Thermodynamics and Chemistry

CHAPTER 12 EQUILIBRIUM CONDITIONS IN MULTICOMPONENT SYSTEMS

12.8 LIQUID–GASEQUILIBRIA 404

possible because both fugacity curves have two inflection points instead of the usual one.
Other types of unusual nonideal behavior are possible.^13

12.8.3 The Duhem–Margules equation

When we divide both sides of Eq.12.8.6by dxA, we obtain theDuhem–Margules equa-
tion:

xA

fA

dfA

dxA

D

xB

fB

dfB

dxA

(12.8.18)

(binary liquid mixture equilibrated

with gas, constantTandp)

If we assume the gas mixture is ideal, the fugacities are the same as the partial pressures,
and the Duhem–Margules equation then becomes

xA

pA

dpA

dxA

D

xB

pB

dpB

dxA

(12.8.19)

(binary liquid mixture equilibrated

with ideal gas, constantTandp)

Solving Eq.12.8.19for dpB=dxA, we obtain

dpB

dxA

D

xApB

xBpA

dpA

dxA

(12.8.20)

To a good approximation, by assuming an ideal gas mixture and neglecting the effect
of total pressure on fugacity, we can apply Eq.12.8.20to a liquid–gas system in which the
total pressure isnotconstant, but instead is the sum ofpAandpB. Under these conditions,
we obtain the following expression for the rate at which the total pressure changes with the
liquid composition at constantT:

dp

dxA

D

d.pACpB/

dxA

D

dpA

dxA

xApB

xBpA

dpA

dxA

D

dpA

dxA



1

xA=xB

pA=pB



D

dpA

dxA



1

xA=xB

yA=yB



(12.8.21)

HereyAandyBare the mole fractions of A and B in the gas phase given byyADpA=p
andyBDpB=p.
We can use Eq.12.8.21to make several predictions for a binary liquid–gas system at
constantT.

 If the ratioyA=yBis greater thanxA=xB(meaning that the mole fraction of A is

greater in the gas than in the liquid), then.xA=xB/=.yA=yB/is less than 1 and

dp=dxAmust have the same sign as dpA=dxA, which is positive.

 Conversely, ifyA=yBis less thanxA=xB(i.e., the mole fraction of B is greater in the

gas than in the liquid), then dp=dxAmust be negative.

(^13) Ref. [ 112 ].