Page 344 - MODERN ELECTROCHEMISTRY

280 CHAPTER 3

The physical significance of is that at very low concentrations the ion
atmosphere has such a large radius compared with that of the ion that one need not
consider the ion as having a finite size a. Considering is tantamount to
reverting to the point-charge model.
One can now proceed rapidly to compare this theoretical expression for with
experiment; but what value of the ion size parameter should be used? The time has
come to worry about the precise physical meaning of the parameter a that was
introduced to allow for the finite size of ions.

3.5.4. The Ion Size Parameter a
One can at first try to speculate on what value of the ion size parameter is
appropriate. A lower limit is the sum of the crystallographic radii of the positive and
negative ions present in solution; ions cannot come closer than this distance [Fig.
3.31 (a)]. But in a solution the ions are generally solvated (Chapter 2). So perhaps the
sum of the solvated radii should be used [Fig. 3.3 l(b)]. However when two solvated
ions collide, is it not likely [Fig. 3.31 (c)] that their hydration shells are crushed to some
extent? This means that the ion size parameter a should be greater than the sum of the
crystallographic radii and perhaps less than the sum of the solvated radii. It should best
be called the mean distance of closest approach, but beneath the apparent wisdom of
this term there lies a measure of ignorance. For example, an attempted calculation of
just how crushed together two solvated ions are would involve many difficulties.
To circumventtheuncertainty in the quantitativedefinition ofa, it is bestto regard
it as a parameter in Eq. (3.120), i.e., a quantity the numerical value of which is left to
be calibrated or adjusted on the basis of experiment. The procedure (Fig. 3.32) is to
assume that the expression for log [Eq. (3.120)] is correct at one concentration, then
to equate this theoretical expression to the experimental value of log corresponding
to that concentration and to solve the resulting equation for a. Once the ion size
parameter, or mean distance of closest approach, is thus obtained at one concentration,
the value can be used to calculate values of the activity coefficient over a range of
other and higher concentrations. Then the situation is regarded as satisfactory if the
value of a obtained from experiments at one concentration can be used in Eq. (3.120)
to reproduce the results of experiments over a range of concentrations.

3.5.5. Comparison of the Finite-Ion-Size Model with Experiment

After taking into account the fact that ions have finite dimensions and cannot
therefore be treated as point charges, the following expression has been derived for
the logarithm of the activity coefficient: