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