Generalized Butler-Volmer Equation

If the mass transfer is presented by diffusion overpotential and the electron transfer is described using the Butler-Volmer theory, these two approaches can be combined 

A graphical representation of the generalized Butler-Volmer Equation 6.24 is given in Eigure 6.10. It can be seen that when overpotential is highly negative or 

FIGURE6.10 Current density vs. overpotential plotted using the generalized Butler-Vobner equation, Equation 6.24 with following parameters n=, T = 298.15 K, = 10 A cm  

The charge transfer overpotential is described by the Butler-Volmer equation. Two parameters of the Butler-Vohner equation are the exchange current density (/o) the symmetry factor p. The physical meaning of these parameters should be clearly understood. 

If an electrochemical reaction is faster (more reversible), then the exchange current density is larger, can range from 10 A cm (very slow reaction) to 10 A cm (very fast reaction). 

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It is an experimental fact that whenever mass transfer limitations are excluded, the rate of charge transfer for a given electrochemical reaction varies exponentially with the so-called overpotential rj, which is the potential difference between the equilibrium potential F0 and the actual electrode potential E (t) = E — Ed). Since for the electrode reaction Eq. (1) there exists a forward and back reaction, both of which are changed by the applied overpotential in exponential fashion but in an opposite sense, one obtains as the effective total current density the difference between anodic and cathodic partial current densities according to the generalized Butler-Volmer equation ... 

The general Butler-Volmer equation for electrode reaction (5.54) is given by the... 

For the equal charge transfer coefficient on the anode and cathode electrode, that is, Ua = Uc = a, the general Butler-Volmer equation (Equation 11.3) can be written as... 

The reader is encouraged to practice employing the Tafel, Butler-Volmer, and generalized Butler-Volmer equations using a computer code such as Excel or Mathematica to better understand the current density-overpotential dependence. 

Based on the generalized Butler-Volmer equation, when current goes to its anodic limiting value, the potential goes to... 

In practice, the Butler- Volmer equation is only obeyed in any case for potentials close to Et, or, more generally, for small currents, not through any neglect of factors such as anharmonicity but rather because the rate of transport of the ions to the electrode becomes rate-limiting, a problem we turn to next. 

The general relationship between tj and i is obtained by combining Eqns (10.10a) and (10.10b) to give the Butler-Volmer equation ... 

Consider a system in which a potential difference AV, in general different from the equilibrium potential between the two phases A 0, is applied from an external source to the phase boundary between two immiscible electrolyte solutions. Then an electric current is passed, which in the simplest case corresponds to the transfer of a single kind of ion across the phase boundary. Assume that the Butler-Volmer equation for the rate of an electrode reaction (see p. 255 of [18]) can also be used for charge transfer across the phase boundary between two electrolytes (cf. [16, 19]). It is mostly assumed (in the framework of the Frumkin correction) that only the potential difference in the compact part of the double layer affects the actual charge transfer, so that it follows for the current density in our system that... 

This general equation covers charge transfer at electrified interfaces under conditions both of zero excess field, low excess fields, and high excess fields, and of the corresponding overpotentials. Thus the Butler-Volmer equation spans a large range of potentials. At equilibrium, it settles down into the Nernst equation. Near equilibrium it reduces to a linear / vs. T) (Ohm slaw for interfaces), whereas, if T) > (RT/fiF) (i.e., one is 50 mV or more from equilibrium at room temperature), it becomes an exponential /vs. T) relation, the logarithmic version ofwhich is called Tafel s equation. 

The Butler-Volmer equation has yielded much that is essentia] to the first appreciation of electrode kinetics. It has not, however, been mined out. One has to dig deeper, and after electron transfer at one interface has been understood in a more general way, electrochemical systems or cells with two electrode/electrolyte interfaces must be tackled. It is the theoretical descriptions of these systems that provide the basis... 

Both these equations are general forms of the Butler-Volmer equation when U= 1, these equations reduce to (7.136). 

Equation (7.144) is the most general form of the Butler-Volmer equation it is valid for a multistep overall electrodic reaction in which there may be electron transfers in steps other than the rds and in which the rds may have to occur V times per occurrence of the overall reaction. This generalized equation is seen to be of the same form as the simple Butler-Volmer equation for a one-step, single-electron transfer reaction ... 

Activation polarizations can be simulated through the typical equations generally used for SOFCs, i.e. the Butler-Volmer equation (see also Chapter 3) ... 

A somewhat general model is that represented by the Butler-Volmer equation,... 

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). 

Volmerian electrode reaction — This term has been used for electrode reactions in which the - charge transfer coefficient is constant. Reactions for which the latter is potential dependent were called non-volmerian. According to the - Marcus theory there is generally a potential dependence of the charge transfer coefficient, however that is usually very small. The terms Volmerian and non-volmerian refer to the classic Butler-Volmer theory (-> Butler-Volmer equation) where no potential dependence was assumed. See also -> Volmer. 

This is of course also true if we need to consider the general electrochemical reaction Eq. (92). If the applied driving force (cf. electrical experiment) is an electrical potential gradient, Eq. (97) leads to the well-known non-linear Butler-Volmer equation.79 We will become acquainted with equally important kinetic equations for the cases of the tracer and the chemical experiment.172... 

As already shown in Fig. 1, a general feature of electrocatalysis is that the current passing through an electrode-electrolyte interface depends exponentially on overpotential, as described by the Butler-Volmer equation discussed in Sect. 2.4.1, so that logi versus r] U — C/rev) gives straight lines, termed Tafel plots (Fig. 1). On this basis, one would expect an exponential-type dependence of current on overpotential in Fig. 12 (curve labeled 7ac). Such a curve would correspond to pure activation control, that is, to infinitely fast mass-transport rates of reactants and products to and from the two electrodes. 

The dependence of the rate of elementary electron transfer reactions on the applied potential, , is governed by the Butler-Volmer equation. For irreversibly adsorbed redox active species, this rate can be expressed, without loss of generality, in terms of the surface concentration of the reduced form of the species, Tred, as follows ... 

The theoretical interpretation of the first voltage drop at low current is based on the Butler-Volmer equation, which is derived by an analysis of electrode kinetics and provides a general description of the relationship between current density and surface overpotential for an electrochemical converter [46] ... 

The treatment by Pritzker and Fahidy [6] also involves anodic dissolution of the metal by a generalized expression of the activation overpotential by a mass transport modified Butler-Volmer equation. In both cases, anodic dissolution and cathodic deposition and stability and instability are possible. Some other studies have been presented with fewer terms that contribute to the surface stability such as those in Refs. [1,10] but with some erroneous conclusions. Anyhow, the limitations of Fahidy s work [6] arise from the fact that only a linear perturbation is considered, and so it can be applied only to the early stages of the instability. 

We will not illustrate the polarization curves because a more generalized treatment and a more complete expression of the current vs. potential profile comprising the entire mass transfer-modified Butler-Volmer equation are available in the literature [7], We only discuss here the most determining features of the curves that are deduced from our detailed comments presented above ... 

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