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230 CHAPTER 3

3.3. THE DEBYE–HÜCKEL (OR ION-CLOUD) THEORY OF ION–ION
INTERACTIONS

3.3.1. A Strategy for a Quantitative Understanding of Ion–Ion
Interactions

The first task in thinking in detail about ion–ion interactions is to evolve a
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quantitative measure of these interactions. One approach is to follow a procedure
similar to that used in the discussion of ion–solvent interactions (Section 2.4). Thus,
one can consider an initial state in which ion–ion interactions do not exist (are
“switched off”) and a final state in which the interactions are in play (are “switched
on”). Then, the free-energy change in going from the initial state to the final state can
be considered the free energy of ion–ion interactions (Fig. 3.3).
The final state is obvious; it is ions in solution. The initial state is not so
straightforward; one cannot take ions in vacuum, because then there will be
ion–solvent interactions when these ions enter the solvent. The following approach is
therefore adopted. One conceives of a hypothetical situation in which the ions are there
in solution but are nevertheless not interacting. Now, if ion–ion interactions are
assumed to be electrostatic in origin, then the imaginary initial state of noninteracting
ions implies an assembly of discharged ions.
Thus, the process of going from an initial state of noninteracting ions to a final
state of ion–ion interactions is equivalent to taking an assembly of discharged ions,
charging them up, and setting the electrostatic charging work equal to the free energy
of ion–ion interactions (Fig. 3.4).
One point about the above procedure should be borne in mind. Since, in the
charging process, both the positively charged and negatively charged ionic species are
charged up, one obtains a free-energy change that involves all the ionic species

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The question of how one obtains an experimental measure of ion–ion interactions is discussed in Section
3.4.

Fig. 3.3. The free energy of ion–ion
interactions is the free-energy change in
going from a hypothetical electrolytic solu-
tion, in which ion-ion interactions do not
operate, to a real solution, in which these
interactions do operate.