Physical Chemistry , 1st ed.

probably be different. The total chemical potential of the ionic solution de-
pends, of course, on the number of moles of each ion, which are given by the
ionic formula variables n and n. The total chemical potential is

(n  ) (n  ) (8.42)

Substituting for  and  from above:

(n ° ) (n ° ) (^) n RTln  m
m
°
 (^) n RTln  m
m
°

This equation is simplified by defining the mean ionic molality mand the
mean ionic activity coefficientas
m(m n^ m n^ )1/(n^ n^ ) (8.43)
( n^  n^ )1/(n^ n^ ) (8.44)
Further, if we define nn n and °n ° n ° , we can rewrite
the expression for total chemical potential as
° nRTln 
m
m

°

 (8.45)

By analogy to equation 8.37, using the properties of logarithms we can define
the mean ionic activity aof an ionic solute An Bn as

a

m

m



°



n

(8.46)

These equations indicate how ionic solutions will really behave.

Example 8.9

Determine the mean ionic molality and activity for a 0.200-molal solution of

Cr(NO 3 ) 3 if its mean activity coefficient is 0.285.

Solution

For chromium(III) nitrate, the coefficients n and n are 1 and 3, re-

spectively, so that nis 4. The ideal molality of Cr^3 (aq) is 0.200 m, and

the ideal molality of NO 3

 (aq) is 0.600 m. The mean ionic molality is

therefore

m(0.200^1 0.600^3 )1/4m

m0.456 m

Using this and the given mean activity coefficient, we can determine the

mean activity of the solution:

a0.285 

0

1

.4

.0

5

0

6

m

m



4

a2.85     10

4

The behavior of this solution is based on a mean activity of 2.85   10

4 ,

rather than a molality of 0.200. This makes a big difference in the expected

behavior of the solution.

8.6 Ions in Solution 227