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ION–ION INTERACTIONS 295

more than half of the water in the solution is associated with the ions, and a sharp
increase of activity coefficient, somewhat of a doubling in fact, would be expected to
express the increase in effective concentration of the ions. To what extent can this
rough sketch be turned into a quantitative model?

3.6.2. Quantitative Theory of the Activity of an Electrolyte as a
Function of the Hydration Number
The basic thought here has to be similar to that which lay beneath the theory of
electrostatic interactions: to calculate the work done in going from a state in which the
ions are too far apart to feel any interionic attraction to the state at a finite concentration
c at which part of the ions’ behavior is due to this. This work was then [Eq. (3.59)]
placed equal to RT ln f, where f is the activity coefficient, which was thereby calculated.
When one realizes that a reversal of the direction in which the activity coefficient
varies with concentration has to be explained by taking into account the removal of
some of the solvent from effectively partaking in the ionic solution’s activity, the
philosophy behind the calculation of this effect on the activity coefficient becomes
clear. One must calculate the work done in the changes caused by solvent removal and
add this to the work done in building up the ionic atmosphere. What, then, is the
contribution due to these water-removing processes? [Note that ions are hydrated at
all times in which they are in the solution. One is not going to calculate the heat of
hydration; that was done in Chapter 2. Here, the task is to calculate the work done as
a consequence of the fact that when water molecules enter the solvation sheath, they
are, so to speak, no longer operating as far as the solvent is concerned. It is to be an
type of calculation.]
As a device for the calculation, let it be assumed that the interionic attraction is
switched off. (Reasons for employing this artifice will be given.) Thus, there are two
kinds of work that must be taken into account.

1. The kind of work done when there is a change in concentration
of the free solvent water caused by introducing ions of a certain concentration
into the solution.
2. The kind of work done when there is a corresponding change in
concentration of the ions due to the removal of the water to their sheaths.
The work done for process 1 is easy to calculate. Before the ions have been added,
the concentration of the water is unaffected by anything; it is the concentration of pure
water and its activity, the activity of a pure substance, can be regarded as unity. After
the ions are there, the activity of the water is, say, Then, the work when the activity
of the water goes from 1 to is RT ln
However, one wishes to know the change of activity caused in the electrolyte by
this change of activity of the water. Furthermore, the calculation must be reduced to
that for 1 mole of electrolyte. Let the sum of the moles of water in the primary sheath
per liter of solution for both ions of the imagined 1:1 electrolyte be (for 1-molar