Physical Chemistry Third Edition

244 6 The Thermodynamics of Solutions

The composition of the vapor phase at equilibrium with a liquid solution is not the

same as the composition of the liquid solution. If an ideal gas mixture is at equilibrium

with a two-component ideal solution, the mole fraction of component 1 in the gaseous

phase is given by Dalton’s law of partial pressures:

y 1 

P 1

Ptot



P 1 ∗x 1

P∗ 1 x 1 +P 2 ∗x 2



P 1 ∗x 1

P 2 ∗+(P 1 ∗−P∗ 2 )x 1

(ideal solution) (6.1-24)

where we denote mole fractions in the vapor phase byy 1 andy 2 and mole fractions in

the solution phase byx 1 andx 2.

EXAMPLE 6.3

The vapor pressure of pure benzene (component 1) at 20.0◦C is equal to 74.9 torr, and that of

pure toluene at this temperature is 21.6 torr. Assuming ideality, find the partial vapor pressure

of each component, the total vapor pressure, and the mole fractions in the vapor at equilibrium

with the solution of Example 6.2.

Solution

Call benzene component 1 and toluene component 2:

P 1 (0.480)(74.9 torr) 36 .0 torr

P 2 (0.520)(21.6 torr) 11 .2 torr

PtotP 1 +P 2  47 .2 torr

y 1 

P 1

Ptot



36 .0 torr

47 .2 torr

 0. 763

y 2 

P 2

Ptot



11 .2 torr

47 .2 torr

 0. 237

The formula giving the total pressure as a function ofy 1 is

Ptot

P∗ 1 P 2 ∗

P 1 ∗+y 1 (P∗ 2 −P 1 ∗)

(6.1-25)

The derivation of this formula is assigned in Problem 6.1. Figure 6.2 shows the liquid–

vapor pressure–composition phase diagram of benzene and toluene at a constant tem-

perature of 80◦C. The lower curve represents the total pressure as a function of the

mole fraction of benzene in the vapor phase at equilibrium with the liquid phase. The

area below this curve represents possible equilibrium intensive states of the system

when it is a one-phase vapor. The upper curve (a line segment) represents Eq. (6.1-24),

giving the total pressure as a function of the benzene mole fraction in the liquid. The

area above this line represents possible equilibrium states of the system when it is a

one-phase liquid.

Tie

line

area

P

/ torr

Area of

liquid states

700

600

500

400

300

200

100

0

0 0.2 0.4 0.6 0.8 1

Points

representing

coexisting

phases

Area of

vapor states

Mole fraction of benzene

Figure 6.2 The Liquid–Vapor Pres-

sure–Composition Phase Diagram of

Benzene and Toluene at 80◦C.Drawn

from data of M. A. Rosanoff, C. W. Bacon,

and F. W. Schulze,J. Am. Chem. Soc., 36 ,

1993 (1914).

With two phases and two substances there are two independent intensive variables.

In Figure 6.2 the temperature is fixed, so there is only one additional independent

variable. A horizontal line segment, ortie line, between the two curves connects the

state points for the two phases at equilibrium with each other at a given pressure. If

the mole fraction in one phase is an independent variable, the pressure is a dependent

variable given by the height of the curve for that phase, and the mole fraction of the

other phase is a dependent variable given by the other end of the tie line at that pressure.