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component. For example, if P* P*, then x x . The curves are calculated from
B C B B Section 9.6
Eqs. (9.52) and (9.54); see Prob. 9.40. Thermodynamic
Properties of Ideal Solutions
In deriving Raoult’s law, we neglected the effect of a pressure change on G* of the liquid
m
components and we assumed ideal gases. At the pressures of 0 to 1 atm at which solutions
are usually studied, the effect of pressure changes on G* of liquids is negligible; the effect
m
of nonideality of the vapor, although small, is usually not negligible and should be in-
cluded in precise work. See Sec. 10.10, V. Fried, J. Chem. Educ., 45, 720 (1968), and
McGlashan, sec. 16.7.

EXAMPLE 9.6 Raoult’s law

The vapor pressure of benzene is 74.7 torr at 20°C, and the vapor pressure of
toluene is 22.3 torr at 20°C. A certain solution of benzene and toluene at 20°C
has a vapor pressure of 46.0 torr. Find the benzene mole fraction in this solution
and in the vapor above this solution.
Benzene (b) and toluene (t) molecules resemble each other closely, so it is a
good approximation to assume an ideal solution and use Raoult’s law (9.51). The
vapor pressure of the solution is
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46.0 torr P P x P* x P* x 174.7 torr2 11 x 2122.3 torr2
t
b
b
t
b
t
b
b
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Solving, we find x 0.452. The benzene partial vapor pressure is P x P*
l
B
b
b
b
v
0.452(74.7 torr) 33.8 torr. The benzene vapor-phase mole fraction is x
b
P /P 33.8/46.0 0.735 [Eq. (1.23)].
b
Exercise
A solution at 20°C is composed of 1.50 mol of benzene and 3.50 mol of toluene.
Find the pressure and the benzene mole fraction for the vapor in equilibrium
with this solution. In this exercise and the next, use data in the above example.
(Answer: 38.0 torr, 0.589.)
Exercise
The vapor in equilibrium with a certain solution of benzene and toluene at 20°C
has a benzene mole fraction of 0.300. Find the benzene mole fraction in the liquid
solution and find the vapor pressure of the solution. (Answer: 0.113, 28.2 torr.)
For a two-component solution, vapor-pressure problems involve four mole frac-
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tions and five pressures. Two of the four mole fractions x , x , x , and x can be elim-
C
B
C
B
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inated using x x 1 and x x 1. The five vapor pressures are the vapor
C
B
C
B
pressures P* and P* of the pure liquids, the vapor pressure P of the solution, and the
C
B
partial pressures P and P in the vapor in equilibrium with the solution. The pressures
B
C
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satisfy the relations P x P and P x P (from which it follows that P P P)
C
B
C
B
C
B
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and if the solution is ideal, the pressures obey the Raoult’s law equations P x P* and
B
B
B
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P x P*. We have seven unknowns (the five unknown pressures and two unknown
C
C
C
independent mole fractions) and four independent equations. To solve the problem, we
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need three pieces of information; for example, the values of P*, P*, and x (or P or x ).
C
B
B
B
Partial Molar Properties
Expressions for partial molar properties of an ideal solution are easily derived from
the chemical potentials m m*(T, P) RT ln x by using Eqs. (9.30), (9.31), and
i
i
i
(9.28). See Prob. 9.44.