Physical Chemistry Third Edition

240 6 The Thermodynamics of Solutions

Vapor

pressure

of benzene

Mole fraction of benzene

Vapor

pressure

of toluene

Total

vapor

pressure

P

/ torr

700

600

500

400

300

200

100

0

0 0.2 0.4 0.6 0.8 1

Figure 6.1 The Partial Vapor Pres-

sures of Benzene and Toluene in a

Solution at 80◦C.Drawn from data of

M. A. Rosanoff, C. W. Bacon, and F. W.

Schulze,J. Am. Chem. Soc., 36 , 1993

(1914).

Substances with similar molecules usually form nearly ideal solutions. Toluene and

benzene are an example. Figure 6.1 shows the partial vapor pressures of benzene and

toluene and the total vapor pressure in a solution at 80◦C, plotted as functions of the

mole fraction of benzene. The line segments in the figure correspond to Raoult’s law,

which is very nearly obeyed.

The Thermodynamic Variables of an Ideal Solution

The values of thermodynamic properties are usually specified with reference to a stan-

dard state. We have already defined the standard state of gases and pure liquids and

solids to correspond to a pressure ofP◦(exactly 1 bar). We now choose the standard

state for a component of an ideal liquid solution to be the pure liquid substance at

pressureP◦. For a component of an ideal solid solution the standard state is the pure

solid at pressureP◦. We will define all of our standard states to correspond to a pressure

P◦. Since we assume that the chemical potential of a pure liquid or solid substance is

nearly pressure-independent, Eq. (6.1-1) becomes, to a good approximation,

μi(T,P)μ◦i(T)+RTln(xi) (6.1-8)

The thermodynamic functions of a solution are usually expressed in terms of the

quantities of mixing. These quantities are defined as the change in the variable for

producing the solution from the unmixed components at the same temperature and

pressure. TheGibbs energy change of mixingis

∆−GmixG(solution)−G(unmixed)

∑c

i 1

niμi−

∑c

i 1

niμ∗i (6.1-9)

where we have used Euler’s theorem, Eq. (5.6-4), to express G(solution). From

Eq. (6.1-1) the Gibbs energy of an ideal solution is

G(solution)

∑c

i 1

ni

[

μ∗i+RTln(xi)

]

(ideal solution) (6.1-10)

The Gibbs energy change of mixing is

∆GmixRT

∑c

i 1

niln(xi) (ideal solution) (6.1-11)

This equation is identical to that for the Gibbs energy change of mixing of an ideal gas

mixture.

Exercise 6.2

Write a formula for the Gibbs energy change of mixing for a mixture of ideal gases, and show

that it is the same as Eq. (6.1-11).