Physical Chemistry Third Edition

5.4 The Gibbs Energy and Phase Transitions 219

EXAMPLE 5.8

Estimate the critical temperature of water by finding the temperature at which∆vapHm

vanishes. Assume that∆CP,mis temperature-independent so that

∆vapHm(T)∆vapHm(T 1 )+∆CP,m(T−T 1 )

LetT 1  373 .15 K and use the value of∆CP, mthat applies at 500 K. Compare your answer

with the correct value, 647.2 K, and explain any significant difference.

Solution

∆vapHm(T 1 )+∆CP,m(T−T 1 )40669 J mol−^1 +(− 41 .761 J K−^1 mol−^1 )

×(T− 373 .15 K)

We set this expression equal to zero: and divide by 1 J mol−^1

0  40699 −(41.761 K−^1 )T+ 15583

T

56252

41 .761 K−^1

1347 K

The agreement is poor, probably because the assumption that∆CP, mis constant is poor.

Exercise 5.12

Estimate the critical temperature of water by finding the temperature at which∆vapSmvanishes.

Assume that∆CP,mis temperature-independent.

The Maxwell Equal-Area Construction

As shown in Figures 1.7 and 1.8, a single surface represents the pressure of both the

liquid and the gas as a function ofTandVm. A completely successful equation of state

would represent both phases, including the area of tie lines below the critical point.

Figure 5.15a schematically shows the pressure as a function of molar volume at a fixed

subcritical temperature as described by an approximate equation of state such as the

van der Waals equation. Instead of the tie line that actually describes the behavior of

the fluid as in Figure 1.4, there is an S-shaped portion of the curve (a “loop”).

P

Vm

P

V

m Area 1

Area 2

a

b

c

d

e

(a)

(b)

Figure 5.15 The Pressure and the
Molar Volume for a Fluid Obeying an
Equation of State Such as the van
der Waals Equation (Schematic).(a)
The pressure as a function of molar
volume. (b) The molar volume as a
function of pressure.

If we assume that the liquid portion of the curve and the gas part of the curve

represent the behavior of the system to some approximation, we can replace the “loop”

by a tie-line segment that connects the points at which the liquid and the solid have the

same value of the molar Gibbs energy. To do this we exchange the roles of the variables

in Figure 5.15a to obtain Figure 5.15b. We want to find two points, labeledaande,

that correspond to equal values of the molar Gibbs energy (chemical potential) in the

two phases.

Since the curve corresponds to fixed temperature, the−SmdTterm indGmvanishes,

and

dμdGmVmdP (constant temperature) (5.4-6)

In order to have the molar Gibbs energy at pointsaandeequal to each other, the integral

ofdGmalong the curve from pointato pointemust vanish. We write this integral in