Tafel’s equation

Tafel s equation (eqn (30)) is accurate at large overpotentials, but fails as q approaches zero. The Tafel plot is obtained by plotting rj vs. log i, with b referred to as the Tafel slope. The Tafel slope is a function of the transfer coefficients and temperature, where... 

Butler27 and Volmer28 advanced Tafel s equation by relating overpotentials to activation barriers. The quantitative relationship between current and overpotential is called the Butler-Volmer equation (eqn (32)), and is valid for electrochemical reactions that are rate limited by charge transfer. 

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. 

Knowing the constant current imposed on the circuit and the final potential in the C-D region, the value of the overpotential corresponding to the current can be obtained. Repeating the measurement at a series of constant current densities allows determination of the t0 and the a value of Tafel s equation. If is available, the corresponding rate constant k0 can be calculated from the equation i0 = Fk0%c0, where X is the thickness of the reacting layer. 

Arrhenius law17 for the variation of the velocity of reactions with temperature was followed in 1905 by Tafel s equation for the variation of the electrochemical reaction rate with potential. The two laws may be compared ... 

Closely connected with the rate of passage of current at a steady overpotential is the rate of decay of overpotential after the current has been suddenly cut off. Apart from the observation of Bowden and Rideal that the rate of decay is enormously accelerated by dissolved oxygen, the data seem insufficient for generalization. Baars gave the equation for decay, Es — a —b log t, b being the same constant as the b in Tafel s equation but his own data do not bear this out very exactly Bowden and Rideal gave the equation... 

A Century of Tafel s Equation A Commemorative Issue of Corrosion Science, Corrosion Science 47(12), December 2005. 

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... 

Because the metal dissolution is an anodic process, for example, Fe(s) Fe +(aq) + 2e , the current of the process is assumed to be positive. When potential increases from Mez+zMe lo f (passivation or Flade potential), the current is increasing exponentially due to the electron transfer reaction, for example, Fe(s) -> Fe +(aq) + 2e", and can be described using Tafel s equation. At a E the formation of an oxide layer (passive film) starts. When the metal surface is covered by a metal oxide passive film (an insulator or a semiconductor), the resistivity is sharply increasing, and the current density drops down to the rest current density, 7r. This low current corresponds to a slow growth of the oxide layer, and possible dissolution of the metal oxide into solution. In the region of transpassivation, another electrochemical reaction can take place, for example, H20(l) (l/2)02(g) + 2H+(aq) + 2e, or the passive film can be broken down due to a chemical interaction with environment and mechanical instability. Clearly, a three-electrode cell and a potentiostat should be used to obtain the current density-potential curve shown in Figure 9.3. 

The general relation between current density and electrode potential was given by Tafel in 1905 (Tafel s equation) ... 

What conditions would be necessary for (9.38) to give Tafel s law (9.36) and replicate the Butler-Volmer equation (Section 7.2.3) Suppose (as with isotopic reactions) AG° = 0, then,... 

Of all these jumps in electrochemistry, each separated by around a century, there is one that best of all shows how electrochemistry is both deep-rooted and at the frontier of the twenty-first century. It was the pedant Julius Tafel who found, in 1905, that electric currents passing aaoss metal-solution interfaces could be made to increase exponentially by changing the electric potential of the electrode aCTOSS the surface of which they passed. In this way, he complemented the finding in ordinary kinetics made by Arrhenius 16 years earlier. Arrhenius s equation tells us that an increase of temperature inaeases the rate of chemical reaction exponentially ... 

Solution On the basis of the reaction model (equations (10.81) and (10.82)), the impedance can be derived under the assumption that the adsorbate Mg obeys a Langmuir isotherm and that the rate constants of electrochemical reactions are exponentially dependent on potential (e.g., following Tafel s law). Each reaction with index i has a normalized rate constant K, corresponding to its rate constant hi by... 

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. 

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. 

One of the first analytical models of CCL was developed by Springer and Gottesfeld [6], based on Pick s equation for oxygen transport and Tafel law for the rate of ORR. A similar approach was then used by Perry, Newman and Cairns [5] and by Eikerling and Komyshev [7]. 

Equation (15), called the Tafel relation (after Tafel s work in 1905), can be recast as ... 

Equation 6 is referred to as the Butler-Volmer equation. Normally, for significant overpotentials, either one or the other of the two terms is dominant, so that the current-density exponentially increases with r, i.e. In i is proportional to 3tiF/RT in the case, for example, of appreciable positive t] values. Here the significance of Tafel s b coefficient (Equation 1) is seen b = dn/d In i = RT/3F for a simple, single-electron charge-transfer process. 

Tafel equation Tafel kinetics Tafel slope Taffy process Taft s SV function Tagamet [51481-61-9] d-Tagatose... 

As the corrosion rate, inclusive of local-cell corrosion, of a metal is related to electrode potential, usually by means of the Tafel equation and, of course, Faraday s second law of electrolysis, a necessary precursor to corrosion rate calculation is the assessment of electrode potential distribution on each metal in a system. In the absence of significant concentration variations in the electrolyte, a condition certainly satisfied in most practical sea-water systems, the exact prediction of electrode potential distribution at a given time involves the solution of the Laplace equation for the electrostatic potential (P) in the electrolyte at the position given by the three spatial coordinates (x, y, z). 

Barkey, Tobias and Muller formulated the stability analysis for deposition from well-supported solution in the Tafel regime at constant current [48], They used dilute-solution theory to solve the transport equations in a Nernst diffusion layer of thickness S. The concentration and electrostatic potential are given in this approximation... 

Equation 1.7 for the reduction of protons at a mercury surface in dilute sulphuric add is followed with a high degree of accuracy over the range -9 Tafel plot i.s shown in Figure 1.5. At large values of the overpotential, one reaction dominates and the polarization curve shows linear behaviour. At low values of the overpotential, both the forward and back reactions are important in determining the overall current density and the polarization curve is no longer linear. 

The quantitative treatment for i as a function of a varying T f was first solved analytically by Sevdk in 1948. The solution involves Laplace transformation and the error function complement expressions applied in Vol. I, Section (4.2.11). It is better to quote here the rather simpler equations that can be found if one takes the entire surface as available for the exchange of electrons, i.e., the easy case of 0 = 0. Then (Gileadi, 1993),22 with this assumption, the peak potential is related to the rate constant (Ay) for the interfacial reaction, to the Tafel constant b, and to the sweep rate s, by the equation ... 

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