Physical Chemistry , 1st ed.

According to equation 4.58, the chemical potential of a real gas is related to

some standard chemical potential plus a correction factor in terms of the fu-

gacity of the gas:

i(g) °(g) i RTln 

p

f

°

 (7.6)

where Rand Thave their usual thermodynamic definitions,fis the fugacity of

the gas, and p° represents the standard condition of pressure (1 bar or 1 atm).

For liquids (and, in appropriate systems, solids as well) there is an equivalent

expression. However, instead of using fugacity, we will define the chemical po-

tential of a liquid in terms of its activity, ai, as introduced in Chapter 5:

i() °(i ) RTln ai (7.7)

At equilibrium, the chemical potentials of the liquid and the vapor phases

must be equal. From the above two equations,

i(g)i()

°(g) i RTln 

p

f

°

i°() RTln ai i1, 2 (7.8)

for each component i. (At this point, it is important to keep track of which

terms refer to which phase, g or .) If we assume that the vapors are acting as

ideal gases, then we can substitute the partial pressure,pi, for the fugacity,f,on

the left side. Making this substitution into equation 7.8:

i°(g) RTln 

p

p

°

i

i°() RTln ai (7.9)

If the system were composed of a purecomponent, then the liquid phase would

not need the second corrective term that includes the activity. For a single-

component system, equation 7.9 would be

°(g) i RTln 

p

p

i*

°

i°() (7.10)

where pi* is the equilibrium vapor pressure of the pure liquid component.

Substituting for °(i ) from equation 7.10 into the right side of equation 7.9,

we get

i° (g) RTln 

p

p

°

i 

i°(g) RTln 

p

p

i*

°

RTln ai

The standard chemical potential ° (g) cancels. Moving both RTterms to one

side, this equation becomes

RTln 

p

p

°

iRTln p

p

i*

°

RTln ai

We can cancel Rand Tfrom the equation, and then combine the logarithms

on the left side. When we do this, the p°’s cancel. We get

ln 

p

p

i*

iln a

i

Taking the inverse logarithm of both sides, we find an expression for the ac-

tivity of the liquid phase of the component labeled i:

ai

p

p

i*

i i1, 2 (7.11)

170 CHAPTER 7 Equilibria in Multiple-Component Systems