Physical Chemistry , 1st ed.

a.Pressure is increased on the equilibrium H 2 O (s,V19.64 mL) H 2 O

(,V18.01 mL).

b.Temperature is decreased on the equilibrium glycerol () glycerol (s).

c.Pressure is decreased on the equilibrium CaCO 3 (aragonite,V34.16 mL)

CaCO 3 (calcite,V36.93 mL).

d.Temperature is increased on the equilibrium CO 2 (s) CO 2 (g).

Solution

a.The change in molar volume for the reaction as written is 1.63 mL. Since

pis positive and a spontaneous process is accompanied by a negative ,

the expression /pwill be negative overall. Therefore, the equilibrium will

move in the direction of the negative V, so the equilibrium will go toward

the liquid phase.

b.Since Tis negative and a spontaneous process is accompanied by a neg-

ative , the expression /Twill be positive. Therefore, the reaction will

proceed in the direction that provides a negative S(as a consequence of the

negative sign in equation 6.21). The equilibrium will move in the direction

of the solid glycerol.

c.pis 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.Tis positive, and for a spontaneous transition must be negative, so

the equilibrium moves in the direction of increased entropy: toward the gas

phase.

Let us interpret these expressions in terms of phase diagrams and the

phase transitions that they represent. First, we recognize the general magni-

tudes of the entropy of the various phases as SsolidSliquidSgas. We also

recognize the general magnitude of the volumes of the various phases as

VsolidVliquidVgas. (However, see our discussion of water below.)

In considering the change in chemical potential as temperature changes but

at constant pressure (equation 6.21), we are moving across the horizontal line

in Figure 6.15, from point A to point B. The derivative in equation 6.21, which

describes this line, suggests that as Tincreases, the chemical potential must de-

crease so that the entropy change,S, is negative. For a phase transition that

involves solid to liquid (melting), solid to gas (sublimation), or liquid to gas

(boiling), the entropy alwaysincreases. Therefore, the negative ofSfor these

processes will alwayshave a negative value. In order to satisfy equation 6.21,

phase transitions accompanying an increase in temperature must always occur

with a simultaneous decrease in the chemical potential. Since chemical poten-

tial is ultimately an energy—it was originally defined in terms of the Gibbs free

energy—what we are saying is that the system will tend toward a state of min-

imum energy. This is consistent with the idea from the last chapter that sys-

tems tend toward the state of minimum (free) energy. We have two different

statements pointing to the same conclusion, so there is self-consistency in ther-

modynamics. (All good theories must be self-consistent in such situations.)

But the basic statement, one that agrees with common experience, is sim-

ple. At low temperatures, substances are solids; as you heat them, they melt into

liquids; as you heat them more, they become gases. Such common experiences

are consistent with the equations of thermodynamics. [You should recognize

by now that the existence of the liquid phase depends on the pressure. If the

pressure of the system is lower than the critical pressure, the solid will sublime

QP

QP

QP

QP

6.7 Natural Variables and Chemical Potential 161

Temperature

Pressure

Solid

Liquid

B

Gas

A

D

C

Figure 6.15 The lines A →B and C →D re-
flect changes in conditions, and the phase transi-
tions along each line are related to the differences
in the chemical potentials of the component, as
given by equations 6.21 and 6.22. See the text for
details.