Diffusion potential step, reversible case

In order to demonstrate the mathematical approch to describing the electrode process we can consider the potential-step experiment for a reversible charge transfer without kinetic complications. In this case there are two diffusing species, A and B in (1). However, if the potential of the electrode is sufficiently greater than the reversible potential for reaction (1), the reverse reaction can be neglected so that only the diffusion of A contributes to the current. The equation to be solved results from Pick s second law and is given by (7). The... 

A simple first order reaction following reversible charge transfer is one of the few cases for which an analytical solution to the diffusion-kinetic differential equations can be obtained. For reactions (1) and (2) under diffusion-controlled charge-transfer conditions after a potential step, the partial differential equations which must be solved are (18) and (19). After Laplace transforma-... 

If a substance is reduced or oxidized reversibly, then its half-wave potential wiU be near the standard potential for the redox reaction. If it is reduced or oxidized irreversibly, the mechanism of electron transfer at the electrode surface involves a slow step with a high energy of activation. Therefore, extra energy must be applied to the electrode for the electrolysis to occur at an appreciable rate. This is in the form of increased applied potential and is called the activation overpotential. Therefore, Exa will be more negative than the standard potential in the case of a reduction, or if will be more positive in the case of an oxidation. An irreversible wave is more drawn out than a reversible wave. Nevertheless, an S-shaped wave is still obtained, and its diffusion current will be the same as if it were... 

Within the electrochemical framework of this classical example of a redox process whose rate is limited by the transport by diffusion, it was shown that, even for a reversible redox process, the derivation of the current response in the time domain is far from simple. In contrast, the impedance approach allows the more difficult case of an irreversible (finite reaction rate constants) redox process to be derived. Using the same approach, we will now examine the case of a multistep reaction, which is very difficult to investigate using techniques of potential step cyclic voltammetry. 

A second type of transient SECM experiment is double potential step chronoam-perometry in which the species generated at the tip under diffusion control in the first potential step reacts at or is transferred across a nearby interface. As the rate of the interfacial reaction increases, the fraction collected in the reverse potential step is reduced. At short times, the current during the reverse step varies linearly with and there are two limiting slopes corresponding to the cases of no reaction at the interface to that of diffusion-controlled reaction at the interface. For intermediate cases, the interfacial kinetics may be extracted [88]. 

The boundary conditions for solving the diffusion equation for linear potential sweep are really the same as those written for the potential step experiment, as discussed in Section 14.2. because in both cases the potential is the externally controlled parameter. As before, we can distinguish between the reversible case, in which it is assumed that the concentrations at the surface are determined by the potential via... 

To impose the diffusion-controlled conversion of O to R as described earlier, the potential E impressed across the electrode-solution interface must be a value such that the ratio Cr/Cq is large. Table 3.1 shows the potentials that must be applied to the electrode to achieve various ratios of C /Cq for the case in which Eq R = 0. For practical purposes, C /C = 1000 is equivalent to reducing the concentration of O to zero at the electrode surface. According to Table 3.1, an applied potential of -177 mV (vs. E° ) for n = 1 (or -88.5 mV for n = 2) will achieve this ratio. Similar arguments apply to the selection of the final potential. On the reverse step, a small C /Cq is desired to cause diffusion-controlled oxidation of R. Impressed potentials of +177 mV beyond the E° for n = 1 (and +88.5 mV for n = 2) correspond to Cr/Cq = 10"3. These calculations are valid only for reversible systems. Larger potential excursions from E° are necessary for irreversible systems. Also, the effects of iR drop in both the electrode and solution must be considered and compensated for as described in Chapter 6. 

A special case of interest involves stepping in the forward phase to a potential on the diffusion plateau of the reduction wave (O 0, Cq 0), then reversing to a potential on the diffusion plateau for reoxidation (6" Cr 0). In that instance, (5.7.14) simplifies to the result first obtained by Kambara (37) ... 

In the previous chapter we have introduced the case of multiple-electron transfers (multi-E mechanisms). As discussed then, depending on the formal potentials of the different electrochemical steps comproportiona-tion/disproportionation reactions may be thermodynamically favourable and may affect the voltammetry if the electron transfers are not reversible, the diffusion coefficients of the species are different, there is mass transport by migration or other chemical reactions take place. For example, let us consider the case of two consecutive reduction processes (the EE mechanism) where the formal potential of the second step is much more negative than the first one ... 

As the rate of the electron transfer process increases it must eventually be fast compared with the maximum rate of mass transport and the surface concentration will then become zero. Diffusion is then the rate-determining step (of sequence (1.16)—(1.18)) and the current becomes independent of potential with the value given by equation (1.53). A similar argument applies to the oxidation reaction although the limiting current is about one-third of the plateau reduction current because of the ratio Cp /Cr employed. Figure 1.10 shows the I-E curves for both the reversible and the irreversible cases. In the former, the I-E response arises directly from equation (1.20) while for an irreversible couple we need an overpotential to drive both the oxidation and reduction processes, see equations (1.35) and (1.37). 

Cyclic voltammetry can be used to investigate if the mediator has a suitable redox potential, if the redox process is reversible, and to investigate if the rate-limiting step of the electron transfer is the chemical reaction at the electrode surface or the diffusion rate of the mediator to the electrode. In the most cases freely diffusing mediators are used. Efficient electron transfer has been demonstrated with mediators immobilized on lysine residues in glucose oxidase, and the... 

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