Thermodynamics and Chemistry

CHAPTER 12 EQUILIBRIUM CONDITIONS IN MULTICOMPONENT SYSTEMS

12.8 LIQUID–GASEQUILIBRIA 402

fB

fA

bc

bc

0

2

4

6

8

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

xB

fA

=kPa and

fB

=kPa

Figure 12.10 Fugacities in a gas phase equilibrated with a binary liquid mixture of

H 2 O (A) and ethanol (B) at 25 C and 1 bar.a The dashed lines show Raoult’s law

behavior. The dotted lines illustrate the inequality.dfB=dxB/ < .fB=xB/.

aBased on data in Ref. [ 45 ].

2.If one component of a binary liquid mixture obeys Raoult’s law at all compositions,
so also must the other component. This is the definition of an ideal binary liquid
mixture (Sec.9.4.2).
Suppose we have a nonideal binary liquid mixture in which component B exhibits posi-
tive deviations from Raoult’s law. An example of this behavior for the water–ethanol system
is shown in Fig.12.10. At each point on the curve offBversusxB, the slope dfB=dxBis
less than the slopefB=xBof a line drawn from the origin to the point (as illustrated by
the open circles and dotted lines in the figure), except that the two slopes become equal at
xBD 1 :
dfB
dxB



fB

xB

(12.8.12)

As we can see from the figure, this relation must apply to any component whose fugacity
curve exhibits a positive deviation from Raoult’s law and has only one inflection point.
Algebraic operations on an inequality must be carried out with care: multiplying both
sides by a quantity that can be negative may change the inequality to one with the wrong
sign. In order to simplify manipulation of the inequality of Eq.12.8.12, it helps to convert
it to the following equality:^11
dfB
dxB

CDD

fB

xB

(12.8.13)

HereDrepresents the difference betweenfB=xBand dfB=dxB; its value is a function
ofxBand is, according to Eq.12.8.12, either positive or zero. We make the substitution
dxBDdxAand rearrange to

xBdfB

fBCDxB

DdxA (12.8.14)

(^11) This procedure is similar to the rectification procedure described on page 146.