Electrochemical rate constants, variations

In this equation, aua represents the product of the coefficient of electron transfer (a) by the number of electrons (na) involved in the rate-determining step, n the total number of electrons involved in the electrochemical reaction, k the heterogeneous electrochemical rate constant at the zero potential, D the coefficient of diffusion of the electroactive species, and c the concentration of the same in the bulk of the solution. The initial potential is E/ and G represents a numerical constant. This equation predicts a linear variation of the logarithm of the current. In/, on the applied potential, E, which can easily be compared with experimental current-potential curves in linear potential scan and cyclic voltammetries. This type of dependence between current and potential does not apply to electron transfer processes with coupled chemical reactions [186]. In several cases, however, linear In/ vs. E plots can be approached in the rising portion of voltammetric curves for the solid-state electron transfer processes involving species immobilized on the electrode surface [131, 187-191], reductive/oxidative dissolution of metallic deposits [79], and reductive/oxidative dissolution of insulating compounds [147,148]. Thus, linear potential scan voltammograms for surface-confined electroactive species verify [79]... 

The Butler-Volmer rate law has been used to characterize the kinetics of a considerable number of electrode electron transfers in the framework of various electrochemical techniques. Three figures are usually reported the standard (formal) potential, the standard rate constant, and the transfer coefficient. As discussed earlier, neglecting the transfer coefficient variation with electrode potential at a given scan rate is not too serious a problem, provided that it is borne in mind that the value thus obtained might vary when going to a different scan rate in cyclic voltammetry or, more generally, when the time-window parameter of the method is varied. 

FIGURE 1.22. Solvent reorganization energies derived from the standard rate constants of the electrochemical reduction of aromatic hydrocarbons in DMF (with n-Bu4N+ as the cation of the supporting electrolyte) uncorrected from double-layer effects. Variation with the equivalent hard-sphere radii. Dotted line, Hush s prediction. Adapted from Figure 4 in reference 13, with permission from the American Chemical Society. 

Another strategy consists in the application of convolution in the same manner as depicted in Section 1.4.3 for outer-sphere electron transfers. The activation-driving force law is then obtained directly from the variation of the rate constant, k(E), with the electrode potential. An example of the successful application of this strategy is provided by the electrochemical reduction of alkyl peroxides7 ... 

The variations of the symmetry factor, a, with the driving force are much more difficult to detect in log k vs. driving force plots derived from homogeneous experiments than in electrochemical experiments. The reason is less precision on the rate and driving force data, mostly because the self-exchange rate constant of the donor couple may vary from one donor to the other. It nevertheless proved possible with the reaction shown in Scheme 3.3.11... 

FIGURE 3.12. Potential energy profiles for the concerted and stepwise mechanism in the case of a thermal reductive process (E is the electrode potential for an electrochemical reaction and the standard potential of the electron donor for a homogeneous reaction) and variation of the rate constant and the symmetry factor when passing from the concerted to the stepwise mechanism. 

FIGURE 5.7. Effect of changing the cosubstrate and the pH on the kinetics of an homogeneous redox enzyme reaction as exemplified by the electrochemical oxidation of glucose by glucose oxidase mediated by one-electron redox cosubstrates, ferricinium methanol ( ), + ferricinium carboxylate ( ), and (dimethylammonio)ferricinium ( ). Variation of the rate constant, k3, with pH. Ionic strength, 0.1 M temperature 25°C. Adapted from Figure 3 in reference 11, with permission from the American Chemical Society. 

Fig. 14 Reduction of PhjCSPh by electrochemically generated aromatic anion radicals (in DMF at 25°C). Variation of the rate-determining step rate constant, A , with the standard potential of the aromatic anion radical, p,g (from left to right azobenzene, benzo[c]cinnoline, 4-dimethylaminoazobenzene, terephthalonitrile, naphthacene, phlhalonitrile, perylene, fluoranthene, 9,10-diphenylanthracene). The dotted lines are the theoretical limiting behaviours corresponding to the concerted (right) and stepwise (left) pathways. (Adapted from Severin et al 1988.)...

The charge transfer kinetics of azobenzene at the mercury electrode is slower than that of methylene blue, thus the frequency interval provided by modem instra-mentation (10 < //Hz < 2000) allows variation of the electrochemical reversibility of the electrode reaction over a wide range [79]. The quasireversible maxima measured by the reduction of azobenzene in media at different pH ate shown in Fig. 2.47 in the previous Sect. 2.5.1. The position of the quasireversible maximum depends on pH hence the estimated standard rate constant obeys the following dependence A sur = (62-12pH) S- for pH < 4. These results confirm the quasite-versible maximum can be experimentally observed for a single electrode reaction by varying the frequency, as predicted by analysis in Fig. 2.75. 

There are a number of problems associated with measuring bimolecular rate constants for ET. Only a small set of data can be obtained. In addition, since a wide range of donor anion radicals is used, there are variations in the reorganization energies that influence local curvature (and thus intrinsic barrier equation 53) for each point. In principle, electrochemical measurements such as those described in Section 2 can provide similar information. 

Branching mechanisms involve both consecutive and parallel electron transfers. The most important application of the RRDE in this context has been to the electrochemical reduction of oxygen [175], on which a large amount of research has been done. Different mechanistic models give rise to different expressions linking the rate constants, which can be compared with experimental data as in previous sections, the most important is the variation of (iD / h ) with rotation speed. A summary of different models has recently appeared [176] the conclusion of which is that, at platinum, the model of Damjanovic et al. [177] is correct diagnostic criteria to test the model have been developed. 

In the system considered it is assumed that there is no variation in the rate constant locally in the structure. For electrochemical systems this is equivalent to assuming a constant potential in the structure and is an approach used in plane or agglomerate models of electrocatalysts. 

In the following we consider a simple electron transfer mechanism in order to discuss quantitatively the variations in the potential location of the steady-state voltammogram of the system according to the kinetics of the heterogeneous electron transfer. In the derivation of the kinetics we consider that the solution contains only the reactant at concentration C before the electrochemical experiment. Let E°, k, and a be the standard reduction potential, the standard heterogeneous rate constant, and the transfer coefficient of the electron transfer in Eq. (176). 

Figure 20. Steady-state electrochemical method, (a) Concentration profiles of the product obtained upon electron transfer in the EC sequence in Eqs. (190) to (191) as a function of the dimensionless chemical rate constant k5 /D (numbers on the solid curves). The reactant concentration is shown for comparison as the dashed line, (b) Variations in the product electrode concentration as a function of k5 /D. The dashed curve corresponds to the approximation in Eq. (206).

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