Page 381 - MODERN ELECTROCHEMISTRY

ION–ION INTERACTIONS 317

This is a difficulty that occurs with many problems involving Coulombic interaction.
Mayer then conceived a most helpful stratagem. Instead of taking the potential of
an ion at a distance r as he took it as [where is the same
of Debye–Hückel; Eq. (3.20)]. With this approach he found that the integrals in his
theory, which diverged earlier [Eq. (3.165)] now converged. Hence, the calculation of
the interionic interaction energy—the interaction of a representative ion with both
negative and positive ions surrounding it—could yield manageable results.
There are many writers who would continue here with an account of the degree
to which experiments agree with Mayer’s theory and pay scant attention to the
25
justification of the move that underlies it. However, it is possible to give some basis
to Mayer’s equation (Friedman, 1989) because in a mixture of ions, interaction at a
distance occurs through many other ions. Such intervening ions might be perceived as
screening the interaction of one ion from its distant sister, and one feels intuitively that
this screening might well be modeled by multiplying the simple by for the
new term decreases the interaction at a given distance and avoids the catastrophe of
Eq. (3.165).
Does Mayer’s theory of calculating the virial coefficients in equations such as Eq.
(3.165) (which gives rise directly to the expression for the osmotic pressure of an ionic
solution and less directly to those for activity coefficients) really improve on the second
and third generations of the Debye–Hückel theory—those involving, respectively, an
accounting for ion size and for the water removed into long-lived hydration shells?
Figure 3.48 shows two ways of expressing the results of Mayer’s virial coefficient
26
approach using the osmotic pressure of an ionic solution as the test quantity. Two
versions of the Mayer theory are indicated. In the one marked the authors
have taken the Debye–Hückel limiting-law theory, redone for osmotic pressure instead
of activity coefficient, and then added to it the results of Mayer’s calculation of the
second virial coefficient, B. In the upper curve of Fig. 3.48, the approximation within
the Mayer theory used in summing integrals (the one called hypernetted chain or HNC)
is indicated. The former replicates experiment better than the latter. The two approxi-

25
One of the reasons for passing over the physical basis of the modified equation for the potential due to an
ion in Mayer’s view is that several mathematical techniques are still needed to obtain final answers in
Mayer’s evaluation of an activity coefficient. (To replace the ionic cloud, he calculates the distribution
of ions around each other and from this the sum of their interactions.) Among these occur equations that
are approximation procedures for solving sums of integrals. To a degree, the mathematical struggle seems
to have taken attention away from the validity of the modified equation for the potential due to an ion at
distance r. These useful approximations consist of complex mathematical series (which is too much detail
for us here) but it may be worthwhile noting their names (which are frequently mentioned in the relevant
literature) for the reader sufficiently motivated to delve deeply into calculations using them. They are, in
the order in which they were first published, the Ornstein–Zernicke equation, the Percus–Yevich equation,
and the “hypernetted chain” approach.
26
Since Mayer’s theory originated in a theory for imperfect gases, it naturally tends to calculate the nearest
analogue of gas pressure that an ionic solution exhibits—osmotic pressure.