Dependence on the electrode potential Tafel plots

In deriving Eqs (17.11) and (17.12) we did not address the imposed electrode potential but the information must implicitly exist in the electronic energies that appear in these equations. To make this information explicit we redefine the electronic energies Ea, E, Ej, as the values at equilibrium relative to some specified fixed 

Taking the integration limits to infinity rests on the assumption that the integrand is well contained within the metallic band. 

This argumenl is used in the electrochemistry literature, but it is only qualitative since it disregards the role of the reorganization energy in determining the free energy. Indeed, if we use the zero-temperature approximation forthe Fermi functions in (17.14) we find that the equality kb a = Ei— t-which must be satisfied at equilibrium, leads to Eab = only for 0. 

If the reorganization energy Er is large relative to keT while — erj is small/ we can use the fact that the integrand vanishes quickly when E — er) increases beyond keT to make the expansion Er + E — = E + 2Er E — erj), which leads to 

The current at the electrode where oxidation a —r b takes place is referred to as the anodic current. If the density Ca of the reduced species a is kept constant near the electrode, the current is = ekb aCa- The result (17.16) predicts that under the specified conditions (large Er, small z/) a logarithmic plot of the current with respect to erf/iksT increases linearly with the overpotential z , with a slope 1/2. This behavior is known in electrochemistry as TafeTs law, and the corresponding slope is related to the so called TafeTs slope. An example where this law is quantitatively observed is shown in Fig. 17.2. In fact, a linear dependence of log(/) on the overpotential is often seen, however the observed slope can considerably deviate from and is sometimes temperature-dependent. Observed deviations are usually associated with the approximations made in deriving (17.16) from (17.15) and may be also related to the assumption made above that all the overpotential is realized as a potential drop between the molecule and the metal, an assumption that is better satisfied when the ionic strength of the solution increases. 

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