Physical Chemistry Third Edition

220 5 Phase Equilibrium

the following way since the curve does not represent a single-valued function:

Gm(e)−Gm(a) 0 

∫Pb

Pa

VmdP+

∫Pc

Pb

VmdP+

∫Pd

Pc

VmdP+

∫Pe

Pd

VmdP (5.4-7)

The area to the right of the vertical line segment between pointsaandcis called area 1

and is equal to

area 1

∫Pb

Pa

VmdP−

∫Pb

Pc

VmdP

∫Pb

Pa

VmdP+

∫Pc

Pb

VmdP (5.4-8)

The area to the left of the line segment between pointscandeis called area 2 and is

equal to

area 2

∫Pc

Pd

VmdP−

∫Pe

Pd

VmdP−

∫Pd

Pc

VmdP−

∫Pe

Pd

VmdP (5.4-9)

Comparison of Eq. (5.4-7) with Eqs. (5.4-8) and (5.4-9) shows that whenGm(e)−
Gm(a)0, area 1 and area 2 are equal to each other. The adjustment of the locations
of pointsaandeto make these areas equal gives a tie line between the coexisting liquid
This constructionisnamedfor the same and gas states and is known as theMaxwell equal-area construction.
James Clerk Maxwell whodevisedthe
Maxwell equations of electrodynamics
andthe Maxwell relationsof
thermodynamics andcontributedto the
founding of gas kinetic theory.

The van der Waals equation of state provides a qualitatively correct description

of the liquid–vapor transition when the equal-area construction is applied to it. The

other common equations of state provide varying degrees of accuracy in describing the

liquid–vapor transition when the equal-area construction is applied to them. Gibbons

and Laughton obtained good agreement with experiment using their modification of

the Redlich–Kwong equation of state (see Table 1.1).^5

The Temperature Dependence of the Gibbs Energy Change

The temperature derivative of the Gibbs energy is given by Eq. (4.2-20):

(

∂G

∂T

)

P,n

−S (5.4-10)

However, we cannot use this equation to calculate a value of∆Gfor a temperature

change because the value of the entropy can always have an arbitrary constant added to

it without any physical effect. We can write an analogous equation for the temperature

dependence of∆Gfor an isothermal process:

(

∂∆G

∂T

)

P,n

−∆S (5.4-11)

The interpretation of this equation is that although∆Gpertains to an isothermal process,

it gives the comparison of isothermal processes at different temperatures.

A useful version of this equation can be written

(

∂(∆G/T)

∂T

)

P,n



1

T

(

∂∆G

∂T

)

P,n

−

∆G

T^2

−

∆S

T

−

∆G

T^2

−

∆H

T^2

(5.4-12)

(^5) R. M. Gibbons and A. P. Laughton,J. Chem. Soc., Faraday Trans. 2, 80 , 1019 (1984).