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ION–ION INTERACTIONS 237

concentration change. Since the potential energy U relates to the time average of the
forces between ions rather than to the actual forces for a given distribution, it is also
known as the potential of averageforce.
If there are no ion–ion interactional forces, U = 0; then, which means that
the local concentration would be equal to the bulk concentration. If the forces are
attractive, then the potential-energy change U is negative (i.e., negative work is done
by the hypothetical external agency) and there is a local accumulation of ions
in excess of their bulk concentrations. If the forces are repulsive, the potential-energy
change is positive (i.e., the work done by the external agency is positive) and
there is local depletion of ions.
In the first instance, and as a first approximation valid for very dilute solutions,
one may ignore all types of ion–ion interactions except those deriving from simple
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Coulombic forces. Thus, short-range interactions (e.g., dispersion interactions) are
excluded. This is a fundamental assumption of the Debye–Hückel theory. Then the
potential of average force U simply becomes the Coulombic potential energy of an ion
of charge in the volume element dV, i.e., the charge on the ion times the
electrostatic potential in the volume element dV. That is,

The Boltzmann distribution law (3.7) thus assumes the form

Now that the concentration of the ionic species i in the volume element dV, has
been related to its bulk concentration the expression (3.6) for the excess charge
density in the volume element dV becomes

3.3.5. A Vital Step in the Debye–Hückel Theory of the Charge
Distribution around Ions: Linearization of the Boltzmann
Equation
At this point of the theory, Debye and Hückel made a move that was not only
mathematically expedient but also turned out to be wise. They decided to carry out the

4
In this book, the term Coulombic is restricted to forces (with dependence on distance) which are based
directly on Coulomb’s law. More complex forces, e.g., those which vary as or may occur as a net
force from the combination of several different Coulombic interactions. Nevertheless, such more complex
results of the interplay of several Coulombic forces will be called non-Coulombic.