CHAPTER 9 MIXTURES
9.6 EVALUATION OFACTIVITYCOEFFICIENTS 266
coefficient of a solution of nonelectrolyte solutes is defined by
mdefD
A A
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
.1 xA/gives the power series lnxAD .1 xA/ .1 xA/^2 =2 .1 xA/^3 =3 .
Thus, in a very dilute solution we have lnxA .1 xA/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
lnxAD MA
X
i§A
mi (9.6.12)
(infinite dilution)
and the solvent chemical potential is related to solute molalities by
ADA RTMA
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:
ADA mRTMA
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 measuresA A, 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
A ADRTln
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.