Tafel-type equation

The product distribution of CO2 reduction at a Cu electrode changes with the potential. Such a potential dependence of the product distribution is derived from the transfer coefficients. The rate of formation of a product P is presented by the partial current ip in a Tafel type equation. 

The majority of the models discussed have in common that the electrochemical reaction at the electrodes is described by a Butler-Volmer- or Tafel-type equation. Therefore, all of these models share a very similar set of fundamental parameters, which are summarized in Table 29.2. These can be compared in order to check for model consistency. Nevertheless, aU of these models are based on slightly different assumptions. The parameters that are obtained by curve-fitting procedures are... 

Table 29.2 Commonly used parameters for Butler-Volmer- and Tafel-type equations.

EXAMPLE 3.1. Determination of the Constants of a Tafel-Type Equation... 

THE PROBLEM The polarization data in Table 3.3 have been obtained for the hydrogen evolution reaction on a rotating disk electrode. Calculate the constants of the resulting Tafel-type equation. 

THE SOLUTION First we write the Tafel-type equation (see Section 3.2.1 in general and Eq. (3.148), in particular) ... 

Equations (39)-(43) indicate that a Tafel-type relationship between the current and the field should be expected, i.e., from Eq. (40),... 

Assuming that the main reaction, (reaction 21.3), is governed by a simple Tafel type expression, we combine Equations 21.10, 21.11 and 21.13 to get... 

Equation (62) indicates that two regions of the E-i relationship are to be expected, both being Tafel-type functions. 

The type of analysis described above requires that both halves of the redox couple are stable and available (a known finite concentration of each must be present in solution to define an equilibrium potential), and that the equilibrium potential can either be measured, or calculated from the Nernst equation (i.e. the standard potential is known). This is often not the case, e.g. particularly in organic electrochemistry, one half of the redox couple may be unstable. However, Tafel type plots may still prove useful. Current-potential data are analysed using the Tafel equation in the fonn... 

Let us develop a typical expression for a reaction model by assuming that we have a primary reaction Awhich is governed by a simple Tafel-type expression (see also Section 3.2.1.1). Equation (3.145) becomes ... 

Kinetics Tafel-type expressions, Butler-Volmer equation, or complex kinetic equations... 

The shape of polarization curves for metals with low polarizability depends primarily on concentration polarization. In the case of highly polarizable metals, where activation polarization can be measured sufficiently accurately, the polarization curve can usually be described by an equation of the type (6.3) (i.e., by a Tafel equation). For metals forming polyvalent ions, slope b in this equation often has values between 30 and 60 mV. 

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... 

The Tafel equation implies that the overvoltage is a measure of the thermodynamic irreversibility of the electrode reaction, and it is associated with the slow step of the process. We distinguish some types of overvoltage depending on the type of slow reaction. 

This form is equivalent to the Tafel equation. It closely resembles the general type of an Eq. (1). 

Despite these successes, important process parameters, like bath agitation, bath constituents and particle type are disregarded. The constants k, 0 and B inherently account for these constants, but they have to be determined separately for every set of process parameters. Moreover, the postulated current density dependence of the particle deposition rate, that is Eq. (2), is not correct. A peak in the current density against the particle composite content curve, as often observed (Section III.3.H), can not be described. The fact that the peak is often accompanied by a kink in the polarization curve indicates that also the metal deposition behavior can not be accounted for by the Tafel equation (Eq. 4). Likewise, the (1-0 term in this equation signifies a polarization of the metal deposition reaction, whereas frequently the opposite is observed (Section 111.3,(0 It can be concluded that Guglielmi s mechanism... 

This equation, which has, e. g., been proposed in ref. [59], predicts a Tkfel slope of 0.03 V for p-type electrodes in indifferent electrolyte (see Sec. 2.1). In reality, to our knowledge Tafel slopes below 0.06 V have not been observed. Actually, in the case of GaAs in an acidic medium, in which the kinetic law (33) has been found to hold, the Tafel slope is found to be almost exactly 0.06 V (see Sec. 2.1). 

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. 

In the general case the above considerations predict deviations from the Tafel equation which have sometimes been observed They may arise from either the nonlinear E (different types of electrode processes to the general current equations (110,1V) and (111,IV) is possible by introducing of special models which allow, in particular, an estimation of the role of the dynamical factor ([Pg.297]

Often, the exponential dependence of the dark current at semiconductor-electrolyte contacts is interpreted as Tafel behavior [49], since the Tafel approximation of the Butler-Volmer equation [50] also shows an exponential increase of the current with applied potential. One should, however, be aware of the fundamental differences of the situation at the metal-electrolyte versus the semiconductor-electrolyte contact. In the former, applied potentials result in an energetic change of the activated complex [51] that resides between the metal surface and the outer Helmholtz plane. The supply of electrons from the Fermi level of the metal is not the limiting factor rather, the exponential behavior results from the Arrhenius-type voltage dependence of the reaction rate that contains the Gibbs free energy in the expraient It is therefore somewhat misleading to refer to Tafel behavior at semiconductor-electrolyte contacts. 

Regardless of fuel cell type, in a certain range of polarization voltages this rate is well approximated by the Butler-Volmer or Tafel equations. A better (non-Tafel) approximation of the reaction rate for DMFC anode is considered in Section 2.7. 

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