Physical Chemistry , 1st ed.

(Darren Dugan) #1
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

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