Thermodynamics and Chemistry

CHAPTER 7 PURE SUBSTANCES IN SINGLE PHASES

7.8 CHEMICALPOTENTIAL ANDFUGACITY 184

By integrating dDVmdpfrom pressurep^00 to pressurep^0 , we obtain

^0 ^00 D

Z 0

^00

dD

Zp 0

p^00

Vmdp (7.8.10)

Equating the two expressions for^0 ^00 and dividing byRTgives

ln

f^0

f^00

D

Zp 0

p^00

Vm

RT

dp (7.8.11)

In principle, we could use the integral on the right side of Eq.7.8.11to evaluatef^0 by
choosing the lower integration limitp^00 to be such a low pressure that the gas behaves as
an ideal gas and replacingf^00 byp^00. However, because the integrandVm=RT becomes
very large at low pressure, the integral is difficult to evaluate. We avoid this difficulty by
subtracting from the preceding equation the identity

ln

p^0

p^00

D

Zp 0

p^00

dp

p

(7.8.12)

which is simply the result of integrating the function1=pfromp^00 top^0. The result is

ln

f^0 p^00

f^00 p^0

D

Zp 0

p^00



Vm

RT

1

p



dp (7.8.13)

Now we take the limit of both sides of Eq.7.8.13asp^00 approaches zero. In this limit, the gas
at pressurep^00 approaches ideal-gas behavior,f^00 approachesp^00 , and the ratiof^0 p^00 =f^00 p^0
approachesf^0 =p^0 :

ln

f^0

p^0

D

Zp 0

0



Vm

RT

1

p



dp (7.8.14)

The integrand.Vm=RT1=p/of this integral approaches zero at low pressure, making it
feasible to evaluate the integral from experimental data.
Thefugacity coefficientof a gas is defined by

 defD

f

p

or f Dp (7.8.15)

(pure gas)

The fugacity coefficient at pressurep^0 is then given by Eq.7.8.14:

ln.p^0 /D

Zp 0

0



Vm

RT

1

p



dp (7.8.16)

(pure gas, constantT)

The isothermal behavior of real gases at low to moderate pressures (up to at least 1 bar)
is usually adequately described by a two-term equation of state of the form given in Eq.
2.2.8:

Vm

RT

p

CB (7.8.17)