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 h ω 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 .