Thermodynamics and Chemistry

CHAPTER 10 ELECTROLYTE SOLUTIONS

10.6 MEANIONICACTIVITYCOEFFICIENTS FROMOSMOTICCOEFFICIENTS 300

From Eq.10.3.9on page 292 , we obtain

dBDRT 



d ln (^) C
dmB
mB



(10.6.3)

Substitution of these expressions in the Gibbs–Duhem equationnAdACnBdB D 0 ,
together with the substitutionnAMADnB=mB, yields

d ln (^) DdmC
m 1
mB
dmB (10.6.4)
Then integration frommBD 0 to any desired molalitym^0 Bgives the result
ln (^) .m^0 B/Dm.m^0 B/ 1 C
Zm (^0) B
0
m 1
mB
dmB (10.6.5)
The right side of this equation is the same expression as derived for ln (^) m;Bfor a nonelec-
trolyte (Eq.9.6.20on page 267 ).
The integrand of the integral on the right side of Eq.10.6.5approaches1asmB
approaches zero, making it difficult to evaluate the integral by numerical integration starting
atmBD 0. (This difficulty does not exist when the solute is a nonelectrolyte.) Instead, we
can split the integral into two parts
Zm (^0) B
0
m 1
mB
dmBD
Zm (^00) B
0
m 1
mB
dmBC
Zm (^0) B
m^00 B
m 1
mB
dmB (10.6.6)
where the integration limitm^00 Bis a low molality at which the value ofmis available and
at which (^) can either be measured or estimated from the Debye–Huckel equation. ̈
We next rewrite Eq.10.6.5withm^0 Breplaced withm^00 B:
ln (^) .m^00 B/Dm.m^00 B/ 1 C
Zm (^00) B
0
m 1
mB
dmB (10.6.7)
By eliminating the integral with an upper limit ofm^00 Bfrom Eqs.10.6.6and10.6.7, we obtain
Zm (^0) B
0
m 1
mB
dmBDln (^) .m^00 B/m.m^00 B/C 1 C
Zm (^0) B
m^00 B
m 1
mB
dmB (10.6.8)
Equation10.6.5becomes
ln (^) .m^0 B/Dm.m^0 B/m.m^00 B/Cln (^) .m^00 B/C
Zm (^0) B
m^00 B
m 1
mB
dmB (10.6.9)
The integral on the right side of this equation can easily be evaluated by numerical integra-
tion.