Thermodynamics and Chemistry

CHAPTER 9 MIXTURES

9.6 EVALUATION OFACTIVITYCOEFFICIENTS 266

coefficient of a solution of nonelectrolyte solutes is defined by

mdefD

AA

RTMA

X

i§A

mi

(9.6.11)

(nonelectrolyte solution)

The definition ofmin Eq.9.6.11has the following significance. The sum

P

i§Amiis

the total molality of all solute species. In an ideal-dilute solution, the solvent chemical

potential isADACRTlnxA. The expansion of the function lnxAin powers of

.1xA/gives the power series lnxAD.1xA/.1xA/^2 =2.1xA/^3 =3


.

Thus, in a very dilute solution we have lnxA.1xA/D

P

i§Axi. In the limit

of infinite dilution, the mole fraction of soluteibecomesxiDMAmi(see Eq.9.1.14).

In the limit of infinite dilution, therefore, we have

lnxADMA

X

i§A

mi (9.6.12)

(infinite dilution)

and the solvent chemical potential is related to solute molalities by

ADARTMA

X

i§A

mi (9.6.13)

(infinite dilution)

The deviation ofmfrom unity is a measure of the deviation ofAfrom infinite-

dilution behavior, as we can see by comparing the preceding equation with a rear-

rangement of Eq.9.6.11:

ADAmRTMA

X

i§A

mi (9.6.14)

The reasonmis called the osmotic coefficient has to do with its relation to the osmotic

pressureof the solution: The ratio=mBis equal to the product ofmand the

limiting value of=mBat infinite dilution (see Sec.12.4.4).

Evaluation ofm

Any method that measuresAA, the lowering of the solvent chemical potential caused
by the presence of a solute or solutes, allows us to evaluatemthrough Eq.9.6.11.
The chemical potential of the solvent in a solution is related to the fugacity in an equili-
brated gas phase byADrefA(g)CRTln.fA=p/(from Eq.9.5.11). For the pure solvent,
this relation isADrefA(g)CRTln.fA=p/. Taking the difference between these two
equations, we obtain

AADRTln

fA

fA

(9.6.15)

which allows us to evaluatemfrom fugacity measurements.
Osmotic coefficients can also be evaluated from freezing point and osmotic pressure
measurements that will be described in Sec.12.2.