Thermodynamics and Chemistry

CHAPTER 12 EQUILIBRIUM CONDITIONS IN MULTICOMPONENT SYSTEMS

12.8 LIQUID–GASEQUILIBRIA 403

fB

fA

0

10

20

30

40

0 0:2 0:4 0:6 0:8 1:0

xB

fA

=kPa and

fB

=kPa

Figure 12.11 Fugacities in a gas phase equilibrated with a binary liquid mixture of

chloroform (A) and ethanol (B) at 35 C (Ref. [ 149 ]).

WhenDis zero, this equation becomesxBdfB=fB DdxA. WhenDis positive, the
left side of the equation is less thanxBdfB=fBand is equal to dxA, so that dxAis less
thanxBdfB=fB. SinceDcannot be negative, Eq.12.8.14is equivalent to the following
relation:

xB
fB

dfBdxA (12.8.15)

A substitution from Eq.12.8.6gives us

xA

fA

dfAdxA or

dfA

fA



dxA

xA

(12.8.16)

We can integrate both sides of the second relation as follows:^12

ZfA 0

fA

dfA

fA



Zx (^0) A
1
dxA
xA
ln
fA^0
fA
lnxA^0 fAxAfA (12.8.17)
Thus,if the curve of fugacity versus mole fraction for one component of a binary liquid
mixture exhibits only positive deviations from Raoult’s law, with only one inflection point,
so also must the curve of the other component. In the water–ethanol system shown in
Fig.12.10, both curves have positive deviations from Raoult’s law, and both have a single
inflection point.
By the same method, we find that if the fugacity curve of one component has only
negativedeviations from Raoult’s law with a single inflection point, the same is true of the
other component.
Figure12.11illustrates the case of a binary mixture in which component B has only
positive deviations from Raoult’s law, whereas component A has both positive and negative
deviations (fAis slightly less thanxAfAforxBless than 0.3). This unusual behavior is
(^12) The equalities are the same as Eqs.12.8.10and12.8.11, with the difference that herexAis not restricted to
the ideal-dilute region.