Page 368 - MODERN ELECTROCHEMISTRY

304 CHAPTER 3

Using the rigorous solutions of the unlinearized P–B equation, one gets the cloud
contribution to the electrostatic potential at the central ion and when this value of the
electrostatic potential is used in the two charging processes to get the chemical-poten-
tial change arising from ion–ion interactions, it is found that the Guntelberg and
Debye charging processes give discordant results. As shown by Onsager, this discrep-
ancy is not due to the invalidity of either of the two charging processes; it is a symptom
of the logical inconsistency intrinsic in the unlinearized P–B equation.
This discussion of rigorous solutions has thus brought out an important point: The
disagreement between the chemical-potential change calculated by the Debye
and Guntelberg charging processes cuts off one approach to an improved theory of
higher concentrations for it prevents our using the unlinearized P–B equation, which
is needed when the concentration is too high for the use of

3.8. TEMPORARY ION ASSOCIATION IN AN ELECTROLYTIC
SOLUTION: FORMATION OF PAIRS, TRIPLETS
3.8.1. Positive and Negative ions Can Stick Together: Ion-Pair
Formation
The Debye–Hückel model assumed the ions to be in almost random thermal
motion and therefore in almost random positions. The slight deviation from random-
ness was pictured as giving rise to an ionic cloud around a given ion, a positive ion
(of charge being surrounded by a cloud of excess negative charge
However, the possibility was not considered that some negative ions in the cloud would
get sufficiently close to the central positive ion in the course of their quasi-random
solution movements so that their thermal translational energy would not be sufficient
for them to continue their independent movements in the solution. Bjerrum suggested
that a pair of oppositely charged ions may get trapped in each other’s Coulombic field.
An ion pair may be formed.
The ions of the pair together form an ionic dipole on which the net charge is zero.
Within the ionic cloud, the locations of such uncharged ion pairs are completely
random, since, being uncharged, they are not acted upon by the Coulombic field of
the central ion. Furthermore, on the average, a certain fraction of the ions in the
electrolytic solution will be stuck together in the form of ion pairs. This fraction will
now be evaluated.

3.8.2. Probability of Finding Oppositely Charged Ions near Each Other

Consider a spherical shell of thickness dr and of radius r from a reference positive
ion (Fig. 3.42). The probability that a negative ion is in the spherical shell is