Implications of extended heterogeneous electron transfer: Part I. Coupling of semi-infinite linear diffusion and heterogeneous electron transfer with a decaying exponential dependence upon distance from the outer Helmholtz plane

Abstract

For a simple redox couple, ▪ the presumption is made that heterogeneous electron transfer can occur over some distance, x, from the outer Helmholtz plane (OHP) with a probability: k s, x, E = k f, x, E + k b, x, E = k s,0, E exp[− x/δ] The implications of this extended heterogeneous electron transfer (EHET) coupled to semi-infinite linear diffusion are analyzed. For a controlled potential perturbation from equilibrium, Δ E eq, the system response is characterized by the variables k*= k s,0, E δ 2/ D (where D = the diffusion coefficient of species A and B) and t q *= Dt/δ 2. The net effect of EHET is to establish electrochemical equilibrium over a region, Δ x, adjacent to the OHP (classical heterogeneous electron transfer establishes the equilibrium only at the OHP). When k*<0.1 or k* 2 t* q <0.01 there is virtually no distinction between classical heterogeneous electron transfer (occurring only at the OHP) or EHET. When k* > 1 and t* q > 100 the entire concentration perturbation is shifted away from the OHP by Δ x where Δ x=δ(1+ln[ k*]). The result of this shift is the addition of a capacitive component, C EHET= eΔ c eqΔ x/Δ E eq, to the double layer capacity, C d1, where Δ c eq is the maximum change in the concentration of A or B at or near the OHP. For a given set of experimental conditions (i.e., fixed Δ E eq, [B] eq/[A] eq) a plot of the measured total capacitance, C T(= C EHET+ C d1), vs. c b (=[A] eq+[B] eq) is shown to be linear with intercept C d1. When k* > 1 (a prerequisite for the observation of the effects of EHET) practical considerations of electrochemical methodology and of possible values for k s,0, E , δ, and D, suggest that the effects of EHET may be experimentally observed when: (1) c b > 10 −4 mol cm −3 thereby effecting C EHET∼ C d1; and (2) D < ∼ 10 −10 cm 2 s −1 thereby increasing the probability that k* > 1 and facilitating the (experimental) separation of diffusive and capacitive current (or charge) components.