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ION–SOLVENT INTERACTIONS 205

used to express that part of the free energy of the solvation of ions which arises from
interactions outside the first, oriented, layers of dipoles near the ion. Thus, sufficiently
near the ion, the structure of the water is fairly definite and can be used to write
equations that express simple models close enough to reality to be credible (see
Appendices 2.2 and 2.3). A molecular picture is more difficult to sustain outside these
first one or two layers. It is argued that there it is better to work in terms of continuum
electrostatics and to suppress questions concerned with structure in the solution, etc.
The basic model upon which Born’s equation rests involves a mental image of a
metallic sphere. It is argued that when such a sphere (at first grounded and charge free)
is given an electric charge q, this charging process must be equivalent to some amount
of energy.
The reasoning is that when a series of small amounts of charge are brought upon
the ion, some work has to be done to put them there because after the first charge
arrives, the rest of the charge bits (all positive, say) have to push against the repelling
interaction between the positive charges themselves and the positive charge already
building up on the metallic sphere.
Now, from electrostatics, the work done, W, when there is a change of charge
of a body of potential, is given by

In the case of the conducting sphere upon which charge is building, the potential
depends upon the charge and so to avoid conceptual trouble [what to use in Eq.
(A2.1.1) as q changes], we take an infinitesimally small change of charge dq and argue
that for such very very small changes of charge, will be very very nearly constant.
To find the work W done in a real finite buildup of charge, one has to overcome
a problem—that itself depends on the degree of charge—and hence express in
terms of q.
It is easy to show that for a conducting sphere, the value of is given by

With this (and the assumptions) as background, one may write for the work to
build up a charge q on the sphere:

Now, solvation energy—and Born’s equation is usually proposed as giving at
least some part of that—is the difference of free energy of an ion in vacuo and that of
an ion in solution. If this work of charging which has been calculated above is then