Thermodynamics and Chemistry

CHAPTER 9 MIXTURES

9.6 EVALUATION OFACTIVITYCOEFFICIENTS 267

Use ofm

Suppose we have a solution of a nonelectrolyte solute B whose activity coefficient (^) m;Bwe
wish to evaluate as a function ofmB. For a binary solution, Eq.9.6.11becomes
mD

AA

RTMAmB

(9.6.16)

(binary nonelectrolyte solution)

Solving forAand taking its differential at constantTandp, we obtain

dADRTMAd.mmB/DRTMA.mdmBCmBdm/ (9.6.17)

FromBDrefm;BCRTln.
m;BmB=m/(Eq.9.5.18), we obtain

dBDRTd ln

(^) m;BmB
m

DRT



d ln (^) m;BC
dmB
mB



(9.6.18)

We substitute these expressions for dAand dBin the Gibbs–Duhem equation in the form
given by Eq.9.2.26,nAdACnBdBD 0 , make the substitutionnAMADnB=mB, and
rearrange to

d ln (^) m;BDdmC
m 1
mB
dmB (9.6.19)
We integrate both sides of this equation for a composition change at constantTandpfrom
mBD 0 (where lnxBis 0 andmis 1 ) to any desired molalitym^0 B, with the result
ln (^) m;B.m^0 B/Dm.m^0 B/ 1 C
Zm (^0) B
0
m 1
mB
dmB (9.6.20)
(binary
nonelectrolyte solution)
When the solute is a nonelectrolyte, the integrand.m1/=mBis found to be a slowly
varying function ofmBand to approach a finite value asmBapproaches zero.
Oncemhas been measured as a function of molality from zero up to the molality
of interest, Eq.9.6.20can be used to evaluate the solute activity coefficient (^) m;Bat that
molality.
Figure9.11(a) on the next page shows the function.m1/=mBfor aqueous sucrose
solutions over a wide range of molality. The dependence of the solute activity coefficient on
molality, generated from Eq.9.6.20, is shown in Fig.9.11(b). Figure9.11(c) is a plot of the
effective sucrose molality (^) m;BmBas a function of composition. Note how the activity co-
efficient becomes greater than unity beyond the ideal-dilute region, and how in consequence
the effective molality (^) m;BmBbecomes considerably greater than the actual molalitymB.

9.6.4 Fugacity measurements

Section9.6.1described the evaluation of the activity coefficient of a constituent of a liquid
mixture from its fugacity in a gas phase equilibrated with the mixture. Section9.6.3men-
tioned the use of solvent fugacities in gas phases equilibrated with pure solvent and with a
solution, in order to evaluate the osmotic coefficient of the solution.