Thermodynamics and Chemistry

CHAPTER 12 EQUILIBRIUM CONDITIONS IN MULTICOMPONENT SYSTEMS

12.7 MEMBRANEEQUILIBRIA 394

If we assume as before that the activity coefficients andxìBare close to 1, and that the gas
phase behaves ideally, the solubility of B is given byxBípB=kH,Bí , wherepBis the vapor
pressure of the pure solute.

12.6.3 Solute distribution between two partially-miscible solvents

Consider a two-component system of two equilibrated liquid phases,íandì. If we add a
small quantity of a third component, C, it will distribute itself between the two phases. It is
appropriate to treat C as a solute inbothphases. The thermodynamic equilibrium constant
for the equilibrium C.ì/ïC.í/, with solute standard states based on mole fraction, is

KD

aíx;C

aìx;C

D

x;íC (^) x;íCxíC
x;ìC (^) x;ìCxìC

(12.6.7)

We defineK^0 as the ratio of the mole fractions of C in the two phases at equilibrium:

K^0 defD

xíC

xCì

D

x;ìC (^) x;ìC
x;íC (^) x;íC

K (12.6.8)

At a fixedT andp, the pressure factors and equilibrium constant are constants. IfxCis

low enough in both phases for (^) x;íCand (^) x;ìCto be close to unity,K^0 becomes a constant for
the givenT andp. The constancy ofK^0 over a range of dilute composition is theNernst
distribution law.
Since solute molality and concentration are proportional to mole fraction in dilute solu-
tions, the ratiosmíC=mìCandcíC=cCìalso approach constant values at a givenTandp. The
ratio of concentrations is called thepartition coefficientordistribution coefficient.
In the limit of infinite dilution of C, the two phases have the compositions that exist
when only components A and B are present. As C is added andxCíandxCìincrease beyond
the region of dilute solution behavior, the ratiosxBí=xíAandxBì=xìAmay change. Continued
addition of C may increase the mutual solubilities of A and B, resulting, when enough C
has been added, in a single liquid phase containing all three components. It is easier to
understand this behavior with the help of a ternary phase diagram such as Fig.13.17on
page 443.

12.7 Membrane Equilibria

A semipermeable membrane used to separate two liquid phases can, in principle, be per-
meable to certain species and impermeable to others. A membrane, however, may not be
perfect in this respect over a long time period (see page 373 ). We will assume that during
the period of observation, those species to which the membrane is supposed to be permeable
quickly achieve transfer equilibrium, and only negligible amounts of the other species are
transferred across the membrane.
Section12.2.2sketched a derivation of the conditions needed for equilibrium in a two-
phase system in which a membrane permeable only to solvent separates a solution from
pure solvent. We can generalize the results for any system with two liquid phases separated
by a semipermeable membrane: in an equilibrium state, both phases must have the same