In the beginning. . . there were experimental kinetics studies. “It was found that isotopic exchange between ions differing only in their valency are generally slow if single cations are involved and fast if the ions are relatively large, such as complex ions” [26].
W. Libby [27] surmised that this behavior was dependent on the orientation of the solvent dipoles around the ions. Immediately after an ET event, the charges on the ions change and the solvent dipoles around them are not anymore in electrostatic equilibrium with the charges (vide infra). This means that the new state, not being in stable equilibrium, is of higher energy than the original and this fact could explain the slow reaction velocity due to the high activation energy barrier for the small ions’ reactions, being more highly solvated than the big ones. The insight of Libby, that the barrier depended on the nonequilibrium polarization of the solvation molecules, was correct. Moreover, he was right in thinking that the quantitative explanation should have been found in applying the Franck–Condon (FC) principle. But he was wrong in the way he applied it [28].
At this point, a brief reminder on molecular polarization is in order.
An electric field—that of an ion in particular—induces in a molecule electronic, atomic and orientational polarization. The electronic polarization is due to the shift of electrons relative to the nuclei, atomic polarization means that atoms are displaced relative to one another [29], with consequent variation of interatomic distances, bond lengths, and angles. The orientational polarization is due to the orienting effect on the molecular dipole by the directing electric field. Marcus uses the symbol Pe for the electronic polarization that he designated as of “E type,” while the symbol Pu is collectively used for atomic and orientation polarization, which is of “U type.”
The relaxation times are of the order of 10−15 sec for electronic polarization, 10−13 sec for the atomic and 10−11 sec or slower for orientational polarization [26].
The potential energy of orientation between an ion and a dipole [30] is given by:
where θ is the angle between the direction of the electric field E of the ion and that of the molecular dipole p. The ion–dipole system is in electrical equilibrium when θ = 0, that is, when the dipole is lined up with the field. In this case, the direction of the dipole is the one dictated by the electric field and U is at its minimum value.
When a molecule is in a medium, the orienting effect of the electric field is counteracted by the thermal agitation of the molecules, by their mutual interactions and orientation correlation of neighboring molecules [29] so that at temperature T the average equilibrium θ of the solvation molecules in the ion’s field may be different from zero. Moreover—and this fact is of paramount importance for the ET theory—the instantaneous polarization continuously fluctuates around its equilibrium value because of thermal agitation, a situation reminiscent of the vibrations of a harmonic oscillator around its equilibrium geometry.
In the first model used by Marcus for the reacting ions, they are supposed to be spheres of radii a1 and a2. The spheres are rigid, formed by the bare ions and possibly by a spherical region of saturated dielectric made up by solvent molecules fully oriented in the ions’ fields. Outside these saturated spheres are the molecules whose Pu polarization is determined, as described above, by the counteracting ordering and disordering effects of electric fields and thermal agitation.
Initially, under the influence of Born’s description of the charging of ions in solution, M. considered the spheres as conducting. This restrictive hypothesis was dropped later on.
1.3.Solvation Molecules’ Contribution to the Barrier in ET Reactions
In order to understand—on a qualitative level—the barrier to ET reactions due to the orientation of the solvent molecules around the ions, let us first consider the barrier to reaction for the most simple threecenter atom transfer reaction involving a linear activated complex, that is the H exchange reaction:
The energy barrier to reaction is due to the fact that an activation energy is necessary to go from the reagents to the products because a chemical bond between two hydrogen atoms is to be broken while another one forms. If we imagine, for simplicity, that the reaction happens on a line, we may represent schematically the process as:
Fig. 2∗. (Adapted from Ref. [31]).
where R stands for the reagents, P for the products and the double dagger symbol “‡” for the transition state at the nuclear configuration intermediate between that for R and P.
The energy barrier to reaction is represented in the simplest possible way as in Fig. 2* [31–33]. The potential energy at the maximum along the reaction coordinate corresponds to the nuclear configuration of the TS. Note though that, contrary to the gradual process in Eq. (1.19), the jump of the electron in the case of ET is an abrupt process.
Imagine now having a system of two ions of charges +3 and +2, for example, of iron, in a polar solvent, and that an ET reaction happens between them. The distribution of orientations of solvation dipoles around ions with different charge is different because the directing electric field is greater for more highly charged ions and so the average angle θ is smaller for the dipoles around the ion with charge +3 than for those around the one of charge +2. After the charge exchange reaction, the equilibrium orientational distribution of the dipoles around the ions is also switched. This is described in the ET literature by means of figures where the authors either pictorially show the different orientation of the solvent molecules around the ions [34] or more simply represent the dipoles by swarms of small ellipses [35] or arrows [36–38]. The number of arrows varies from many to three used by Krishtalik and two by Kuznetsov. It is possible to give the reader the sense of the shift in dipole orientation using a single arrow, as suggested by Eq. (1.18), to represent the average dipoles distribution.
Fig. 3∗.
In