Butler-Volmer potential dependence

As in the case of differential double potential pulse techniques like DDPV, slow electrochemical reactions lead to a decrease in the peak height and a broadening of the response of differential multipulse and square wave voltammetries as compared with the response obtained for a Nemstian process. Moreover, the peak potential depends on the rate constant and is typically shifted toward more negative potentials (when a reduction is considered) as the rate constant or the pulse length decreases. SWV is the most interesting technique for the analysis of non-reversible electrochemical reactions since it presents unique features which allow us to characterize the process (see below). Hereinafter, unless expressly stated, a Butler-Volmer potential dependence is assumed for the rate constants (see Sect. 1.7.1). 

The following problem arises in the interpretation of such semicircles in the complex plane impedance plots every parallel combination of a constant resistance and constant capacity leads to a semicircle in the Nyquist plot of the impedance. To verify a charge transfer, for instance, the potential dependence of the charge-transfer resistance should be investigated to demonstrate the Butler-Volmer potential dependence of the exchange current. 

Both the frequency of the well and its depth cancel, so that the free energy of activation is determined by the height of the maximum in the potential of mean force. The height of this maximum varies with the applied overpotential (see Fig. 13). To a first approximation this dependence is linear, and a Butler-Volmer type relation should hold over a limited range of potentials. Explicit model calculation gives transfer coefficients between zero and unity there is no reason why they should be close to 1/2. For large overpotentials the barrier disappears, and the rate will then be determined by ion transport. 

The current-potential relationship predieted by Eqs. (49) and (50) differs strongly from the Butler-Volmer law. For y 1 the eurrent density is proportional to the eleetro-static driving force. Further, the shape of the eurrent-potential curves depends on the ratio C1/C2 the curve is symmetrical only when the two bulk concentrations are equal (see Fig. 19), otherwise it can be quite unsymmetrieal, so that the interface can have rectifying properties. Obviously, these current-potential eurves are quite different from those obtained from the lattice-gas model. 

The non-steady-state optical analysis introduced by Ding et al. also featured deviations from the Butler-Volmer behavior under identical conditions [43]. In this case, the large potential range accessible with these techniques allows measurements of the rate constant in the vicinity of the potential of zero charge (k j). The potential dependence of the ET rate constant normalized by as obtained from the optical analysis of the TCNQ reduction by ferrocyanide is displayed in Fig. 10(a) [43]. This dependence was analyzed in terms of the preencounter equilibrium model associated with a mixed-solvent layer type of interfacial structure [see Eqs. (14) and (16)]. The experimental results were compared to the theoretical curve obtained from Eq. (14) assuming that the potential drop between the reaction planes (A 0) is zero. The potential drop in the aqueous side was estimated by the Gouy-Chapman model. The theoretical curve underestimates the experimental trend, and the difference can be associated with the third term in Eq. (14). 

The previous analysis indicates that although the voltammetiic behavior suggests that the aqueous phase behaves as a metal electrode dipped into the organic phase, the interfacial concentration of the aqueous redox couple does exhibit a dependence on the Galvani potential difference. In this sense, it is not necessary to invoke potential perturbations due to interfacial ion pairing in order to account for deviations from the Butler Volmer behavior [63]. This phenomenon has also been discarded in recent studies of the same system based on SECM [46]. In this work, the authors observed a potential independent ket for the reaction sequence. 

The ET reaction between aqueous oxidants and decamethylferrocene (DMFc), in both DCE and NB, has been studied over a wide range of conditions and shown to be a complex process [86]. The apparent potential-dependence of the ET rate constant was contrary to Butler-Volmer theory, when the interfacial potential drop at the ITIES was adjusted via the CIO4 concentration in the aqueous phase. The highest reaction rate was observed with the smallest concentration of CIO4 in the aqueous phase, which corresponded to the lowest driving force for the oxidation process. In contrast, the ET rate increased with driving force when this was adjusted via the redox potential of the aqueous oxidant. Moreover, a Butler-Volmer trend was found when TBA was used as the potential-determining ion, with an a value of 0.38 [86]. 

The voltammograms at the microhole-supported ITIES were analyzed using the Tomes criterion [34], which predicts ii3/4 — iii/4l = 56.4/n mV (where n is the number of electrons transferred and E- i and 1/4 refer to the three-quarter and one-quarter potentials, respectively) for a reversible ET reaction. An attempt was made to use the deviations from the reversible behavior to estimate kinetic parameters using the method previously developed for UMEs [21,27]. However, the shape of measured voltammograms was imperfect, and the slope of the semilogarithmic plot observed was much lower than expected from the theory. It was concluded that voltammetry at micro-ITIES is not suitable for ET kinetic measurements because of insufficient accuracy and repeatability [16]. Those experiments may have been affected by reactions involving the supporting electrolytes, ion transfers, and interfacial precipitation. It is also possible that the data was at variance with the Butler-Volmer model because the overall reaction rate was only weakly potential-dependent [35] and/or limited by the precursor complex formation at the interface [33b]. 

There axe indications that in a number of systems a Butler-Volmer-type law holds in the phenomenological sense that is, the partial current seems to depend exponentially on the potential difference between the two phases. 

Hydrogen evolution, the other reaction studied, is a classical reaction for electrochemical kinetic studies. It was this reaction that led Tafel (24) to formulate his semi-logarithmic relation between potential and current which is named for him and that later resulted in the derivation of the equation that today is called "Butler-Volmer-equation" (25,26). The influence of the electrode potential is considered to modify the activation barrier for the charge transfer step of the reaction at the interface. This results in an exponential dependence of the reaction rate on the electrode potential, the extent of which is given by the transfer coefficient, a. 

The cyclic voltammetric responses depend on the manner in which the rate constants are related to the electrode potential. We start with cases in which the Butler-Volmer law applies ... 

The dimensionless time (t), potential ( ), and current (i/0 are all as defined in equations (1.4). The exact characteristics of the voltammograms depend on the rate law. In the case of Butler-Volmer kinetics,... 

If the kinetics of electron transfer does not obey the Butler-Volmer law, as when it follows a quadratic or quasi-quadratic law of the MHL type, convolution (Sections 1.3.2 and 1.4.3) offers the most convenient treatment of the kinetic data. A potential-dependent apparent rate constant, kap(E), may indeed be obtained derived from a dimensioned version of equation (2.10) ... 

The Butler-Volmer approach is not empirical, so it therefore helps to explain the observed Tafel behaviour. Here, we start by saying that the observed current always has two components, i.e. oxidative and reductive, with both components depending strongly on the applied potential, V. The oxidative (anodic) current is... 

Equation (25) is general in that it does not depend on the electrochemical method employed to obtain the i-E data. Moreover, unlike conventional electrochemical methods such as cyclic or linear scan voltammetry, all of the experimental i-E data are used in kinetic analysis (as opposed to using limited information such as the peak potentials and half-widths when using cyclic voltammetry). Finally, and of particular importance, the convolution analysis has the great advantage that the heterogeneous ET kinetics can be analyzed without the need of defining a priori the ET rate law. By contrast, in conventional voltammetric analyses, a specific ET rate law (as a rule, the Butler-Volmer rate law) must be used to extract the relevant kinetic information. 

In writing out the Butler-Volmer equation, it has been assumed that, apart from factors concerning the potential-energy barrier, the current density depends only on the concentrations of reactants on the solution side of the interface. The metal surface was always considered empty, i.e., not blocked with any species, intermediate radical, or products. 

The third exponential term in eqn. (187) is identical to the exponential term in the Butler—Volmer equation, eqn. (80), in the absence of specific adsorption. The first two exponential factors in eqn. (187) corresponding to the variation in the electrical part of the free energy of adsorption of R and O with and without specific adsorption A(AGr) and A(AG0), respectively. The explicit form of as, the activity of the adsorption site, and potential dependence of as, A(AGr) and A(AG0) is necessary for a complete description of the electrode kinetics. 

The example treated above is of appreciable relevance for several parts of this chapter, as will be seen later. The potential dependence of c 0 and Cr can be derived explicitly if it is known how kf depends on E. As an illustration in Fig. 2(a), c 0 vs. E curves are given assuming the commonly accepted Butler—Volmer model... 

It is useful at this stage to consider the potential dependency of l — ktDqV1 (1 + exp f)- Assuming Butler—Volmer behaviour for kf, i.e. kf = ksh exp (— oap) with constant a, it is found that l has a minimum at a potential near E° and increases to infinity at both sides of this minimum. 

In Fig. 3.14a, the dimensionless limiting current 7j ne(t)/7j ne(tp) (where lp is the total duration of the potential step) at a planar electrode is plotted versus 1 / ft under the Butler-Volmer (solid line) and Marcus-Hush (dashed lines) treatments for a fully irreversible process with k° = 10 4 cm s 1, where the differences between both models are more apparent according to the above discussion. Regarding the BV model, a unique curve is predicted independently of the electrode kinetics with a slope unity and a null intercept. With respect to the MH model, for typical values of the reorganization energy (X = 0.5 — 1 eV, A 20 — 40 [4]), the variation of the limiting current with time compares well with that predicted by Butler-Volmer kinetics. On the other hand, for small X values (A < 20) and short times, differences between the BV and MH results are observed such that the current expected with the MH model is smaller. In addition, a nonlinear dependence of 7 1 e(fp) with 1 / /l i s predicted, and any attempt at linearization would result in poor correlation coefficient and a slope smaller than unity and non-null intercept. 

In the above equations, kIt(i/l and k(txh are the first-order heterogeneous rate constants (in s ) at potential Eh of a pulse sequence, for the electro-reduction and electro-oxidation reactions, respectively. It will be assumed that the potential dependence of the rate constants is in agreement with the Butler-Volmer formalism (see Eq. (1.101)), i.e.,... 

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