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ION–ION INTERACTIONS 235

Hence, the complicated problem of the time-averaged distribution of ions inside
an electrolytic solution reduces, in the Debye–Hückel model, to the mathematically
simpler problem of finding out how the excess charge density varies with distance
r from the central ion.
An objection may be raised at this point. The electrolytic solution as a whole is
electroneutral, i.e., the netchargedensity is zero.Thenwhy isnot everywhere?
So as not to anticipate the detailed discussion, an intuitive answer will first be
given. If the central ion is, for example, positive, it will exert an attraction for negative
ions; hence, there should be a greater aggregation of negative ions than of positive ions
in the neighborhood of the central positive ion, i.e., An analogous situation, but
with a change in sign, obtains near a central negative ion. At the same time, the thermal
forces are knocking the ions about in all directions and trying to establish electroneu-
trality, i.e., the thermal motions try to smooth everything to Thus, the time
average of the electrostatic forces of ordering and the thermal forces of disordering is
a local excess of negative charge near a positive ion and an excess of positive charge
near a negative ion. Of course, the excess positive charge near a negative ion
compensates for the excess negative charge near a positive ion, and the overall effect
is electroneutrality, i.e., a of zero for the whole solution.

3.3.3. Charge Density near the Central Ion Is Determined by
Electrostatics: Poisson’s Equation

Consider an infinitesimally small volume element dV situated at a distance r from
the arbitrarily selected central ion, upon which attention is to be fixed during the
discussion (Fig. 3.8), and let the net charge density inside the volume element be
3
Further, let the average electrostatic potential in the volume element be The
question is: What is the relation between the excess density in the volume element
and the time-averaged electrostatic potential
One relation between and is given by Poisson’s equation (Appendix 3.1).
There is no reason to doubt that there is spherically symmetrical distribution of positive
and negative charge and therefore excess charge density around a given central ion.
Hence, Poisson’s equation can be written as

where is the dielectric constant of the medium and is taken to be that of bulk water,
an acceptable approximation for dilute solutions.

3
Actually, there are discrete charges in the neighborhood of the central ion and therefore discontinuous
variations in the potential. But because in the Debye–Hückel model the charges are smoothed out, the
potential is averaged out.