Physical Chemistry , 1st ed.

Similar substitutions can be made into equation 7.20 to get an expression like

equation 7.19. As an example, for the 0.5850/0.4150 mixture of hexane and

heptane in Example 7.4, the gas-phase mole fractions are 0.790 and 0.210, re-

spectively, using equations 7.19 and 7.20. Note how different the mole fractions

in the gas phase are from the mole fractions in the liquid phase.

We can take a slightly different perspective and derive an expression for the

total pressure ptotabove the solution in terms ofvapor-phase composition. For

ideal gases, the partial pressure of a gas in a mixture is equal to the total pres-

sure times the gas’s mole fraction:

piyiptot (7.21)

We can combine this with Raoult’s law and its definition of the partial pres-

sure of a gas-phase component to get

yiptotxipi*

This equation relates the total pressure ptot, the vapor pressure of the ith com-

ponent pi*, and the mole fractions of the ith component in the liquid phase (xi)

and the gas phase (yi). Solving for ptot:

ptot

xi

y

p

i

i* (7.22)

To be consistent with Figure 7.4, let us assume that i1. If we solve equation

7.19 for x 1 ,we get

x 1 

p 1 *(

y

p

1

2 *

p



2 *

p 1 *)y 1

 (7.23)

We do this because we want to be able to express ptotin terms of the mole frac-

tions of the vapor,not the liquid, so we need to eliminate x 1. Substituting equa-

tion 7.23 into equation 7.22, we find that

ptot

ptot

The y 1 terms in the numerator and denominator cancel, and we have for our

final expression

ptot

p 1 *(

p

p

2 *

2 *

p



1 *

p 1 *)y 1

 (7.24)

A similar expression can be determined in terms ofy 2 instead ofy 1.

There is a key point about equation 7.24. It is similar to equation 7.17 in

that we can plot the total pressure of the vapor phase with respect to the mole

fraction of one component,y 1. However, it is not an equation for a straight

line! Instead, it is an equation for a curved line, and ifptotis plotted versus y 1

on the same scale as Figure 7.4, this line typically lies underneath the straight

line ofptotversus x 1. Figure 7.5 shows what this plot ofptotversus y 1 looks

like relative to ptotversus x 1. The plot ofptotversus x 1 , the liquid mole frac-

tion, is called the bubble point linewhereas the plot ofptotversus y 1 , the va-

por mole fraction, is called the dew point line.Diagrams like Figure 7.5, which

plot vapor pressure versus mole fraction, are called pressure-composition phase

diagrams.

y^1 p^2 *p^1 *

[p 1 * (p 2 * p 1 *)y 1 ]y 1



p 1 *(

y

p

1

2 *

p



2 *

p 1 *)y 1

pi*



y 1

174 CHAPTER 7 Equilibria in Multiple-Component Systems

p* 2

p* 1

Partial pressure

0.5

x 1 , y 1

ptot vs. x 1 : the bubble point line

ptot vs. y 1 : the dew point line

0.0 1.0

Figure 7.5 The mole fractions in the vapor
phase are not the same as in the liquid phase.
The bubble point line gives total pressure versus
liquid-phase mole fraction,xi. The dew point line
gives total pressure versus vapor-phase mole frac-
tion,yi. The two lines would coincide only if both
components had the same pure vapor pressure.