Physical Chemistry , 1st ed.

where piis the equilibrium vapor pressure above the solution and pi* is the

equilibrium vapor pressure of the pure liquid. Equation 7.11 lets us determine

the activities of liquids using equilibrium vapor pressures of gases.

Referring back to equation 7.9, the right side of the equation is simply the

chemical potential,i(). For a two-component liquid in equilibrium with its

vapor, each component must satisfy an expression like equation 7.9:

i() °(g) i RTln 

p

p

°

i i1, 2 (7.12)

where we have reversed equation 7.9 as well as substituted i(). If the solu-

tion were ideal, then the amounts of vapor piof each component in the vapor

phase would be determined by how much of each component was in the liq-

uid phase. The more of one component in the liquid mixture, the more of its

vapor would be in the vapor phase, going from pi0 (corresponding to hav-

ing no component iin the system) to pipi* (corresponding to all compo-

nent iin the system).Raoult’s lawstates that for an ideal solution, the partial

pressure of a component,pi, is proportional to its mole fraction of the com-

ponent in the liquidphase. The proportionality constant is the vapor pressure

of the pure component pi*:

pixipi* i1, 2 for binary solution (7.13)

Figure 7.3 shows a plot of the partial pressures of two components of a solu-

tion that follows Raoult’s law. The straight lines between zero partial pressure

and pi* are characteristic Raoult’s-law behavior. (As required by the straight-

line form of equation 7.13, the slope of each line is equal to the equilibrium

vapor pressure of each component. The intercepts also equal pi* because the

x-axis is mole fraction, which ranges from 0 to 1.) The following of Raoult’s

law is one requirement for defining an ideal solution; other requirements of an

ideal solution will be presented at the end of this section.

If the solution is ideal, we can use Raoult’s law to understand chemical po-

tentials for liquids in equilibrium with their vapors in two-component sys-

tems. We rewrite equation 7.12 by substituting into the numerator for pi:

i() i°(g) RTln 

x

p

ip

i°

i* (7.14)

We can rearrange the logarithm term, isolating the characteristic values pi* (the

equilibrium vapor pressure) and pi° (the standard pressure):

i() ° (g) i RTln 

p

p

i*

i°

RTln xi

The first two terms on the right side are characteristic of the component and

are constant at a given temperature; we group them together into a single con-

stant term i(g):

i(g) i°(g) RTln 

p

p

i*

i°

 (7.15)

Substituting, we find a relationship for the chemical potential of a liquid in an

ideal solution:

i() i(g) RTln xi i1, 2 (7.16)

Chemical potentials of liquids are thus related to their mole fractions in

multiple-component systems.

7.3 Two Components: Liquid/Liquid Systems 171

p* 2

p* 1

Partial pressure

0.5

Mole fraction of component 1 (x 1 )

0.0 1.0

Partial pressure

of component 1

Partial pressure

of component 2

Figure 7.3 Raoult’s law states that the partial
pressure of a component in the gas phase that is
in equilibrium with the liquid phase is directly
proportional to the mole fraction of that compo-
nent in the liquid. Each plot of partial pressure is
a straight line. The slope of the straight line is pi*,
the equilibrium vapor pressure of the pure liquid
component.