Theory of electrochemical hydrogen evolution reactions: Part II. Model of harmonic proton terms of different frequencies

Abstract

The theory of simultaneous electron and proton transfer in polar media developed in Part I [1] is extended to the case of harmonic proton terms with different frequencies in the initial, ω 1, and final, ω 2, states. The dependence of the transfer probability w on the free energy of the reaction Δ G is characterized by the transfer coefficient α, which decreases from unity to zero as Δ G decreases, i.e. the overvoltage increases. The relations derived in ref. 1 are shown to be unconnected with the special choice of the terms. The dependence of w on Δ G that is obtained reduces to the relations derived in ref. 1 in a certain range of Δ G; in another range of Δ G it reduces to the generalized Marcus formula with a pre-exponential factor weighted by the Frank-Condon factor of the proton transfer. The latter is valid for the limited range of Δ G lying within the normal discharge region. Outside this region the relations derived in ref. 1 are valid. The results of the theoretical calculations for ω 2/ω 1=0.25–0.30 and hω 2/ kT=3–4 are in good agreement with the experimental data on hydrogen ion discharge on a mercury electrode. The relations derived in ref. 1 are shown to be valid in three discharge regions with Δ G also varying over 150 kJ mol −1.