Thermodynamics and Chemistry

CHAPTER 9 MIXTURES

9.6 EVALUATION OFACTIVITYCOEFFICIENTS 261

interactions weak, we expect the local solute concentration to be proportional to the

macroscopic solute mole fraction. Thus, the partial molar quantitiesUBandVBof the

solute should be approximately linear functions ofxBin a dilute solution at constant

Tandp.

From Eqs.9.2.46and9.2.50, the solute chemical potential is given byBDUBC

pVBTSB. In the dilute solution, we assumeUBandVBare linear functions ofxB

as explained above. We also assume the dependence ofSBonxBis approximately the

same as in an ideal mixture; this is a prediction from statistical mechanics for a mixture

in which all molecules have similar sizes and shapes. Thus we expect the deviation

of the chemical potential from ideal-dilute behavior,BDrefx;BCRTlnxB, can be

described by adding a term proportional toxB:BDrefx;BCRTlnxBCkxxB, where

kxis a positive or negative constant related to solute-solute interactions.

If we equate this expression forBwith the one that defines the activity coeffi-

cient,BDrefx;BCRTln.

x;BxB/(Eq.9.5.16), and solve for the activity coefficient,

we obtain the relation^9 x;BDexp.kxxB=RT /. An expansion of the exponential in

powers ofxBconverts this to

(^) x;BD 1 C.kx=RT /xBC


(9.5.25)
Thus we predict that at constantTandp, (^) x;Bis a linear function ofxBat lowxB.
An ideal-dilute solution, then, is one in whichxBis much smaller thanRT=kxso that
(^) x;Bis approximately 1. An ideal mixture requires the interaction constantkxto be
zero.
By similar reasoning, we reach analogous conclusions for solute activity coef-
ficients on a concentration or molality basis. For instance, at lowmBthe chemical
potential of B should be approximatelyrefm;BCRTln.mB=m/CkmmB, wherekmis
a constant at a givenTandp; then the activity coefficient at lowmBis given by
(^) m;BDexp.kmmB=RT /D 1 C.km=RT /mBC


(9.5.26)
The prediction from the theoretical argument above, that a solute activity coefficient in a
dilute solution is a linear function of the composition variable, is borne out experimentally
as illustrated in Fig.9.10on page 264. This prediction applies only to a nonelectrolyte
solute; for an electrolyte, the slope of activity coefficient versus molality approaches1
at low molality (page 290 ).

9.6 Evaluation of Activity Coefficients

This section describes several methods by which activity coefficients of nonelectrolyte sub-
stances may be evaluated. Section9.6.3describes an osmotic coefficient method that is also
suitable for electrolyte solutes, as will be explained in Sec.10.6.

9.6.1 Activity coefficients from gas fugacities

Suppose we equilibrate a liquid mixture with a gas phase. If componenti of the liquid
mixture is a volatile nonelectrolyte, and we are able to evaluate its fugacityfiin the gas

phase, we have a convenient way to evaluate the activity coefficient (^) iin the liquid. The
relation between (^) iandfiwill now be derived.
(^9) This is essentially the result of the McMillan–Mayer solution theory from statistical mechanics.