Tafel representation

In this equation, SLm represents the Tafel slope for the mixture of X plus Y, and Slx,Siy represents the Tafel slopes for the individual components. This equation enables a determination of / from Tafel representations, providing that the quotients between the individual electrochemical rate constants, kx and ky, and the electron transfer coefficients, ocxnax,ocYnaY, are known. 

To study the kinetics of the catalytic process, Tafel representations from the current-potential data in the rising portion of the voltammetric curve can be used (Doherty and Vos, 1992). Under several conditions, plots of log i vs. E lead to straight lines whose slope (TSL) satisfies (Lyons et al., 1994) ... 

The initiation step can be studied by cyclic voltammetry at a low sweep rate (1 mV s ). A clean electrode in a monomeric solution is submitted to a potential sweep, starting at a potential where no faradaic processes occur, below the monomer oxidation potential, until an overpotential of 200-250 mV The plot of log i versus the overpotential gives a straight line the Tafel representation. 

Figure 1.8 A schematic representation or a Tafel plot of loge / vs. if. showing linearity at high overpotentials. At values of the overpotential < ryL, the current shows a linear dependence on...

This equation again fits to a linear dependence of Ini on E. This means that the sample containing a mixture of X plus Y should give a linear Tafel plot. Remarkably, both the slope and the ordinate at the origin of that representation should be intermediate between those obtained for the X and Y components separately (Eq. 3.15). 

Typical Tafel plots for different copper materials are shown in Fig. 3.11. In all cases, an excellent linearity was obtained for n i/ip) on E representations in terms of the correlation coefficient for linear fitting. Similar results were obtained for binary or ternary mixtures of such materials where highly overlapping peaks were recorded, both using linear potential scan and square-wave voltammetries. 

The hydrogen evolution reaction is studied on two metals, A and B, in sulfuric acid solutions of a = 1. The graphic representation of the corresponding Tafel... 

Fig. 16.4. A Tafel plot for the situation represented in Fig. 16.3 (in normal representation rotated 90° anticlockwise).

The Tafel plot permits the calculation of the rate constants of the reactions from the intersections, and the charge transfer coefficient from the slope. Corrosion researchers use the parameter b (the inverse of the slope of the Tafel plot) extensively in their studies, the reason being the representation of the Tafel plot in the way illustrated in Fig. 16.4. In fact,... 

While the calculations presented here were performed in terms of solution of Laplace s equation for a disk geometry, the nature of the electrode-electrolyte interface can be imderstood in the context of the schematic representation given in Figure 13.5. Under linear kinetics, both Co and Rt can be considered to be independent of radial position, whereas, for Tafel kinetics, 1/Rf varies with radial position in accordance with the current distribution presented in Figure 5.10. The calculated results for global impedance, local impedance, local interfacial impedance, and both local and global Ohmic impedances are presented in this section. 

Figure 13.6 Calculated representation of the impedance response for a disk electrode under assumption of Tafel kinetics with / as a parameter. The value / = 0 corresponds to an ideally capacitive blocking electrode a) real part and b) imaginary part.

Figure 13.8 Calculated representation of the local impedance response for a disk electrode as a function of dimensionless frequency K under assumptions of Tafel kinetics with 7 = 1. (Taken from Huang et al. and reproduced with permission of The Electrochemical Society.)...

Fig. 12 Log-log plot of normalized current density versus exponent of normalized voltage losses incurred by the cathode catalyst layer (Uc = rjo), in the limit of fast oxygen diffusion (Sect. 8.2.3.4.3). This representation reveals the transition from the simple Tafel kinetics at jo

The Tafel equation rj = a b ni, where fc, the so-called Tafel slope, conventionally written in the form b = RT/aF, where a is a charge transfer coefficient, has formed the basis of empirical and theoretical representations of the potential dependence of electrochemical reaction rates, in fact since the time of Tafel s own work. It will be useful to recall here, at the outset, that the conventional representation of the Tafel slope as RT/aF arises in a simple way from the supposition that the free energy of activation AG becomes modified in an electrochemical reaction by some fraction, 0.5, of the applied potential expressed as a relative electrical energy change rjF, and that the resulting combination of AG and 0.5tjF are subject to a Boltzmann distribution in an electrochemical Arrhenius equation involving an exponent n I/RT. Hence we have the conventional role of T in b = RT/aF, as will be discussed in more detail later. 

Tafel s original work in 1905 was concerned with organic reactions and H2 evolution at electrodes, and Eq. (1) was written as an empirical representation of the behavior he first observed. A particular value o b = RT/2F has come to be associated specifically with Tafel s name for the behavior of the cathodic H2 evolution reaction (h.e.r.) when under kinetic control by the recombination of two (adsorbed) H atoms following their discharge from or H2O in a prior step. Such kinetic behavior of the h.e.r. is observed under certain conditions at active Pt electrodes and in anodic CI2 evolution at Pt. (We note here, in parentheses, that an alternative origin for a Tafel slope of RT/2F for the h.e.r. at Pt has been discussed by Breiter and by Schuldiner in terms of a quasiequilibrium diffusion potential for H2 diffusing away from a very active Pt electrode at which H2 supersaturation arises). 

As referred to earlier, the usual representation of effects of electrode potential on electrochemical reaction rates is through a modulating term pVF operating on AG° [Eq. (11)]. [Taking account of doublelayer structure, this term is written as )8( V - il/i)F where il/i is the diffuse-layer potential, but this is a trivial difference in the present context.] Change of potential, V, modifies the Fermi level energy by eV as in Eq. (10). In the usual transition-state treatment, this quantity, modified by the factor p, appears in the Arrhenius-Boltzmann exponent [-(AG - iSVF)/RT]. It is from this exponent that the conventional Tafel-slope quantity b arises, linear in T. 

From the above, it seems clear that a Fermi-type distribution for the electronic energy levels in the metal should certainly be employed in the representation of electrode reaction rates as a function of potential and hence enters into how the Tafel slope or transfer coefficient is expressed, including its relation to T. 

Fig. 4.19 Schematic representation of polarization curves for the analysis of galvanic coupling when the coupled metals have similar electrochemical parameters. Tafel polarization is represented.

A numerical study of the influence of the ohmic drop on the evaluation of electrochemical quantities has been conducted, for example, over the AE interval [-20, 20] mV by means of the IRCOM program, which makes use of a polynomial of the sixth degree, considering some experimental polarization curves and taking the values of the electrochemical parameters obtained by the NOLI method. The examples examined have shown that the representation of experimental data by a polynomial of the sixth degree is very good and that the evaluation of the correct order of magnitude of the corrosion current density, in the presence of an ohmic contribution to the electrode potential, requires that the actual values of the Tafel slopes be known. 

Fig. 17 Schematic representation of how the /R drop affects galvanic couple potentials (a) equal anodic and cathodic Tafel slopes and (b) unequal anodic and cathodic Tafel slopes.

From a phenomenological point of view, the first quantitative representation of electrolysis as a kinetic process arose from the work of Tafel in 1905 at Erlangen(12) on cathodic H2 evolution. 

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. 

In electrode kinetics, as empirically represented by Tafel s equation, a basic feature is the potential-dependence of the reaction rate (current-density). This effect arises in Gurney s representation in a fundamental and general way as the electric potential V, of the electrode metal is changed by AV relative to that of the solution (in practice, measured relative to the potential of a reference electrode at open-circuit), the effective value of the electron work function 4> of the metal is changed according to... 

Electrochemical corrosion systems can be characterized using the kinetic parameters previously described as Tafel slopes, exchange and limiting current densities. However, the mixed potential theory requires a mixed electrode system. This is shown in Eigure 5.1 for the classical pure zinc (Zn) electrode immersed in hydrochloric (NCl)acid solution [1,8-9]. This type of graphical representation of electrode potential and current density is known as Evans Diagram for representing the electrode kinetics of pure zinc. 

FIGURE 2.2 Schematic representation of a Tafel plot showing curvature at low overvoltage and indicating significance of parameters a and b. 

Show for what values of q the Tafel equation is a good representation of the BV equation. 

Figure 4. Schematic representation of Tafel lines for the cathodic and anodic directions of the reaction in Eq. (1), occurring in an isothermal WE-RE cell without transference, at various temperatures. It is assumed that Pc=Pa = 0-5 and that both symmetry factors are independent of temperature. EARO, Experimentally accessible region of overpotential.

This error was also investigated by the numerical simulation technique. The error increases with increasing Rs/Rp ratio and with increasing Tafel-slope ratio ( >iarge/ smaii) The error also depends slightly on the individual Tafel slope values, but this introduces only an uncertainty of 20% (relative) of the representation given in Fig. 15. This technique, as well as the three-point technique, is more sensitive to the solution resistance than the... 

A(l/6 )-l-A(l/6 ). In earlier treatments of this problem,the error was considered to be a function of four separate variables. Consequently, only selected results could be presented in several graphs. The recent representation is completely general and depicts the error over a wide range of all variables in one graph as a function of two parameters. This is accomplished with two combined parameters, one representing both Tafel slopes, and the other representing the error of both Tafel slopes. While a complete... 

In order to simplify the calculation of the Butler-Volmer equation, the main relationship is linearized to the form of the Tafel equation. At sufficiently small (in practice <0.01 V), the Butler-Volmer equation can be developed into a series and then, taking into account only the first two components, a linear representation of it is obtained ... 

In the Tafel regime, we consider the Butler-Volmer equation in terms of two subregions, one for high anodic polarization (rjs aAp/RT), the second for high cathodic polarization -rjs cicF/RT). It can be shown that these approximations are valid when / avg io- The mathematical representations for the anodic and cathodic Tafel regions are similar ... 

---