Butler-Volmer approximation

FIGURE 1.13. Free-energy profiles in outer-sphere electron transfer according to the Butler-Volmer approximation (a) and to the Marcus-Hush model (b). 

In contrast with the Butler-Volmer approximation, the activation-driving force relationships [equations (1.32)] are quadratic rather than linear. This... 

As another example of solving the nonlinear model by ADM, we assume that at high overpotentials the electrochemical kinetics can be represented by the following high field Butler-Volmer approximation, the current density in Eq. (227) is rewritten as ... 

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. 

Previous SECM studies have suggested that a Butler-Volmer type approximation could be used for the ITIES at low driving forces [83], For a system where an ET reaction occurs between an aqueous electron donor and an oxidant in the organic phase, the free energy barrier is given by... 

BUTLER-VOLMER EQUATION HIGH-OVERPOTENTIAL APPROXIMATION... 

The Butler-Volmer (BV) approximation is the simplest approach to model and capture the essential features of the empirical Tafel equation. It considers an electrochemical half-cell reaction as an activated process, with the forward and backward reaction rates following an Arrhenius type law according to... 

In the high-overpotential case (cf. Section 7.2.3b.2), the first exponential term can be neglected for n 0, i.e., for net electronation, and the second exponential term for T) 0, i.e., for net deelectronation. In the low-field approximation, where both exponential terms in the Butler-Volmer equation can be linearized, Eq. (7.136) becomes... 

Under these conditions the expotentials in the Butler-Volmer equation (6.45) can be approximated using a power series... 

The expression (12.27) for the corrosion potential can be introduced into Eq. (12.22) and thus, an explicit result for the corrosion current can be obtained. But the resulting equation is quite cumbersome and therefore a simpler equation will be derived by assuming that overpotentials are sufficiently large that the high-field approximation of the Butler-Volmer equation can be used for the electronation- and deelectronation-current densities. Thus, Eqs. (12.22) and (12.23) become... 

In -> Butler-Volmer equation describing the charge transfer kinetics, the transfer coefficient a (or sometimes symbol jS is also used) can range from 0 to 1. The symmetrical energy barrier results in a = 0.5. Typically, a is in the range of 0.3 to 0.7. In general, a is a potential-dependent factor (which is a consequence of the harmonic oscillator approximation, see also - Marcus theory) but, in practice, one can assume that a is potential-independent, as the potential window usually available for determination of kinetic parameters is rather narrow (usually not more than 200 mV). 

This equation (Tafel equation) is of fundamental importance in studies of electrode kinetics. It is actually an approximation of the - Butler-Volmer equation at... 

This mechanism is denoted as an EC mechanism (Testa and Reinmuth, 1961 Bott, 1997). Thus homogeneous kinetic terms may be combined with the expressions for diffusion and convection [i.e. a modified version of (18)] to give the temporal variation of the concentration of a species in an electrode reaction mechanism. In order to model the voltammetric response associated with this mechanism, a knowledge of , a, ko and k is required, or deduced from a theoretical-experimental comparison, and the set of concentrationtime equations for species A, B and C must be solved subject to the constraints of the Butler-Volmer equation and the experimental design. Considerable simplification of the theory is achieved if the kinetics for the forward and reverse processes associated with the E step are fast, which is a good approximation for many organic reactions. Section 7 describes the approaches used to solve the equations associated with electrode reaction mechanisms, thus enabling theoretical simulation of voltammetric responses to be achieved. 

When Ip I >100 mV then the Butler-Volmer equation reduces to its high field approximation form ... 

The Butler-Volmer equations simplify in the case where one of the exponential terms dominate, which will happen if the loss potential in question is large. In this case the electrode potential is linearly related to the logarithm of the corresponding current, and if the losses occur equally at both electrodes, then the total potential becomes linearly related to log(f). This is called the Tafel relation, and the slope of the line is called the "Tafel slope". The following sections will give many examples of potential-current relationships, and except for the interval of smallest currents, the Tafel approximation is often a valid one. 

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