Simple electron-transfer processes

Initially a simple reversible one-electron oxidation process is examined [see (19)], such as the oxidation of ferrocene to the ferricinium cation in acetonitrile/0.1 M (C4H9)4NC104 (Sharp et al., 1980 Kadish et al., 1984). In (19), initially only A is present in solution. At the usual macrodisc electrode (radius in the millimetre range), material reaches the electrode by linear diffusion which is perpendicular to its surface (x-direction), and the concentrations of A and B may be obtained as a function of time by solving Pick s second law of diffusion as applied to species A and B, (30) and (31). 

However, the problem is subject to a number of boundary conditions which are defined in Table 4 (the symbols are described in the appendix). 

The time variation of the electrode potential is given by (28) and (29). Details of the solution of (30) and (31) are beyond the scope of this review, although a general approach for solving voltammetric problems is discussed in Section 7. 

The cyclic voltammogram shown in Fig. 14 is obtained for the reversible system described in (19) on scanning from an initial potential ( i) which is 

Time coordinate Spatial coordinate Species A Species B Reason for boundary condition 

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The basic assumption of the Marcus theory of electron transfer is that only a weak interaction between the reactants is needed in order for a simple electron-transfer process to occur. Marcus theory considers... 

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 shown in Fig. 7.44, in the absence of coupled reactions (the so-called E mechanism), the voltammetry is insensitive to changes in the concentration of species L provided that this does not lead to significant changes in the ionic strength. Therefore, the study of the position of the voltammograms in the presence of different concentrations of species L offers a simple criterion to discriminate between simple electron transfer processes and those complicated by coupled chemical equilibria. 

In these equations occ and oca are the cathodic and anodic charge transfer coefficients and are a measure of the symmetry of the activation barrier, being close to 0.5 for a metallic electrode and a simple electron transfer process. As mentioned above, the standard rate constant is the rate constant at E = E... 

In Sections 6.3-6.5 expressions for the analysis of the voltammograms corresponding to the simple electron transfer process O + ne-— R, obtained for uniformly accessible electrodes such as the rotating disc electrode, were presented. In this section these expressions will be applied to hydrodynamic electrodes in general. 

For a CE mechanism, the electrode product of interest is formed via an initial chemical reaction. Consequently, the measured limiting current will directly correlate with the amount of electroactive product formed on the time-scale of the experiment. Thus, sufficiently slow rates of mass transport result in complete conversion of bulk material to electroactive product and under this condition the limiting current will be identical to that calculated from the expressions described in Table 5 for a simple electron transfer process (see Fig. 28a). As the electrode angular velocity (a>) or flow rate (Tf) increases, less of the material reaching the electrode will have converted into... 

The fact that US influences the mechanism of chemical reactions via the action of highly reactive radicals such as OH- and H- formed during solvent sonolysis is well known (see Chapter 7). Solvents sensitive to thermolysis or sonolysis (e.g. dimethylformamide [158], dimethylsulphoxide [159]) decompose slowly in the presence of intense US. Thus, radical species formed by cavitation are highly reactive and may participate as activators or enhancers in the electrode process [160]. In fast, qt/asr-reversible or irreversible systems, however, the only effect of US is to enhance mass transport without any direct effect on the rate of simple electron transfer processes. 

The two major classes of voltammetric technique 4 Evaluation of reaction mechanisms 6 General concepts of voltammetry 6 Electrodes roles and experimental considerations 8 The overall electrochemical cell experimental considerations 12 Presentation of voltammetric data 14 Faradaic and non-Faradaic currents 15 Electrode processes 17 Electron transfer 22 Homogeneous chemical kinetics 22 Electrochemical and chemical reversibility 25 Cyclic voltammetry 27 A basic description 27 Simple electron-transfer processes 29 Mechanistic examples 35... 

The peak current ratio —ip /ip is unity and independent of v. For a simple electron transfer process, measurements of the peak current ratio serve to control the assumption that B is indeed stable in the solution on the time scale of the experiment. 

Equation (11) is the transition-state equation for electrochemical rates (i is, of course, proportional to d and concentration of reactants in the double layer at the electrode interface in the usual way ) and is obviously equivalent to the Tafel equation in exponential form [Eq. (4)]. From Eq. (11) it is seen that the Tafel slope for a simple electron transfer process is RT/pF, i.e., b is linear in temperature. We shall return later to a more critical examination of Eq. (11) insofar as energy and entropy components of the free energy of activation are concerned. 

Fig. 17. Theoretical chronoamperometric response for a simple electron transfer process at a channel electrode to a potential step, from a region in which no current flows to one at which...

The results obtained with different mixing devices led Haber and Weiss (3) to the formulation of the reaction as a radical and chain reaction. If Fen-salt is at all times in excess the mechanism can be represented by the two simple electron transfer processes ... 

There is evidence that some cross-reactions involving Co /Co couples have values of Kei< 1. Rate constants for the simple electron transfer processes (only the electron transfer step is shown, not the subsequent substitution of the ligands) in Equations (55) and (56) are altered by intense magnetic fields ... 

When the electroactive species or an intermediate adsorbs on the electrode surface, the adsorption process usually becomes an integral part of the charge transfer process and therefore cannot be studied without the interference of a faradaic current. In this situation, surface coverages cannot be measured directly and the role of an adsorbate must be inferred from a kinetic investigation. Tafel slopes and reaction orders will deviate substantially from those for a simple electron transfer process when an adsorbed intermediate is involved. Moreover the kinetic parameters, exchange current or standard rate constant, are likely to become functions of the electrode material and even the final products may change. These factors will be discussed further in the section on electrocatalysis (Section 1.4). 

The electrolytic formation of chlorate is dependent on complex chemistry coupled to a simple electron transfer process, i.e. 

Two limiting cases are apparent. If the chemical step is fast relative to the rate of mass transport, then the reaction will behave like a simple electron transfer process, and A will be reduced but via a mechanism where it is first converted to 0. On the other hand, if Reaction (2.47) is slow, a steady state current controlled by the conversion of A to 0 will be observed in response to a potential step. (Note in the extreme case where the reaction is very slow and no conversion of A to O occurs on the timescale of the experiment, then all that is observed is a diffusion controlled current due to the reduction of the low equilibrium concentration of O). The major region of interest is, of course, where the chemical step and mass transport are of comparable rate. To investigate reactions of this type it is again necessary to solve Pick s 2nd Law. For planar diffusion the equations to be solved are ... 

This is a very useful equation in experimental and applied dectrochemistry and shows that the measured current density is a function of (1) overpotential (2) exchange current density, 1 and (3) the transfer coefficients, and The transfer coefficients are, at least for simple electron transfer processes, not independent variables. In general ... 

One of the currently most promising approaches to a quantitative theory of electron transfer at an electrode is that of Marcus, whose fundamental assumption is that only a weak electronic interaction of the two reactants is required for a simple electron transfer process to occur. Interesting and significant deductions have been made quantum mechanically for simple electrode reaction in which no rupture or formation of chemical bonds occurs in the electron transfer step. The elaboration of the theory to include bond rupture is of obvious importance for the treatment of organic electrode processes. 

As with microelectrodes, diffusive transport to nanoelectrodes on conventional voltammetric timescales is dominated by convergent, as opposed to planar, diffusion. Therefore, for a simple electron transfer process, the voltammetric response at steady state is characterised by a sigmoidal shape. Simulation of such voltammetry requires solution of the diffusion equation typically with a Nemstian or Butler-Vofiner boundary condition for the rate of electron transfer at the electrode surface, depending on its reversibility. For simple, uniformly accessible, electrode geometries analytical solutions of these equations are available, and so for a disk electrode we obtain the familiar equation for the current (iiim) in the limit of diffusion control ... 

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