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associate with X ions to form ion pairs. If no ion pairing occurred, the number of moles Section 10.8
z
of M in solution would be n n , where n is the number of moles of M X used to pre- Ionic Association
z
i
i
n
n
pare the solution. With ion pairing, the number of moles of M in solution is n an n .
z
i

There are a total of n n moles of M in the solution, present partly as M ions and partly
z
i
in MX z z ion pairs [Eq. (10.71)]. Hence the number of moles of the ion pair is
n n n n n n an n 11 a2n n (10.73)
i

i
IP
i
i
A total of n n moles of X is present, partly as X and partly in the ion pairs. Hence
z
i
the number of moles of X is
z
n n n n 3n 11 a2n 4n i (10.74)

IP
i
Dividing the equations for n and n by the solvent mass, we get as the molalities

m an m , m 3n 11 a2n 4m i (10.75)

i
Substitution of these molalities into Eq. (10.46) for m and the use of (10.45) and
i
(10.50) for n and n gives (after a bit of algebra—Prob. 10.43)

†
m m° nRT ln 1n g m >m°2 strong electrolyte (10.76)

i
i
i
†
†
q
g a n >n 31 11 a21n >n 24 n >n g , 1g 2 1 (10.77)

†
where the dagger indicates that g allows for ion pairing. The infinite-dilution value
† q
of g follows from its definition in (10.77) and from g 1 [Eq. (10.47)] and the fact
that the extent of ion pairing goes to zero at infinite dilution (a 1). If n n ,
q

†

then g ag (Prob. 10.42).
Equation (10.76) differs from (10.51) for an electrolyte with no ion pairing by the
†

replacement of g with g . Comparison of (10.51) and (10.76) shows that
†
g g if no ion pairing

This result also follows by putting a 1 in (10.77).
The degree of ion pairing is not always known, so a in (10.77) may not be known.
†

One therefore measures (see Sec. 10.6) and tabulates g , rather than g , for strong
†
electrolytes. The activity coefficient g deviates from 1 because of (a) deviations of
the solution from ideally dilute behavior and (b) ion-pair formation, which makes a in
†
(10.77) less than 1. Although it is actually g that is tabulated for strong electrolytes,
† †

tables (such as Table 10.2) use the symbol g for g . Strictly speaking g g only
for no ion-pair formation.
Taking ionic association into account improves the accuracy of the Debye–Hückel
equation. Formation of ion pairs reduces the number of ions in the solution. In calcu-
lating the ionic strength, one does not include ions that are associated to form neutral
ion pairs. Also, one uses (10.77) to relate the activity coefficient g calculated by the

†
Debye–Hückel theory to the experimentally observed activity coefficient g . These
procedures should also be followed when using the Davies equation. (See Prob.
10.46.) The Davies-equation example (Example 10.3) ignored ion pairing. This is a
reasonably good approximation for the dilute solutions in that example.
Taking ion pairing into account considerably complicates calculations in a solu-
tion of several electrolytes since many ion-pairing equilibria have to be taken into ac-
count and solved for. The equilibrium constants for ion-pairing reactions are often not
accurately known. Moreover, ion pairing cannot completely account for all effects of
ionic association because at high concentrations of ions with z 1, three ions may
associate with one another to form triple ions. To avoid the complications introduced
by ion pairing, some workers prefer to ignore ion pairing. Thus, the Pitzer model of
Sec. 10.7 assumes there are no ion pairs. The Pitzer parameters are chosen to fit