Extended electron transfer and the Frumkin correction

It is now well established that the rate constant for electron transfer between a metal electrode and a given redox moiety (at distance x from the electrode) decays according to exp[ - betax]; values of beta are of the order of 10(8) cm(-1) and depend primarily upon the nature of the intervening medium. The present work demonstrates how this 'extended' electron transfer can modify the Frumkin correction for diffuse layer effects in the measurement of heterogeneous rate constants. We quantify the effect of this 'extended' electron transfer by comparing the electron transfer rate constants deduced with (k'(eff)) and without (k(eff)) extended electron transfer. Butler-Volmer kinetics and a single electron transfer are assumed. The ratio k'(eff)/k(eff) is deduced as a function of beta/kappa, where kappa is the reciprocal Debye length; (F/RT)phi (PCA) where phi (PCA) is the diffuse layer potential at the plane of closest approach; and z(ox) - alpha, where z(ox) is the charge of the oxidized redox species and alpha is the Butler-Volmer transfer coefficient. The deviation of k'(eff)/k(eff) from unity is greatest when the diffuse layer potential, phi (x), decays over distances of a few angstroms from the plane of closest approach, as is the case with higher concentrations of supporting electrolyte and/or large values of the diffuse layer potential. When /Z(ox)/ less than or equal to /z(se)/ and /Z(red)/ less than or equal to /Z(se)/ the classical Frumkin expression is applicable as long as -ln[1 + beta/kappa] less than or equal to F/RT phi (PCA)(z(ox) - alpha) less than or equal to ln[1 + beta/kappa] When /Z(ox)/ > /Z(se)/ or /Z(red)/ > /Z(se)/ and (F/RT)phi (PCA)(Z(ox) - alpha)< 0 redox ions can become a significant component of the diffuse layer and the classical Gouy-Chapman theory is no longer applicable. To avoid this complication and apply the classical Frumkin correction the condition to be met is 0 (F/RT)phi (PCA)(z(ox) - alpha) less than or equal to ln[1 + beta/kappa] thus ensuring that the redox species is not disproportionately accumulated in the diffuse layer. We also show that when beta/kappa less than or equal to 0.5, phi (PCA)Z(ox) greater than or equal to 0 and phi (PCA)z(red) greater than or equal to 0 then virtually all ET occurs outside the diffuse layer and there is no Frumkin effect. However, this condition is not easily achieved experimentally. (C) 2000 Elsevier Science B.V. All rights reserved.