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ION–SOLVENT INTERACTIONS 169

Now, the reader will probably be able to see there is a flaw here. Where? The error
is easily recognized if one recalls the Debye argument for the average moment of a
gas dipole. What is the guarantee that a water dipole far from the ion is aligned parallel
to the ionic field? What about the thermal motions that tend to knock dipoles out of
alignment? So what matters is the average dipole moment of the molecules in the
direction of the ionic field. Thus, one has to follow the same line of reasoning as in
the treatment of the dielectric constant of a polar liquid and think in terms of the average
moment of the individual molecule, which will depend in Debye’s treatment on
the interplay of electrical and thermal forces, and in Kirkwood’s treatment also on
possible short-range interactions and associations of dipoles. One has therefore

Further, with α now the orientation polarizability,

(2.142)
so that

This expression for the work of replacing a water molecule by a nonelectrolyte
molecule at a distance r from an ion can now be introduced into Eq. (2.138) to give
(in number of nonelectrolyte molecules per unit volume)

The exponent of Eq. (2.144) is easily shown to be less than unity at 298 K for
most ions. Thus, for distances outside the primary hydration shell of nearly all ions,
the field X will be sufficiently small (because of the large dielectric constant of bulk
water), so one can expand the exponential and retain only the first two terms, i.e.,

Thus, the excess number per unit volume of nonelectrolyte molecules at a distance
r from the ion is (in number of molecules per unit volume)