DG 5 2 278 J Would this be considered a spontaneous process? Because the pressure is not kept
6.7 Natural Variables and Chemical Potential
We have implied previously that the conditions of the phase equilibrium depend on the state variables of the system, namely, volume, temperature, pressure, and amount. Usually we deal with changes in systems as temperature and pressure vary.
It would therefore be useful to know how the chemical potential varies with respect to temperature and pressure. That is, we want to know ('m/'T) and ('m/'p).
The chemical potential is the change in the Gibbs energy with respect to amount.
For a pure substance, the total Gibbs energy of a system is G5 m#n
where n is the number of moles of the material having chemical potential m. [This expression comes directly from the definition of m, which is ('G/'n)T,p.] From the relationship between G and m presented in Chapter 4, and knowing how G itself varies with T and p (given in equations 4.24 and 4.25), we can get
a'm 'Tb
p,n5 2S (6.20) and
a'm 'pb
T,n
5V (6.21)
Temperature
Pressure
Solid Gas
Liquid
Figure 6.15 If all you know about a system is that H2O is solid, then any set of pressure and temperature conditions in the shaded area would be possible con- ditions of the system. You will need to specify two degrees of freedom to describe your system.
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6.7 | Natural Variables and Chemical Potential 175
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The natural variable equation for dm is
dm 5 2S dT1V dp (6.22)
This is similar to the natural variable equation for G. We can also write the deriva- tives from equations 6.20 and 6.21 in terms of the change in chemical potential, Dm.
This will be more relevant when we consider phase transitions. Equations 6.20 and 6.21 can be rewritten as
c'(Dm) 'T d
p5 2DS (6.23) c'(Dm)
'p d
T5 DV (6.24) We can use these equations to predict what direction an equilibrium will move if conditions of T or p are changed. Consider the solid-to-liquid phase transition.
Liquids typically have greater entropy than solids, so going from solid to liquid is an increase in entropy, and the negative sign on the right of equation 6.23 implies that the slope of the m versus T plot is negative. Thus, as temperature increases, the chemical potential decreases. Because chemical potential is defined in terms of an energy—here, the Gibbs energy—and because spontaneous changes have negative changes in the Gibbs energies, as the temperature increases the system will tend toward the phase with the lower chemical potential: the liquid. Equation 6.23 explains why substances melt when the temperature is increased.
The same argument applies for the liquid-to-gas phase transition. In this case, the slope of the curve is usually higher because the difference in entropy between liquid and gas phases is much larger in magnitude than the difference in S between solid and liquid phases. However, the reasoning is the same, and equation 6.23 explains why liquids change to gas when the temperature is increased.
The effects of pressure on the equilibrium depend on the molar volumes of the phases. Again, the magnitude of the effect depends on the relative change in the molar volume. Between solid and liquid, volume changes are usually very small. That is why pressure changes do not substantially affect the position of solid-liquid equilibria, unless the change in pressure is very large. However, for liquid-gas (and solid-gas, for sublimation) transitions, the change in molar vol- ume can be on the order of hundreds or thousands of times. Pressure changes have substantial effects on the relative positions of phase equilibria involving the gas phase.
Equation 6.24 is consistent with the behavior of the solid and liquid phases of water. Water is one of the few substances whose solid molar volume is larger than its liquid molar volume.* Equation 6.24 implies that an increase in pres- sure (Dp is positive) would drive a phase equilibrium toward the phase that has the lower molar volume (because for spontaneous changes, the Gibbs energy goes down). For most substances, an increase in pressure would drive the equi- librium toward the solid phase. But water is one of the few chemical substances (elemental bismuth is another) whose liquid is denser than its solid. Its DV term for equation 6.24 is positive when going from liquid to solid, so for a spontane- ous process (that is, Dm negative), an increase in pressure translates into going from solid to liquid. This is certainly unusual behavior—but it is consistent with thermodynamics.
*Another way to say this is that a given volume of liquid is denser than the same amount of solid.
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176 Chapter 6 | Equilibria in Single-Component Systems
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Let us interpret these expressions in terms of phase diagrams and the phase transitions that they represent. First, we recognize the general magnitudes of the entropy of the various phases as Ssolid,Sliquid,Sgas. We also recognize the general magnitude of the volumes of the various phases as Vsolid,Vliquid,Vgas. (How- ever, see our discussion of water below.)
In considering the change in chemical potential as temperature changes but at constant pressure (equation 6.23), we are moving across the horizontal line in Figure 6.16, from point A to point B. The derivative in equation 6.23, which de- scribes this line, suggests that as T increases, the chemical potential must decrease so that the entropy change, DS, is negative. For a phase transition that involves solid to liquid (melting), solid to gas (sublimation), or liquid to gas (boiling), the entropy always increases. Therefore, the negative of DS for these processes will always have a negative value. In order to satisfy equation 6.23, phase transitions accompanying an increase in temperature must always occur with a simultane- ous decrease in the chemical potential. Because chemical potential is ultimately an energy—it was originally defined in terms of the Gibbs energy—what we are say- ing is that the system will tend toward a state of minimum energy. This is con- sistent with the idea from the last chapter that systems tend toward the state of minimum (Gibbs) energy. We have two different statements pointing to the same conclusion, so there is self-consistency in thermodynamics. (All good theories must be self-consistent in such situations.)
But the basic statement, one that agrees with common experience, is simple. At low temperatures, substances are solids; as you heat them, they melt into liquids; as In terms of the variables in equations 6.23 and 6.24, state what happens to the fol- lowing equilibria when the given changes in conditions are imposed. Assume all other conditions are kept constant.
a. Pressure is increased on the equilibrium H2O (s, V519.64 mL) L H2O (,, V 5 18.01 mL).
b. Temperature is decreased on the equilibrium glycerol (O) L glycerol (s).
c. Pressure is decreased on the equilibrium CaCO3 (aragonite, V534.16 mL) L CaCO3 (calcite, V536.93 mL).
d. Temperature is increased on the equilibrium CO2 (s) L CO2 (g).
Solution
a. The change in molar volume for the reaction as written is 21.63 mL. Because Dp is positive and a spontaneous process is accompanied by a negative Dm, the expression Dm/Dp will be negative overall. Therefore, the equilibrium will move in the direction of the negative DV, and the equilibrium will go toward the liquid phase.
b. Because DT is negative and a spontaneous process is accompanied by a negative Dm, the expression Dm/DT will be positive. Therefore, the reaction will proceed in the direction that provides a negative DS (as a consequence of the negative sign in equation 6.23). The equilibrium will move in the direction of the solid glycerol.
c. Dp is negative, so the reaction will spontaneously move in the direction of the positive change in volume. The equilibrium will move toward the calcite phase.
d. DT is positive, and Dm for a spontaneous transition must be negative, so the equilibrium moves in the direction of increased entropy: toward the gas phase.
Temperature
Pressure
Solid
Liquid
B Gas A
D
C
Figure 6.16 The lines ASB and CSD reflect changes in conditions, and the phase transitions along each line are related to the differences in the chemical potentials of the component, as given by equations 6.23 and 6.24. See the text for details.
ExamplE 6.12
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you heat them more, they become gases. Such common experiences are consistent with the equations of thermodynamics. [You should recognize by now that the ex- istence of the liquid phase depends on the pressure. If the pressure of the system is lower than the critical pressure, the solid will sublime (as CO2 commonly does). If the temperature is higher than the critical temperature, then the solid will “melt”
into a supercritical fluid. The ASB line in Figure 6.16 was intentionally selected to sample all three phases.]
Equation 6.24 is related to the vertical line in Figure 6.16 that connects points C and D. As the pressure is increased at constant temperature, the chemical potential also increases because for (almost) all substances, the relation Vsolid,Vliquid,Vgas applies. That is, the volume of the solid is smaller than the volume of the liquid, which is in turn smaller than the volume of the gas. Therefore, as one increases the pressure, one tends to go to the phase that has the smaller volume: This is the only way for the partial derivative in equation 6.24 to remain negative. If systems tend to go to lower chemical potential, then the numerator '(Dm) is negative. But if 'p is positive—the pressure is increased—then the overall fraction on the left side of equation 6.24 represents a negative number. Therefore, systems tend to go to phases that have smaller volumes when the pressure is increased. Because solids have lower volumes than liquids, which have smaller volumes than gases, increasing the pres- sure at constant temperature takes a component from gas to liquid to solid: exactly what is experienced.
This is not the case for H2O. Because of the crystal structure of solid H2O, the solid phase of H2O has a larger volume than the equivalent amount of liquid-phase H2O. This is reflected in the negative slope of the solid-liquid equilibrium line in the phase diagram of H2O, Figure 6.3. When the pressure is increased (at certain temperatures), the liquid phase is the stable phase, not the solid phase. H2O is the exception, not the rule. It’s just that water is so common, and its behavior so ac- cepted by us, that we tend to forget the thermodynamic implications.
There is also a Maxwell relationship that can be derived from the natural variable equation for chemical potential m. It is
a'S 'pb
T5 2a'V 'Tb
p (6.25)
However, because this is the same relation as equation 4.37 from the natural vari- able equation for G, it does not provide any new, usable relationships.
Single-component systems are useful for illustrating some of the concepts of equi- librium. Using the concept that the chemical potential of two phases of the same component must be the same if they are to be in equilibrium in the same system, we were able to use thermodynamics to determine first the Clapeyron and then the Clausius-Clapeyron equation. Plots of the pressure and temperature conditions for phase equilibria are the most common form of phase diagram. We use the Gibbs phase rule to determine how many conditions we need to know in order to specify the exact state of our system.
For systems with more than one chemical component, there are additional considerations. Solutions, mixtures, and other multicomponent systems can be described using some of the tools described in this chapter, but because of the pres- ence of multiple components, more information is necessary to describe the exact state. We will consider some of those tools in the next chapter.
6 . 8 S u m m a r y
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6.8 | SUMMARY 177
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178 Chapter 6 | Equilibria in Single-Component Systems
K E y E q u a t i o n S
phasesa mphase#dnphase50 (requirement for phase equilibrium)
q 5 m#DtransH or q 5 n#DtransH (heat for phase change) DtransS5 DtransH
Ttrans (entropy for phase change)
dp dT 5 DS
DV< Dp
DT (Clapeyron equation)
Dp5 DH DV ln Tf
Ti (alternate form of Clapeyron equation)
ln p2
p1 5 2DH R a1
T2 2 1
T1b (Clausius-Clapeyron equation) ln p*
p 5V(,)DP
RT or p* 5peV(,)DP/RT (change in vapor pressure due to change in pressure on liquid)
degrees of freedom 5 3 2 P (Gibbs phase rule for one component) a'm
'Tb
p,n5S and a'm 'pb
T,n5V (natural variable relationships for chemical potential)
a'S 'pb
T5 2a'V 'Tb
p (Maxwell relation from chemical
potential)