Differential equation diffusion impedance

Example 2.6 Foundation for Warburg Impedance The following differential equation appears in the treatment of impedance associated with diffusion ... 

In this case the differential equation corresponds to diffusion normal and radial (two-dimensional) to the electrode. They obtained the following equations describing the faradaic impedance in the case of a slow charge transfer when only Ox is initially present in the solution, its concentration being Cq ... 

Cylindrical diffusion is observed, for example, during diffusion to a thin metallic wire or a carbon fiber electrode. Impedance of cylindrical electrodes was studied by Fleischmann et al. [165,166] and Jacobsen and West [167]. The partial differential equation describing diffusion to a cylinder, written for the oscillation concentration of an Ox form, is [144]... 

From the point of view of impedance spectroscopy, solutions of the diffusion equation are required in the frequency domain. The Laplace transform of Eq. (121) is an ordinary differential equation... 

An example of a transfer function based on a physical model is the Nemst impedance of a transport controlled electrode reaction. The impedance spectra in Fig. 7-14, which were obtained on a rotating platinum disk electrode at the equilibrium potential of the iron hexacyanoferrate redox system, exhibit the typical shape of a transport-controlled process. The transfer function cannot be described by a limited number of electrical circuit elements but must be derived from the differential equations of Fick s 2nd law and the appropriate boundary conditions. For finite linear diffusion, the so-called Nemst impedance Z can be derived theoretically... 

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