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314 CHAPTER 3

3.8.5. Activity Coefficients, Bjerrum’s Ion Pairs, and Debye’s Free Ions

What direct role do the ion pairs have in the Debye–Hückel electrostatic theory
of activity coefficients? The answer is simply: None. Since ion pairs carry no net
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charge, they are ineligible for membership in the ion cloud, where the essential
qualification is charge. Hence, ion pairs are dismissed from a direct consideration in
the Debye–Hückel theory.
This does not mean that the Debye–Hückel theory gives the right answer when
there is ion-pair formation. The extent of ion-pair formation decides the value of the
concentration to be used in the ionic-cloud model. By removing a fraction of the
total number of ions, only a fraction of the ions remain for the Debye–Hückel
treatment, which interests itself only in the free charges. Thus, the Debye–Hückel
expression for the activity coefficient [Eq. (3.120)] is valid for the free ions, with two
important modifications: (1) Instead of there being a concentration c of ions, there is
only the remainder is not reckoned with owing to association. (2) The
distance of closest approach of free ions is q and not a. These modifications yield

This calculated mean activity coefficient is related to the measured mean activity
coefficient of the electrolyte by the relation (for the derivation, see Appendix
3.6)

or

This equation indicates how the activity coefficient depends on the extent of ion
association. In fact, this equation constitutes the bridge between the treatment of
solutions of true electrolytes and that of solutions of potential electrolytes.

3.8.6. From Ion Pairs to Triple Ions to Clusters of Ions

The Coulombic attractive forces given by are large when the dielectric
constant is small. When nonaqueous solvents of low dielectric constant are used, the
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Remember that the equations for the Bjerrum theory as presented here are correct only for electrolytes
yielding ions of the same valence z., i.e., only for symmetrical 1:1- or 2:2-valent electrolytes.