m_{I}=1$. The microwave quantum $h \nu_{\mathrm{mw}}$ has constant energy, since the microwave frequency $\nu_{\mathrm{mw}}$ is constant. During a magnetic field sweep, resonance is observed when the energy $h \nu_{\mathrm{mw}}$ matches the energy difference of the levels of an allowed transition. The three transitions correspond to the three possible ${ }^{14} \mathrm{~N}$ magnetic quantum numbers $m_{I}=-1,0,1$. (b) In a solid, each orientation gives a three-line spectrum, but the splitting $A\left({ }^{14} \mathrm{~N}\right)$ and the center field $h \nu_{\mathrm{mw}} / g \mu_{\mathrm{B}}$ depend on orientation, since $A$ and $g$ are anisotropic. To a good approximation, the hyperfine tensor has axial symmetry with the unique $z$ axis corresponding to the direction of the $p_{\pi}$ orbital lobes on the ${ }^{14} \mathrm{~N}$ atom. The $g$ tensor is orthorhombic, i.e., the spectra in the $x y$ plane of the molecular frame, which all have the same hyperfine splitting, have different center fields. The $\mathrm{N}-\mathrm{O}$ bond direction, which corresponds to the maximum $g$ value, is the molecular frame $x$ axis.
couplings to other nuclei, such as the methyl protons, are not usually resolved and contribute only to line broadening. The hyperfine coupling to the $s p^{2}$ hybridized ${ }^{14} \mathrm{~N}$ atom has a significant isotropic Fermi contact contribution from spin density in the $2 s$ orbital and a significant anisotropic contribution from spin density in the $p_{\pi}$ orbital that combines with a $p_{\pi}$ orbital on the oxygen atom to give the N-O bond partial double bond character. The direction of the lobes of the $p_{\pi}$ orbital is chosen as the molecular $z$ axis (Fig. 10.2(b)). The ${ }^{14} \mathrm{~N}$ hyperfine tensor has nearly axial symmetry with $z$ being the unique axis. The hyperfine coupling is much larger along $z$ (on the order of $90 \mathrm{MHz}$ ) than in the $x y$ plane (on the order of $15 \mathrm{MHz}$ ).
The spin-orbit coupling, which induces $g$ anisotropy, arises mainly at the $\mathrm{O}$ atom, where a lone pair energy level is very close to the SOMO. The $g$ tensor is orthorhombic with nearly maximal asymmetry. The largest $g$ shift is positive and observed along the N-O bond, which is the molecular frame $x$ axis $\left(g_{x} \approx 2.009\right)$. An intermediate $g$ shift is observed along the $y$ axis $\left(g_{y} \approx 2.006\right)$, whereas the $g_{z}$ value is very close to $g_{e}=2.0023$. At X-band frequencies, where $\nu_{\mathrm{mw}} \approx 9.5 \mathrm{GHz}, g$ anisotropy corresponds to only $1.13 \mathrm{mT}$ dispersion in resonance fields, while hyperfine anisotropy corresponds to $6.5 \mathrm{mT}$ dispersion. At W-band frequencies, where $\nu_{\mathrm{mw}} \approx 95 \mathrm{GHz}$, hyperfine anisotropy is still the same but $g$ anisotropy contributes a ten times larger dispersion of $11.3 \mathrm{mT}$, which now dominates.
The field-swept CW EPR spectrum for a single orientation can be understood by considering the selection rule that the magnetic quantum number $m_{S}$ of the electron spin must change by 1 , whereas the magnetic quantum number $m_{I}$ of the ${ }^{14} \mathrm{~N}$ nuclear spin must not change. Each transition can thus be assigned to a value of $m_{I}$. For $I=1$ there are three such values, $m_{I}=-1,0$, and 1 (Fig. $10.2(\mathrm{a})$ ). The microwave frequency $\nu_{\mathrm{mw}}$ is fixed and resonance is observed at fields where the energy of the microwave quantum $h \nu_{\mathrm{mw}}$ matches the energy of a transition.
$a$
<figure>$1<figcaption>$2</figcaption></figure>
$b$
<figure>$1<figcaption>$2</figcaption></figure>
Figure 10.3: Construction of the solid-state EPR spectrum of a nitroxide at X band. (a) The absorption spectrum of each transition is considered separately. For $m_{I}=0$, the hyperfine contribution vanishes and only $g$ anisotropy contributes. This line is the narrowest one at $\mathrm{X}$ band. For $m_{I}=+1$ the dispersion by $g$ anisotropy subtracts from the larger dispersion by hyperfine anisotropy. This line has intermediate width. For $m_{I}=-1$ the dispersion from $g$ anisotropy adds to the dispersion from hyperfine anisotropy. This line has the largest width. (b) The three contributions from individual $m_{I}$ values add to the total EPR absorption spectrum (top). In CW EPR, the derivative of this absorption spectrum is observed (bottom). Because hyperfine anisotropy dominates, the separation between the outer extremities is $2 A_{z z}$.
In order to construct the solid-state spectrum, orientation dependence of the three transitions must be considered (Fig. 10.3(a)). At each individual orientation, the $m_{I}=0$ line is the center line. Since the hyperfine contributions scales with $m_{I}$, it vanishes for this line and only $g$ anisotropy is observed. At X band, where hyperfine anisotropy dominates by far, this line is the narrowest one. The lineshape is the one for pure $g$ anisotropy (see Fig. 3.4). For $m_{I}=+1$, the orientation with the largest $g$ shift of the resonance field coincides with the one of smallest hyperfine shift. Hence, the smaller resonance field dispersion by $g$ anisotropy subtracts from the larger dispersion by hyperfine anisotropy. For $m_{I}=-1$, the situation is opposite and the two dispersions add. Hence, the $m_{I}=-1$ transition, which at any given orientation is the high-field line, has the largest resonance dispersion, whereas the low-field $m_{I}=+1$ transition has intermediate resonance field dispersion. The central feature in the total absorption spectrum (Fig. $10.3(\mathrm{~b})$ ) is strongly dominated by the $m_{I}=0$ transition, whereas the outer shoulders correspond to the $m_{I}=+1$ (low field) and $m_{I}=-1$ (high field) transitions at the $z$ orientation. Therefore, the splitting between the outer extremities in the CW EPR spectrum, which correspond to these shoulders in the absorption spectrum, is $2 A_{z z}$.
Influence of dynamics on the nitroxide spectrum
In liquid solution, molecules tumble stochastically due to Brownian rotational diffusion. In the following we consider isotropic rotational diffusion, where the molecule tumbles with the same average rate about any axis in its molecular frame. This is a good approximation for nitroxide spin probes with small substituents $\mathrm{R}$. For instance, TEMPO $(\mathbf{1}$ with $\mathrm{R}=\mathrm{H})$ is almost spherical with a van-der-Waals radius of $3.43 \AA \AA$. In water at ambient temperature, the $\tau_{\mathrm{r}}$ rotational correlation time for TEMPO is of the order of $10 \mathrm{ps}$. The product $\tau_{\mathrm{r}} \Delta \omega$ with the maximum anisotropy $\Delta \omega$ of the nitroxide EPR spectrum on an angular frequency axis is much smaller than unity. In this situation, anisotropy averages and three narrow lines of equal width and intensity are expected. The spectrum in Fig. 10.2(a) corresponds to this situation and a closer look reveals that the high-field line has somewhat lower amplitude. This can be traced back to a larger linewidth than for the other two lines, which indicates a shorter $T_{2}$ for the $m_{I}=-1$ transition than for the other transitions. Indeed, transverse relaxation is dominated by the effect from combined hyperfine and $g$ anisotropy, which is largest for the $m_{I}=-1$ transition that has the largest anisotropic dispersion of resonance frequencies. With increasing rotational correlation time $\tau_{r}$, one expects this relaxation process to become stronger, which should lead to more line broadening that is strongest for the high-field line and weakest for the central line. This is indeed observed in the simulation for $\tau_{\mathrm{r}}=495$ ns shown in the bottom trace of Fig. $10.4$.
<figure>$1<figcaption>$2</figcaption></figure>
Figure 10.4: Simulation of X-band CW EPR spectra of an isotropically tumbling nitroxide radical for different rotational correlation times $\tau_{\mathrm{r}}$. A rotational correlation time of $1 \mu$ s at $190 \mathrm{~K}$ and an activated process with activation energy of $22.9 \mathrm{~kJ} \mathrm{~mol}^{-1}$ were assumed, close to parameters observed for TEMPO in a synthetic polymer.
According to Kivelson relaxation theory, the ratio of the line width of one of the outer lines to the line width of the central line is given by
$\frac{T_{2}^{-1}\left(m_{I}\right)}{T_{2}^{-1}(0)}=1+B m_{I}+C m_{I}^{2}$
where
$B=-\frac{4}{15} b \Delta \gamma B_{0} T_{2}(0) \tau_{\mathrm{r}}$
and
$C=\frac{1}{8} b^{2} T_{2}(0) \tau_{\mathrm{r}}$
with the hyperfine anisotropy parameter
$b=\frac{4 \pi}{3}\left[A_{z z}-\frac{A_{x x}+A_{y y}}{2}\right]$
and the electron Zeeman anisotropy parameter $\Delta \gamma$
$\Delta \gamma=\frac{\mu_{\mathrm{B}}}{\hbar}\left[g_{z z}-\frac{g_{x x}+g_{y y}}{2}\right]$
The relaxation time $T_{2}(0)$ for the central line can be computed from the corresponding peak-topeak linewidth in field domain $\Delta B_{p p}(0)$ as
$T_{2}(0)=\frac{2}{\sqrt{3} g_{\text {iso }} \mu_{\mathrm{B}} \Delta B_{\mathrm{pp}}(0)}$
Thus, Eqs. (10.1-10.3) can be solved for the only remaining unknown $\tau_{\mathrm{r}}$. In practice, ratios of peak-to-peak line amplitudes $I\left(m_{I}\right)$ are analyzed rather than linewidth ratios, as they can be measured with higher precision. The linewidth ratio is related to the amplitude ratio $I(0) / I(-1)$ (see bottom trace in Fig. 10.4) in a first derivative spectrum by
$\frac{T_{2}^{-1}\left(m_{I}\right)}{T_{2}^{-1}(0)}=\sqrt{\frac{I(0)}{I\left(m_{I}\right)}}$
since the integral intensity of the absorption line (double integral of the derivative lineshape) is the same for each of the three transitions. The rotational correlation time can thus be determined by, e.g.,
$\tau_{\mathrm{r}}=\frac{\sqrt{3}}{2 b}\left[\frac{b}{8}-\frac{4 \Delta \gamma B_{0}}{15}\right]^{-1} \frac{g_{\mathrm{iso}} \mu_{\mathrm{B}}}{\hbar} \Delta B_{\mathrm{pp}}(0)\left[\sqrt{\frac{I(0)}{I(-1)}}-1\right]$
where $\Delta B_{0}$ is the peak-to-peak linewidth of the central line. This equation can be applied in the fast tumbling regime, where the three individual lines for $m_{I}=-1,0$, and $+1$ can still be clearly recognized and have the shape of symmetric derivative absorption lines.
For slower tumbling with $\tau_{\mathrm{r}}>1.5 \mathrm{~ns}$, the line shape becomes more complex and approaches the rigid limit (solid-state spectrum) at about $\tau_{\mathrm{r}}=1 \mu \mathrm{s}$ (Fig. 10.4). These lineshapes can be simulated by considering multi-site exchange between different orientations of the molecule with respect to the magnetic field. Unlike for two-site exchange, which is discussed in the NMR part of the lecture course (see Section 3 of the NMR lecture notes), no closed expressions can be obtained for multi-site exchange. Nevertheless we can estimate the time scale where the spectral features are broadest and transverse relaxation times are shortest. Coalescence for two-site exchange is observed at $\Delta \Omega / k=2 \sqrt{2}$. By substituting $k$ by $1 / \tau_{\text {r and }} \Delta \Omega$ by the maximum anisotropy of $7.6 \mathrm{mT}$, corresponding to $213 \mathrm{MHz}$, we find a "coalescence time" $2 \sqrt{2} / \Delta \Omega \approx 2.1$ ns. The simulations in Fig. $10.4$ show indeed that around this rotational correlation time, the
<figure>$1<figcaption>$2</figcaption></figure>
Figure 10.5: Plot of the outer extrema separation $2 A_{z z}^{\prime}$ as a function of temperature $T$ for nitroxide spectra simulated under the same assumptions as in Fig. 10.4.
character of the spectrum changes from fast orientation exchange (liquid-like spectrum with three distinct peaks) to slow orientation exchange (solid-like spectrum).
A simple way of analyzing a temperature dependence, such as the one shown in Fig. 10.4, is to plot the outer extrema separation $2 A_{z z}^{\prime}$ as a function of temperature (Fig. 10.5). The "coalescence time" in such a plot corresponds to the largest gradient $\mathrm{d} A_{z z}^{\prime} / \mathrm{d} T$, which coincides with the mean between the $2 A_{z z}^{\prime}$ values in the fast tumbling limit and rigid limit, which is $5 \mathrm{mT}$. In the case at hand, this coalescence time is $3.5 \mathrm{~ns}$ and is observed at a temperature $T_{5 \mathrm{mT}}=312$ $\mathrm{K}$. The $T_{5 \mathrm{mT}}$ temperature is the temperature where the material becomes "soft" and molecular conformations can rearrange. Nitroxide spectra in the slow tumbling regime can reveal more details on dynamics, for instance, whether there are preferred rotation axes, whether motion is restricted due to covalent linkage of the nitroxide to a large molecule, or whether there is local order, such as in a lipid bilayer.
<figure>$1<figcaption>$2</figcaption></figure>
Figure 10.6: Influence of polarity of the environment and of hydrogen bonding on $g_{x x}$ shift and hyperfine coupling. (a) In the mesomeric structure where the unpaired electron is on the oxygen atom (left), five valence electrons are formally assigned to $\mathrm{N}$ and six to $\mathrm{O}$, which corresponds to electroneutrality. In the mesomeric structure where the unpaired electron is on the nitrogen atom (right), only four valence electrons are formally assigned to $\mathrm{N}$ and seven to $\mathrm{O}$, which corresponds to a positive charge at $\mathrm{N}$ and to a negative charge at $\mathrm{O}$. (b) Admixture of the charge-separated mesomeric structure generates partial charges and is favored in a polar environment that screens Coulomb attraction of the two charges. Hydrogen bonding to oxygen lowers energy of the lone pair, making excitation of a lone pair electron to the SOMO less likely, and thus decreasing $g_{x x}$ shift.
Polarity and proticity
Delocalization of the unpaired electron in the $\mathrm{N}-\mathrm{O}^{\bullet}$ group can be understood by considering mesomeric structures (Fig. 10.6). If the unpaired electron resides on oxygen, the formal number of valence electrons is five on nitrogen and six on oxygen, corresponding to the nuclear charge that is not compensated by inner shell electrons. Hence, both atoms are formally neutral in this limiting structure. If, on the other hand, the unpaired electron resides on the nitrogen atom, only four valence electrons are assigned to this atom, whereas seven valence electrons are assigned to the oxygen atom. This corresponds to charge separation with the formal positive charge on nitrogen and the formal negative charge on oxygen. The charge-separated form is favored in polar solvents, which screen Coulomb attraction between the two charges, whereas the neutral form is favored in nonpolar solvents. Hence, for a given nitroxide radical in a series of solvents, the ${ }^{14} \mathrm{~N}$ hyperfine coupling, which stems from spin density on the nitrogen atom, is expected to increase with increasing solvent polarity. This effect has indeed been found. It is most easily seen in the solid state for $A_{z z}$ but can also be discerned in the liquid state for $A_{\text {iso }}$.
The change in $A_{z z}$ is expected to be anti-correlated to the $g_{x x}$ shift, because this shift arises from SOC at the oxygen atom and, the higher spin density on the nitrogen atom is, the lower it is on the oxygen atom. This effect has also been found and is most easily detected by high-field/high-frequency EPR at frequencies of W-band frequencies of $\approx 95 \mathrm{GHz}$ or even higher frequencies. How $A_{z z}$ is correlated to $g_{x x}$ depends on proticity of the solvent. Protic solvents form hydrogen bonds with the lone pairs on the oxygen atom of the N-O ${ }^{\bullet}$ group. This lowers energy of the lone pair orbitals, making excitation of an electron from these orbitals to the SOMO less likely. Since this excitation provides the main contribution to SOC and thus to $g_{x x}$ shift, hydrogen bonding to oxygen reduces $g_{x x}$ shift. If two nitroxides have the same hyperfine coupling $A_{z z}$ in an aprotic and protic environment, $g_{x x}$ will be lower in the protic environment. This effect has also been found. In some cases it was possible to discern nitroxide labels with zero, one, and two hydrogen bonds by resolution of their $g_{x x}$ features in W-band CW EPR spectra. Slopes of $-1.35 \mathrm{~T}^{-1}$ for aprotic at $-2 \mathrm{~T}^{-1}$ for protic environments have been found for the correlation between $A_{z z}$ and $g_{x x}$ for MTSL in spin-labeled bacteriorhodopsin in lipid bilayers [Ste+00].
$10.1 .6$ Water accessibility
Polarity and proticity are proxy parameters for water accessibility of spin-labeled sites in proteins. Two other techniques provide complementary information. First, water can be replaced by deuterated water and the modulation depth of deuterium ESEEM can be measured. Because of the $r^{-6}$ dependence of modulation depth (see Eq. (8.7)) the technique is most sensitive to deuterium nuclei in the close vicinity of the spin label. As long as $k \ll 1$, modulation depth contributions of individual nuclei add, so that the total deuterium modulation depth is a measure for local deuterium concentration close to the label. Data can be processed in a way that removes the contribution from directly hydrogen-bonded nuclei. Strictly speaking, this technique measures the concentration of not only water protons but also the one of any exchangeable protons near the label, but only to the extent that these exchangeable protons are water accessible during sample preparation or measurement.
A second, more direct technique that is applicable at ambient temperature measures the proton NMR signal as a function of irradiated microwave power with the microwave frequency being on-resonant with the central transition of a nitroxide spin label. Such irradiation transfers electron spin polarization to water protons by the Overhauser effect. This Overhauser dynamic nuclear polarization (DNP) is highly specific to water, as it critically depends on the water proton NMR signal being narrow and on fast diffusion of water. In biomolecules, water accessibility of spin labels is high at water-exposed surfaces of soluble and membrane proteins and low inside the proteins and at lipid-exposed surfaces. For transporters, water accessibility can change with state in the transport process. <figure>$1<figcaption>$2</figcaption></figure>
Figure 10.7: Characterization of oxygen accessibility at spin-labeled site V229C in major plant light harvesting complex LHCII by CW progressive power saturation. (a) Ribbon model of LHCII with green with its carotenoid cofactors (yellow, violet) and space-filling model of residue 229 (red, marked by an arrow). The pink planes correspond to the lipid headgroup layer of the thylakoid membrane in chloroplasts. (b) Progressive power saturation curves in the absence (blue) and presence (red) of oxygen.
Oxygen accessibility
Since collision of paramagnetic triplet oxygen with spin probes enhances relaxation (Fig. $7.4)$, the saturation parameter $S=\omega_{1}^{2} T_{1} T 2$ is smaller for oxygen-accessible spin labels than for spin labels not accessible to oxygen. This change can be detected by CW progressive power saturation measurements (Section 7.2.2). The experiment is most conveniently performed with capillary tubes made of the gas permeable plastic TPX. A reference measurement is performed in a nitrogen atmosphere, which causes deoxygenation of the sample on the time scale of 15 min. The gas stream is then changed to air (20% oxygen) or pure oxygen and the measurement is repeated. Such data are shown in Fig. $10.7$ for residue 229 in major plant light harvesting complex LHCII. This residue is lipid exposed. As a nonpolar molecule, oxygen dissolves well in the alkyl chain region of a lipid bilayer. Accordingly, the signal saturates at higher power in an air atmosphere than in a nitrogen atmosphere. Oxygen accessibility can be quantified by a normalized $P_{1 / 2}$ parameter (Section 7.2.2).
$10.1 .8$ Local pH measurements
The ${ }^{14} \mathrm{~N}$ hyperfine coupling of nitroxide spin probes becomes $\mathrm{pH}$ sensitive if the heterocycle that contains the $\mathrm{N}-\mathrm{O}^{\bullet}$ group also contains a nitrogen atom that can be protonated in the desired $\mathrm{pH}$ range. This applies, for instance, to the imidazolidine nitroxide 3 in Fig. $10.1$, which has a pK value of $\approx 4.7$ and exhibits a change in isotropic ${ }^{14}$ N hyperfine coupling of $0.13 \mathrm{mT}$ between the protonated (1.43 mT) and deprotonated $(1.56 \mathrm{mT})$ form, which can be resolved easily in liquid solution. By modifying the probe to a label, local $\mathrm{pH}$ can be measured near a residue of interest in a protein.
$10.2$ Spin traps
Many radicals are very reactive. This fact makes their detection during chemical reactions and in living cells very important, but it also makes their concentration very low, since often their formation reaction is slower than the reactions that destroy them again. For instance, concentration of the hydroxyl radical ${ }^{\circ} \mathrm{OH}$, a reactive oxygen species (ROS) in living cells, is too low for EPR detection even under conditions where ${ }^{\bullet} \mathrm{OH}$ leads to cell damage or cell death. The situation is somewhat better for the superoxide anion radical $\mathrm{O}_{2}^{2-}$, but physiologically relevant concentrations are hard to detect also for this species. <figure>$1<figcaption>$2</figcaption></figure>
Figure 10.8: Reaction of the commonly used spin traps phenylbutylnitrone (PBN) and 5,5-dimethyl-1pyrroline N-oxide (DMPO) with unstable radicals $\mathrm{R}^{\bullet}$. Hyperfine couplings of the ${ }^{14} \mathrm{~N}$ and $\mathrm{H}^{\alpha}$ atom of the formed nitroxide (red) as well as the $g$ value of the nitroxide provide fingerprint information on the type of radical $\mathrm{R}^{\bullet}$.
ROS and some other highly reactive radicals of interest are most easily detected by spin trapping. A spin trap (Fig. 10.8) is a diamagnetic compound that is primed to form a stable radical by reaction with an unstable radical. The most frequently used spin traps are nitrones that form nitroxide radicals by addition of the unstable radical to the $\mathrm{C}$ atom in $\alpha$ position of the nitrone group. The formed nitroxide radicals are not as stable as the ones used as spin labels, mainly because they contain a hydrogen atom in $\alpha$ position to the $\mathrm{N}-\mathrm{O}$ group. Their lifetime is usually on the minute time scale, which is sufficient for detection. The hyperfine coupling of the $\mathrm{H}^{\alpha}$ atom is sensitive to the type of primary radical $\mathrm{R}^{\bullet}$, i.e. to the nature of the other substituent at the $C^{\alpha}$ atom. Furthermore, these nitrones are less sterically crowded than the ones that would yield more stable nitroxides and thus the nitrones are more reactive and trap radicals $\mathrm{R}^{\bullet}$ more easily. In addition to the $\mathrm{H}^{\alpha}$ hyperfine coupling, the hyperfine coupling of the ${ }^{14} \mathrm{~N}$ atom of the $\mathrm{N}-\mathrm{O}^{\bullet}$ group is sensitive to the nature of $\mathrm{R}^{\bullet}$. A database of experimental results supports assignment of $R^{\bullet}$ in difficult cases: https://tools.niehs.nih.gov//stdb/index.cfm 1
${ }^{1}$ Look a the "Hints for Using the Spin Trap Database" before you start your search. The keyword format is powerful, but not very intuitive.
Books
[CCM16] V. Chechik, E. Carter, and D. M. Murphy. Electron Paramagnetic Resonance. $1^{\text {st }}$ Ed. Oxford: Oxford University Press, 2016 (cited on pages 8, 26).
[KBE04] M. Kaupp, M. Buhl, and V. G. Malkin (Eds.) Calculation of NMR and EPR Parameters: Theory and Applications. $1^{\text {st }}$ Ed. Weinheim: Wiley-VCH, $2004($ cited on page 16).
[Rie07] Philip Rieger. Electron Spin Resonance. Analysis and Interpretation. The Royal Society of Chemistry, 2007, P001-173. ISBN: 978-0-85404-355-2. DOI: 10.1039/ 9781847557872 . URL: http://dx. doi .org/10.1039/9781847557872 (cited on page 37).
[WBW94] J. A. Weil, J. R. Bolton, and J. E. Wertz. Electron Paramagnetic Resonance. $1^{\text {st }}$ Ed. New York: John Wiley & Sons, Inc., 1994 (cited on page 8).
Articles
[Cas+60] Theodore Castner et al. "Note on the Paramagnetic Resonance of Iron in Glass". In: J. Chem. Phys. $32.3(1960)$, pages 668-673. Dor: http://dx. doi . org/10. $1063 / 1.1730779$ (cited on page 40$)$.
[KM85] A. K. Koh and D. J. Miller. "Hyperfine coupling constants and atomic parameters for electron paramagnetic resonance data". In: Atomic Data and Nuclear Data Tables 33 (1985), pages 235-253 (cited on pages 22,23 ).
[Lef67] R. Lefebvre. "Pseudo-hyperfine interactions in radicals". In: Molecular Physics $12.5$ (1967), pages 417-426. Dor: $10.1080 / 00268976700100541$ (cited on page 22).
[Ste+00] Heinz-Jürgen Steinhoff et al. "High-field EPR studies of the structure and conformational changes of site-directed spin labeled bacteriorhodopsin". In: Biochim. Biophys. $\operatorname{Acta}(B B A)$ - Bioenergetics 1457 (2000), pages 253-262. DOI: 10. 1016/S0005 $2728(\theta \theta) 00106-7$ (cited on page 82 ).
<figure>$1<figcaption>$2</figcaption></figure>
Index
<figure>$1<figcaption>$2</figcaption></figure>
Davies ENDOR ..................... 59
dead time $\ldots \ldots \ldots \ldots \ldots \ldots . \ldots \ldots . \ldots 49$
DEER .............................. 68
form factor $\ldots \ldots \ldots \ldots \ldots \ldots \ldots .71$
DNP ............................. 82
dynamic nuclear polarization............82
<figure>$1<figcaption>$2</figcaption></figure>
Fermi contact interaction $\ldots \ldots \ldots \ldots \ldots .22$
<figure>$1<figcaption>$2</figcaption></figure>
g value
free electron $\ldots . \ldots \ldots \ldots \ldots \ldots \ldots . \ldots 9$
H
high-field approximation $\ldots . \ldots .10,17$
high-field approximation $\ldots \ldots \ldots . \ldots 10,17$
homogeneous linewidth $\ldots \ldots \ldots \ldots \ldots .54$
Hund’s rule......................37
hyperfine contrast selectivity $\ldots . \ldots . \ldots 61$
<figure>$1<figcaption>$2</figcaption></figure>
Kramers ions ..................... 37
Kramers theorem $\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .37$
<figure>$1<figcaption>$2</figcaption></figure>
$\mathrm{L}$ curve $\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots 73$
level energies
first order $\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \ldots . \ldots \ldots \ldots . \ldots . \ldots$
linear regime $\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .54$
M
<figure>$1<figcaption>$2</figcaption></figure>
modulation depth
ESEEM ....................62 molecular orbital
$\mathrm{~ s i n g l y ~ o c c u p i e d ~ . . . . . . . . . . . . . . .}$
<figure>$1<figcaption>$2</figcaption></figure>
orientation selection $\ldots \ldots \ldots \ldots \ldots \ldots \ldots . \ldots 19$
<figure>$1<figcaption>$2</figcaption></figure>
PELDOR ........................68
progressive power saturation ............55
<figure>$1<figcaption>$2</figcaption></figure>
quantum number
$\operatorname{good} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \ldots . \ldots . \ldots . \ldots . \ldots . \ldots$
magnetic $\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \ldots . \ldots$
spin $\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \ldots . \ldots \ldots \ldots . \ldots \ldots . \ldots \ldots . \ldots$
$\mathbf{R}$
rapid scan $\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .53$
reactive oxygen species $\ldots . \ldots \ldots . \ldots .84$
regularization parameter ............ 72
$\operatorname{ROS} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \ldots \ldots . \ldots \ldots$
S
saturation curve $\ldots \ldots \ldots \ldots \ldots \ldots \ldots . \ldots .55$
selection rule $\ldots . \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \ldots$
site-directed spin labeling ............75
SOMO .......................11
spin packet $\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \ldots . \ldots 3$
spin-orbit coupling $\ldots \ldots \ldots \ldots \ldots \ldots .15,37$
T
Tikhonov regularization $\ldots \ldots \ldots \ldots \ldots .72$
two-pulse ESEEM $\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .47$
$Z$
zero-field splitting. | textbooks/chem/Physical_and_Theoretical_Chemistry_Textbook_Maps/Electron_Paramagnetic_Resonance_(Jenschke)/01%3A_Introduction/1.01%3A_General_Remarks.txt |
3. Part 2: Electron
Gunnar Jeschke
25
20
Title image: HYSCORE spectrum of a Ti(III) surface species (in collaboration with C. Copéret, F. Allouche, V. Kalendra)
Chapter 2 Stern-Gerlach memory plaque: Pen (GNU Free Documentation License)
Chapter 3 Zeeman effect on spectral lines (photograph by Pieter Zeeman, 1897)
Chapter 4 Hyperfine interaction by spin polarization (own work)
Chapter 5 Dipole-dipole interaction (own work)
Chapter 6 Energy level scheme of an spin system (own work)
Chapter 7 Field modulation in CW EPR (own work)
Chapter 8 Deuterium ESEEM trace of a spin-labelled protein (own work)
Chapter 9 Distance distribution measurement in a spin-labelled RNA construct (collaboration with O. Duss, M. Yulikov, F. H.-T. Allain)
Chapter 10 Electronic structure and molecular frame of a nitroxide spin label (own work)
5. PUBLISHED BY GUNNAR JESCHKE
http://WWW . epr. ethz . ch
Licensed under the Creative Commons Attribution-NonCommercial Unported License (the "License"). You may not use this file except in compliance with the License. You may obtain a copy of the License at http://creativecommons. org/licenses/by-nc/3.0.
Design and layout of the lecture notes are based on the Legrand Orange Book available at http://latextemplates. com/template/the-legrand-orange-book.
First printing, October 2016
6. Contents
Introduction
Electron spin
Magnetic resonance of the free electron
2.1.1 The magnetic moment of the free electron
2.2 Interactions in electron-nuclear spin systems 10
2.2.1 General consideration on spin interactions
2.2.2 The electron-nuclear spin Hamiltonian . . .......................... 12
3 Electron Zeeman Interaction .......................... 15
3.1 Physical origin of the shift 15
3.2 Electron Zeeman Hamiltonian 17
3.3 Spectral manifestation of the electron Zeeman interaction 18
3.3.1 Liquid solution
Solid state
Hyperfine Interaction
4.1 Physical origin of the hyperfine interaction 21
Fermi contact interaction
Spin polarization .................................... 23
4.2 Hyperfine Hamiltonian 24 4.3 Spectral manifestation of the hyperfine interaction 25
4.3.2 Liquid-solution nuclear frequency spectra
Solid-state EPR spectra ................................. 28
Solid-state nuclear frequency spectra
5 Electron-Electron Interactions .......................
5.1 Exchange interaction 31
Exchange Hamiltonian ................................... 32
5.1.3 Spectral manifestation of the exchange interaction . ....................
5.2 Dipole-dipole interaction 33
Physical picture
Dipole-dipole Hamiltonian
Spectral manifestation of the dipole-dipole interaction . . . . . . . . . . . . 35
5.3.1 Physical picture
6 Forbidden Electron-Nuclear Transitions . ................. 43
6.1 Physical picture 43
6.1.1 The
Local fields at the nuclear spin ........................................ 43
6.2 Product operator formalism with pseudo-secular interactions 45
Transformation of to the eigenbasis
6.2.2 General product operator computations for a non-diagonal Hamiltonian ......... 46
6.3 Generation and detection of nuclear coherence by electron spin excitation 47
Nuclear coherence generator
7 CW EPR Spectroscopy ................................ 49
7.1 Why and how CW EPR spectroscopy is done 49
7.1.1 Sensitivity advantages of CW EPR spectroscopy .........................................
7.1.3 Considerations on sample preparation
7.2 Theoretical description of CW EPR 53
8 Measurement of Small Hyperfine Couplings . .......... . . . 57
8.1 ENDOR 57
8.1.1 Advantages of electron-spin based detection of nuclear frequency spectra
8.1.2 Types of ENDOR experiments ............................ 57
8.1.3 Davies ENDOR 8.2 ESEEM and HYSCORE
8.2.1 ENDOR or ESEEM?
8.2.2 Three-pulse ESEEM
HYSCORE
Distance Distribution Measurements
9.1.1 The four-pulse DEER experiment
9.1.2 Sample requirements
9.2.1 Expression for the DEER signal
Background correction
9.2.3 Tikhonov regularization with non-negativity constraint . ................... 72
Spin Probes and Spin Traps
10.1.1 Spin probes and labels
Nitroxide radicals
The nitroxide EPR spectrum
Influence of dynamics on the nitroxide spectrum
Polarity and proticity
Water accessibility
Oxygen accessibility
Local measurements
Spin traps 83
Books 85
Articles 85
Index
7. 1 - Introduction
7.1. General Remarks
Electron Paramagnetic Resonance (EPR) spectroscopy is less well known and less widely applied than NMR spectroscopy. The reason is that EPR spectroscopy requires unpaired electrons and electron pairing is usually energetically favorable. Hence, only a small fraction of pure substances exhibit EPR signals, whereas NMR spectroscopy is applicable to almost any compound one can think of. On the other hand, as electron pairing underlies the chemical bond, unpaired electrons are associated with reactivity. Accordingly, EPR spectroscopy is a very important technique for understanding radical reactions, electron transfer processes, and transition metal catalysis, which are all related to the 'reactivity of the unpaired electron'. Some species with unpaired electrons are chemically stable and can be used as spin probes to study systems where NMR spectroscopy runs into resolution limits or cannot provide sufficient information for complete characterization of structure and dynamics. This lecture course introduces the basics for applying EPR spectroscopy on reactive or catalytically active species as well as on spin probes.
Many concepts in EPR spectroscopy are related to similar concepts in NMR spectroscopy. Hence, the lectures on EPR spectroscopy build on material that has been introduced before in the lectures on NMR spectroscopy. This material is briefly repeated and enhanced in this script and similarities as well as differences are pointed out. Such a linked treatment of the two techniques is not found in introductory textbooks. By emphasizing this link, the course emphasizes understanding of the physics that underlies NMR and EPR spectroscopy instead of focusing on individual application fields. We aim for understanding of spectra at a fundamental level and for understanding how parameters of the spin Hamiltonian can be measured with the best possible sensitivity and resolution.
Chapter 2 of the script introduces electron spin, relates it to nuclear spin, and discusses, which interactions contribute to the spin Hamiltonian of a paramagnetic system. Chapter 3 treats the electron Zeeman interaction, the deviation of the value of a bound electron from the value of a free electron, and the manifestation of anisotropy in solid-state EPR spectra. Chapter 4 introduces the hyperfine interaction between electron and nuclear spins, which provides most information on electronic and spatial structure of paramagnetic centers. Spectral manifestation in the liquid and solid state is considered for spectra of the electron spin and of the nuclear spins. Chapter 5 discusses phenomena that occur when the hyperfine interaction is so large that the high-field approximation is violated for the nuclear spin. In this situation, formally forbidden transitions become partially allowed and mixing of energy levels leads to changes in resonance frequencies. Chapter 6 discusses how the coupling between electron spins is described in the spin Hamiltonian, depending on its size. Throughout Chapters , the introduced interactions of the electron spin are related to electronic and spatial structure.
Chapter 7-9 are devoted to experimental techniques. In Chapter 7 , continuous-wave EPR is introduced as the most versatile and sensitive technique for measuring EPR spectra. The requirements for obtaining well resolved spectra with high signal-to-noise ratio are derived from first physical principles. Chapter 8 discusses two techniques for measuring hyperfine couplings in nuclear frequency spectra, where they are better resolved than in EPR spectra. Electron nuclear double resonance (ENDOR) experiments use electron spin polarization and detection of electron spins in order to enhance sensitivity of such measurements, but still rely on direct excitation of the nuclear spins. Electron spin echo envelope modulation (ESEEM) experiments rely on the forbidden electron-nuclear spin transitions discussed in Chapter 5. Chapter 9 treats the measurement of distance distributions in the nanometer range by separating the dipole-dipole coupling between electron spins from other interactions.
The final Chapter 10 introduces spin probing and spin trapping and, at the same time, demonstrates the application of concepts that were introduced in earlier Chapters.
At some points (dipole-dipole coupling, explanation of CW EPR spectroscopy in terms of the Bloch equations) this lecture script significantly overlaps with the NMR part of the lecture script. This is intended in order to make the EPR script reasonably self-contained. Note also that this lecture script serves two purposes. First, it should serve as a help in studying the subject and preparing for the examination. Second, it is reference material when you later encounter paramagnetic species in your own research and need to obtain information on them by EPR spectroscopy.
7.2. Suggested Reading & Electronic Resources
There is no textbook on EPR spectroscopy that treats all material of this course on a basic level. However, many of the concepts are covered by a title from the Oxford Chemistry Primer series by Chechik, Carter, and Murphy [CCM16]. Physically minded students may also appreciate the older standard textbook by Weil, Bolton, and Wertz [WBW94].
For some of the simulated spectra and worked examples in these lecture notes, Matlab scripts or Mathematica notebooks are provided on the lecture homepage. Part of the numerical simulations is based on EasySpin by Stefan Stoll (http://wWW. easyspin.org/) and another part on SPIDYAN by Stephan Pribitzer (http://www. epr . ethz. ch/software. html). Computations with product operator formalism require the Mathematica package SpinOp.m by Serge Boentges, which is available on the course homepage. An alternative larger package for such analytical computations is SpinDynamica by Malcolm Levitt (http://Www. spindynamica. soton. ac. uk/). Last, but not least the most extensive package for numerical simulations of magnetic resonance experiments is SPINACH by Ilya Kuprov et al. (http://spindynamics. org/ Spinach.php). For quantum-chemical computations of spin Hamiltonian parameters, the probably most versatile program is the freely available package ORCA (https://orcaforum . cec.mpg.de/).
8. Magnetic resonance of the free electron
The magnetic moment of the free electron Differences between EPR and NMR spectroscopy
Interactions in electron-nuclear spin systems General consideration on spin interactions The electron-nuclear spin Hamiltonian
10. Magnetic resonance of the free electron
10.2.1. The magnetic moment of the free electron
As an elementary particle, the electron has an intrinsic angular momentum called spin. The spin quantum number is , so that in an external magnetic field along , only two possible values can be observed for the component of this angular momentum, , corresponding to magnetic quantum number state and , corresponding to magnetic quantum number ( state . The energy difference between the corresponding two states of the electron results from the magnetic moment associated with spin. For a classical rotating particle with elementary charge , angular momentum and mass , this magnetic moment computes to
The charge-to-mass ratio is much larger for the electron than the corresponding ratio for a nucleus, where it is of the order of , where is the proton mass. By introducing the Bohr magneton and the quantum-mechanical correction factor , we can rewrite Eq. (2.1) as
Dirac-relativistic quantum mechanics provides , a correction that can also be found in a non-relativistic derivation. Exact measurements have shown that the value of a free electron deviates slightly from . The necessary correction can be derived by quantum electrodynamics, leading to . The energy difference between the two spin states of a free electron in an external magnetic field is given by
so that the gyromagnetic ratio of the free electron is . This gyromagnetic ratio corresponds to a resonance frequency of at a field of , which is by a factor of about 658 larger than the nuclear Zeeman frequency of a proton.
10.2.2. Differences between EPR and NMR spectroscopy
Most of the differences between NMR and EPR spectroscopy result from this much larger magnetic moment of the electron. Boltzmann polarization is larger by this factor and at the same magnetic field the detected photons have an energy larger by this factor. Relaxation times are roughly by a factor shorter, allowing for much faster repetition of EPR experiments compared to NMR experiments. As a result, EPR spectroscopy is much more sensitive. Standard instrumentation with an electromagnet working at a field of about and at microwave frequencies of about (X band) can detect about spins, if the sample has negligible dielectric microwave losses. In aqueous solution, organic radicals can be detected at concentrations down to in a measurement time of a few minutes.
Due to the large magnetic moment of the electron spin the high-temperature approximation may be violated without using exotic equipment. The spin transition energy of a free electron matches thermal energy at a temperature of and a field of about corresponding to a frequency of about (W band). Likewise, the high-field approximation may break down. The dipole-dipole interaction between two electron spins is by a factor of larger than between two protons and two unpaired electrons can come closer to each other than two protons. The zero-field splitting that results from such coupling can amount to a significant fraction of the electron Zeeman interaction or can even exceed it at the magnetic fields, where EPR experiments are usually performed . The hyperfine coupling between an electron and a nucleus can easily exceed the nuclear Zeeman frequency, which leads to breakdown of the high-field approximation for the nuclear spin.
11. Interactions in electron-nuclear spin systems
11.2.3. General consideration on spin interactions
Spins interact with magnetic fields. The interaction with a static external magnetic field is the Zeeman interaction, which is usually the largest spin interaction. At sufficiently large fields, where the high-field approximation holds, the Zeeman interaction determines the quantization direction of the spin. In this situation, is a good quantum number and, if the high-field approximation also holds for a nuclear spin , the magnetic quantum number is also a good quantum number. The energies of all spin levels can then be expressed by parameters that quantify spin interactions and by the magnetic quantum numbers. The vector of all magnetic quantum numbers defines the state of the spin system.
Spins also interact with the local magnetic fields induced by other spins. Usually, unpaired electrons are rare, so that each electron spin interacts with several nuclear spins in its vicinity, whereas each nuclear spin interacts with only one electron spin (Fig. 2.1). The hyperfine interaction between the electron and nuclear spin is usually much smaller than the electron Zeeman interaction, with exceptions for transition metal ions. In contrast, for nuclei in the close vicinity of the electron spin, the hyperfine interaction may be larger than the nuclear Zeeman interaction at the fields where EPR spectra are usually measured. In this case, which is discussed in Chapter 6, the high-field approximation breaks down and is not a good quantum number. Hyperfine couplings to nuclei are relevant as long as they are at least as large as the transverse relaxation rate of the coupled nuclear spin. Smaller couplings are unresolved.
In some systems, two or more unpaired electrons are so close to each other that their coupling exceeds their transverse relaxation rates . In fact, the isotropic part of this coupling can by far exceed the electron Zeeman interaction and often even thermal energy if two unpaired electrons reside in different molecular orbitals of the same organic molecule (triplet state molecule) or if several unpaired electrons belong to a high-spin state of a transition metal or rare earth metal ion. In this situation, the system is best described in a coupled representation with an
Figure 2.1: Scheme of interactions in electron-nuclear spin systems. All spins have a Zeeman interaction with the external magnetic field . Electron spins (red) interact with each other by the dipole-dipole interaction through space and by exchange due to overlap of the singly occupied molecular orbitals (green). Each electron spin interacts with nuclear spins (blue) in its vicinity by hyperfine couplings (purple). Couplings between nuclear spins are usually negligible in paramagnetic systems, as are chemical shifts. These two interactions are too small compared to the relaxation rate in the vicinity of an electron spin.
electron group spin . The isotropic coupling between the individual electron spins does not influence the sublevel splitting for a given group spin quantum number . The anisotropic coupling, which does lead to sublevel splitting, is called the zero-field or fine interaction. If the electron Zeeman interaction by far exceeds the spin-spin coupling, it is more convenient to describe the system in terms of the individual electron spins . The isotropic exchange coupling , which stems from overlap of two singly occupied molecular orbitals (SOMOs), then does contribute to level splitting. In addition, the dipole-dipole coupling through space between two electron spins also contributes.
Concept - Singly occupied molecular orbital (SOMO). Each molecular orbital can be occupied by two electrons with opposite magnetic spin quantum number . If a molecular orbital is singly occupied, the electron is unpaired and its magnetic spin quantum number can be changed by absorption or emission of photons. The orbital occupied by the unpaired electron is called a singly occupied molecular orbital (SOMO). Several unpaired electrons can exist in the same molecule or metal complex, i.e., there may be several SOMOs.
Nuclear spins in the vicinity of an electron spin relax much faster than nuclear spins in diamagnetic substances. Their transverse relaxation rates thus exceed couplings between nuclear spins and chemical shifts. These interactions, which are very important in NMR spectroscopy, are negligible in EPR spectroscopy. For nuclear spins no information on the chemical identity of a nucleus can be obtained, unless its hyperfine coupling is understood. The element can be identified via the nuclear Zeeman interaction. For nuclear spins , information on the chemical identity is encoded in the nuclear quadrupole interaction, whose magnitude usually exceeds .
An overview of all interactions and their typical magnitude in frequency units is given in Figure 2.2. This Figure also illustrates another difference between EPR and NMR spectroscopy. Several interactions, such as the zero-field interaction, the hyperfine interaction, larger dipole-dipole and exchange couplings between electron spins and also the anisotropy of the electron Zeeman interaction usually exceed the excitation bandwidth of the strongest and shortest microwave pulses
There is an exception. If the electron spin longitudinal relaxation rate exceeds the nuclear Zeeman interaction by far, nuclear spin relaxation is hardly affected by the presence of the electron spin. In this situation, EPR spectroscopy is impossible, however. that are available. NMR pulses sequences that rely on the ability to excite the full spectrum of a certain type of spins thus cannot easily be adapted to EPR spectroscopy.
11.2.4. The electron-nuclear spin Hamiltonian
Considering all interactions discussed in Section 2.2.1, the static spin Hamiltonian of an electron-nuclear spin system in angular frequency units can be written as
where index runs over all nuclear spins, indices and run over electron spins and the symbol denotes the transpose of a vector or vector operator. Often, only one electron spin and one nuclear spin have to be considered at once, so that the spin Hamiltonian simplifies drastically. For electron group spins , terms with higher powers of spin operators can be significant. We do not consider this complication here.
The electron Zeeman interaction is, in general, anisotropic and therefore parametrized by tensors . It is discussed in detail in Chapter 3 . In the nuclear Zeeman interaction , the nuclear Zeeman frequencies depend only on the element and isotope and thus can be specified without knowing electronic and spatial structure of the molecule. The hyperfine interaction is again anisotropic and thus characterized by tensors . It is discussed in detail in Chapter 4. All electron-electron interactions are explained in Chapter 5 . The zero-field interaction is purely anisotropic and thus characterized by traceless tensors . The exchange interaction is often purely isotropic and any anisotropic contribution cannot be experimentally distinguished from the purely anisotropic dipole-dipole interaction . Hence, the former interaction is characterized by scalars and the latter interaction by tensors . Finally, the nuclear quadrupole interaction is characterized by traceless tensors .
Electron Zeeman interaction
Zero field interaction
Dipole-dipole interaction
between weakly coupled electron spins
Homogeneous EPR linewidths
Figure 2.2: Relative magnitude of interactions that contribute to the Hamiltonian of electron-nuclear spin systems.
Physical origin of the shift Electron Zeeman Hamiltonian Spectral manifestation of the electron Zeeman interaction
Liquid solution Solid state
13. Physical origin of the shift
Bound electrons are found to have values that differ from the value for the free electron. They depend on the orientation of the paramagnetic center with respect to the magnetic field vector . The main reason for this value shift is coupling of spin to orbital angular momentum of the electron. Spin-orbit coupling is a purely relativistic effect and is thus larger if orbitals of heavy atoms contribute to the SOMO. In most molecules, orbital angular momentum is quenched in the ground state. For this reason, SOC leads only to small or moderate shifts and can be treated as a perturbation. Such a perturbation treatment is not valid if the ground state is degenerate or near degenerate.
The perturbation treatment considers excited states where the unpaired electron is not in the SOMO of the ground state. Such excited states are slightly admixed to the ground state and the mixing arises from the orbital angular momentum operator. For simplicity, we consider a case where the main contribution to the shift arises from orbitals localized at a single, dominating atom and by single-electron SOC. To second order in perturbation theory, the matrix elements of the tensor can then be expressed as
where is a Kronecker delta, the factor in the shift term is the spin-orbit coupling constant for the dominating atom, and the matrix elements are computed as
where indices and run over the Cartesian directions , and . The operators , and are Cartesian components of the angular momentum operator, designates the orbital where the unpaired electron resides in an excited-state electron configuration, counted from for the SOMO of the ground state configuration. The energy of that orbital is .
Since the product of the overlap integrals in the numerator on the right-hand side of Eq. (3.2) is usually positive, the sign of the shift is determined by the denominator. The denominator is positive if a paired electron from a fully occupied orbital is promoted to the ground-state SOMO and negative if the unpaired electron is promoted to a previously unoccupied orbital (Figure Ground state
Excitation of a paired electron
Figure 3.1: Admixture of excited states by orbital angular momentum operators leads to a shift by spin-orbit coupling. The energy difference in the perturbation expression is positive for excitation of a paired electron to the ground-state SOMO and negative for excitation of the paired electron to a higher energy orbital.
3.1). Because the energy gap between the SOMO and the lowest unoccupied orbital (LUMO) is usually larger than the one between occupied orbitals, terms with positive numerator dominate in the sum on the right-hand side of Eq. (3.2). Therefore, positive shifts are more frequently encountered than negative ones.
The relevant spin-orbit coupling constant depends on the element and type of orbital. It scales roughly with , where is the nuclear charge. Unless there is a very low lying excited state (near degeneracy of the ground state), contributions from heavy nuclei dominate. If their are none, as in organic radicals consisting of only hydrogen and second-row elements, shifts of only are observed, typical shifts are . Note that this still exceeds typical chemical shifts in NMR by one to two orders magnitude. For first-row transition metals, shifts are of the order of .
For rare-earth ions, the perturbation treatment breaks down. The Landé factor can then be computed from the term symbol for a doublet of levels
where is the quantum number for total angular momentum and the quantum number for orbital angular momentum. The principal values of the tensor are , and , where the with are differences between the eigenvalues of for the two levels.
If the structure of a paramagnetic center is known, the tensor can be computed by quantum chemistry. This works quite well for organic radicals and reasonably well for most first-row transition metal ions. Details are explained in [KBE04].
The tensor is a global property of the SOMO and is easily interpretable only if it is dominated by the contribution at a single atom, which is often, but not always, the case for transition metal and rare earth ion complexes. If the paramagnetic center has a symmetry axis with , the tensor has axial symmetry with principal values . For cubic or tetrahedral symmetry the value is isotropic, but not necessarily equal to . Isotropic values are also encountered to a very good approximation for transition metal and rare earth metal ions with half-filled shells, such as in Mn(II) complexes ( electron configuration) and Gd(III) complexes .
14. Electron Zeeman Hamiltonian
We consider a single electron spin and thus drop the sum and index in in Eq. (2.4). In the principal axes system (PAS) of the tensor, we can then express the electron Zeeman Hamiltonian as
where is the magnetic field, , and are the principal values of the tensor and the polar angles and determine the orientation of the magnetic field in the PAS.
This Hamiltonian is diagonalized by the Bleaney transformation, providing
with the effective value at orientation
If anisotropy of the tensor is significant, the axis in Eq. (3.5) is tilted from the direction of the magnetic field. This effect is negligible for most organic radicals, but not for transition metal ions or rare earth ions. Eq. (3.6) for the effective values describes an ellipsoid (Figure ).
Figure 3.2: Ellipsoid describing the orientation dependence of the effective value in the PAS of the tensor. At a given direction of the magnetic field vector (red), corresponds to distance between the origin and the point where intersects the ellipsoid surface.
Concept 3.2.1 - Energy levels in the high-field approximation. In the high-field approximation the energy contribution of a Hamiltonian term to the level with magnetic quantum numbers and can be computed by replacing the operators by the corresponding magnetic quantum numbers. This is because the magnetic quantum numbers are the eigenvalues of the operators, all operators commute with each other, and contributions with all other Cartesian spin operators are negligible in this approximation. For the electron Zeeman energy contribution is . If the high-field approximation is slightly violated, this expression corresponds to a first-order perturbation treatment. The selection rule for transitions in EPR spectroscopy is and it applies strictly as long as the high-field approximation applies strictly to all spins. This selection rule results from conservation of angular momentum on absorption of a microwave photon and from the fact that the microwave photon interacts with electron spin transitions. It follows that the first-order contribution of the electron Zeeman interaction to the frequencies of all electron spin transitions is the same, namely . As we shall see in Chapter 7 , EPR spectra are usually measured at constant microwave frequency by sweeping the magnetic field . The resonance field is then given by
For nuclear spin transitions, , the electron Zeeman interaction does not contribute to the transition frequency.
14.1. Spectral manifestation of the electron Zeeman interaction
14.1.1. Liquid solution
In liquid solution, molecules tumble due to Brownian rotational diffusion. The time scale of this motion can be characterized by a rotational correlation time that in non-viscous solvents is of the order of for small molecules, and of the order of to 100 ns for proteins and other macromolecules. For a globular molecule with radius in a solvent with viscosity , the rotational correlation time can be roughly estimated by the Stokes-Einstein law
If this correlation time and the maximum difference between the transition frequencies of any two orientations of the molecule in the magnetic field fulfill the relation , anisotropy is fully averaged and only the isotropic average of the transition frequencies is observed. For somewhat slower rotation, modulation of the transition frequency by molecular tumbling leads to line broadening as it shortens the transverse relaxation time . In the slow-tumbling regime, where , anisotropy is incompletely averaged and line width attains a maximum. For , the solid-state spectrum is observed. The phenomena can be described as a multi-site exchange between the various orientations of the molecule (see Section 10.1.4), which is analogous to the chemical exchange discussed in the NMR part of the lecture course.
For the electron Zeeman interaction, fast tumbling leads to an average resonance field
with the isotropic value . For small organic radicals in non-viscous solvents at X-band frequencies around , line broadening from anisotropy is negligible. At W-band frequencies of for organic radicals and already at X-band frequencies for small transition metal complexes, such broadening can be substantial. For large macromolecules or in viscous solvents, solid-state like EPR spectra can be observed in liquid solution.
14.1.2. Solid state
For a single-crystal sample, the resonance field at any given orientation can be computed by Eq. (3.7). Often, only microcrystalline powders are available or the sample is measured in glassy frozen solution. Under such conditions, all orientations contribute equally. With respect to the
Figure 3.3: Powder line shape for a tensor with axial symmetry. (a) The probability density to find an orientation with polar angle is proportional to the circumference of a circle a angle on a unit sphere. (b) Probability density . The effective value at angle is . (c) Schematic powder line shape. The pattern corresponds to for a field sweep and to for a frequency sweep. Because of the frame tilting, the isotropic value is not encountered at the magic angle, although the shift is small if .
polar angles, this implies that is uniformly distributed, whereas the probability to encounter a certain angle is proportional to (Figure 3.3). The line shape of the absorption spectrum is most easily understood for axial symmetry of the tensor. Transitions are observed only in the range between the limiting resonance fields at and . The spectrum has a global maximum at and a minimum at .
In CW EPR spectroscopy we do not observe the absorption line shape, but rather its first derivative (see Chapter 7). This derivative line shape has sharp features at the line shape singularities of the absorption spectrum and very weak amplitude in between (Figure ).
Concept 3.3.1 - Orientation selection. The spread of the spectrum of a powder sample or glassy frozen solution allows for selecting molecules with a certain orientation with respect to the magnetic field. For an axial tensor only orientations near the axis of the tensor PAS are selected when observing near the resonance field of . In contrast, when observing near the resonance field for , orientations withing the whole plane of the PAS contribute. For the case of orthorhombic symmetry with three distinct principal values , and , narrow sets of orientations can be observed at the resonance fields corresponding to the extreme values and (see right top panel in Figure 3.4). At the intermediate principal value a broad range of orientations contributes, because the same resonance field can be realized by orientations other than and . Such orientation selection can enhance the resolution of ENDOR and ESEEM spectra (Chapter 8) and simplify their interpretation. axial symmetry
orthorhombic
Figure 3.4: Simulated X-band EPR spectra for systems with only anisotropy. The upper panels show absorption spectra as they can be measured by echo-detected field-swept EPR spectroscopy. The lower panels show the first derivative of the absorption spectra as they are detected by continuous-wave EPR. The unit-sphere pictures in the right upper panel visualize the orientations that are selected at the resonance fields corresponding to the principal values of the tensor.
14.2. Physical origin of the hyperfine interaction
The magnetic moments of an electron and a nuclear spin couple by the magnetic dipole-dipole interaction; similar to the dipole-dipole interaction between nuclear spins discussed in the NMR part of the lecture course. The main difference to the NMR case is that, in many cases, a point-dipole description is not a good approximation for the electron spin, as the electron is distributed over the SOMO. The nucleus under consideration can be considered as well localized in space. We now picture the SOMO as a linear combination of atomic orbitals. Contributions from spin density in an atomic orbital of another nucleus (population of the unpaired electron in such an atomic orbital) can be approximated by assuming that the unpaired electron is a point-dipole localized at this other nucleus.
For spin density in atomic orbitals on the same nucleus, we have to distinguish between types of atomic orbitals. In orbitals, the unpaired electron has finite probability density for residing at the nucleus, at zero distance to the nuclear spin. This leads to a singularity of the dipole-dipole interaction, since this interaction scales with . The singularity has been treated by Fermi. The contribution to the hyperfine coupling from spin density in orbitals on the nucleus under consideration is therefor called Fermi contact interaction. Because of the spherical symmetry of orbitals, the Fermi contact interaction is purely isotropic.
For spin density in other orbitals ( orbitals) on the nucleus under consideration, the dipole-dipole interaction must be averaged over the spatial distribution of the electron spin in these orbitals. This average has no isotropic contribution. Therefore, spin density in orbitals does not influence spectra of fast tumbling radicals or metal complexes in liquid solution and neither does spin density in orbitals of other nuclei. The isotropic couplings detected in solution result only from the Fermi contact interaction.
Since the isotropic and purely anisotropic contributions to the hyperfine coupling have different physical origin, we separate these contributions in the hyperfine tensor that describes the interaction between electron spin and nuclear spin :
where is the isotropic hyperfine coupling and the purely anisotropic coupling. In the following, we drop the electron and nuclear spin indices and .
14.2.1. Dipole-dipole hyperfine interaction
The anisotropic hyperfine coupling tensor of a given nucleus can be computed from the ground state wavefunction by applying the correspondence principle to the classical interaction between two point dipoles
Such computations are implemented in quantum chemistry programs such as ORCA, ADF, or Gaussian. If the SOMO is considered as a linear combination of atomic orbitals, the contributions from an individual orbital can be expressed as the product of spin density in this orbital with a spatial factor that can be computed once for all. The spatial factors have been tabulated [KM85]. In general, nuclei of elements with larger electronegativity have larger spatial factors. At the same spatial factor, such as for isotopes of the same element, the hyperfine coupling is proportional to the nuclear value and thus proportional to the gyromagnetic ratio of the nucleus. Hence, a deuterium coupling can be computed from a known proton coupling or vice versa.
A special situation applies to protons, alkali metals and earth alkaline metals, which have no significant spin densities in , or -orbitals. In this case, the anisotropic contribution can only arise from through-space dipole-dipole coupling to centers of spin density at other nuclei. In a point-dipole approximation the hyperfine tensor is then given by
where the sum runs over all nuclei with significant spin density (summed over all orbitals at this nucleus) other than nucleus under consideration. The are distances between the nucleus under consideration and the centers of spin density, and the are unit vectors along the direction from the considered nucleus to the center of spin density. For protons in transition metal complexes it is often a good approximation to consider spin density only at the central metal ion. The distance from the proton to the central ion can then be directly inferred from the anisotropic part of the hyperfine coupling.
Hyperfine tensor contributions computed by any of these ways must be corrected for the influence of if the tensor is strongly anisotropic. If the dominant contribution to arises at a single nucleus, the hyperfine tensor at this nucleus can be corrected by
The product g may have an isotropic part, although is purely anisotropic. This isotropic pseudocontact contribution depends on the relative orientation of the tensor and the spin-only dipole-dipole hyperfine tensor . The correction is negligible for most organic radicals, but not for paramagnetic metal ions. If contributions to arise from several centers, the necessary correction cannot be written as a function of the tensor.
14.2.2. Fermi contact interaction
The Fermi contact contribution takes the form
Most literature holds that the correction should be done for all nuclei. As pointed out by Frank Neese, this is not true. An earlier discussion of this point is found in [Lef67] where is the spin density in the orbital under consideration, the nuclear value and the nuclear magneton . The factor denotes the probability to find the electron at this nucleus in the ground state with wave function and has been tabulated [KM85].
Figure 4.1: Transfer of spin density by the spin polarization mechanism. According to the Pauli principle, the two electrons in the C-H bond orbital must have opposite spin state. If the unpaired electron resides in a orbital on the atom, for other electrons on the same atom the same spin state is slightly favored, as this minimizes electrostatic repulsion. Hence, for the electron at the atom, the opposite spin state (left panel) is slightly favored over the same spin state (right panel). Positive spin density in the orbital on the atom induces some negative spin density in the orbital on the atom.
14.2.3. Spin polarization
The contributions to the hyperfine coupling discussed up to this point can be understood and computed in a single-electron picture. Further contributions arise from correlation of electrons in a molecule. Assume that the orbital on a carbon atom contributes to the SOMO, so that the spin state of the electron is preferred in that orbital (Fig. 4.1). Electrons in other orbitals on the same atom will then also have a slight preference for the state (left panel), as electrons with the same spin tend to avoid each other and thus have less electrostatic repulsion. In particular, this means that the spin configuration in the left panel of Fig. is slightly more preferable than the one in the right panel. According to the Pauli principle, the two electrons that share the bond orbital of the bond must have antiparallel spin. Thus, the electron in the orbital of the hydrogen atom that is bound to the spin-carrying carbon atom has a slight preference for the state. This corresponds to a negative isotropic hyperfine coupling of the directly bound proton, which is induced by the positive hyperfine coupling of the adjacent carbon atom. The effect is termed "spin polarization", although it has no physical relation to the polarization of electron spin transitions in an external magnetic field.
Spin polarization is important, as it transfers spin density from orbitals, where it is invisible in liquid solution and from carbon atoms with low natural abundance of the magnetic isotope to orbitals on protons, where it can be easily observed in liquid solution. This transfer occurs, both, in radicals, where the unpaired electron is localized on a single atom, and in radicals, where it is distributed over the system. The latter case is of larger interest, as the distribution of the orbital over the nuclei can be mapped by measuring and assigning the isotropic proton hyperfine couplings. This coupling can be predicted by the McConnell equation
where is the spin density at the adjacent carbon atom and is a parameter of the order of , which slightly depends on structure of the system.
This preference for electrons on the same atom to have parallel spin is also the basis of Hund's rule.
Figure 4.2: Mapping of the LUMO and HOMO of an aromatic molecule via measurements of hyperfine couplings after one-electron reduction or oxidation. Reduction leads to an anion radical, whose SOMO is a good approximation to the lowest unoccupied molecular orbital (LUMO) of the neutral parent molecule. Oxidation leads to an cation radical, whose SOMO is a good approximation to the highest occupied molecular orbital (HOMO) of the neutral parent molecule.
The McConnell equation is mainly applied for mapping the LUMO and HOMO of aromatic molecules (Figure 4.2). An unpaired electron can be put into these orbitals by one-electron reduction or oxidation, respectively, without perturbing the orbitals too strongly. The isotropic hyperfine couplings of the hydrogen atom directly bound to a carbon atom report on the contribution of the orbital of this carbon atom to the orbital. The challenges in this mapping are twofold. First, it is hard to assign the observed couplings to the hydrogen atoms unless a model for the distribution of the orbital is already available. Second, the method is blind to carbon atoms without a directly bonded hydrogen atom.
14.3. Hyperfine Hamiltonian
We consider the interaction of a single electron spin with a single nuclear spin and thus drop the sums and indices and in in Eq. (2.4). In general, all matrix elements of the hyperfine tensor will be non-zero after the Bleaney transformation to the frame where the electron Zeeman interaction is along the axis (see Eq. 3.5). The hyperfine Hamiltonian is then given by
Note that the axis of the nuclear spin coordinate system is parallel to the magnetic field vector whereas the one of the electron spin system is tilted, if anisotropy is significant. Hence, the hyperfine tensor is not a tensor in the strict mathematical sense, but rather an interaction matrix.
In Eq. (4.7), the term is secular and must always be kept. Usually, the high-field approximation does hold for the electron spin, so that all terms containing or operators are non-secular and can be dropped. The truncated hyperfine Hamiltonian thus reads
The first two terms on the right-hand side can be considered as defining an effective transverse coupling that is the sum of a vector with length along and a vector of length along . The length of the sum vector is . The truncated hyperfine Hamiltonian simplifies if we take the laboratory frame axis for the nuclear spin along the direction of this effective transverse hyperfine coupling. In this frame we have
where quantifies the secular hyperfine coupling and the pseudo-secular hyperfine coupling. The latter coupling must be considered if and only if the hyperfine coupling violates the high-field approximation for the nuclear spin (see Chapter 6).
If anisotropy is very small, as is the case for organic radicals, the axes of the two spin coordinate systems are parallel. In this situation and for a hyperfine tensor with axial symmetry, and can be expressed as
where is the angle between the static magnetic field and the symmetry axis of the hyperfine tensor and is the anisotropy of the hyperfine coupling. The principal values of the hyperfine tensor are and . The pseudo-secular contribution vanishes along the principal axes of the hyperfine tensor, where is either or or for a purely isotropic hyperfine coupling. Hence, the pseudo-secular contribution can also be dropped when considering fast tumbling radicals in the liquid state. We now consider the point-dipole approximation, where the electron spin is well localized on the length scale of the electron-nuclear distance and assume that arises solely from through-space interactions. This applies to hydrogen, alkali and earth alkali ions. We then find
For the moment we assume that the pseudo-secular contribution is either negligible or can be considered as a small perturbation. The other case is treated in Chapter 6 . To first order, the contribution of the hyperfine interaction to the energy levels is then given by . In the EPR spectrum, each nucleus with spin generates electron spin transitions with that can be labeled by the values of . In the nuclear frequency spectrum, each nucleus exhibits transitions with . For nuclear spins in the solid state, each transition is further split into transitions by the nuclear quadrupole interaction. The contribution of the secular hyperfine coupling to the electron transition frequencies is , whereas it is for nuclear transition frequencies. In both cases, the splitting between adjacent lines of a hyperfine multiplet is given by .
14.4. Spectral manifestation of the hyperfine interaction
14.4.1. Liquid-solution EPR spectra
Since each nucleus splits each electron spin transition into transitions with different frequencies, the number of EPR transitions is . Some of these transitions may coincide if hyperfine couplings are the same or integer multiples of each other. An important case, where hyperfine couplings are exactly the same are chemically equivalent nuclei. For instance, two nuclei can have spin state combinations , and . The contributions to the transition frequencies are , and
Figure 4.3: Hyperfine splitting in the EPR spectrum of the phenyl radical. The largest hyperfine coupling for the two equivalent ortho protons generates a triplet of lines with relative intensities . The medium coupling to the two equivalent meta proton splits each line again into a pattern, leading to 9 lines with an intensity ratio of . Finally, each line is split into a doublet by the small hyperfine coupling of the para proton, leading to 18 lines with intensity ratio .
. For equivalent nuclei with only three lines are observed with hyperfine shifts of , and with respect to the electron Zeeman frequency. The unshifted center line has twice the amplitude than the shifted lines, leading to a pattern with splitting . For equivalent nuclei with the number of lines is and the relative intensities can be inferred from Pascal's triangle. For a group of equivalent nuclei with arbitrary spin quantum number the number of lines is . The multiplicities of groups of equivalent nuclei multiply. Hence, the total number of EPR lines is
where index runs over the groups of equivalent nuclei.
Figure illustrates on the example of the phenyl radical how the multiplet pattern arises. For radicals with more extended systems, the number of lines can be very large and it may become impossible to fully resolve the spectrum. Even if the spectrum is fully resolved, analysis of the multiplet pattern may be a formidable task. An algorithm that works well for analysis of patterns with a moderate number of lines is given in [CCM16].
14.4.2. Liquid-solution nuclear frequency spectra
As mentioned in Section the secular hyperfine coupling can be inferred from nuclear frequency spectra as well as from EPR spectra. Line widths are smaller in the nuclear frequency spectra, since nuclear spins have longer transverse relaxation times . Another advantage of nuclear frequency spectra arises from the fact that the electron spin interacts with all nuclear spins whereas each nuclear spin interacts with only one electron spin (Figure 4.4). The number of lines in nuclear frequency spectra thus grows only linearly with the number of nuclei, whereas
Figure 4.4: Topologies of an electron-nuclear spin system for EPR spectroscopy (a) and of a nuclear spin system typical for NMR spectroscopy (b). Because of the much larger magnetic moment of the electron spin, the electron spin "sees" all nuclei, while each nuclear spin in the EPR case sees only the electron spin. In the NMR case, each nuclear spin sees each other nuclear spin, giving rise to very rich, but harder to analyze information.
it grows exponentially in EPR spectra. In liquid solution, each group of equivalent nuclear spins adds lines, so that the number of lines for such groups is
The nuclear frequency spectra in liquid solution can be measured by CW ENDOR, a technique that is briefly discussed in Section 8.1.2.
Figure 4.5: Energy level schemes (a,c) and nuclear frequency spectra (b,d) in the weak hyperfine coupling and strong hyperfine coupling (c,d) cases for an electron-nuclear spin system , . Here, is assumed to be negative and is assumed to be positive. (a) In the weak-coupling case, , the two nuclear spin transitions (green) have frequencies . (b) In the weak-coupling case, the doublet is centered at frequency and split by . (c) In the strong-coupling case, , levels cross for one of the electron spin states. The two nuclear spin transitions (green) have frequencies . (d) In the strong-coupling case, the doublet is centered at frequency and split by .
A complication in interpretation of nuclear frequency spectra can arise from the fact that the hyperfine interaction may be larger than the nuclear Zeeman interaction. This is illustrated in Figure 4.5. Only in the weak-coupling case with the hyperfine doublet in nuclear frequency spectra is centered at and split by . In the strong-coupling case, hyperfine sublevels cross for one of the electron spin states and the nuclear frequency becomes negative. As the sign of the frequency is not detected, the line is found at frequency instead, i.e., it is "mirrored" at the zero frequency. This results in a doublet centered at frequency and split by . Recognition of such cases in well resolved liquid-state spectra is simplified by the fact that the nuclear Zeeman frequency can only assume a few values that are known if the nuclear isotopes in the molecule and the magnetic field are known. Figure illustrates how the nuclear frequency spectrum of the phenyl radical is constructed based on such considerations. The spectrum has only 6 lines, compared to the 18 lines that arise in the EPR spectrum in Figure 4.3.
Figure 4.6: Schematic ENDOR (nuclear frequency) spectrum of the phenyl radical at an X-band frequency where . (a) Subspectrum of the two equivalent ortho protons. The strong-coupling case applies. (b) Subspectrum of the two equivalent meta protons. The weak-coupling case applies. (c) Subspectrum of the para proton. The weak-coupling case applies. (d) Complete spectrum.
14.4.3. Solid-state EPR spectra
In the solid state, construction of the EPR spectra is complicated by the fact that the electron Zeeman interaction is anisotropic. At each individual orientation of the molecule, the spectrum looks like the pattern in liquid state, but both the central frequency of the multiplet and the hyperfine splittings depend on orientation. As these frequency distributions are continuous, resolved splittings are usually observed only at the singularities of the line shape pattern of the interaction with the largest anisotropy. For organic radicals at X-band frequencies, often hyperfine anisotropy dominates. At high frequencies or for transition metal ions, often electron Zeeman anisotropy dominates. The exact line shape depends not only on the principal values of the tensor and the hyperfine tensors, but also on relative orientation of their PASs. The general case is complicated and requires numerical simulations, for instance, by EasySpin.
However, simple cases, where the hyperfine interaction of only one nucleus dominates and the PASs of the and hyperfine tensor coincide, are quite often encountered. For instance, Cu(II) complexes are often square planar and, if all four ligands are the same, have a symmetry axis. The tensor than has axial symmetry with the axis being the unique axis. The hyperfine tensors of and have the same symmetry and the same unique axis. The two isotopes both have spin and very similar gyromagnetic ratios. The spectra can thus be understood by considering one electron spin and one nuclear spin with axial and hyperfine tensors with a coinciding unique axis.
In this situation, the subspectra for each of the nuclear spin states , and take on a similar form as shown in Figure . The resonance field can be computed by solving
where is the angle between the symmetry axis and the magnetic field vector . The singularities are encountered at and and correspond to angular frequencies and .
Figure 4.7: Construction of a solid-state EPR spectrum for a copper(II) complex with four equivalent ligands and square planar coordination. The and principal axes directions coincide with the symmetry axis of the complex (inset). (a) Subspectra for the four nuclear spin states with different magnetic spin quantum number . (b) Absorption spectrum. (c) Derivative of the absorption spectrum.
The construction of a Cu(II) EPR spectrum according to these considerations is shown in Figure 4.7. The values of and can be inferred by analyzing the singularities near the low-field edge of the spectrum. Near the high-field edge, the hyperfine splitting is usually not resolved. Here, corresponds to the maximum of the absorption spectrum and to the zero crossing of its derivative.
14.4.4. Solid-state nuclear frequency spectra
Again, a simpler situation is encountered in nuclear frequency spectra, as the nuclear Zeeman frequency is isotropic and chemical shift anisotropy is negligibly small compared to hyperfine anisotropy. Furthermore, resolution is much better for the reasons discussed above, so that smaller hyperfine couplings and anisotropies can be detected. If anisotropy of the hyperfine coupling is dominated by through-space dipole-dipole coupling to a single center of spin density, as is often the case for protons, or by contribution from spin density in a single or orbital, as is often the case for other nuclei, the hyperfine tensor has nearly axial symmetry. In this case, one can infer from the line shapes whether the weak-or strong-coupling case applies and whether the isotropic hyperfine coupling is positive or negative (Figure 4.8). The case with corresponds to the Pake pattern discussed in the NMR part of the lecture course.
Figure 4.8: Solid-state nuclear frequency spectra for cases with negative nuclear Zeeman frequency . (a) Weak-coupling case with and . (b) Weak-coupling case with and . (a) Strong-coupling case with and . (b) Strong-coupling case with and .
14.5. Exchange interaction
14.5.1. Physical origin and consequences of the exchange interaction
If two unpaired electrons occupy SOMOs in the same molecule or in spatially close molecules, the wave functions and of the two SOMOs may overlap. The two unpaired electrons can couple either to a singlet state or to a triplet state. The energy difference between the singlet and triplet state is the exchange integral
There exist different conventions for the sign of and the factor 2 may be missing in parts of the literature. With the sign convention used here, the singlet state is lower in energy for positive . Since the singlet state with spin wave function is antisymmetric with respect to exchange of the two electrons and electrons are Fermions, it corresponds to the situation where the two electrons could also occupy the same orbital. This is a bonding orbital overlap, corresponding to an antiferromagnetic spin ordering. Negative correspond to a lower-lying triplet state, i.e., antibonding orbital overlap and ferromagnetic spin ordering. The triplet state has three substates with wave functions for the state, for the state, and for the state. The and state are eigenstates both in the absence and presence of the coupling. The states and are eigenstates for , where is the difference between the electron Zeeman frequencies of the two spins. For the opposite case of , the eigenstates are and . The latter case corresponds to the high-field approximation with respect to the exchange interaction.
For strong exchange, , the energies are approximately for the singlet state and and for the triplet substates , and , respectively, where is the electron Zeeman interaction, which is the same for both spins within this approximation. If , microwave photons with energy cannot excite transitions between the singlet and triplet subspace of spin Hilbert space. It is then convenient to use a coupled representation and consider the two subspaces separately from each other. The singlet subspace corresponds to a diamagnetic molecule and does not contribute to EPR spectra. The triplet subspace can be described by a group spin of the two unpaired electrons. In the coupled representation, does not enter the spin Hamiltonian, as it shifts all subspace levels by the same energy. For , the triplet state is the ground state and is always observable by EPR spectroscopy. However, usually one has and the singlet state is the ground state. As long as does not exceed thermal energy by a large factor, the triplet state is thermally excited and observable. In this case, EPR signal amplitude may increase rather than decrease with increasing temperature. For organic molecules, this case is also rare. If , the compound does not give an EPR signal. It may still be possible to observe the triplet state transiently after photoexcitation to an excited singlet state and intersystem crossing to the triplet state.
Weak exchange coupling is observed in biradicals with well localized SOMOs that are separated on length scales between and . In such cases, exchange coupling decreases exponentially with the distance between the two electrons or with the number of conjugated bonds that separate the two centers of spin density. If the two centers are not linked by a continuous chain of conjugated bonds, exchange coupling is rarely resolved at distances larger than . In any case, at such long distances exchange coupling is much smaller than the dipole-dipole coupling between the two unpaired electrons if the system is not conjugated. For weak exchange coupling, the system is more conveniently described in an uncoupled representation with two spins and .
Exchange coupling is also significant during diffusional encounters of two paramagnetic molecules in liquid solution. Such dynamic Heisenberg spin exchange can be pictured as physical exchange of unpaired electrons between the colliding molecules. This causes a sudden change of the spin Hamiltonian, which leads to spin relaxation. A typical example is line broadening in EPR spectra of radicals by oxygen, which has a paramagnetic triplet ground state. If radicals of the same type collide, line broadening is also observed, but the effects on the spectra can be more subtle, since the spin Hamiltonians of the colliding radicals are the same. In this case, exchange of unpaired electrons between the radicals changes only spin state, but not the spin Hamiltonian.
14.5.2. Exchange Hamiltonian
The spin Hamiltonian contribution by weak exchange coupling is
This Hamiltonian is analogous to the coupling Hamiltonian in NMR spectroscopy. If the two spins have different values and the field is sufficiently high , the exchange Hamiltonian can be truncated in the same way as the coupling Hamiltonian in heteronuclear NMR:
14.5.3. Spectral manifestation of the exchange interaction
In the absence of hyperfine coupling, the situation is the same as for coupling in NMR spectroscopy. Exchange coupling between like spins (same electron Zeeman frequency) does not influence the spectra. For radicals in liquid solution, hyperfine coupling is usually observable. In this case, exchange coupling does influence the spectra even for like spins, as illustrated in Figure for two exchange-coupled electron spins and with each of them coupled exclusively to only one nuclear spin and , respectively) with the same hyperfine coupling . If the exchange coupling is much smaller than the isotropic hyperfine coupling, each of the individual lines of the hyperfine triplet further splits into three lines. If the splitting is very small, it may be noticeable only as a line broadening. At very large exchange coupling, the electron spins are uniformly distributed over the two exchange-coupled moieties. Hence, each of them has the same hyperfine coupling to both nuclei. This coupling is half the original hyperfine coupling, since, on average, the electron spin has only half the spin density in the orbitals of a given nucleus as compared to the case without exchange coupling. For intermediate exchange couplings, complex splitting patterns arise that are characteristic for the ratio between the exchange and hyperfine coupling.
Figure 5.1: Influence of the exchange coupling on EPR spectra with hyperfine coupling in liquid solution (simulation). Spectra are shown for two electron spins and with the same isotropic value and the same isotropic hyperfine coupling to a nuclear spin or , respectively. In the absence of exchange coupling, a triplet with amplitude ratio is observed. For small exchange couplings, each line splits into a triplet. At intermediate exchange couplings, complicated patterns with many lines result. For very strong exchange coupling, each electron spin couples to both nitrogen nuclei with half the isotropic exchange coupling. A quintuplet with amplitude ratio is observed.
15. Dipole-dipole interaction
15.5.4. Physical picture
The magnetic dipole-dipole interaction between two localized electron spins with magnetic moments and takes the same form as the classical interaction between two magnetic point dipoles. The interaction energy
generally depends on the two angles and that the point dipoles include with the vector between them and on the dihedral angle (Figure 5.2). The dipole-dipole interaction scales with the inverse cube of the distance between the two point dipoles.
In general, the two electron spins are spatially distributed in their respective SOMOs. The point-dipole approximation is still a good approximation if the distance is much larger than the spatial distribution of each electron spin. Further simplification is possible if anisotropy is much smaller than the isotropic value. In that case, the two spins are aligned parallel to the magnetic field and thus also parallel to each other, so that and . Eq. (5.4) then simplifies to
which is the form known from NMR spectroscopy.
Figure 5.2: Geometry of two magnetic point dipoles in general orientation. Angles and are included between the respective magnetic moment vectors or and the distance vector between the point dipoles. Angle is the dihedral angle.
15.5.5. Dipole-dipole Hamiltonian
For two electron spins that are not necessarily aligned parallel to the external magnetic field, the dipole-dipole coupling term of the spin Hamiltonian assumes the form
If the electrons are distributed in space, the Hamiltonian has to be averaged (integrated) over the two spatial distributions, since electron motion proceeds on a much faster time scale than an EPR experiment.
If the two unpaired electrons are well localized on the length scale of their distances and their spins are aligned parallel to the external magnetic field, the dipole-dipole Hamiltonian takes the form
with the terms of the dipolar alphabet
Usually, EPR spectroscopy is performed at fields where the electron Zeeman interaction is much larger than the dipole-dipole coupling, which has a magnitude of about at a distance of and of at a distance of . In this situation, the terms , and are non-secular and can be dropped. The term is pseudo-secular and can be dropped only if
Figure 5.3: Explanation of dipole-dipole coupling between two spins in a local field picture. At the observer spin (blue) a local magnetic field is induced by the magnetic moment of the coupling partner spin (red). In the secular approximation only the component of this field is relevant, which is parallel or antiparallel to the external magnetic field . The magnitude of this component depends on angle between the external magnetic field and the spin-spin vector . For the (left) and (right) states of the partner spin, the local field at the observer spin has the same magnitude, but opposite direction. In the high-temperature approximation, both these states are equally populated. The shift of the resonance frequency of the observer spin thus leads to a splitting of the observer spin transition, which is twice the product of the local field with the gyromagnetic ratio of the observer spin.
the difference between the electron Zeeman frequencies is much larger than the dipole-dipole coupling 1 . In electron electron double resonance (ELDOR) experiments, the difference of the Larmor frequencies of the two coupled spins can be selected via the difference of the two microwave frequencies. It is thus possible to excite spin pairs for which only the secular part of the spin Hamiltonian needs to be considered,
with
The dipole-dipole coupling then has a simple dependence on the angle between the external magnetic field and the spin-spin vector and the coupling can be interpreted as the interaction of the spin with the component of the local magnetic field that is induced by the magnetic dipole moment of the coupling partner (Figure 5.3). Since the average of the second Legendre polynomial over all angles vanishes, the dipole-dipole interaction vanishes under fast isotropic motion. Measurements of this interaction are therefore performed in the solid state.
The dipole-dipole tensor in the secular approximation has the eigenvalues . The dipole-dipole coupling at any orientation is given by
15.5.6. Spectral manifestation of the dipole-dipole interaction
The energy level scheme and a schematic spectrum for a spin pair with fixed angle are shown in Figure and b, respectively. The dipole-dipole couplings splits the transition of either coupled spin by . If the sample is macroscopically isotropic, for instance a microcrystalline powder or a glassy frozen solution, all angles occur with probability . Each line of the dipolar doublet
Hyperfine coupling of the electron spins can modify this condition.
Figure 5.4: Energy level scheme (a) and schematic spectrum (b) for a dipole-dipole coupled spin pair at fixed orientation with respect to the magnetic field. The electron Zeeman frequencies of the two spins are and , respectively. Weak coupling is assumed. The dipolar splitting is the same for both spins. Depending on homogeneous linewidth , the splitting may or may not be resolved. If and are distributed, for instance by anisotropy, resolution is lost even for .
is then broadened to a powder pattern as illustrated in Figure 3.3. The powder pattern for the state of the partner spin is a mirror image of the one for the state, since the frequency shifts by the local magnetic field have opposite sign for the two states. The superposition of the two axial powder patterns is called Pake pattern (Figure 5.5). The center of the Pake pattern corresponds to the magic angle . The dipole-dipole coupling vanishes at this angle.
Figure 5.5: Pake pattern observed for a dipole-dipole coupled spin pair. (a) The splitting of the dipolar doublet varies with angle between the spin-spin vector and the static magnetic field. Orientations have a probability . (b) The sum of all doublets for a uniform distribution of directions of the spin-spin vector is the Pake pattern. The "horns" are split by and the "shoulders" are split by . The center of the pattern corresponds to the magic angle.
The Pake pattern is very rarely observed in an EPR spectrum, since usually other anisotropic interactions are larger than the dipole-dipole interaction between electron spins. If the weakcoupling condition is fulfilled for the vast majority of all orientations, the EPR lineshape is well approximated by a convolution of the Pake pattern with the lineshape in the absence of dipole-dipole interaction. If the latter lineshape is known, for instance from measuring analogous samples that carry only one of the two electron spins, the Pake pattern can be extracted by deconvolution and the distance between the two electron spins can be inferred from the splitting by inverting Eq. (5.15).
15.1. Zero-field interaction
15.1.1. Physical picture
If several unpaired spins are very strongly exchange coupled, then they are best described by a group spin . The concept is most easily grasped for the case of two electron spins that we have already discussed in Section 5.1.1. In this case, the singlet state with group spin is diamagnetic and thus not observable by EPR. The three sublevels of the observable triplet state with group spin correspond to magnetic quantum numbers , and at high field. These levels are split by the electron Zeeman interaction. The transitions and are allowed electron spin transitions, whereas the transition is a forbidden double-quantum transition.
At zero magnetic field, the electron Zeeman interaction vanishes, yet the three triplet sublevels are not degenerate, they exhibit zero-field splitting. This is because the unpaired electrons are also dipole-dipole coupled. Integration of Eq. (5.6) over the spatial distribution of the two electron spins in their respective SOMOs provides a zero-field interaction tensor that can be cast in a form where it describes coupling of the group spin with itself [Rie07]. At zero field, the triplet sublevels are not described by the magnetic quantum number , which is a good quantum number only if the electron Zeeman interaction is much larger than the zero-field interaction. Rather, the triplet sublevels at zero field are related to the principal axes directions of the zero-field interaction tensor and are therefore labeled , and , whereas the sublevels in the high-field approximation are labeled , and .
This concept can be extended to an arbitrary number of strongly coupled electron spins. Cases with up to 5 strongly coupled unpaired electrons occur for transition metal ions (d shell) and cases with up to 7 strongly coupled unpaired electrons occur for rare earth ions (f shell). According to Hund's rule, in the absence of a ligand field the state with largest group spin is the ground state. Kramers ions with an odd number of unpaired electrons have a half-integer group spin . They behave differently from non-Kramers ions with an even number of electrons and integer group spin . This classification relates to Kramers' theorem, which states that for a time-reversal symmetric system with half-integer total spin, all eigenstates occur as pairs (Kramers pairs) that are degenerate at zero magnetic field. As a consequence, for Kramers ions the ground state at zero field will split when a magnetic field is applied. For any microwave frequency there exists a magnetic field where the transition within the ground Kramers doublet is observable in an EPR spectrum. The same does not apply for integer group spin, where the ground state may not be degenerate at zero field. If the zero-field interaction is larger than the maximum available microwave frequency, non-Kramers ions may be unobservable by EPR spectroscopy although they exist in a paramagnetic high-spin state. Typical examples of such EPR silent non-Kramers ions are high-spin and high-spin . In rare cases, non-Kramers ions are EPR observable, since the ground state can be degenerate at zero magnetic field if the ligand field features axial symmetry. Note also that "EPR silent" non-Kramers ions can become observable at sufficiently high microwave frequency and magnetic field.
For transition metal and rare earth ions, zero-field interaction is not solely due to the dipole-dipole interaction between the electron spins. Spin-orbit coupling also contributes, in many cases even stronger than the dipole-dipole interaction. Quantum-chemical prediction of the zero-field interaction is an active field of research. Quite reasonable predictions can be obtained for transition metal ions, whereas only order-of-magnitude estimates are usually possible for rare earth ions.
15.1.2. Zero-field interaction Hamiltonian
The zero-field interaction Hamiltonian is often given as
where denotes the transpose of the spin vector operator. In the principal axes system of the zero-field splitting (ZFS) tensor, the Hamiltonian simplifies to
where and . The reduction to two parameters is possible, since is a traceless tensor. In other words, the zero-field interaction is purely anisotropic. The notation presumes that is the principal value with the largest absolute value can be negative). Together with the absence of an isotropic component, this means that , which is always the intermediate value, is either closer to than to or exactly in the middle between these two values. Accordingly, . At axial symmetry . Axial symmetry applies if the system has a symmetry axis with . At cubic symmetry, both and are zero. For group spin , the leading term of the is then a hexadecapolar contribution that scales with the fourth power of the spin operators .
In the high-field approximation the ZFS contribution to the Hamiltonian is a term. In other words, to first order in perturbation theory the contribution of the ZFS to the energy of a spin level with magnetic quantum number scales with . For an allowed transition , this contribution is . This contribution vanishes for the central transition of Kramers ions. More generally, because of the scaling of the level energies with to first-order, the contribution of ZFS to transition frequencies vanishes for all transitions.
Figure 5.6: Schematic CW EPR spectra for triplet states at high field. Simulations were performed at an X-band frequency of . (a) Axial symmetry . The spectrum is the derivative of a Pake pattern. (b) Orthorhombic symmetry .
15.1.3. Spectral manifestation of zero-field splitting
Spectra are most easily understood in the high-field approximation. Quite often, deviations from this approximation are significant for the ZFS (see Fig. 2.2), and such deviations are discussed later. The other limiting case, where the ZFS is much larger than the electron Zeeman interaction (Fe(III) and most rare earth ions), is discussed in Section 5.3.4.
For triplet states with axial symmetry of the ZFS tensor, the absorption spectrum is a Pake pattern (see Section 5.2.3). With continuous-wave EPR, the derivative of the absorption spectrum is detected, which has the appearance shown in Fig. 5.6(a). A deviation from axial symmetry leads to a splitting of the "horns" of the Pake pattern by , whereas the "shoulders" of the pattern are not affected (Fig. 5.6(b)). Triplet states of organic molecules are often observed after optical excitation of a singlet state and intersystem crossing. Such intersystem crossing generally leads to different population of the zero-field triplet sublevels , and . In this situation the spin system is not at thermal equilibrium, but spin polarized. Such spin polarization affects relative intensity of the lineshape singularities in the spectra and even the sign of the signal may change. However, the singularities are still observed at the same resonance fields, i.e., the parameters and can still be read off the spectra as indicated in Fig. .
Even if the populations of the triple sublevels have relaxed to thermal equilibrium, the spectrum may still differ from the high-field approximation spectrum, as is illustrated in Fig. for the excited naphtalene triplet (simulation performed with an example script of the software package EasySpin http://WWW . easyspin.org/). For at a field of about (electron Zeeman frequency of about ) the high-field approximation is violated and is no longer a good quantum number. Hence, the formally forbidden double-quantum transition becomes partially allowed. To first order in perturbation theory, this transition is not broadened by the ZFS. Therefore it is very narrow compared to the allowed transitions and appears with higher amplitude.
Figure 5.7: CW EPR spectrum of the excited naphtalene triplet at thermal equilibrium (simulation at an -band frequency of ). The red arrow marks the half-field signal, which corresponds to the formally forbidden double-quantum transition .
For Kramers ions, the spectra are usually dominated by the central transition, which is not ZFS-broadened to first order. To second order in perturbation theory, the ZFS-broadening of this line scales inversely with magnetic field. Hence, whereas systems with anisotropy exhibit broadening proportional to the magnetic field , central transitions of Kramers ions exhibit narrowing with . The latter systems can be detected with exceedingly high sensitivity at high fields if they do not feature significant anisotropy. This applies to systems with half-filled shells (e.g. ). In the case of Mn(II) (Figure 5.8) the narrow central transition is split into six lines by hyperfine coupling to the nuclear spin of (nuclear spin natural abundance). Because of the scaling of anisotropic ZFS broadening of transitions, satellite transitions become the broader the larger is for the involved levels. In the high-temperature approximation, the integral intensity in the absorption spectrum is the same for all transitions. Hence, broader transitions make a smaller contribution to the amplitude in the absorption spectrum and in its first derivative that is acquired by CW EPR.
Figure 5.8: CW EPR spectrum of a Mn(II) complex (simulation at a W-band frequency of ). . The six intense narrow lines are the hyperfine multiplet of the central transition .
The situation can be further complicated by and strain, which is a distribution of the and parameters due to a distribution in the ligand field. Such a case is demonstrated in Fig. for Gd(III) at a microwave frequency of where second-order broadening of the central transition is still rather strong. In such a case, lineshape singularities are washed out and ZFS parameters cannot be directly read off the spectra. In CW EPR, the satellite transitions may remain unobserved as the derivative of the absorption lineshape is very small except for the central transition.
15.1.4. Effective spin in Kramers doublets
For some systems, such as Fe(III), ZFS is much larger than the electron Zeeman interaction at any experimentally attainable magnetic field. In this case, the zero-field interaction determines the quantization direction and the electron Zeeman interaction can be treated as a perturbation [Cas+60]. The treatment is simplest for axial symmetry , where the quantization axis is the axis of the ZFS tensor. The energies in the absence of the magnetic field are
which for high-spin Fe(III) with gives three degenerate Kramers doublets corresponding to , and . If the magnetic field is applied along the axis of the ZFS tensor, is a good quantum number and there is simply an additional energy term with being the value for the half-filled shell, which can be approximated as . Furthermore, in this situation only the transition is allowed. The Zeeman term leads to a splitting of the Kramers doublet that is proportional to and
Figure 5.9: Echo-detected EPR spectrum (absorption spectrum) of a Gd(III) complex with GHz, a Gaussian distribution of with standard deviation of and a correlated distribution of (simulation at a Q-band frequency of courtesy of Dr. Maxim Yulikov). (a) Total spectrum (black) and contributions of the individual transitions (see legend). The signal from the central transition (blue) dominates. (b) Contributions of the satellite transitions scaled vertically for clarity.
corresponds to . This Kramers doublet can thus be described as an effective spin with .
If the magnetic field is perpendicular to the tensor axis, the and Kramers doublets are not split, since the and operator does not connect these levels. The operator has an off-diagonal element connecting the levels that is . Since the levels are degenerate in the absence of the electron Zeeman interaction, they become quantized along the magnetic field and is again a good quantum number of this Kramers doublet. The energies are , so that the transition frequency is again proportional to , but now with an effective value . Intermediate orientations can be described by assuming an effective tensor with axial symmetry and . This situation is encountered to a good approximation for high-spin Fe(III) in hemoglobins .
For the non-axial case , the magnetic field will split all three Kramers doublets. To first order in perturbation theory the splitting is proportional to , meaning that each Kramers doublet can be described by an effective spin with an effective tensor. Another simple case is encountered for extreme rhombicity, . By reordering principal axes (exchanging with either or ) one can the get rid of the term in Eq. (5.18), so that the ZFS Hamiltonian reduces to with . The level pair corresponding to the new direction of the tensor has zero energy at zero magnetic field and it can be shown that it has an isotropic effective value . Indeed, signals near are very often observed for high-spin Fe(III).
15.2. Physical picture
15.2.1. The spin system
The basic phenomena can be well understood in the simplest possible electron-nuclear spin system consisting of a single electron spin with isotropic value that is hyperfine coupled to a nuclear spin with a magnitude of the hyperfine coupling that is much smaller than the electron Zeeman interaction. In this situation the high-field approximation is valid for the electron spin, so that the hyperfine Hamiltonian can be truncated to the form given by Eq. (4.9). Because of the occurrence of an operator in this Hamiltonian, we cannot simply transform the Hamiltonian to the rotating frame for the nuclear spin . However, we don't need to, as we shall consider only microwave irradiation. For the electron spin , we transform to the rotating frame where this spin has a resonance offset . Hence, the total Hamiltonian takes the form
in the rotating frame for the electron spin and the laboratory frame for the nuclear spin. Such a Hamiltonian is a good approximation, for instance, for protons in organic radicals.
The Hamiltonian deviates from the Hamiltonian that would apply if the high-field approximation were also fulfilled for the nuclear spin. The difference is the pseudo-secular hyperfine coupling term . As can be seen from Eq. (4.10), this term vanishes if the hyperfine interaction is purely isotropic, i.e. for sufficiently fast tumbling in liquid solution, and along the principal axes of the hyperfine tensor. Otherwise, the term can only be neglected if , corresponding to the high-field approximation of the nuclear spin. Within the approximate range the pseudo-secular interaction may affect transition frequencies and makes formally forbidden transitions with partially allowed, as is no longer a good quantum number.
15.2.2. Local fields at the nuclear spin
The occurrence of forbidden transitions can be understood in a semi-classical magnetization vector picture by considering local fields at the nuclear spin for the two possible states and
The product of rotational correlation time and hyperfine anisotropy must be much smaller than unity
Figure 6.1: Local fields (multiplied by the gyromagnetic ratio of the nuclear spin) at the nuclear spin in the two states and of an electron spin . The quantization axes are along the effective fields and and are, thus, not parallel.
of the electron spin. These local fields are obtained from the parameters , and of the Hamilton operator terms that act on the nuclear spin. When divided by the gyromagnetic ratio of the nuclear spin these terms have the dimension of a local magnetic field. The local field corresponding to the nuclear Zeeman interaction equals the static magnetic field and is the same for both electron spin states, since the expectation value of does not depend on the electron spin state. It is aligned with the direction of the laboratory frame (blue arrow in Figure 6.1). Both hyperfine fields arise from Hamiltonian terms that contain an factor, which has the expectation value for the state and for the state. The term is aligned with the axis and directed towards in the state and towards in the state, assuming (violet arrows). The term is aligned with the axis and directed towards in the state and towards in the state, assuming (green arrows).
The effective fields at the nuclear spin in the two electron spins states are vector sums of the three local fields. Because of the component along , they are tilted from the direction by angle in the state and by angle in the state. The length of the sum vectors are the nuclear transition frequencies in these two states and are given by
For , the hyperfine splitting is given by
and the sum frequency is given by
For , the nuclear frequency doublet is centered at Fig. ). The sum frequency is always larger than twice the nuclear Zeeman frequency. None of the nuclear frequencies can become zero, the minimum possible value is attained in one of the electron spin states for matching of the nuclear Zeeman and hyperfine interaction at . For the nuclear frequency doublet is split by and centered at Fig. . The tilt angles and (Figure 6.1) can be inferred from trigonometric relations and are given by
Consider now a situation where the electron spin is in its state. The nuclear magnetization from all radicals in this state at thermal equilibrium is aligned with . Microwave excitation causes transitions to the state. In this state, the local field at the nuclear spin is directed along . Hence, the nuclear magnetization vector from the radicals under consideration is tilted by angle (Figure 6.1) with respect to the current local field. It will start to precess around this local field vector. This corresponds to excitation of the nuclear spin by flipping the electron spin, which is a formally forbidden transition. Obviously, such excitation will occur only if angle differs from and from . The case of corresponds to the absence of pseudo-secular hyperfine coupling and is also attained in the limit . The situation is attained in the limit of very strong secular hyperfine coupling, . Forbidden transitions are observed for intermediate hyperfine coupling. Maximum excitation of nuclear spins is expected when the two quantization axes are orthogonal with respect to each other, .
Figure 6.2: Electron-nuclear spin system in the presence of pseudo-secular hyperfine coupling. (a) Level scheme. In EPR, transitions are allowed (red), in NMR transitions are allowed (blue), and the zero- and double-quantum transitions with are formally forbidden. (b) EPR stick spectrum. Allowed transitions have transition probability and forbidden transitions probability . The spectrum is shown for . For , the forbidden transitions lie inside the allowed transition doublet. (c) NMR spectrum for . (d) NMR spectrum for .
15.3. Product operator formalism with pseudo-secular interactions
15.3.1. Transformation of to the eigenbasis
Excitation and detection in EPR experiments are described by the and operators in the rotating frame. These operators act only on electron spin transitions and thus formalize the spectroscopic selection rules. If the spin Hamiltonian contains off-diagonal terms, such as the pseudo-secular term in Eq. (6.1), the eigenbasis deviates from the basis of the electron spin rotating frame/nuclear spin laboratory frame in which the Hamiltonian is written and in which the excitation and detection operators are linear combinations of and . In order to understand which transitions are driven and detected with what transition moment, we need to transform to the eigenbasis (the transformation of is analogous). This can be done by product operator formalism and can be understood in the local field picture.
The Hamiltonian in the eigenbasis has no off-diagonal elements, meaning that all quantization axes are along . Thus, we can directly infer from Fig. that, in the state, we need a counterclockwise (mathematically positive) rotation by tilt angle about the axis, which is pointing into the paper plane. In the state, we need a clockwise (mathematically negative) rotation by tilt angle about the axis. The electron spin states can be selected by the projection operators and , respectively. Hence, we have to apply rotations and . These two rotations commute, as the and subspaces are distinct when is a good quantum number. For the rotation into the eigenbasis, we can write a unitary matrix
where and . Note that the definition of angle corresponds to the one given graphically in Fig. 6.1.2 The two new rotations about and also commute. Furthermore, commutes with and ), so that the transformation of to the eigenbasis reduces to
The transition moment for the allowed transitions that are driven by is multiplied by a factor , i.e. it becomes smaller when . In order to interpret the second term, it is best rewritten in terms of ladder operators and . We find
In other words, this term drives the forbidden electron-nuclear zero- and double-quantum transitions (Fig. 6.2(a)) with a transition proportional to .
In a CW EPR experiment, each transition must be both excited and detected. In other words, the amplitude is proportional to the square of the transition moment, which is the transition probability. Allowed transitions thus have an intensity proportional to and forbidden transitions a transition probability proportional to (Fig. 6.2(b)).
15.3.2. General product operator computations for a non-diagonal Hamiltonian
In a product operator computation, terms of the Hamiltonian can be applied one after the other if and only if they pairwise commute. This is not the case for the Hamiltonian in Eq. (6.1). However, application of diagonalizes the Hamiltonian:
This provides a simple recipe for product operator computations in the presence of the pseudosecular hyperfine coupling. Free evolution and transition-selective pulses are computed in the
We have used and . eigenbasis, using the Hamiltonian on the right-hand side of relation (6.9). For application of non-selective pulses, the density operator needs to be transformed to the electron spin rotating frame/nuclear spin laboratory frame basis by applying . In product operator formalism this corresponds to a product operator transformation . After application of non-selective pulses, the density operator needs to be backtransformed to the eigenbasis. Detection also needs to be performed in the electron spin rotating frame/nuclear spin laboratory frame basis.
This concept can be extended to any non-diagonal Hamiltonian as long as one can find a unitary transformation that transforms the Hamiltonian to its eigenbasis and can be expressed by a single product operator term or a sum of pairwise commuting product operator terms.
15.4. Generation and detection of nuclear coherence by electron spin excitation
15.4.1. Nuclear coherence generator
We have seen that a single microwave pulse can excite coherence on forbidden electron-nuclear zero- and double-quantum transitions. In principle, this provides access to the nuclear frequencies and , which are differences of frequencies of allowed and forbidden electron spin transitions, as can be inferred from Fig. 6.2(a,b). Indeed, the decay of an electron spin Hahn echo echo as a function of is modulated with frequencies and as well as with and ). Such modulation arises from forbidden transitions during the refocusing pulse, which redistribute coherence among the four transitions. The coherence transfer echoes are modulated by the difference of the resonance frequencies before and after the transfer by the pulse, in which the resonance offset cancels, while the nuclear spin contributions do not cancel. This two-pulse ESEEM experiment is not usually applied for measuring hyperfine couplings, as the appearance of the combination frequencies and complicates the spectra and linewidth is determined by electron spin transverse relaxation, which is much faster the nuclear spin transverse relaxation.
Better resolution and simpler spectra can be obtained by indirect observation of the evolution of nuclear coherence. Such coherence can be generated by first applying a pulse to the electron spins, which will generate electron spin coherence on allowed transitions with amplitude proportional to and on forbidden transitions with amplitude proportional to sin . After a delay a second pulse is applied. Note that the block is part of the EXSY and NOESY experiments in NMR. The second pulse will generate an electron spin magnetization component along for half of the existing electron spin coherence, i.e., it will "switch off" half the electron spin coherence and convert it to polarization. However, for the coherence on forbidden transitions, there is a chance that the nuclear spin is not flipped, i.e. that the coherent superposition of the nuclear spin states survives. For electron spin coherence on allowed transitions, there is a chance that the "switching off" of the electron coherence will lead to a "switching on" of nuclear coherences. Hence, in both these pathways there is a probability proportional to that nuclear coherence is generated. The delay is required, since at the different nuclear coherence components have opposite phase and cancel.
The nuclear coherence generated by the block can be computed by product operator formalism as outlined in Section 6.2.2. We find
This expression can be interpreted in the following way. Nuclear coherence is created with a phase as if it had started to evolve as at time (last cosine factors on the right-hand side of each line). It is modulated as a function of the electron spin resonance offset and zero exactly on resonance (first factor on each line). The integral over an inhomogeneously broadened, symmetric EPR line is also zero, since . However, this can be compensated later by applying another pulse. The amplitude of the nuclear coherence generally scales with , since one allowed and one forbidden transfer are required to excite it and (second factor). The third factor on the right-hand side of lines 1 and 2 tells that the amplitude of the coherence with frequency is modulated as a function of with frequency . Likewise, the amplitude of the coherence with frequency is modulated as a function of with frequency (lines 3 and 4 ). At certain values of no coherence is created at the transition with frequency , at other times maximum coherence is generated. Such behavior is called blind-spot behavior. In order to detect all nuclear frequencies, an experiment based on the nuclear coherence generator has to be repeated for different values of . Why and how CW EPR spectroscopy is done Sensitivity advantages of CW EPR spectroscopy The CW EPR experiment
Considerations on sample preparation Theoretical description of CW EPR Spin packet lineshape Saturation
16. 7 - CW EPR Spectroscopy
16.1. Why and how CW EPR spectroscopy is done
16.1.1. Sensitivity advantages of CW EPR spectroscopy
In NMR spectroscopy, CW techniques have been almost completely displaced by Fourier transform (FT) techniques, except for a few niche applications. FT techniques have a sensitivity advantage if the spectrum contains large sections of baseline and the whole spectrum can be excited simultaneously by the pulses. Neither condition is usually fulfilled in EPR spectroscopy. For two reasons, FT techniques lose sensitivity in EPR spectroscopy compared to the CW experiment. First, while typical NMR spectra comfortably fit into the bandwidth of a welldesigned critically coupled radiofrequency resonance circuit, EPR spectra are much broader than the bandwidth of a microwave resonator with high quality factor. Broadening detection bandwidth and proportionally lowering the quality factor of the resonator reduces signal-to-noise ratio unless the absorption lineshape is infinitely broad. A quality factor of the order of 10 ' 000 , which can be achieved with cavity resonators, corresponds to a bandwidth of roughly at X-band frequencies around . The intrinsic high sensitivity of detection in such a narrow band can be used only in a CW experiment. Second, even if the resonator is overcoupled to a much lower quality factor or resonators with intrinsically lower are used (the sensitivity loss can partially be compensated by a higher filling factor of such resonators), residual power from a high-power microwave pulse requires about in order to decay below the level of an EPR signal. This dead time is often a significant fraction of the transverse relaxation time of electron spins, which entails signal loss by relaxation. In contrast, in NMR spectroscopy the dead time is usually negligibly short compared to relaxation times. In many cases, the dead time in pulsed EPR spectroscopy even strongly exceeds . In this situation FT EPR is impossible, even with echo refocusing, while CW EPR spectra can still be measured. This case usually applies to transition metal complexes at room temperature and to many rare earth metal complexes and high-spin Fe(III) complexes even down to the boiling point of liquid helium at normal pressure (4.2 K). For these reasons, any unknown potentially paramagnetic sample should first be characterized by CW EPR spectroscopy. Pulsed EPR techniques are required if the resolution of CW EPR spectroscopy provides insufficient information to assign a structure. This applies mainly to small hyperfine couplings in organic radicals and of ligand nuclei in transition metal complexes (see Chapter 8) and to the measurement of distances between electron spins in the nanometer range (see Chapter 9). At temperatures where pulse EPR signals can be obtained, measurement of relaxation times is also easier and more precise with pulsed EPR techniques.
Figure 7.1: Scheme of a CW EPR spectrometer. Microwave from a fixed-frequency source is passed through an attenuator for adjusting its power and then through a circulator to the sample. Microwave that comes back from the sample passes on a different way through the same circulator and is combined with reference microwave of adjustable power (bias) and phase before it is detected by a microwave diode. The output signal of this diode enters a phase-sensitive detector (PSD) where it is demodulated with respect to the field modulation frequency (typically ) and at the same time amplified. The output signal of the PSD is digitized and further processed in a computer. The spectrum is obtained by sweeping the static magnetic field at constant microwave frequency.
16.1.2. The CW EPR experiment
Since the bandwidth of an optimized microwave resonator is much smaller than the typical width of EPR spectra, it is impractical to sweep the frequency at constant magnetic field in order to obtain a spectrum. Instead, the microwave frequency is kept constant and coincides with the resonator frequency at all times. The resonance condition for the spins is established by sweeping the magnetic field . Another difficulty arises from the weak magnetic coupling of the spins to the exciting electromagnetic field. Only a very small fraction of the excitation power is therefore observed. This problem is solved as follows. First, direct transmission of excitation power to the detector is prevented by a circulator (Figure 7.1). Power that enters port 1 can only leave to the sample through port 2 . Power that comes from the sample through port 2 can only leave to the detector diode through port 3. Second, the resonator is critically coupled. This means that all microwave power coming from the source that is incident to the resonator enters the resonator and is converted to heat by the impedance (complex resistance) of the resonator. If the sample is off resonant and thus does not absorb microwave, no microwave power leaves the resonator through port 3. If now the magnetic field is set to the resonance condition and the sample resonantly absorbs microwave, this means that the impedance of resonator + sample has changed. The resonator is no longer critically coupled and some of the incoming microwave is reflected. This microwave leaves the circulator through port 3 and is incident on the detector diode.
This reflected power at resonance absorption can be very weak at low sample concentration. It is therefore important to detect it sensitively. A microwave diode is only weakly sensitive to a change in incident power at low power (Fig. 7.2, input voltage is proportional to the square root of power). The diode is most sensitive to amplitude changes near its operating point, marked green in Fig. 7.2. Hence, the diode must be biased to its operating point by adding constant power from a reference arm. The phase of the reference arm must be adjusted so that microwave coming from the resonator and microwave coming from the reference arm interfere constructively.
Figure 7.2: Characteristic curve of a microwave detection diode. At small input voltage, the diode is rather insensitive to changes in input voltage. At the operating point (green), dependence of output current on input voltage is linear and has maximum slope. This corresponds to output current. If input voltage is too large, the diode is destroyed (red point).
A further problem arises from the fact that microwave diodes are broadband detectors. On the one hand, this is useful, since samples can significantly shift resonator frequency. On the other hand, broadband detectors also collect noise from a broad frequency band. This decreases signal-to-noise ratio and must be countered by limiting the detection bandwidth to the signal bandwidth or even below. Such bandwidth limitation can be realized most easily by effect modulation and phase sensitive detection. By applying a small sinusoidal magnetic field modulation with typical frequency of and typical amplitude of , the signal component at detector diode output becomes modulated with the same frequency, whereas noise is uncorrelated to the modulation. Demodulation with a reference signal from the field modulation generator (Figure 7.1) by a phase-sensitive detector amplifies the signal and limits bandwidth to the modulation frequency.
Effect modulation with phase-sensitive detection measures the derivative of the absorption lineshape, as long as the modulation amplitude is much smaller than the width of the EPR line (Fig. 7.3). Since signal-to-noise ratio is proportional to , one usually measures at , where lineshape distortion is tolerable for almost all applications. Precise lineshape analysis may require , whereas maximum sensitivity at the expense of significant artificial line broadening is obtained at . The modulation frequency should not be broader than the linewidth in frequency units. However, with the standard modulation frequency of that corresponds on a magnetic field scale to only at , this is rarely a problem.
16.1.3. Considerations on sample preparation
Since electron spins have a much larger magnetic moment than nuclear spins, electron-electron couplings lead to significant line broadening in concentrated solutions. Concentrations of paramagnetic centers should not usually exceed in order to avoid such broadening. For organic radicals in liquid solution it may be necessary to dilute the sample to in order to achieve ultimate resolution. For paramagnetic metal dopants in diamagnetic host compounds, at most of the diamagnetic sites should be substituted by paramagnetic centers. Such concentrations can be detected easily and with good signal-to-noise ratio. For most samples, good spectra can be obtained down to the range in solution and down to the 100 ppm dopant range in solids.
Line broadening in liquid solution can also arise from diffusional collision of paramagnetic
Figure 7.3: Detection of the derivative lineshape by field modulation. The situation is considered at the instantaneous field during a field sweep (vertical dashed line) that is slow compared to the field modulation frequency of . Modulation of the magnetic field with amplitude (blue) causes a modulation of the output signal (red) with the same frequency and an amplitude . Phase-sensitive detection measures this amplitude , which is proportional to the derivative of the grey absorption lineshape and to , as long as is much smaller than the peak-to-peak linewidth of the line. In practice, is usually acceptable. For precise lineshape analysis, is recommended.
Figure 7.4: Relaxation enhancement by collisional exchange with oxygen in solution. (a) Situation before diffusional encounter. As an example, triplet oxygen is assumed to be in a state (red), whereas the spin of a nitroxide radical is assumed to be (green). (b) The oxygen molecule and nitroxide radical have collided during diffusional encounter. Their wavefunctions overlap and the three unpaired electrons cannot be distinguished from each other (grey). (c) After separation, the three unpaired electrons have been redistributed arbitrarily to the two molecules. For example, oxygen may now be in the state (red) and the nitroxide in the state (green). The electron spin of the nitroxide radical has flipped.
species with paramagnetic triplet oxygen (Figure 7.4). During such a collision, wavefunctions of the two molecules overlap and, since electrons are undistinguishable particles, spin states of all unpaired electrons in both molecules are arbitrarily redistributed when the two molecules separate again. The stochastic diffusional encounters thus lead to additional flips of the observed electron spins, which corresponds to relaxation and shortens longitudinal relaxation time . Since the linewidth is proportional to and cannot be longer than , frequent collisional encounters of paramagnetic species lead to line broadening. Such line broadening increases with decreasing viscosity (faster diffusion) and increasing oxygen concentration. The effect is stronger in apolar solvents, where oxygen solubility is higher than in polar solvents, but it is often significant even in aqueous solution. Best resolution is obtained if the sample is free of oxygen. The same mechanism leads to line broadening at high concentration of a paramagnetic species in liquid solution. In the solid state, line broadening at high concentration is mainly due to dipole-dipole coupling. Often, the anisotropically broadened EPR spectrum in the solid state is of interest, as it provides information on anisotropy and anisotropic hyperfine couplings. This may require freezing of a solution of the species of interest. Usually, the species will precipitate if the solvent crystallizes, which may cause line broadening and, in extreme cases, even collapse of the hyperfine structure and averaging of anisotropy by exchange between neighboring paramagnetic species. These problems are prevented if the solvent forms a glass, as is often the case for solvents that have methyl groups or can form hydrogen bonds in very different geometries. Typical glass-forming solvents are toluene, 2-methyltetrahydrofuran, ethanol, ethylene glycol, and glycerol. Aqueous solutions require addition of at least glycerol as a cryoprotectant. In most cases, crystallization will still occur on slow cooling. Samples are therefore shock frozen by immersion of the sample tube into liquid nitrogen. Glass tubes would break on direct immersion into liquid nitrogen, but EPR spectra have to be measured in fused silica sample tubes anyway, since glass invariably contains a detectable amount of paramagnetic iron impurities.
16.2. Theoretical description of CW EPR
This section overlaps with Section of the NMR lecture notes.
16.2.1. Spin packet lineshape
All spins in a sample that have the same resonance frequency form a spin packet. In the following we also assume that all spins of a spin packet have the same longitudinal and transverse relaxation times and , respectively. If the number of spins in the spin packet is sufficiently large, we can assign a magnetization vector to the spin packet. Dynamics of this magnetization vector with equilibrium magnetization during microwave irradiation is described by the Bloch equations in the rotating frame. In EPR spectroscopy, it is unusual to use the gyromagnetic ratio. Hence, we shall denote the resonance offset by
where is the microwave frequency in frequency units. The rotating-frame Bloch equations for the three components of the magnetization vector can then be written as
where is the microwave field amplitude in angular frequency units and is the mean value in the plane perpendicular to the static magnetic field. The apparent sign difference for the and terms arises from the different sense of spin precession for electron spins compared to nuclear spins with a positive gyromagnetic ratio.
If the spin packet is irradiated at constant microwave frequency, constant microwave power, and constant static magnetic field for a sufficiently long time (roughly ), the magnetization vector attains a steady state. Although the static field is swept in a CW EPR experiment, assuming a steady state is a good approximation, since the field sweep is usually slow compared to and . Faster sweeps correspond to the rapid scan regime that is not treated in this lecture course. In the steady state, the left-hand sides of the differential equations (7.2) for the magnetization vector components must all be zero,
This linear system of equations has the solution
where is not usually detected, is in phase with the exciting microwave irradiation and corresponds to the dispersion signal, and is out of phase with the exciting irradiation and corresponds to the absorption line. Unperturbed lineshapes are obtained in the linear regime, where the saturation parameter
fulfills . One can easily ascertain from Eq. (7.4) that in the linear regime increases linearly with increasing , which corresponds to proportionality of the signal to the square root of microwave power. is very close to the equilibrium magnetization. Within this regime, a decrease of in microwave attenuation, i.e., a power increase by , increases signal amplitude by a factor of 2 . Lineshape does not depend on in the linear regime. Therefore, it is good practice to measure at the highest microwave power that is still well within the linear regime, as this corresponds to maximum signal-to-noise ratio. For higher power the line is broadened.
Within the linear regime, takes the form of a Lorentzian absorption line
with linewidth in angular frequency units. The peak-to-peak linewidth of the first derivative of the absorption line is . Since EPR spectra are measured by sweeping magnetic field, we need to convert to magnetic field units,
The linewidth of a spin packet is called homogeneous linewidth. If is the same for all spin packets, this homogeneous linewidth is proportional to , a fact that needs to be taken into account in lineshape simulations for systems with large anisotropy. For most samples, additional line broadening arises from unresolved hyperfine couplings and, in the solid state, anisotropy. Therefore, cannot usually be obtained by applying Eq. (7.7) to the experimentally observed peak-to-peak linewidth.
16.2.2. Saturation
For microwave power larger than in the linear regime, the peak-to-peak linewidth increases by a factor . If a weak signal needs to be detected with maximum signal-to-noise ratio it is advantageous to increase power beyond the linear regime, but not necessarily to the maximum available level. For very strong irradiation, , the term 1 can be neglected in the denominator of Eqs. (7.4) for the magnetization vector components. The on-resonance amplitude of the absorption line is then given by
i.e., it is inversely proportional to . In this regime, the amplitude decreases with increasing power of the microwave irradiation.
Figure 7.5: Progressive saturation measurement on the membrane protein LHCII solubilized in detergent micelles in nitrogen atmosphere. Residue V229 was mutated to cysteine and spin-labelled by iodoacetamidoPROXYL. Experimental data points (red) were obtained at microwave attenuations of , and with a full power of . The fit by Eq. (7.9) (black line) provides and .
Semi-quantitative information on spin relaxation can be obtained by the progressive power saturation experiment, where the EPR spectrum is measured as a function of microwave power . Usually, the peak-to-peak amplitude of the larges signal in the spectrum is plotted as a function of . Such saturation curves can be fitted by the equation
where the inhomogeneity parameter takes the value in the homogeneous limit and in the inhomogeneous limit. Usually, is not known beforehand and is treated as a fit parameter. The other fit parameters are , which is the slope of the amplitude increase with the square root of microwave power in the linear regime, and , which is the half-saturation power. More precisely, is the incident mw power where is reduced to half of its unsaturated value. Figure shows experimental data from a progressive saturation measurement on spin-labelled mutant V229C of major plant light harvesting complex LHCII solubilized in detergent micelles in a nitrogen atmosphere and a fit of this data by Eq. (7.9).
16.3. ENDOR
16.3.1. Advantages of electron-spin based detection of nuclear frequency spectra
Nuclear frequency spectra in the liquid (Section 4.3.2) and solid states (4.3.4) exhibit much better hyperfine resolution than EPR spectra, because the former spectra feature fewer and narrower lines. In fact, small hyperfine couplings to ligand nuclei in metal complexes are not usually resolved in EPR spectra and only the largest hyperfine couplings may be resolved in solid-state EPR spectra. The nuclear frequency spectra cannot be measured by a dedicated NMR spectrometer because they extend over several Megahertz to several tens of Megahertz, whereas NMR spectrometers are designed for excitation and detection bandwidths of a few tens of kilohertz. Furthermore, electron spin transitions have 660 times more polarization than proton transitions and more than that for other nuclei. Their larger magnetic moment also leads to higher detection sensitivity. It is thus advantageous to transfer polarization from electron spins to nuclear spins and to backtransfer the response of the nuclear spins to the electron spins for detection. Two classes of experiments can achieve this, electron nuclear double resonance (ENDOR) experiments, discussed in Section and electron spin echo envelope modulation (ESEEM) experiments discussed in Section 8.2.
16.3.2. Types of ENDOR experiments
An ENDOR experiment can be performed with strong CW irradiation of both electron and nuclear spins. In this CW ENDOR experiment, an electron spin transition is partially saturated, in Eq. (7.5). By driving a nuclear spin transition that shares an energy level with the saturated transition, additional relaxation pathways are opened up. The electron spin transition under observation is thus partially desaturated, and an increase in the EPR signal is observed. The experiment is performed at constant magnetic field with strong microwave irradiation at a maximum of the first-derivative absorption spectrum (i.e. the CW EPR spectrum) and the EPR signal is recorded as a function of the frequency of additional radiofrequency irradiation, which must fulfill the saturation condition for the nuclear spins. Usually, the radiofrequency irradiation is frequency modulated and the response is detected with another phase-sensitive detector, which leads to observation of the first derivative of the nuclear frequency spectrum. The CW ENDOR experiment depends critically on a balance of relaxation times, so that in the solid state sufficient sensitivity may only be achieved in a certain temperature range. Furthermore, simultaneous strong continuous irradiation by both microwave and radiofrequency, while keeping resonator frequency and temperature constant, is experimentally challenging. Therefore, CW ENDOR has been largely replaced by pulsed ENDOR techniques. However, for liquid solution samples CW ENDOR is usually the only applicable ENDOR technique.
The conceptually simplest pulsed ENDOR experiment is Davies ENDOR (Section 8.1.3), where saturation of the EPR transition is replaced by inversion by a pulse (Fig. 8.1(a)). A subsequent radiofrequency pulse, which is on-resonant with a transition that shares a level with the inverted EPR transition, changes population of this level and thus polarization of the EPR observer transition. This polarization change as a function of the radiofrequency is observed by a Hahn echo experiment on the observer transition. The approach works well for moderately large hyperfine couplings , in particular for nuclei directly coordinated to a transition metal ion or for protons at hydrogen-bonding distance or distances up to about Å. As we shall see in Section 8.1.3, the experiment is rather insensitive for very small hyperfine couplings.
Figure 8.1: Pulsed ENDOR sequences. (a) Davies ENDOR. A selective inversion pulse on the electron spins is followed by a delay and Hahn echo detection (red). During microwave interpulse delay , a frequency-variable radiofrequency pulse is applied (blue). If this pulse is on resonant with a nuclear transition, the inverted echo recovers (pale blue). (b) Mims ENDOR. An non-selective stimulated echo sequence with interpulse delays and is applied to the electron spins (red). During microwave interpulse delay , a frequency-variable radiofrequency pulse is applied (blue). If this pulse is on resonant with a nuclear transition, the stimulated echo is attenuated (pale blue).
The smallest hyperfine couplings can be detected with the Mims ENDOR experiment that is based on the stimulated echo sequence Fig. b) ). The preparation block creates a polarization grating of the functional form , where is the EPR absorption spectrum as a function of the resonance offset and is the delay between the two microwave pulses. A radiofrequency pulse with variable frequency is applied during time when the electron spin magnetization is aligned with the axis. If this pulse is on resonant with a nuclear transition that shares a level with the observer EPR transition, half of the polarization grating is shifted by the hyperfine splitting , as will also become apparent in Section 8.1.3. For with integer the polarization grating is destroyed by destructive interference. Since the stimulated echo is the free induction decay (FID) of this polarization grating, it is canceled by a radiofrequency pulse that is on resonant with a nuclear transition. It is apparent that the radiofrequency pulse has no effect for with integer , where the original and frequency-shifted gratings interfere constructively. Hence, the Mims ENDOR experiment features blind spots as a function of interpulse delay . These blind spots are not a serious problem for very small hyperfine couplings . Note however that the first blind spot corresponds to . Hence, long interpulse delays are required in order to detect very small hyperfine couplings, and this leads to strong echo attenuation by a factor due to electron spin transverse relaxation. It can be shown that maximum sensitivity for very small couplings is attained approximately at .
Figure 8.2: Polarization transfer in Davies ENDOR. (a) Level populations at thermal equilibrium, corresponding to green label 0 in Fig. 8.1(a). The electron transitions (red, pale red) are much more strongly polarized than the nuclear transitions (blue, pale blue). (b) Level populations after a selective mw inversion pulse on resonance with the transition (dark red), corresponding to green label 1 in Fig. 8.1(a). A state of two-spin order is generated, where the two electron spin transitions are polarized with opposite sign and the same is true for the two nuclear spin transitions. (c) Level populations after a selective rf inversion pulse on resonance with the transition (dark blue), corresponding to green label 2 in Fig. 8.1(a). The electron spin observer transition is no longer inverted, but only saturated.
16.3.3. Davies ENDOR
The Davies ENDOR experiment is most easily understood by looking at the polarization transfers. At thermal equilibrium the electron spin transitions (red and pale red) are much more strongly polarized than the nuclear spin transitions (Fig. 8.2(a)). Their frequencies differ by an effective hyperfine splitting to a nuclear spin that is color-coded blue. The first microwave pulse is transition-selective, i.e., it has an excitation bandwidth that is smaller than . Accordingly, it inverts only one of the two electron spin transitions. We assume that the transition (red) is inverted and the transition (pale red) is not inverted; the other case is analogous. Such transition-selective inversion leads to a state of two-spin order, where all individual transitions in the two-spin system are polarized. However, the two electron spin transitions are polarized with opposite sign and the two nuclear transitions are also polarized with opposite sign (Fig. 8.2(b)). Now a radiofrequency pulse is applied. If this pulse is not resonant with a nuclear transition, the state of two-spin order persists and the observer electron spin transition (red) is still inverted. The radiofrequency pulse is also transition-selective. We now assume that this pulse inverts the transition (blue); the other case is again analogous. After such a resonant radiofrequency pulse, the two nuclear transitions are polarized with equal sign and the two electron spin transitions are saturated with no polarization existing on them (Fig. 8.2(c)). After the radiofrequency pulse a microwave Hahn echo sequence is applied resonant with the observer transition (Fig. 8.1(a)). If the radiofrequency pulse was off resonant (situation as in Fig. 8.2(b)), an inverted echo is observed. If, on the other hand, the radiofrequency pulse was on resonant (situation as in Fig. 8.2(c)) no echo is observed. In practice, polarization transfers are not complete and a weak echo is still observed. However, an on-resonant radiofrequency pulse causes some recovery of the inverted echo. If the radiofrequency is varied, recovery of the inverted echo is observed at all frequencies where the radiofrequency pulse is resonant with a nuclear transition.
C
Figure 8.3: Spectral hole burning explanation of Davies ENDOR. (a) An inhomogeneously broadened EPR line with width (red) consists of many narrower homogeneously broadened lines with linewidth . (b) Long weak microwave irradiation saturates the on-resonant spin packet and does not significantly affect off-resonant spin packets. A spectral hole is burnt into the inhomogeneously broadened line, which can be as narrow as . (c) A selective microwave pulse burns an inversion hole into the EPR line whose width is approximately the inverse width of the pulse. (d) Situation after applying an on-resonant radiofrequency pulse. For the spin packet, where the microwave pulse was on-resonant with the transition, half of the spectral hole is shifted by to lower EPR frequencies. For the spin packet where the microwave pulse was on-resonant with the transition, half of the spectral hole is shifted by to higher EPR frequencies. Considering both cases, half of the hole persists, corresponding to saturation. Two side holes with a quarter of the depth of the inversion hole are created at . These side holes do not contribute to the echo signal, as long as they are outside the detection window (pale red) whose width is determined by the excitation bandwidth of the Hahn echo detection sequence.
Further understanding of Davies ENDOR is gained by considering an inhomogeneously broadened EPR line (Fig. 8.3). In such a line with width , each individual spin packet with much narrower width can, in principle, be selectively excited. A long rectangular pulse inverts the on-resonant spin packet and partially inverts spin packets roughly over a bandwidth corresponding to the inverse length of the pulse. In Davies ENDOR, pulse lengths between 50 and , corresponding to excitation bandwidths between 20 and are typical. Such a pulse creates an inversion hole centered at the microwave frequency . In an electron-nuclear spin system, two on-resonant spin packets exist, those where is the frequency of the transition and those where it is the frequency of the transition. For the former spin packet, inversion of the nuclear spin from the to the state increases the EPR frequency by the effective hyperfine splitting , whereas for the latter packet, inversion from the to the state decreases it by . In both cases half of the inversion hole is shifted to a side hole, leaving a saturation hole at and creating a saturation side hole. The saturation center holes of the two spin packets coincide in frequency and combine to a saturation hole in the inhomogeneously broadened line. At the side hole frequencies , only one of the two spin packets contributes to the hole, so that the side holes are only half as deep. The Hahn echo subsequence in the Davies ENDOR sequence must have a detection bandwidth that covers only the central hole (pale red in Fig. ), since no ENDOR effect would be observed if the side hole would also be covered. For this purpose, the detection bandwidth of the Hahn echo sequence can be limited either by using sufficiently long microwave pulses or by using a sufficiently long integration gate for the inverted echo.
In any case, a Davies ENDOR effect is only be observed if exceeds the width of the original inversion hole. The smaller , the longer the first inversion pulse needs to be and the fewer spin packets contribute to the signal. In general, hyperfine splittings much smaller than the homogeneous linewidth in the EPR spectrum cannot be detected. In practice, Davies ENDOR becomes very insensitive for pulse lengths exceeding 400 ns. If broadening of the inversion hole by electron spin relaxation is negligible, the suppression of signals with small hyperfine couplings in Davies ENDOR can be described by a selectivity parameter
where the length of the first mw pulse. Maximum absolute ENDOR intensity is obtained for . As a function of , the absolute ENDOR intensity is given by
The hyperfine contrast selectivity described by Eq. (8.2) can be used for spectral editing. For instance, ENDOR signals of directly coordinated ligand nitrogen atoms in transition metal complexes with of the order of overlap with ENDOR signals of weakly coupled ligand protons at X-band frequencies. At an inversion pulse length of about ENDOR signals are largely suppressed.
The sensitivity advantage of Mims ENDOR for very small hyperfine couplings can also be understood in the hole burning picture. Instead of a single center hole, a preparation block with nonselective microwave pulses creates a polarization grating that can be imagined as a comb of many holes that are spaced by frequency difference . The width of each hole is approximately . The width of the comb of holes is determined by the inverse length of the non-selective pulses, which are typically long. For small couplings, where in Davies ENDOR needs to be very long, more than an order of magnitude more spin packets take part in a Mims ENDOR experiment than in a Davies ENDOR experiment. The Mims ENDOR effect arises from the shift of one quarter of the polarization grating by frequency difference and one quarter of the grating by . The shifted gratings interfer with the grating at the center frequency. Depending on and on the periodicity of the grating, this interference is destructive (ENDOR effect) or constructive (no ENDOR effect).
16.4. ESEEM and HYSCORE
16.4.1. ENDOR or ESEEM?
In ESEEM experiments, polarization transfer from electron spins to nuclear spins and detection of nuclear frequencies on electron spin transitions are based on the forbidden electron-nuclear transitions discussed in Chapter 6. Much of the higher polarization of the electron spin transitions is lost in such experiments, since the angle between the quantization axes of the nuclear spin in the two electron spin states is usually small and the depth of nuclear echo modulations is sin . Furthermore, modulations vanish along the principal axes of the hyperfine tensor, where and thus . Therefore, lineshape singularities are missing in one-dimensional ESEEM spectra, which significantly complicates lineshape analysis. For this reason, one-dimensional ESEEM experiments are not usually competitive with ENDOR experiments, at least if the ENDOR experiments can be performed at -band frequencies or even higher frequencies. An exception arises for weakly coupled "remote" nuclei in transition metal complexes where exact cancellation between the nuclear Zeeman and the hyperfine interactions can be achieved for one of the electron spin states at X-band frequencies or slightly below. In this situation, pure nuclear quadrupole frequencies are observed, which leads to narrow lines and easily interpretable spectra. One-dimensional ESEEM data are also useful for determining local proton or deuterium concentrations around a spin label, which can be used as a proxy for water accessibility (Section 10.1.6).
The main advantage of ESEEM compared to ENDOR spectroscopy is the easier extension of ESEEM to a two-dimensional correlation experiment. Hyperfine sublevel correlation (HYSCORE) spectroscopy 8.2.3 resolves overlapping signal from different elements, simplifies peak assignment, and allows for direct determination of hyperfine tensor anisotropy even if the lineshape singularities are not observed.
16.4.2. Three-pulse ESEEM
The HYSCORE experiment is a two-dimensional extension of the three-pulse ESEEM experiment that we will treat first. In this experiment, the amplitude of a stimulated echo after is observed with the pulse sequence as a function of the variable interpulse delay at fixed interpulse delay (Fig. 8.4). The block serves as a nuclear coherence generator, as discussed in Section 6.3.1 and, simultaneously, creates the polarization grating discussed in the context of the Mims ENDOR experiment (Section 8.1.2). In fact, most of the thermal equilibrium magnetization is converted to the polarization grating whose FID after the final pulse is the stimulated echo, while only a small fraction is transferred to nuclear coherence. The phase of the nuclear coherence determines how much of it contributes to the stimulated echo after back transfer to electron spin coherence by the last pulse. For an electron-nuclear spin system this phase evolves with frequencies or if during interpulse delay the electron spin is in its or state, respectively. Hence, the part of the stimulated echo that arises from back transferred nuclear coherence is modulated as a function of with frequencies and .
An expression for the echo envelope modulation can be derived by product operator formalism using the concepts explained in Section 6.2. Disregarding relaxation, the somewhat lengthy derivation provides
where the terms and correspond to contributions with the electron spin in its or state, respectively, during interpulse delay . These terms are given by
The factors for the term and for the term describe the blind spot behavior of three-pulse ESEEM. The modulation depth is given by
For small hyperfine couplings, , we have , so that Eq. (8.5) reduces to
i.e., the modulation depth is inversely proportional to the square of the magnetic field. Using Eqs. (4.10) and (4.11) we find for protons not too close to a well localized unpaired electron
where is the angle between the electron-proton axis and the static magnetic field .
Because of the star topology of electron-nuclear spin systems (Fig. 4.4(a)), Eq. (8.3) can be easily extended by a product rule to multiple nuclei with spins , where is an index that runs over all nuclei. One finds
In the weak modulation limit, where all modulation depths fulfill the condition , the ESEEM spectrum due to several coupled nuclei is the sum of the spectra of the individual nuclei.
a
Figure 8.4: Pulse sequences for three-pulse ESEEM (a) and HYSCORE (b). In three-pulse ESEEM, time is varied and time is fixed. In HYSCORE, times and are varied independently in order to obtain a two-dimensional data set.
16.4.3. HYSCORE
The HYSCORE experiment is derived from the three-pulse ESEEM experiment by inserting a microwave pulse midway through the evolution of nuclear coherence. This splits the interpulse delay into two interpulse delays and (Fig. 8.4(b)), which are varied independently to provide a two-dimensional data set that depends parametrically on fixed interpulse delay . The inserted pulse inverts the electron spin state. Hence, coherence that has evolved with frequency during interpulse delay evolves with frequency during interpulse delay and vice versa. In the weak modulation limit, the HYSCORE experiment correlates only frequencies and of the same nuclear spin. The full modulation expression for the HYSCORE experiment contains a constant contribution and contributions that vary only with respect to either or . These contributions can be removed by background correction with low-order polynomial functions along both dimensions. The remaining modulation corresponds to only cross peaks and can be expressed as
with
In this representation with unsigned nuclear frequencies, one has for the weak coupling case and for the strong coupling case , as can be inferred from Fig. 6.1. Hence, in the weak coupling case and in the strong coupling case. In the weak coupling case, the cross peaks that correlate nuclear frequencies with the same sign ( terms) are much stronger than those that correlate frequencies with opposite terms) whereas it is the other way around in the strong coupling case. Therefore, the two cases can be easily distinguished in HYSCORE spectra, since the cross peaks appear in different quadrants (Fig. 8.5). Furthermore, disregarding a small shift that arises from the pseudo-secular part of the hyperfine coupling (see below), the cross peaks of a given isotope with spin are situated on parallels to the anti-diagonal that corresponds to the nuclear Zeeman frequency . This frequency in turn can be computed from the nuclear value (or gyromagnetic ratio ) and the static magnetic field . Peak assignment for nuclei is thus straightforward. For nuclei with the peaks are further split by the nuclear quadrupole interaction. Unless this splitting is much smaller than both the hyperfine interaction and the nuclear Zeeman interaction , numerical simulations are required to assign the peaks and extract the hyperfine and nuclear quadrupole coupling.
Figure 8.5: Schematic HYSCORE spectrum for the phenyl radical (compare Fig. 4.6). Note that hyperfine couplings are given here in frequency units, not angular frequency units. Signals from weakly coupled nuclei appear in the right quadrant. To first order, these peaks are situated on a line parallel to the anti-diagonal that intersects the axis at . The doublets are centered at and split by the respective hyperfine couplings. Signals from strongly coupled nuclei appear in the (-,+) quadrant. To first order, these peaks are situated on two lines parallel to the anti-diagonal that intersect the axis at and . The doublets are centered at half the hyperfine coupling and split by .
The small pseudo-secular shift of the correlation peaks with respect to the anti-diagonal contains information on the anisotropy of the hyperfine interaction (Fig. 8.5). In the solid state, the cross peaks from different orientations form curved ridges. For a hyperfine tensor with axial symmetry, as it is encountered for protons not too close to a well-localized unpaired electron, the maximum shift in the diagonal direction corresponds to and is given by . Since is known, , and thus the electron-proton distance can be computed from this maximum shift. If , which is usually the case, the orientation with maximum shift is at the same time the orientation with maximum modulation depth.
The curved ridges end at their intersection with the parallel to the anti-diagonal. These points correspond to the principal values of the hyperfine tensor and modulation depth is zero at these points. However, it is usually possible to fit the theoretical ridge to the experimentally observed ridge, as the curvature near together with the position of the point fully determines the problem.
Figure 8.6: Schematic HYSCORE spectrum for a proton with an axial hyperfine tensor with anisotropy and isotropic component . The correlation peaks from different orientations form curved ridges (red). Curvature is the stronger the larger the anisotropy is and the ratio of squared anisotropy to the nuclear Zeeman frequency determines the maximum shift with respect to the anti-diagonal.
Analysis of HYSCORE spectra requires some precaution due to the blind-spot behavior (factor in Eq. (8.9)) and due to orientation selection by the limited bandwidth of the microwave pulses that is much smaller than spectral width for transition metal complexes. It is therefore prudent to measure HYSCORE spectra at several values of and at several observer positions within the EPR spectrum.
At a distance of between two localized unpaired electrons, splitting between the "horns" of the Pake pattern is about for two electron spins. Even strongly inhomogeneously broadened EPR spectra usually have features narrower than that (about in a magnetic field sweep). Depending on the width of the narrowest features in the EPR spectrum and on availability of an experimental spectrum or realistic simulated spectrum in the absence of dipole-dipole coupling, distances up to can be estimated from dipolar broadening by lineshape analysis. At distances below , such analysis becomes uncertain due to the contribution from exchange coupling between the two electron spins, which cannot be computed by first principles and cannot be predicted with sufficient accuracy by quantum-chemical computations. If the two unpaired electrons are linked by a continuous chain of conjugated bonds, exchange coupling can be significant up to much longer distances.
Distance measurements are most valuable for spin labels or native paramagnetic centers in biomolecules or synthetic macromolecules and supramolecular assemblies. In such systems, if the two unpaired electrons are not linked by a -electron system, exchange coupling is negligible with respect to dipole-dipole coupling for distances longer than . Such systems can often assume different molecular conformations, i.e. their structure is not perfectly defined. Structural characterization thus profits strongly from the possibility to measure distance distributions on length scales that are comparable to the dimension of these systems. This dimension is of the order of 2 to , corresponding to between and . In order to infer the distance distribution, this small dipole-dipole coupling needs to be separated from larger anisotropic interactions.
This separation of interactions is possible by observing the resonance frequency change for one spin in a pair (blue in Fig. 5.3) that is induced by flipping the spin of its coupling partner (red). In Fig. the resonance frequency of the observer spin before the flip of its coupling partner is indicated by a dashed line. If the coupling partner is in its state before the flip (left panel in Fig. 5.3), the local field at the observer spin will increase by upon flipping the coupling partner. This causes an increase of the resonance frequency of the observer spin by the dipole-dipole coupling (see Eq. (5.16)). If the coupling partner is in its state before the flip (right panel in Fig. 5.3), the local field at the observer spin will decrease by upon flipping the coupling partner. This causes an decrease of the resonance frequency of the observer spin by the dipole-dipole coupling . In the high-temperature approximation, both these cases have the
Figure 9.1: Resonance frequency shift of an observer spin (blue transitions) by the change in local magnetic field that arises upon a flip of a second spin that is dipole-dipole coupled to the observer spin. Compare Fig. for the local field picture.
same probability. Hence, half of the observer spins will experience a frequency change and the other half will experience a frequency change . If the observer spin evolves with changed frequency for some time , phases will be gained compared to the situation without flipping the coupling partner. The additional phase can be observed as a cosine modulation for both cases, as the cosine is an even function.
16.5. DEER
16.5.1. The four-pulse DEER experiment
The most commonly used experiment for distance distribution measurements in the nanometer range is the four-pulse double electron electron resonance (DEER) experiment (Figure 9.2), which is sometimes also referred to as pulsed electron electron double resonance (PELDOR) experiment. All interactions of the observer spin are refocused twice by two pulses at times and after the initial pulse. Repeated refocusing is necessary since all spin packets must be in phase at and overlap of the pump pulse with the observer pulse would lead to signal distortion. The first refocusing with interpulse delay restores the situation (1) immediately after the pulse with phase , where the magnetization vectors of all spin packets are aligned with the axis. In practice, coherence is excited on both observer spin transitions (blue in the energy level panels), but for clarity we consider only observer spin coherence that is on the upper transition and is symbolized by a wavy line in panel (1).
During time after the first refocusing, magnetization vectors of spin packets with different resonance offset dephase (panel (2)). Only the on-resonant spin packet, marked dark blue, is still aligned with the direction. The pump pulse flips the coupling partner and thus transfers the coherence to the lower observer spin transition. The resonance frequency of this transition is shifted by the dipole-dipole coupling in all spin packets. Observer spin magnetization further dephases until the time just before application of the second observer pulse (3)) and, in addition, the whole bundle of spin packet magnetization vectors precesses counterclockwise with the frequency shift . The originally on-resonant spin packet thus gains phase before the second observer pulse is applied. The second observer pulse with phase corresponds to a rotation about the axis. This mirrors the bundle of magnetization vectors with respect to the axis, inverting phase of the observer spin coherence (panel (4)). The bundle, which still
the first refocusing and all following magnetization panels are mirrored with respect to the axis.
Figure 9.2: Four-pulse DEER sequence, coherence transfers, and evolution of the observer spin magnetization. Pulses shown in blue are applied to the observer spin, the pump pulse shown in red is applied to its coupling partner. The echo at time (dashed blue line) is not observed. Interpulse delays and are fixed, time is varied, and the echo amplitude is observed as a function of .
precesses counterclockwise with angular frequency now lags the axis by phase . During the final interpulse delay of length the bundle as a whole gains phase (grey arrow in panel (4) and simultaneously realigns along its center due to echo refocusing. However, the center corresponding to the originally on-resonant spin packet does not end up along , as it would have in the absence of the pump pulse. Rather, this spin packet has gained phase with respect to the direction (panel (5)). The magnetization vector component along , which corresponds to the echo signal, is given by .
The distance range of the DEER experiment is limited towards short distances by the requirement that, for echo refocusing, the observer pulses must excite both observer transitions, which are split by and, for coherence transfer, the pump pulse must excite both transitions of the coupling partner, which are also split by . In other words, both the observer refocused echo subsequence and the pump pulse must have an excitation bandwidth that exceeds . This requirement sets a lower distance bound of about at X-band frequencies and of about at -band frequencies. A limit towards long distances arises, since several dipolar oscillations need to be observed for inferring the width or even shape of a distance distribution and at least one oscillation needs to be observed for determining the mean distance. This requires . On the other hand, we have and the fixed interpulse delay cannot be much longer than the transverse relaxation time , since otherwise coherence has completely relaxed and no echos is observed. Electron spin transverse relaxation times are of the order of a few microseconds. Depending on sample type (see Section ), can be chosen between and , corresponding to maximum observable distances between 5 and .
16.5.2. Sample requirements
In the wanted coherence transfer pathway of the DEER experiment, observer pulses exclusively excite observer spins and the pump pulse exclusively excites the coupling partner. The excitation bandwidth must be sufficiently large to cover the dipole-dipole coupling at all orientations, i.e., larger than . If the two coupled spins have the same EPR spectrum, this spectrum must be broader than twice this minimum excitation bandwidth. This condition is fulfilled for nitroxide spin labels (Chapter 10) and transition metal ions at all EPR frequencies, whereas some organic radicals, such as trityl radicals, have spectra that are too narrow at X-band or even Q-band frequencies. Furthermore, must be sufficiently long for at least the observer spins. This condition can be fulfilled for almost all species at temperatures of (transition metal complexes) or (organic radicals), but may require cooling below for some high-spin species. For high-spin species with a half-filled valence shell, such as Mn(II) or measurement temperatures of are also sufficient.
Sample concentration should be sufficiently low for intermolecular distances to be much longer than intramolecular distances. For short distances, concentrations up to are possible, but concentrations of provide better results, if a spectrometer with sufficient sensitivity is available. Depending on distance and , measurements can be performed down to concentrations of . For membrane proteins reconstituted into liposomes, data quality is not only a function of bulk spin concentration, but also of lipid-to-protein ratio. This parameter needs to be optimized for each new protein. Required sample volume varies between a few microliters (W-band frequencies) and with at Q-band frequencies usually being optimal.
If concentration is not too high and the low-temperature limit of transverse relaxation can be attained, depends on the concentration and type of protons around the observer spin. Deuteration of the solvent and cryoprotectant (usually glycerol) usually dramatically improve data quality. If the matrix can be perdeuterated, deuteration of the protein or nuclei acid may further prolong and extend distance range or improve signal-to-noise ratio.
Complications arise if more than two unpaired electrons are found in the same molecule, but these complications can usually be solved. However, none of the spin pairs should have a distance shorter than the lower limit of the accessible distance range.
17. Conversion of dipolar evolution data to distance distributions
17.5.3. Expression for the DEER signal
In Section 9.1.1 we have seen that the echo is modulated with . Usually, this applies only to a fraction of the echo, because the pump pulse excites only a fraction of all spin packets of the coupling partner of the observer spin. Therefore, the echo signal for an isolated pair of electron spins in a fixed orientation with respect to the magnetic field is described by
where the dependence is given by Eqs. (5.16) and (5.15). The dependence cannot be expressed in closed form, but often is so weakly correlated with that it can be assumed as a constant, empirical parameter. In this situation, Eq. (9.1) can be integrated over all orientations
The pump pulse inverts not only the coupling partner of the observer spin in the same molecule, but also electron spins in remote other molecules. If these neighboring spins are homogeneously distributed in space, the background factor that arises from them assumes the form
where the orientation-averaged inversion efficiency is the fraction of spins excited by the pump pulse, is an average value, and is the total concentration of spins. For subtle reasons, differs significantly from the empirical two-spin modulation depth . Homogeneous distributions of neighboring spins that are nearly confined to a plane or a line give rise to a stretched exponential background function , where is a fractional dimension of the distribution that is usually somewhat larger than 2 or 1 for nearly planar or linear distributions, respectively. The total DEER signal is given by
If distance is distributed with normalized probability density , the form factor needs to be replaced by .
Figure 9.3: Background correction in DEER spectroscopy. (a) Primary data (simulation) normalized to . Dipolar modulation decays until a time . An exponentially decay function (red) is fitted to the data in the range , where is the maximum dipolar evolution time. This background function is extrapolated to the range (ochre). (b) The form factor is obtained by normalizing the background function, and dividing the normalized primary data by . It decays to a constant level , where is the modulation depth. The red curve is a simulation corresponding to the distance distribution extracted by Tikhonov regularization with optimum regularization parameter .
17.5.4. Background correction
The information on the distance distribution is contained in , which must thus be separated from . Often, the distribution is sufficiently broad for dipolar oscillations to decay within a time shorter than the maximum dipolar evolution time (Fig. 9.3(a)). For , the primary signal is then given by plus noise. The expression for is fitted to the primary data in this range (red line in Fig. 9.3(a)). In some cases, for instance for soluble proteins, a homogeneous distribution of molecules in three dimensions can be assumed, so that can be fixed. Otherwise, is treated as a fit parameter, as are and . The background function is obtained by extrapolating to the range (ochre line) and dividing it by . According to Eq. (9.4), the form factor results by dividing by . For narrow distance distributions, oscillations in may endure until the longest attainable . This does not create a problem if at least the first oscillation is completed well before . All the following oscillations have very similar amplitude and do not bias the background fit. As a rule of thumb, a good estimate for can be obtained by fitting data at if , i.e., if two full oscillations can be observed. If the data trace is shorter than that, background fitting is fraught with uncertainty. Wrong background correction may suppress long distances or create artificial peaks at long distances.
17.5.5. Tikhonov regularization with non-negativity constraint
In order to extract the distance distribution from the experimental form factor , we need to remove the constant contribution and renormalize to the dipolar evolution function
and invert the integral equation , where the kernel is given by
Here, we have substituted by by and reversed direction of the integration, which compensated for the negative in .
In practice, is digitized and given as a vector at sampling times . Likewise, it is sufficient to compute as a vector at sampling distances . The integral equation is thus transformed to a matrix equation
Unfortunately, this matrix equation cannot easily be inverted, since the rows of kernel are not orthogonal, i.e., the scalar product of dipolar evolution function vectors at different is not zero. The weak linear dependence of the rows makes the problem ill-posed. Small deviations of the experimental from the "true" , for instance due to noise, cause large deviations of from the true distance distribution. This problem can be solved only by taking into account additional information.
First, we know that, as a probability density, at all . Hence, we can impose a non-negativity constraint on . It turns out that this is not sufficient for stabilizing the solution. Noise can be fitted by ragged distance distributions with many narrow peaks, although we know that the distance distribution must be smooth, as it arises from a continuous distribution of molecular conformations. Tikhonov regularization imposes a smoothness restraint by minimizing
where
is the mean square deviation between experimental and simulated data and
is the square norm of the second derivative, which can be computed from by multiplication with the second derivative operator . The regularization parameter determines the relative
Figure 9.4: Tikhonov regularization of the data shown in Fig. 9.3. (a) L curve. The optimum regularization parameter corresponds to the corner (green circle) and provides the simulation shown in Fig. 9.3(b) as well as the extracted distance distribution shown as a black line in panel (c) of the current Figure. The red circle marks a too large regularization parameter that leads to oversmoothing. (b) Input form factor (black) and simulation for the too large regularization parameter corresponding to the red circle in the curve. (c) Theoretical distance distribution used for simulating a noiseless form factor (green) and distance distribution extracted from the noisy form factor with optimum regularization parameter corresponding to the green circle in the curve (black). (d) (c) Theoretical distance distribution used for simulating a noiseless form factor (green) and distance distribution extracted from the noisy form factor with a too large regularization parameter corresponding to the red circle in the curve (black).
weight of the smoothing restraint with respect to mean square deviation between experimental and simulated data. A parametric plot of versus as a function of is approximately L-shaped (Fig. 9.4). For very small , roughness of the distance distribution can be decreased strongly without increasing mean square deviation very much. For large is already smooth and a further increase of will lead only to a small decrease in roughness , but to a large increase in , since the overly broadened distance distribution no longer fits the dipolar oscillations. Hence, in a mathematical sense, the optimum regularization parameter corresponds to the corner of the curve. At this regularization parameter the extracted distance distribution (black line in Fig. 9.4(c)) is only slightly broader than the true distance distribution (green line) and the simulated form factor (red line in Fig. 9.3(b)) agrees with the experimental form factor (black line), except for the white noise contribution. If the regularization parameter is too large (red circle in Fig. 9.4(a)), the simulated form factor is overdamped (red line in Fig. 9.4(b)) and the distance distribution unrealistically broad (black line in Fig. 9.4(d)). For a too small regularization parameter the distance distribution unrealistically splits into several narrow peaks and the simulated form factor fits part of the noise (not shown). This error cannot be as clearly discerned in the simulated form factor as overdamping can be discerned. Undersmoothing is apparent only in the L curve.
Nitroxide spin probes and labels Spin probes and labels
Nitroxide radicals
The nitroxide EPR spectrum Influence of dynamics on the nitroxide spectrum Polarity and proticity
Water accessibility
Oxygen accessibility Local pH measurements
Spin traps
18. 0 - Spin Probes and Spin Traps
18.1. Nitroxide spin probes and labels
18.1.1. Spin probes and labels
Spin probes are stable paramagnetic species that are admixed to a sample in order to obtain structural or dynamical information on their environment and, thus, indirectly on the sample. Spin labels are spin probes that are covalently attached to a molecule of interest, often at a specific site. As compared to more direct characterization of structure and dynamics by other techniques, EPR spectroscopy on spin probes may be able to access other length and time scales or may be applicable in aggregation states or environments where these other techniques exhibit low resolution or do not yield any signal. Site-directed spin labeling (SDSL) has the advantage that assignment of the signal to primary molecular structure is already known and that a specific site in a complex system can be studied without disturbance from signals of other parts of the system. This approach profits from the rarity of paramagnetic centers. For instance, many proteins and most nucleic acids and lipids are diamagnetic. If a spin label is introduced at a selected site, EPR information is specific to this particular site.
In principle, any stable paramagnetic species can serve as a spin probe. Some paramagnetic metal ions can substitute for diamagnetic ions native to the system under study, as they have similar charge and ionic radius or with similar complexation properties as the native ions. This applies to , which can often substitute for without affecting function of proteins or nucleic acids, or Ln(III) lanthanide ions, which bind to Ca(II) sites. Paramagnetic metal ions can also be attached to proteins by engineering binding sites with coordinating amino acids, such as histidine, or by site-directed attachment of a metal ligand to the biomolecule. Such approaches are used for lanthanide ions, in particular Gd(III), and Cu(II).
For many spin probe approaches, organic radicals are better suited than metal ions, since in radicals the unpaired electron has closer contact to its environment (ligands screen environmental access of metal ions, in particular for lanthanide ions) and the EPR spectra are narrower, which allows for some experiments that cannot be performed on species with very broad spectra. Among organic radicals, nitroxides are the most versatile class of spin probes, mainly because of their relatively small size, comparable to an amino acid side group or nucleobase, and because of hyperfine and tensor anisotropy of a magnitude that is convenient for studying dynamics (Section 10.1.4). Triarylmethyl (TAM) radicals are chemically even more inert than nitroxide radicals and have slower relaxation times in liquid solution. Currently they are much less in use than nitroxide radicals, mainly because they are not commercially available and much harder to synthesize than nitroxide radicals.
Figure 10.1: Structures of nitroxide probes. TEMPO derivatives. PROXYL derivatives. pH-sensitive imidazolidine nitroxide. 4 DOXYL derivatives. 5 Methanethiosulfonate spin label (MTSL)
18.1.2. Nitroxide radicals
The nitroxide radical is defined by the group, which is isoelectronic with the carbonyl group and can thus be replaced in approximate force field and molecular dynamics computations by a group. The unpaired electron is distributed over both atoms, which contributes to radical stability, with a slight preference for the oxygen atom. Nitroxide radicals become stable on the time scale of months or years if both positions are sterically protected, for instance by attaching two methyl groups to each of the atoms (Fig. 10.1). Nitroxides of this type are thermally stable up to temperatures of about , but they are easily reduced to the corresponding hydroxylamines, for instance by ascorbic acid, and are unstable at very low and very high pH. Nitroxides with five-membered rings (structures , and ) tend to be chemically more stable than those with six-membered rings (6). The five-membered rings also have less conformational freedom than the six-membered rings.
Spin probes can be addressed to certain environments in heterogeneous systems by choice of appropriate substituents (Fig. 10.1). The unsubstituted species are hydrophobic and partition preferably to nonpolar environments. Preference for hydrogen bond acceptors is achieved by hydroxyl derivatives , whereas ionic environments can be addressed by a carboxylate group at sufficiently high or by a trimethyl ammonium group . Reactive groups are used for SDSL, such as the methanethiosulfonate group in the dehydro-PROXYL derivative MTSL 5 , which selectively reacts with thiol groups under mild conditions. Thiol groups can be introduced into proteins by site-directed point mutation of an amino acid to cysteine and to RNA by replacement of a nucleobase by thiouridine. In DOXYL derivatives , a six-membered ring is spiro-linked to an alkyl chain, which can be part of stearic acid or of lipid molecules. The group in DOXYL derivatives is rigidly attached to the alkyl chain and nearly parallel to the axis of a hypothetical all-trans chain.
18.1.3. The nitroxide EPR spectrum
To a good approximation, the spin system of a nitroxide radical can be considered as an electron spin coupled to the nuclear spin of the atom of the group. Hyperfine
Figure 10.2: The EPR spectrum and molecular frame of nitroxide radicals. (a) The hyperfine sublevels corresponding to the three possible magnetic quantum numbers are shifted by . Allowed transitions are those with and . The microwave quantum has constant energy, since the microwave frequency is constant. During a magnetic field sweep, resonance is observed when the energy matches the energy difference of the levels of an allowed transition. The three transitions correspond to the three possible magnetic quantum numbers . (b) In a solid, each orientation gives a three-line spectrum, but the splitting and the center field depend on orientation, since and are anisotropic. To a good approximation, the hyperfine tensor has axial symmetry with the unique axis corresponding to the direction of the orbital lobes on the atom. The tensor is orthorhombic, i.e., the spectra in the plane of the molecular frame, which all have the same hyperfine splitting, have different center fields. The bond direction, which corresponds to the maximum value, is the molecular frame axis.
couplings to other nuclei, such as the methyl protons, are not usually resolved and contribute only to line broadening. The hyperfine coupling to the hybridized atom has a significant isotropic Fermi contact contribution from spin density in the orbital and a significant anisotropic contribution from spin density in the orbital that combines with a orbital on the oxygen atom to give the N-O bond partial double bond character. The direction of the lobes of the orbital is chosen as the molecular axis (Fig. 10.2(b)). The hyperfine tensor has nearly axial symmetry with being the unique axis. The hyperfine coupling is much larger along (on the order of ) than in the plane (on the order of ).
The spin-orbit coupling, which induces anisotropy, arises mainly at the atom, where a lone pair energy level is very close to the SOMO. The tensor is orthorhombic with nearly maximal asymmetry. The largest shift is positive and observed along the N-O bond, which is the molecular frame axis . An intermediate shift is observed along the axis , whereas the value is very close to . At X-band frequencies, where anisotropy corresponds to only dispersion in resonance fields, while hyperfine anisotropy corresponds to dispersion. At W-band frequencies, where , hyperfine anisotropy is still the same but anisotropy contributes a ten times larger dispersion of , which now dominates.
The field-swept CW EPR spectrum for a single orientation can be understood by considering the selection rule that the magnetic quantum number of the electron spin must change by 1 , whereas the magnetic quantum number of the nuclear spin must not change. Each transition can thus be assigned to a value of . For there are three such values, , and 1 (Fig. ). The microwave frequency is fixed and resonance is observed at fields where the energy of the microwave quantum matches the energy of a transition.
Figure 10.3: Construction of the solid-state EPR spectrum of a nitroxide at X band. (a) The absorption spectrum of each transition is considered separately. For , the hyperfine contribution vanishes and only anisotropy contributes. This line is the narrowest one at band. For the dispersion by anisotropy subtracts from the larger dispersion by hyperfine anisotropy. This line has intermediate width. For the dispersion from anisotropy adds to the dispersion from hyperfine anisotropy. This line has the largest width. (b) The three contributions from individual values add to the total EPR absorption spectrum (top). In CW EPR, the derivative of this absorption spectrum is observed (bottom). Because hyperfine anisotropy dominates, the separation between the outer extremities is .
In order to construct the solid-state spectrum, orientation dependence of the three transitions must be considered (Fig. 10.3(a)). At each individual orientation, the line is the center line. Since the hyperfine contributions scales with , it vanishes for this line and only anisotropy is observed. At X band, where hyperfine anisotropy dominates by far, this line is the narrowest one. The lineshape is the one for pure anisotropy (see Fig. 3.4). For , the orientation with the largest shift of the resonance field coincides with the one of smallest hyperfine shift. Hence, the smaller resonance field dispersion by anisotropy subtracts from the larger dispersion by hyperfine anisotropy. For , the situation is opposite and the two dispersions add. Hence, the transition, which at any given orientation is the high-field line, has the largest resonance dispersion, whereas the low-field transition has intermediate resonance field dispersion. The central feature in the total absorption spectrum (Fig. ) is strongly dominated by the transition, whereas the outer shoulders correspond to the (low field) and (high field) transitions at the orientation. Therefore, the splitting between the outer extremities in the CW EPR spectrum, which correspond to these shoulders in the absorption spectrum, is .
18.1.4. Influence of dynamics on the nitroxide spectrum
In liquid solution, molecules tumble stochastically due to Brownian rotational diffusion. In the following we consider isotropic rotational diffusion, where the molecule tumbles with the same average rate about any axis in its molecular frame. This is a good approximation for nitroxide spin probes with small substituents . For instance, TEMPO with is almost spherical with a van-der-Waals radius of ÅÅ. In water at ambient temperature, the rotational correlation time for TEMPO is of the order of . The product with the maximum anisotropy of the nitroxide EPR spectrum on an angular frequency axis is much smaller than unity. In this situation, anisotropy averages and three narrow lines of equal width and intensity are expected. The spectrum in Fig. 10.2(a) corresponds to this situation and a closer look reveals that the high-field line has somewhat lower amplitude. This can be traced back to a larger linewidth than for the other two lines, which indicates a shorter for the transition than for the other transitions. Indeed, transverse relaxation is dominated by the effect from combined hyperfine and anisotropy, which is largest for the transition that has the largest anisotropic dispersion of resonance frequencies. With increasing rotational correlation time , one expects this relaxation process to become stronger, which should lead to more line broadening that is strongest for the high-field line and weakest for the central line. This is indeed observed in the simulation for ns shown in the bottom trace of Fig. .
Figure 10.4: Simulation of X-band CW EPR spectra of an isotropically tumbling nitroxide radical for different rotational correlation times . A rotational correlation time of s at and an activated process with activation energy of were assumed, close to parameters observed for TEMPO in a synthetic polymer.
According to Kivelson relaxation theory, the ratio of the line width of one of the outer lines to the line width of the central line is given by
where
and
with the hyperfine anisotropy parameter
and the electron Zeeman anisotropy parameter
The relaxation time for the central line can be computed from the corresponding peak-topeak linewidth in field domain as
Thus, Eqs. (10.1-10.3) can be solved for the only remaining unknown . In practice, ratios of peak-to-peak line amplitudes are analyzed rather than linewidth ratios, as they can be measured with higher precision. The linewidth ratio is related to the amplitude ratio (see bottom trace in Fig. 10.4) in a first derivative spectrum by
since the integral intensity of the absorption line (double integral of the derivative lineshape) is the same for each of the three transitions. The rotational correlation time can thus be determined by, e.g.,
where is the peak-to-peak linewidth of the central line. This equation can be applied in the fast tumbling regime, where the three individual lines for , and can still be clearly recognized and have the shape of symmetric derivative absorption lines.
For slower tumbling with , the line shape becomes more complex and approaches the rigid limit (solid-state spectrum) at about (Fig. 10.4). These lineshapes can be simulated by considering multi-site exchange between different orientations of the molecule with respect to the magnetic field. Unlike for two-site exchange, which is discussed in the NMR part of the lecture course (see Section 3 of the NMR lecture notes), no closed expressions can be obtained for multi-site exchange. Nevertheless we can estimate the time scale where the spectral features are broadest and transverse relaxation times are shortest. Coalescence for two-site exchange is observed at . By substituting by by the maximum anisotropy of , corresponding to , we find a "coalescence time" ns. The simulations in Fig. show indeed that around this rotational correlation time, the
Figure 10.5: Plot of the outer extrema separation as a function of temperature for nitroxide spectra simulated under the same assumptions as in Fig. 10.4.
character of the spectrum changes from fast orientation exchange (liquid-like spectrum with three distinct peaks) to slow orientation exchange (solid-like spectrum).
A simple way of analyzing a temperature dependence, such as the one shown in Fig. 10.4, is to plot the outer extrema separation as a function of temperature (Fig. 10.5). The "coalescence time" in such a plot corresponds to the largest gradient , which coincides with the mean between the values in the fast tumbling limit and rigid limit, which is . In the case at hand, this coalescence time is and is observed at a temperature . The temperature is the temperature where the material becomes "soft" and molecular conformations can rearrange. Nitroxide spectra in the slow tumbling regime can reveal more details on dynamics, for instance, whether there are preferred rotation axes, whether motion is restricted due to covalent linkage of the nitroxide to a large molecule, or whether there is local order, such as in a lipid bilayer.
Figure 10.6: Influence of polarity of the environment and of hydrogen bonding on shift and hyperfine coupling. (a) In the mesomeric structure where the unpaired electron is on the oxygen atom (left), five valence electrons are formally assigned to and six to , which corresponds to electroneutrality. In the mesomeric structure where the unpaired electron is on the nitrogen atom (right), only four valence electrons are formally assigned to and seven to , which corresponds to a positive charge at and to a negative charge at . (b) Admixture of the charge-separated mesomeric structure generates partial charges and is favored in a polar environment that screens Coulomb attraction of the two charges. Hydrogen bonding to oxygen lowers energy of the lone pair, making excitation of a lone pair electron to the SOMO less likely, and thus decreasing shift.
18.1.5. Polarity and proticity
Delocalization of the unpaired electron in the group can be understood by considering mesomeric structures (Fig. 10.6). If the unpaired electron resides on oxygen, the formal number of valence electrons is five on nitrogen and six on oxygen, corresponding to the nuclear charge that is not compensated by inner shell electrons. Hence, both atoms are formally neutral in this limiting structure. If, on the other hand, the unpaired electron resides on the nitrogen atom, only four valence electrons are assigned to this atom, whereas seven valence electrons are assigned to the oxygen atom. This corresponds to charge separation with the formal positive charge on nitrogen and the formal negative charge on oxygen. The charge-separated form is favored in polar solvents, which screen Coulomb attraction between the two charges, whereas the neutral form is favored in nonpolar solvents. Hence, for a given nitroxide radical in a series of solvents, the hyperfine coupling, which stems from spin density on the nitrogen atom, is expected to increase with increasing solvent polarity. This effect has indeed been found. It is most easily seen in the solid state for but can also be discerned in the liquid state for .
The change in is expected to be anti-correlated to the shift, because this shift arises from SOC at the oxygen atom and, the higher spin density on the nitrogen atom is, the lower it is on the oxygen atom. This effect has also been found and is most easily detected by high-field/high-frequency EPR at frequencies of W-band frequencies of or even higher frequencies. How is correlated to depends on proticity of the solvent. Protic solvents form hydrogen bonds with the lone pairs on the oxygen atom of the N-O group. This lowers energy of the lone pair orbitals, making excitation of an electron from these orbitals to the SOMO less likely. Since this excitation provides the main contribution to SOC and thus to shift, hydrogen bonding to oxygen reduces shift. If two nitroxides have the same hyperfine coupling in an aprotic and protic environment, will be lower in the protic environment. This effect has also been found. In some cases it was possible to discern nitroxide labels with zero, one, and two hydrogen bonds by resolution of their features in W-band CW EPR spectra. Slopes of for aprotic at for protic environments have been found for the correlation between and for MTSL in spin-labeled bacteriorhodopsin in lipid bilayers [Ste+00].
19. Water accessibility
Polarity and proticity are proxy parameters for water accessibility of spin-labeled sites in proteins. Two other techniques provide complementary information. First, water can be replaced by deuterated water and the modulation depth of deuterium ESEEM can be measured. Because of the dependence of modulation depth (see Eq. (8.7)) the technique is most sensitive to deuterium nuclei in the close vicinity of the spin label. As long as , modulation depth contributions of individual nuclei add, so that the total deuterium modulation depth is a measure for local deuterium concentration close to the label. Data can be processed in a way that removes the contribution from directly hydrogen-bonded nuclei. Strictly speaking, this technique measures the concentration of not only water protons but also the one of any exchangeable protons near the label, but only to the extent that these exchangeable protons are water accessible during sample preparation or measurement.
A second, more direct technique that is applicable at ambient temperature measures the proton NMR signal as a function of irradiated microwave power with the microwave frequency being on-resonant with the central transition of a nitroxide spin label. Such irradiation transfers electron spin polarization to water protons by the Overhauser effect. This Overhauser dynamic nuclear polarization (DNP) is highly specific to water, as it critically depends on the water proton NMR signal being narrow and on fast diffusion of water. In biomolecules, water accessibility of spin labels is high at water-exposed surfaces of soluble and membrane proteins and low inside the proteins and at lipid-exposed surfaces. For transporters, water accessibility can change with state in the transport process.
Figure 10.7: Characterization of oxygen accessibility at spin-labeled site V229C in major plant light harvesting complex LHCII by CW progressive power saturation. (a) Ribbon model of LHCII with green with its carotenoid cofactors (yellow, violet) and space-filling model of residue 229 (red, marked by an arrow). The pink planes correspond to the lipid headgroup layer of the thylakoid membrane in chloroplasts. (b) Progressive power saturation curves in the absence (blue) and presence (red) of oxygen.
19.1.6. Oxygen accessibility
Since collision of paramagnetic triplet oxygen with spin probes enhances relaxation (Fig. , the saturation parameter is smaller for oxygen-accessible spin labels than for spin labels not accessible to oxygen. This change can be detected by CW progressive power saturation measurements (Section 7.2.2). The experiment is most conveniently performed with capillary tubes made of the gas permeable plastic TPX. A reference measurement is performed in a nitrogen atmosphere, which causes deoxygenation of the sample on the time scale of 15 min. The gas stream is then changed to air (20% oxygen) or pure oxygen and the measurement is repeated. Such data are shown in Fig. for residue 229 in major plant light harvesting complex LHCII. This residue is lipid exposed. As a nonpolar molecule, oxygen dissolves well in the alkyl chain region of a lipid bilayer. Accordingly, the signal saturates at higher power in an air atmosphere than in a nitrogen atmosphere. Oxygen accessibility can be quantified by a normalized parameter (Section 7.2.2).
20. Local pH measurements
The hyperfine coupling of nitroxide spin probes becomes sensitive if the heterocycle that contains the group also contains a nitrogen atom that can be protonated in the desired range. This applies, for instance, to the imidazolidine nitroxide 3 in Fig. , which has a pK value of and exhibits a change in isotropic N hyperfine coupling of between the protonated (1.43 mT) and deprotonated form, which can be resolved easily in liquid solution. By modifying the probe to a label, local can be measured near a residue of interest in a protein.
21. Spin traps
Many radicals are very reactive. This fact makes their detection during chemical reactions and in living cells very important, but it also makes their concentration very low, since often their formation reaction is slower than the reactions that destroy them again. For instance, concentration of the hydroxyl radical , a reactive oxygen species (ROS) in living cells, is too low for EPR detection even under conditions where leads to cell damage or cell death. The situation is somewhat better for the superoxide anion radical , but physiologically relevant concentrations are hard to detect also for this species.
Figure 10.8: Reaction of the commonly used spin traps phenylbutylnitrone (PBN) and 5,5-dimethyl-1pyrroline N-oxide (DMPO) with unstable radicals . Hyperfine couplings of the and atom of the formed nitroxide (red) as well as the value of the nitroxide provide fingerprint information on the type of radical .
ROS and some other highly reactive radicals of interest are most easily detected by spin trapping. A spin trap (Fig. 10.8) is a diamagnetic compound that is primed to form a stable radical by reaction with an unstable radical. The most frequently used spin traps are nitrones that form nitroxide radicals by addition of the unstable radical to the atom in position of the nitrone group. The formed nitroxide radicals are not as stable as the ones used as spin labels, mainly because they contain a hydrogen atom in position to the group. Their lifetime is usually on the minute time scale, which is sufficient for detection. The hyperfine coupling of the atom is sensitive to the type of primary radical , i.e. to the nature of the other substituent at the atom. Furthermore, these nitrones are less sterically crowded than the ones that would yield more stable nitroxides and thus the nitrones are more reactive and trap radicals more easily. In addition to the hyperfine coupling, the hyperfine coupling of the atom of the group is sensitive to the nature of . A database of experimental results supports assignment of in difficult cases: http://tools . niehs . nih.gov/stdb/index. .
Look a the "Hints for Using the Spin Trap Database" before you start your search. The keyword format is powerful, but not very intuitive.
23. Books
[CCM16] V. Chechik, E. Carter, and D. M. Murphy. Electron Paramagnetic Resonance. Ed. Oxford: Oxford University Press, 2016 (cited on pages 8, 26).
[KBE04] M. Kaupp, M. Buhl, and V. G. Malkin (Eds.) Calculation of NMR and EPR Parameters: Theory and Applications. Ed. Weinheim: Wiley-VCH, cited on page 16).
[Rie07] Philip Rieger. Electron Spin Resonance. Analysis and Interpretation. The Royal Society of Chemistry, 2007, P001-173. ISBN: 978-0-85404-355-2. DOI: 10.1039/ 9781847557872 . URL: http://dx. doi .org/10.1039/9781847557872 (cited on page 37).
[WBW94] J. A. Weil, J. R. Bolton, and J. E. Wertz. Electron Paramagnetic Resonance. Ed. New York: John Wiley & Sons, Inc., 1994 (cited on page 8).
24. Articles
[Cas+60] Theodore Castner et al. "Note on the Paramagnetic Resonance of Iron in Glass". In: J. Chem. Phys. , pages 668-673. Dor: http://dx. doi . org/10. (cited on page 40 .
[KM85] A. K. Koh and D. J. Miller. "Hyperfine coupling constants and atomic parameters for electron paramagnetic resonance data". In: Atomic Data and Nuclear Data Tables 33 (1985), pages 235-253 (cited on pages 22,23 ).
[Lef67] R. Lefebvre. "Pseudo-hyperfine interactions in radicals". In: Molecular Physics (1967), pages 417-426. Dor: (cited on page 22).
[Ste+00] Heinz-Jürgen Steinhoff et al. "High-field EPR studies of the structure and conformational changes of site-directed spin labeled bacteriorhodopsin". In: Biochim. Biophys. - Bioenergetics 1457 (2000), pages 253-262. DOI: 10. 1016/S0005 (cited on page 82 ).
25. Index
Davies ENDOR ..................... 59
dead time
DEER .............................. 68
form factor
DNP ............................. 82
dynamic nuclear polarization............82
Fermi contact interaction
g value
free electron
H
high-field approximation
high-field approximation
homogeneous linewidth
Hund's rule......................37
hyperfine contrast selectivity
Kramers ions ..................... 37
Kramers theorem
curve
level energies
first order
linear regime
M
modulation depth
ESEEM ....................62 molecular orbital
orientation selection
PELDOR ........................68
progressive power saturation ............55
quantum number
magnetic
spin
26.
rapid scan
reactive oxygen species
regularization parameter ............ 72
27. S
saturation curve
selection rule
site-directed spin labeling ............75
SOMO .......................11
spin packet
spin-orbit coupling
28. T
Tikhonov regularization
two-pulse ESEEM
29.
zero-field splitting.
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