The faradaic impedance for a simple electrode process

The general relation reflecting the charge transfer process, E =/(/, [O], [R] ), leads to 

The three partial derivatives describe the kinetics of the reaction and (dE/dl) is the charge transfer resistance, Rct. It can be shown, using Laplace transformation, that 

Since the a.c. perturbation is small, the linearized relation between current and overpotential, rj (equation (6.50)), considering ara = arc = 0.5, may be used, that is 

In this way, from (11.14), we can identify the Warburg impedance, Zw, of the Randles circuit as consisting of a resistance and a capacitance in series and where the components in phase Zw and out of phase Z w, with Z = Z + iZ , are given by 

A plot of Z and of —Z vs. co 1/2 should give straight lines of slope o and of intercept Rct for the in-phase component (Fig. 11.5) corresponding to infinite frequency. The physical explanation for the intercept is that at very high frequencies the time scale is so short that diffusion cannot influence the current, being dependent only on the kinetics. If the lines obtained are not parallel then either the theory cannot be applied, or the experimental accuracy is bad. 

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