Thermodynamics and Chemistry

CHAPTER 10 ELECTROLYTE SOLUTIONS

10.6 MEANIONICACTIVITYCOEFFICIENTS FROMOSMOTICCOEFFICIENTS 299

between the central ion and the surrounding ions (the ion atmosphere) is the product of the
central ion charge and^0 .a/.
The last step of the derivation is the calculation of the work of a hypothetical reversible
process in which the surrounding ions stay in their final distribution, and the charge of the
central ion gradually increases from zero to its actual valuezie. Letziebe the charge at
each stage of the process, where is a fractional advancement that changes from 0 to 1.
Then the workw^0 due to the interaction of the central ion with its ion atmosphere is^0 .a/
integrated over the charge:

w^0 D

Z D 1

D 0

å.zie=4r 0 /=.1Ca/çd.zi/

D.z^2 ie^2 =8r 0 /=.1Ca/ (10.5.4)

Since the infinitesimal Gibbs energy change in a reversible process is given by dG D
SdTCVdpC∂w^0 (Eq.5.8.6), this reversible nonexpansion work at constantT and
pis equal to the Gibbs energy change. The Gibbs energy change per amount of species
iisw^0 NAD .z^2 ie^2 NA=8r 0 /=.1Ca/. This quantity isÅG=nifor the process in
which a solution of fixed composition changes from a hypothetical state lacking ion–ion
interactions to the real state with ion–ion interactions present.ÅG=nimay be equated to
the difference of the chemical potentials ofiin the final and initial states. If the chemical
potential without ion–ion interactions is taken to be that for ideal-dilute behavior on a mo-
lality basis,iDrefm;iCRTln.mi=m/, then.z^2 ie^2 NA=8r 0 /=.1Ca/is equal to

iårefm;iCRTln.mi=m/çDRTln (^) m;i. In a dilute solution,cican with little error be
set equal toAmi, andIctoAIm. Equation10.4.1follows.

10.6 Mean Ionic Activity Coefficients from Osmotic Coefficients

Recall that (^) is the mean ionic activity coefficient of a strong electrolyte, or the stoichio-
metric activity coefficient of an electrolyte that does not dissociate completely.
The general procedure described in this section for evaluating (^) requires knowledge
of the osmotic coefficientmas a function of molality. mis commonly evaluated by
the isopiestic method (Sec.9.6.4) or from measurements of freezing-point depression (Sec.
12.2).
The osmotic coefficient of a binary solution of an electrolyte is defined by
mdefD

AA

RTMAmB

(10.6.1)

(binary electrolyte solution)

That is, for an electrolyte the sum

P

i§Amiappearing in the definition ofmfor a nonelec-
trolyte solution (Eq.9.6.11on page 266 ) is replaced bymB, the sum of the ion molalities
assuming complete dissociation. It will now be shown thatmdefined this way can be used

to evaluate (^) .
The derivation is like that described in Sec.9.6.3for a binary solution of a nonelec-
trolyte. Solving Eq.10.6.1forAand taking the differential ofAat constantTandp, we
obtain
dADRTMA.mdmBCmBdm/ (10.6.2)