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ION–ION INTERACTIONS 233

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is a particular time-averaged spatial distribution of the ions. The number of particles
involved is enormous, and the situation appears far too complex for mathematical
treatment.
However, there exist conceptual techniques for tackling complex situations. One
of them is model building. This involves conceiving a model that contains only the
essential features of the real situation. All the thinking and mathematical analysis is
done on the (relatively simple) model and then the theoretical predictions are compared
with the experimental behavior of the real system. A good model simulates nature. If
the model yields wrong answers, then one tries again by changing the imagined model
until one arrives at a model, the theoretical predictions of which agree well with
experimental observations.
The genius of Debye and Hückel lay in their formulation of a very simple but
powerful model for the time-averaged distribution of ions in very dilute solutions of
electrolytes. From this distribution they were able to obtain the electrostatic potential
contributed by the surrounding ions to the total electrostatic potential at the reference
ion and hence the chemical-potential change arising from ion–ion interactions
[Eq.(3.3)]. Attention will now be focused on their approach.
The electrolytic solution consists of solvated ions and water molecules. The first
step in the Debye–Hückel approach is to select arbitrarily any one ion out of the
assembly and call it a reference ion or central ion. Only the reference ion is given the
individuality of a discrete charge. What is done with the water molecules and the
remaining ions? The water molecules are looked upon as a continuous dielectric
medium. The remaining ions of the solution (i.e., all ions except the central ion) lapse
into anonymity, their charges being “smeared out” into a continuous spatial distribu-
tion of charge (Fig. 3.6). Whenever the concentration of ions of one sign exceeds that
of the opposite sign, there will arise a net or excess charge in the particular region
under consideration. Obviously, the total charge in the atmosphere must be of opposite
sign and exactly equal to the charge on the reference ion.
Thus, the electrolytic solution is considered to consist of a central ion standing
alone in a continuum. Thanks to the water molecules, this continuum acquires a
dielectric constant (taken to be the value for bulk water). The charges of the discrete
ions that populate the environment of the central ion are thought of as smoothed out
and contribute to the continuum dielectric a net charge density (excess charge per unit
volume). In this way, water enters the analysis in the guise of a dielectric constant
and the ions, except the specific one chosen as the central ion, in the form of an excess
charge density (Fig. 3.7).

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Using an imaginary camera (with exposuretime of ), suppose that it were possible to take snapshots
of the ions in an electrolytic solution. Different snapshots would show the ions distributed differently in
the space containing the solution, but the scrutiny of a large enoughnumber of snapshots (say, ) would
permit one to recognize a certain average distribution characterized by average positions of the ions; this
is the time-averaged spatial distribution of the ions.