The theory presented in this chapter for ET in electrode processes is a continuation of that for homogeneous ET reactions. Like the theory for homogeneous (redox) ET reactions, even this theory “proceeds from first principles plus assumptions that appear reasonable on a priori grounds and is free from arbitrary assumptions and adjustable parameters. As before, the theory is not applicable to atom transfer mechanisms (hydrogen overvoltage, for example)” [2, p. 182].
4.8.The ET Rate Constants
“Equations describing any overall electrochemical process may include terms for
(i)The transport of ions to the electrochemical double layer region at the electrode for any chemical reaction of the electrochemically active species
(ii)The work required to penetrate the double layer (if necessary)
(iii)The actual ET
Only under certain conditions can these facts be disentangled in a relatively simple way and simple ET rates defined: The double layer region should be sufficiently thin that
(1)No chemical reaction occurs in it
(2)Diffusion across it is sufficiently rapid so that the concentration of an ion at any point in the double layer is related to its concentration just outside it by the work required to transport it to that point.
At appreciable salt concentration, the double layer thickness appears to be only of the order of several Ångstroms [14]. Under the earlier conditions, one can describe the entire electrochemical process by force-free differential equations [14] (containing any chemical reaction terms [16] if necessary) outside of the double layer region, while the boundary condition, at a boundary surface S drawn just outside the double layer, contains rate constants which depend only on the ET process itself. This boundary condition is that the flux of any ion through this surface S equals its net rate of disappearance by ET. If A and B represent the electrochemically active ion before and after its electron transfer with the electrode, the ET step may be written as:
where n is the number of electrons lost by the electrode.
If
Net ET per unit area =
4.9.Basic Assumptions
The basic assumptions used in this extension of the Marcus theory to the electrode systems are analogous to those made in the oxidation–reduction theory for homogeneous systems [4].
The assumptions are
(a)“In the activated state of reaction (4.19), the spatial overlap between the electronic orbitals of the two ‘reactants’ is assumed to be small. The ‘reactants’ are now the electrode and the discharging ion or molecule (to be referred as ‘central ion’).” This assumption was previously discussed in Chapter 1.
(b)The central ion in this 1957 early form of the theory is treated as “a sphere within which no changes in interatomic distances occur during reaction (4.19) and outside of which the solvent is treated as a dielectrically unsaturated continuum. For complex ions or hydrated monoatomic cations, the sphere includes the first coordination shell of the central atom [4, 17].” [2, p. 183] The concept of “effective size” for unsymmetric organic molecules has been described in Chapter 3. In a refinement of the theory, changes, if any, in interatomic distances during ET were considered by Marcus, and were briefly mentioned in Chapter 3 and will be further dealt with later in Chapter 6.
4.10.Nature of the Transition State
As in the case of homogeneous ET, a successful ET between the central ion and the electrode proceeds via two successive intermediate states, X* and X, both participating in the ET process. X* represents the state of the reactants just before ET, X that of the products just after ET while a linear combination of the two wave function of the states X* and X represents the TS. “These states have the same atomic configuration and the same total energy but in X* the electronic configuration in the central ion is that of A and in X that of B. The electronic configuration of the electrode undergoes a corresponding change” because of the Franck–Condon principle as was already observed in Chapter 1. “By arguments similar to those employed in the redox theory” dealt with in the preceding Chapters, “it then follows that the actual ET, that is, the formation of X from X*, must be preceded by a reorientation of the solvent molecules in the vicinity of the discharging ion and nearby area of the electrode. For similar reasons, a change in ionic atmosphere in this region also precedes ET. The new configuration of the solvent and ionic atmosphere, which is the same in X* and X, will prove to be intermediate between that of the initial state and that in the final state. As such, it is not that which is predictable by the charge distribution in X* or by that in X. That is, it is not in electrostatic equilibrium with either charge distribution, and its properties cannot be described by the usual electrostatic expressions. Instead, expressions which take this nonequilibrium behavior into account must be used. Once again [4] there are an infinite number of pairs of thermodynamic states, X* and X, satisfying the constant energy-atomic configuration restriction and we are interested in determining the properties of the most probable pair, that is, the one with the minimum free energy of formation from the initial state. This is done by minimizing the expression for the free energy of formation subject to the restriction that X* and X have the same atomic configuration and the same energy” [2, pp. 183–184].