Voltammetry at microelectrodes of reversible electrode reactions with complex stoichiometry: A general analytical theoretical framework

The complex theoretical problem of higher-order reversible electron transfers at electrodes and microelectrodes of different geometries is addressed by analytical mathematical methods. A novel and general approach is established, valid for uniformly and non-uniformly accessible electrodes, and it is particularized for the most common geometries (micro-discs, hemispheres, bands and cylinders) and stoichiometries (2:1, 1:2, 3:1 and 1:3). General and explicit solutions are derived for the surface concentrations, the half-wave potential and the current-potential response in any voltammetric technique under transient and stationary conditions, whatever the initial concentrations of the oxidized and reduced species. With the theoretical results obtained, the response in cyclic voltammetry (CV) is analyzed in detail for a variety of a:b stoichiometries of interest. The influence of the main working parameters (microelectrode shape and size, scan rate and bulk concentration of redox species) is investigated, pointing out similarities and differences between the stoichiometries considered and with respect to the 1:1 case. In relation to the similarities, in all cases the surface concentrations are only potential-dependent and the thickness of the linear diffusion layer takes the same value regardless of the stoichiometric coefficients. Also, the effects of the electrode size and the scan rate (v) are analogous for any a:b values, including the appearance of isopoint(s) between voltammograms of different scan rates, the potential and current of which have a remarkable diagnostic value. Regarding differences between stoichiometries, the peak current, the peak-to-peak separation and the shape of transient CV curves, as well as the slope of the stationary current-potential response, depend on the a:b values. Thus, simple elucidation procedures are proposed based on the peak-to-peak separation, on the current and potential of the isopoint(s), as well as on an extension of the Tomes criterion (vertical bar E-3/4 - E-1/4 vertical bar) to CV under transient and stationary conditions.