Physical Chemistry , 1st ed.

Raoult’s law is useful in understanding the vapor-phase behavior of ideal

solutions. If the vapor phase is treated as an ideal gas, then Dalton’s law of par-

tial pressures says that the total pressure is the sum of the individual partial

pressures. For our two-component system, this becomes

ptotp 1 p 2

From Raoult’s law, this becomes

ptotx 1 p 1 * x 2 p 2 *

However,x 1 and x 2 are not independent: since the sum of the mole fractions

of the liquid phase must equal 1, we have x 1 x 2 1, or x 2  1 x 1. We can

substitute:

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

We can algebraically rearrange this:

ptotp 2 * (p 1 * p 2 *)x 1 (7.17)

This expression has the form of a straight line,ymxb. In this case,x 1

represents the mole fraction of component 1 in the liquidphase. If we plot

total pressure versus mole fraction of component 1, we would get a straight

line as shown in Figure 7.4. The slope would be p 1 * p 2 *, and the y-intercept

would be p 2 *. Figure 7.4 suggests that there is a smooth, linear variation in

total vapor pressure from p 1 * to p 2 * as the composition of the solution varies.

Figure 7.4 also shows, in dotted lines, the individual partial pressures. Compare

this to Figure 7.3.

Example 7.3

An ideal solution can be approximated using the liquid hydrocarbons hexane

and heptane. At 25°C, hexane has an equilibrium vapor pressure of 151.4 mmHg

and heptane has an equilibrium vapor pressure of 45.70 mmHg. What is the

equilibrium vapor pressure of a 5050 molar hexane and heptane solution (that

is,x 1 x 2 0.50) in a closed system? It does not matter which liquid is labeled

1 or 2.

Solution

Using Raoult’s law, we have

p 1 (0.50)(151.4 mmHg) 75.70 mmHg

p 2 (0.50)(45.70 mmHg) 22.85 mmHg

By Dalton’s law, the total vapor pressure in the system is the sum of the two

partial pressures:

ptot75.70 22.85 mmHg 98.55 mmHg

Since boiling of a liquid occurs when the vapor pressure of a liquid equals

the surrounding pressure, liquid solutions will boil at different temperatures

depending on their composition and the vapor pressures of the pure compo-

nents. The next example illustrates how this idea can be used.

172 CHAPTER 7 Equilibria in Multiple-Component Systems

p* 2

p 1

p 2

p* 1

Partial pressure

0.5

Mole fraction of component 1 (x 1 )

Total pressure

of vapor phase

0.0 1.0

Figure 7.4 The total pressure of an ideal liq-
uid solution is a smooth transition from one pure
vapor pressure to the other.