Butler-Volmer formulation

The importance of the Butler-Volmer formulation lies in the possibility of setting up the relationship between an electrolytic current i and Uappl on the electrode. In view of this, we shall first address the question of whether the concept of dynamic equilibria at the electrode interface is realistically correct if so, at Eeforward current ic must be equal to the backward current ia and they will compensate one another, so that the net current i0 - ia = = 0. 

As ksh in this instance is very small, then according to the Butler-Volmer formulation (eqn. 3.5) the reaction rate of the forward reaction, K — 8,he "F(E 0)/flr, even at E = E°, is also very low. Hence Etppl. must be appreciably more negative to reach the half-wave situation than for a reversible electrode process. Therefore, in the case of irreversibility, the polarographic curve is not only shifted to a more negative potential, but also the value of its slope is considerably less than in the case of reversibility (see Fig. 3.21). In... 

For irreversible processes, we can neglect the contribution made by the back reaction. In the Butler—Volmer formulation of electrode kinetics, k i is given as... 

The Butler-Volmer formulation of electrode kinetics [16,17] is the oldest and least complicated model constructed to describe heterogeneous electron transfer. However, this is a macroscopic model which does not explicitly consider the individual steps described above. Consider the following reaction in which an oxidized species, Ox, e.g. a ferricenium center bound to an alkanethiol tether, [Fe(Cp)2]+, is converted to the reduced form, Red, e.g. [Fe(Cp)2], by adding a single electron ... 

The empirical Butler-Volmer formulation of electrode kinetics provides an experimentally accessible theoretical description of the kinetics of these systems. From a plot of ln(fc) versus the overpotential, rj (= E — E0/), a and k° can be obtained from the slope and intercept, respectively. 

Using again the Butler-Volmer formulation of electrode kinetics,... 

This relation is very important. It, or a variation derived from it, is used in the treatment of almost every problem requiring an account of heterogeneous kinetics. Section 3.4 will cover some of its ramifications. These results and the inferences derived from them are known broadly as the Butler-Volmer formulation of electrode kinetics, in honor of the pioneers in this area (17, 18). 

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. 

Where k0 denotes the standard rate constant. The overall Butler-Volmer equation assuming the reactant and product concentrations to be the same, cA = cB = c, can be formulated as... 

This relationship may be inserted back into the general case of eqn. (40) to produce an alternative formulation of the Butler—Volmer relationship. 

Therefore, unlike the empirical Butler-Volmer theory, in the Marcus formulation the heterogeneous electron transfer rate constant is sensitive to both the structure of the redox center and the solvent. 

Obviously, when K = kf/kb 1, the equilibrium lies toward P and does not affect the R/P electron transfer. In the converse situation, and when kf and kb are larger than the mass transfer rate, a rapid equilibrium displaced toward Z establishes, and (P)x=0 (Z)x=o/K. Introduction of this relation in the current rate law [Eq. (109)] then yields Eq. (123), which is identical in its formulation to a Butler-Volmer law but for a... 

Guglielmi applied a Langmuir-type adsorption concept and a Butler-Volmer equation to formulate the final deposition process. But his final equation for the volume percent of co-deposited particles Xy is rather empirical. It can be written in the following form ... 

The Tafel equations (4.41) and (4.44) describe the anodic and cathodic limits of the Butler-Volmer equation. Their discovery by Tafel at the beginning of the twentieth century took place many years before the formulation of the rate equation by Butler and Volmer. 

The theoretical significance of the logarithmic increase of v with current-density was not understood until much later in the present century in terms of activation ideas in chemical kinetics, starting with Arrhenius in 1889 However, the proper representation of this behavior in electrode kinetics was not formulated until the independent works of Butler (15) in England in 1924 and of Volmer (14) in Germany around 1930 (see later), some 20 years after Tafel s empirical relation for electrode-kinetic behavior. 

We know that thermodynamics is a very powerful tool for the study of systems at equilibrium, but electrode processes are systems not at equilibrium when at equilibrium there is no net flow of current and no net reaction. Therefore electrode reactions should be studied using the concepts and formalities of kinetics. Indeed, the same period that saw the flourishing of solution electrochemistry, also saw the formulation of the fundamental theoretical concepts of electrode kinetics the work of Tafel on the relationship of current and potential was published in 1905 those of Butler and Volmer and Erdey Gruz, which formulated the basic equation for electrode kinetics, were published in 1924 and 1930 respectively. Frumkin in 1933 showed the correlation between the structure of the double layer and the kinetics of the electrode process. The first quantum mechanical approach to electrode kinetics was published by Gurney in 1931. 

By considering the kinetic substantiation of the Nernst equation, Audubert[2] and Butler[3] were the first to formulate the semiquantitative concepts concerning the effect of electrode potential on the rate of an electrochemical process. In its present form, the concept of slow discharge was put forward by Erdey-Gruz and Volmer[4]. Their approach was based on a consideration of the effect of potential on the activation energy of an electrode process and led to the introduction of a measure of this effect, viz. the transfer coefficient a. 

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