Semi-infinite linear diffusion, mathematics

In a typical spectroelectrochemical measurement, an optically transparent electrode (OTE) is used and the UV/vis absorption spectrum (or absorbance) of the substance participating in the reaction is measured. Various types of OTE exist, for example (i) a plate (glass, quartz or plastic) coated either with an optically transparent vapor-deposited metal (Pt or Au) film or with an optically transparent conductive tin oxide film (Fig. 5.26), and (ii) a fine micromesh (40-800 wires/cm) of electrically conductive material (Pt or Au). The electrochemical cell may be either a thin-layer cell with a solution-layer thickness of less than 0.2 mm (Fig. 9.2(a)) or a cell with a solution layer of conventional thickness ( 1 cm, Fig. 9.2(b)). The advantage of the thin-layer cell is that the electrolysis is complete within a short time ( 30 s). On the other hand, the cell with conventional solution thickness has the advantage that mass transport in the solution near the electrode surface can be treated mathematically by the theory of semi-infinite linear diffusion. 

The mathematics of semi-infinite linear diffusion were given by eqns. (19a—d). The Laplace transforms of these equations are... 

As in the previous section, we have assumed semi-infinite linear diffusion to a planar electrode throughout the mathematical discussion here. With a reversible dc process, the effects of sphericity and drop growth at the DME are exactly as discussed in Section... 

When dealing with systems with unidirectional geometry, numerous authors would apply the term semi-infinite linear diffusion to the example in question. However, in this book we prefer to use the term unidirectional semi-infinite diffusion because the word linear is ambiguous. It is also often used in opposition to the non-linear diffusion, that is to say, it is applied In the mathematical sense. The second Pick law, as It is usually written (the same as in this document), represents a linear differential equation, and Is based on the assumption that the diffusion coefficient is a constant. When this approximation does not apply, the mass balance is dC/dt=d/dx(DdC/dx) and then the differential equation is usually non-linear. 

For all reactions, the mass transport regime is controlled by the diffusion of the reacting ligand only, as the mercury electrode serves as an inexhaustible source for mercury ions. Hence, with respect to the mathematical modeling, reactions (2.205) and (2.206) are identical. This also holds true for reactions (2.210) and (2.211). Furthermore, it is assumed that the electrode surface is covered by a sub-monomolecular film without interactions between the deposited particles. For reactions (2.207) and (2.209) the ligand adsorption obeys a linear adsorption isotherm. Assuming semi-infinite diffusion at a planar electrode, the general mathematical model is defined as follows ... 

Fick s first and second laws (Equations 6.15 and 6.18), together with Equation 6.17, the Nernst equation (Equation 6.7) and the Butler-Volmer equation (Equation 6.12), constitute the basis for the mathematical description of a simple electron transfer process, such as that in Equation 6.6, under conditions where the mass transport is limited to linear semi-infinite diffusion, i.e. diffusion to and from a planar working electrode. The term semi-infinite indicates that the electrode is considered to be a non-permeable boundary and that the distance between the electrode surface and the wall of the cell is larger than the thickness, 5, of the diffusion layer defined as Equation 6.19 [1, 33] ... 

As in the previous chapter, the semi-infinite diffusion at a planar electrode is considered, where the adsorption is described by a linear adsorption isotherm. The modeling of reaction (2.173) does not require a particular mathematical procedure. The model comprises equation (1.2) and the boundary conditions (2.148) to (2.152) that describe the mass transport and adsorption of the R form. In addition, the diffusion of the O form, affected by an irreversible follow-up chemical reaction, is described by the following equation ... 

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