Thermodynamics and Chemistry

CHAPTER 12 EQUILIBRIUM CONDITIONS IN MULTICOMPONENT SYSTEMS

12.8 LIQUID–GASEQUILIBRIA 401

p. This causes infinitesimal changes in the chemical potentials and fugacities:

dA(l)DRT

dfA

fA

dB(l)DRT

dfB

fB

(12.8.5)

By inserting these expressions in the Gibbs–Duhem equationxAdA D xBdB(Eq.
9.2.43), we obtain

xA

fA

dfAD

xB

fB

dfB (12.8.6)

(binary liquid mixture equilibrated

with gas, constantTandp)

This equation is a relation between changes in gas-phase fugacities caused by a change in
the liquid-phase composition. It shows that a composition change at constantTandpthat
increases the fugacity of A in the equilibrated gas phase must decrease the fugacity of B.
Now let us treat the liquid mixture as a binary solution with component B as the solute.
In the ideal-dilute region, at constantTandp, the solute obeys Henry’s law for fugacity:

fBDkH,BxB (12.8.7)

For composition changes in the ideal-dilute region, we can write

dfB

dxB

DkH,BD

fB

xB

(12.8.8)

With the substitution dxBDdxAand rearrangement, Eq.12.8.8becomes

xB

fB

dfBDdxA (12.8.9)

Combined with Eq.12.8.6, this is.xA=fA/dfADdxA, which we can rearrange and inte-
grate as follows within the ideal-dilute region:

ZfA 0

fA

dfA

fA

D

ZxA 0

1

dxA

xA

ln

fA^0

fA

Dlnx^0 A (12.8.10)

The result is

fADxAfA (12.8.11)

(ideal-dilute binary solution)

HerefAis the fugacity of A in a gas phase equilibrated with pure liquid A at the sameT
andpas the mixture. Equation12.8.11is Raoult’s law for fugacity applied to component
A.
If component B obeys Henry’s law at all compositions, then the Henry’s law constant
kH,Bis equal tofBand B obeys Raoult’s law,fBDxBfB, over the entire range ofxB.
We can draw two conclusions:
1.In the ideal-dilute region of a binary solution, where the solute obeys Henry’s law,
the solvent must obey Raoult’s law. (A similar result was derived in Sec.9.4.6for a
solution with any number of solutes.)