Physical Chemistry , 1st ed.

Example 7.4

In analogy to ice baths, there are vapor baths that are kept at constant tem-

perature by the equilibrium between the liquid and gas phases. A hexane/

heptane solution is used to establish a constant 65°C temperature in a closed

system that has a pressure of 500.0 mmHg. At 65°, the vapor pressures of

hexane and heptane are 674.9 and 253.5 mmHg. What is the composition of

the solution?

Solution

If we are seeking the composition of the solution, we need to determine one

of the mole fractions of the liquid phase, say x 1. We can find x 1 by rearrang-

ing equation 7.17 algebraically:

x 1 

p

p

t

1 *

ot





p

p

2 *

^2 *

We have all the information needed:p 1 * 674.9 mmHg,p 2 * 253.5 mmHg,

and ptot500 mmHg. Substituting and solving:

x 1 

2

4

4

2

6

1

.

.

5

4

m

m

m

m

H

H

g

g



Notice that the units of mmHg will cancel, leaving a unitless value. Mole frac-

tions are unitless, so this is as it should be. We get

x 1 0.5850

which suggests that our liquid mixture is a little over half hexane. The mole

fraction of heptane would be 1 0.5850 0.4150, just less than half.

What are the mole fractions of the two components in the vaporphase?
They are not equal to the mole fractions of the liquid phase. We use the vari-
ables y 1 and y 2 to represent the vapor-phase mole fractions.†They can be de-
termined using Dalton’s law, the idea that the mole fraction of a component in
a gas mixture is equal to its partial pressure divided by the total pressure:

y 1 

p

p

to

1

t



p 1

p



1

p 2



x 1 p 1 *

x



1 p 1 *

x 2 p 2 *

 (7.18)

In the last expression, we used Raoult’s law to substitute in terms ofp 1 and p 2 .
Again, we note that x 2  1 x 1 , so we can substitute into equation 7.18 to get

y 1 

which rearranges to

y 1 

p 2 *(

x

p

1

1 *

p



1 *

p 2 *)x 1

 (7.19)

Similarly, the mole fraction of component 2 is

y 2 

x 1 p 1 *

x 2



p 2 *

x 2 p 2 *

 (7.20)

x 1 p 1 *



x 1 p 1 * (1 x 1 )p 2 *

500.0 mmHg 253.5 mmHg



674.9 mmHg 253.5 mmHg

7.3 Two Components: Liquid/Liquid Systems 173

†For solutions and their vapor phases, the convention is to use xito represent the solu-
tion-phase mole fractions, and yito represent the vapor-phase mole fractions.