Current densities, preexponential

Thus, the recombination theory provided the first theoretical interpretation of the linear relation between polarization and the logarithm of current density that had been established experimentally. It is true, though, that the preexponential factor in Eq. (15.12) [2303(RTI2FI) 0.03 V] is four times smaller than the experimental values of slope b but it has been shown in later work that factors closer to the experimental values can be obtained when an inhomogeneous surface is assumed. 

Figure 6 Sensitivity analysis of the ionic current density (/) on various kinetic parameters. Left activation energy barrier (Ea) right preexponential factor (F0). Reprinted from Reference [120], copyright 2009, with permission from Elsevier.

The Arrhenius plots in Fig. 8 are also characterized by the preexponential current densities and parameters analogous to the preexponential factor in the Arrhenius equation. These preexponential current densities are defined from Eqs. (54), when the limiting condition T = 0 is introduced ... 

As follows from Eqs. (57), the preexponential current densities are dependent on the electrode potential, as schematically presented in Fig. 9. Combination of Eqs. (54) and (57) gives ... 

Figure 9. Schematic dependence of preexponential current densities on electrode potential for the cathodic and anodic directions of the reaction in Eq. (1), characterized by 7. EARP, Experimentally accessible region of potential.

Once the SHE is taken as the electrical reference potential in evaluations of the effect of temperature in electrode kinetics, combinations of kinetic parameters pertinent to the standard equilibrium state of electrode reactions may be related to the corresponding equilibrium data. For instance, taking the difference of the natural logarithms of the preexponential current densities from Eqs. (57) for = assuming cq = cr = and Pc Pa- y one obtains... 

The potential-dependent preexponential current densities are defined from Eqs. (62), for the limiting condition T = 0 ... 

Eq. (2.31) is identical to Eq. (2.18) derived for a majority carrier device (thermionic emission model). Accordingly, the same type of current-voltage curve is expected as that given in Fig. 2.7. The characteristics of the models occur only in the preexponential factors, which indeed are different in both cases (compare Eqs. 2.17 and 2.30). As mentioned before the jo of the majority carrier device is only determined by the barrier height and some physical constants (Eq. 2.19), whereas the y o of the minority carriers depends on material-specific quantities such as carrier density, diffusion constant and diffusion length. 

In the early days of the transition-state theory, there was great hope that this theory would allow quantitative predictions of rate constants, although this requires adequate preexponential factors and energy barriers for the reactions. The theories previously discussed are essentially concerned with the pre-exponential factors of Arrhenius-type equations. Currently energy barriers for unimolecular reactions are estimated by ab initio methods or by density functional theory (DFT). Such quantum-mechanical methods are particularly useful for small systems, but the numerical solution they offer are not easily related to processes of a similar kind. 

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