Electrochemical Electron Transfer: From Marcus Theory to Electrocatalysis - Fuel Cell Catalysis: A Surface Science Approach - page 38

From these potential energy curves, the reaction rate can be calculated with the
aid of Kramers theory. In the limit of a high solvent friction g, the rate is given by
Kramers [1940] and Zusman [1980]

k ¼ gexp E act
k B T

(2 : 10)

where the energy of activation is obtained by finding the saddle point from (2.9). In
contrast to (2.4), this expression does not contain an integral over 1; this has already
be performed in the calculation of the energy from the density of states.
Thus, for weak electronic interactions, the pre-exponential factor is proportional to
the strength of the electron interaction; in contrast, in the adiabatic limit, it is deter-
mined by the solvent friction. These two cases are bridged by a region with a
mixed regime. A typical plot of the dependence of the rate on the interaction strength
D is shown in Fig. 2.5. For small interactions, the rate is proportional to D; with
increasing D, it reaches a plateau, where it becomes almost constant. This is the so-
called weakly adiabatic region, in which the Marcus and Hush theories are valid.
On further increase, the energy of activation is lowered, and the rate increases again.
In typical outer sphere electron transfer on metal electrodes, D is in the weakly adia-
batic region and thus sufficiently large to ensure adiabaticity, but too small to lead to a
noticeable reduction of the activation energy. In this case, the rate is determined by
solvent reorganization, and is independent of the nature of the metal [Iwasita et al.,
1985; Santos et al., 1986].

Figure 2.5 Electron transfer rate as a function of the electronic interaction D. The full line is
the prediction of first-order perturbation theory. The upper points correspond to a solvent with a
low friction; the lower points to a high friction. The data have been taken from Schmickler and
Mohr [2002].