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3. Transport in electrolyte solutions

In the Introduction of this book, we explained that the analysis of transport processes in electrolyte solutions constitutes the salient part of electrochemistry. Because electric current can occur only in a closed circuit, it also flows across the electrolyte solution that contributes to the properties of an electrochemical cell. More importantly, transport of electroactive species provides the boundary condition for the solution of electrochemical problems, as will be seen later.

From the point of view of transport, an electrolyte solution can be roughly divided into two domains: the interior or the bulk solution and the polarization layer close to the electrode surface. The bulk solution is chemically homogeneous, merely determining the total resistance of the cell via its conductivity. The thickness of the polarization layer is of the order of a few microns where concentration changes take place. These changes are due to electrode reactions. Let us consider a simple one-electron transfer reaction

Cu2+   +   e -  \(\ce{ <=> }\)  Cu+ , (3.1)

where Cu2+ reduces to Cu+. As the reaction occurs on the electrode surface, the concentration of Cu2+ on the surface (x = 0) decreases and that of Cu+ simultaneously increases. Transport, in other words diffusion (and migration) tries to compensate for the emerging deficit of Cu2+ and excess of Cu+. The reaction rate, r, is a scalar while the flux of a species, J, is a vector although they both have the unit mol cm2 s-1. These two quantities with different tensorial dimensions are coupled via the mass balance and the Faraday law:

\(\displaystyle r=- \frac{i}{F}=-J_{\text{Cu}^{2+}}|_{x=0}=J_{\text{Cu}^+}|_{x=0} \) (3.2)

Equation (3.2) shows that an electrode reaction, i.e. electric current density, is the boundary condition of a transport problem. This is the specific feature – and the mathematical difficulty – of electrochemical analysis. In an electrochemical experiment, electric current, the boundary condition of a transport problem, is usually measured, not the concentration that is the quantity appearing in the transport equations. This means there is a need for the evaluation of concentration (and potential) gradients at the surface. Often numerical methods must be resorted to in order to solve a transport problem, and the calculation of numerical derivatives is prone to errors in accuracy.

The surface concentrations \( c_i^s \) adjust themselves so that the reaction rate equals to the rate of transport:

\( \displaystyle r=k_{\text{red}}c_{\text{Cu}^{2+}}^s-k_{ox}c_{\text{Cu}^{+}} \) (3.3)

where kred and kox are the reaction rate constants of reduction and oxidation respectively. If a reaction is very fast, i.e. reversible, its rate is limited by the transport of the reactant to the electrode. In that case the surface concentrations are coupled with the Nernst equation instead of Equation (3.3), but in all cases Equation (3.2) must be applied in the analysis.

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