Tafel Slopes and Proton Tunneling

In electrochemical proton transfer, such as may occur as a primary step in the hydrogen evolution reaction (h.e.r.) or as a secondary, followup step in organic electrode reactions or O2 reduction, the possibility exists that nonclassical transfer of the H particle may occur by quantum-mechanical tunneling. In homogeneous proton transfer reactions, the consequences of this possibility were investigated quantitatively by Bernal and Fowler and Bell, while Bawn and Ogden examined the H/D kinetic isotope effect that would arise, albeit on the basis of a primitive model, in electrochemical proton discharge and transfer in the h.e.r. 

More detailed calculations of these effects were given later by Christov and Conway, who calculated proton tunneling probabilities through an Eckart barrier, the height of which was varied with potential. This gave a Tafel relation, as shown in Fig. 13, for proton transfer at a cathode for the case of complete tunneling control. In practice, both classical and nonclassical transfer occur in parallel to relative extents dependent on temperature. 

Arrhenius equation for the rate constant of any process where the condition kT is often involved and (2) in H-bonded solvents, OH vibrations are coupled with librations and intermolecular modes giving a wide manifold of states within which activation can arise—cf. the known heat-capacity behavior of H (aq). 

Unfortunately, the quantum-mechanical tunneling approach does not lead to any better understanding of why a should be temperature dependent in many reactions, if only for the reason that in most of the cases where such behavior is observed, the reactions involve heavy particles for which quantum effects are negligible. O2 reduction, however, could be quantally controlled if 

In the h.e.r. case at Hg, where b is of the form RT/p F + K, it was suggested that the T-independent component, X, of b (the intercept at T = 0 K of a plot of b vs. T) might be connected with a residual, nonclassical, transfer rate that was potential dependent at the absolute zero. However, since the rate, expressed as /, always involves a reciprocal form of b (the Lefat slope), namely, 

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