Implications of Marcus-Hush theory for steady-state heterogeneous electron transfer at an inlaid disk electrode

For an outer-sphere heterogeneous electron transfer, Ox + e = Red, between an electrode and a redox couple, the Butler-Volmer formalism predicts that the operative heterogeneous rate constant, k(red) (cm s(-1)) for reduction (or k(ox) for oxidation) increases without limit as an exponential function of -alpha (E - E(0)) for reduction (or (1 - alpha)(E - E(0)) for oxidation), where E is the applied electrode potential, alpha (~1/2) is the transfer coefficient and E(0) is the formal potential. The Marcus-Hush formalism, as exposited by Chidsey (Chidsey, C. E. D. Science 1991, 215, 919), predicts that the value of k(red) or k(ox) limits at sufficiently large values of -(E - E(0)) or (E - E(0)). The steady-state currents at an inlaid disk electrode obtained for a redox species in solution were computed using both formalisms with the Oldham-Zoski approximation (Oldham, K. B.; Zoski, C. G. J. Electroanal. Chem. 1988, 256, 11). Significant differences are noted for the two formalisms. When k(0)r(0)/D is sufficiently small (k(0) is the standard rate constant, r(0) is the radius of the disk electrode, and D is the diffusion coefficient of the redox species), the Marcus-Hush formalism effects a limiting current that can be significantly smaller than the mass transport limited current. This is easily explained in terms of the limiting values of k(red) and k(ox) predicted by the Marcus-Hush formalism. The experimental conditions that must be met to effect significant differences in behavior are discussed; experimental conditions that effect virtually identical behavior are also discussed. As a caveat for experimentalists, applications of the Butler-Volmer formalism to systems that are more properly described using the Marcus-Hush formalism are shown to yield incorrect values of k(0) and meaningless values of alpha, which serves only as a fitting parameter.