Electron-transfer reactions Tafel equation

Chapter 2, by B. E. Conway, deals with a curious fundamental but hitherto little-examined problem in electrode kinetics the real form of the Tafel equation with regard to the temperature dependence of the Tafel-slope parameter 6, conventionally written as fe = RT/ aF where a is a transfer coefficient. He shows, extending his 1970 paper and earlier works of others, that this form of the relation for b rarely represents the experimental behavior for a variety of reactions over any appreciable temperature range. Rather, b is of the form RT/(aH + ctsT)F or RT/a F + X, where and as are enthalpy and entropy components of the transfer coefficient (or symmetry factor for a one-step electron transfer reaction), and X is a temperature-independent parameter, the apparent limiting... 

For outer-sphere electron transfer, the rate equations (62.IV) and (63.IV) for non-adiabatic and adiabatic reactions may be used by introducing the electrode potential cp through the relation (107.IV) or the overvoltage through equation (114.IV), instead of the reaction heat Q. For inner-sphere redox reactions the expressions (83 IV) and (85.IV) can be used in a similar way for electronically non-adiabatic and adiabatic reactions, respectively. The conditions of validity of the Tafel equation are then given by ( 12.IV) or (116.IV). 

The Temperature and Potential Dependence of Electrochemical Reaction Rates, and th Real Form of the Tafel Equation Theoretical Aspects of Semiconductor Electrochemistry Theories for the Metal in the Metal-Electrolyte Interface Theories of Elementary Homogeneous Electron-Transfer Reactions Theory and Applications of Periodic Electrolysis... 

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 proportionality of n kf) to the applied potential difference (also called Tafel behavior) was observed back in 1975 by Gavach et al and has been corroborated ever since by many groups (e.g.. Ref. 53, 54, 56, 57). The results of Shao and Girault, illustrated in Fig. 7, show beyond any doubts that a Butler-Volmer relationship (as described by Eq. (10)) accounts very well for the experimental data. In the case of the metal-electrolyte solution interface, such an equation is rationalized by the fact that the applied potential difference, A, the driving force for the electron transfer reaction, is located at the interface and that the variation of the activation energy with A is a fraction of the variation of the electrical driving force. 

Thus, in the region of very high anodic or cathodic polarization, the RDS is always the first step in the reaction path. The transfer coefficient of the full reaction which is equal to that of this step is always smaller than unity (for a one-electron RDS), while slope i in the Tafel equation is always larger than 0.06 V. When the potential is outside the region of low polarization, a section will appear in the polarization curve at intermediate values of anodic or cathodic polarization where the transfer coefficient is larger than unity and b is smaller than 0.06 V. This indicates that in this region the step that is second in the reaction path is rate determining. 

The transfer currents of redox electrons and redox holes represented by Eqns. 8-63 and 8-64 are formally in agreement with the Tafel equation given by Eqn. 7-32. However, the Tafel constant (the transfer coefficient) a equals one or zero at semiconductor electrodes in contrast with metal electrodes at which a is close to 0.5. From Eqns. 8-64 and 8-65 for reaction currents, the Tafel constants is obtained as defined in Eqns. 8-66 and 8-67 ... 

The actual current passed / = 2F/4Jt,[H + ]exp[ — J pAE] since two electrons are transferred for every occurrence of reaction I. Equation (1.64) constitutes the fundamental kinetic equation for the hydrogen evolution reaction (her) under the conditions that the first reaction is rate limiting and that the reverse reaction can be neglected. From this equation, we can calculate the two main observables that can be measured in any electrochemical reaction. The first is the Tafel slope, defined for historical reasons as ... 

If the reaction proceeds via an electron transfer from the plane RP (see Fig. 11.2), one could see a factor (/) in the Tafel equation, so that ... 

It is important to remember that some assumptions have been made in the derivation of Fig.l. First, the equations given there are applicable only if the electron transfer is the rate determining step in the partial corrosion reactions. This is important with respect to the calculation of the Tafel slope (RT/anF) or the interpretation of an experimental one. It is further assumed that during the polarization of the test electrode (corroding piece of metal) the composition of the solution in the vicinity of the electrode remains... 

In electrochemical kinetics, this model corresponds to the Butler-Vohner equation widely used for the electrode reaction rate. The latter postulates an exponential (Tafel) dependence of both partial faradaic currents, anodic and cathodic, on the overall interfacial potential difference. This assumption can be rationalized if the electron transfer (ET) takes place between the electrode and the reactant separated by the above-mentioned compact layer, that is, across the whole area of the potential variation within the framework of the Helmholtz model. An additional hypothesis is the absence of a strong variation of the electronic transmission coefficient", for example, in the case of adiabatic reactions. 

From these equations the corresponding equations for Tafel lines, polarization resistance, and electrochemical reaction orders are obtained as was described in Section 6.1 for electron transfer. 

Xe is the electrode length, D is the diffusion coefficient of the electrochemically active species, and n is the number of electrons transferred per mole reactant during the electrode reaction. The applicability of this equation was tested using the known one-electron reduction of fluorescein in aqueous alkali, to yield the semifluorescein radical anion, and it was found that the well-defined current potential curves observed gave rise to hnear Tafel plots, with the expected room temperature value of 59 mV decade" slope. A linear Levich plot allowed inference of a diffusion coefficient of 3.0 X 10 cm s , in agreement with literature values [87]. Thus, the hydrodynamics of this particular flow cell are such that there is the expected parabolic, laminar flow. 

Most electrode reactions encountered in the field of corrosion involve the transfer of more than one electron. Such reactions take place in steps, of which the slowest, called the rate-determining step, abbreviated RDS, determines the overall reaction rate. In simple cases, one can identify the rate-determining step by an analysis of the measured Tafel slopes. In the so-called quasi-equilibrium approach one assumes that with the exception of the rate-limiting step, all other steps are at equilibrium. This greatly simplifies the mathematical equations for the reaction rate. More realistic approaches require numerical simulation and shall not be discussed here. To illustrate the quasi equilibrium approach to the study of multi-step electrode reactions we shall look at proposed mechanisms for the dissolution of copper and of iron. 

Under steady state conditions electron transfer processes with values less than about 5 X lO cm s are regarded as irreversible, i.e. when current flows the electron transfer is insufficiently fast to maintain Nernstian equilibrium at the electrode surface. In such cases kinetic data can be obtained directly from steady state current-voltage measurements analysed on the basis of the Tafel equations (see Fig. 1.4). For a cathodic reaction... 

The last 25 years have seen several attempts to develop a statistical-medmiad theory of electron transfer These tr ttn ts, however, do not predict the simple linear log vs. relationship of the Tafel equations which seems adequate for the description of charge transfer controlled electrode reactions in efec ocli nk technology. Therefore, they will not be discussed here. 

Figure 12.10 shows the CVs for methanol oxidation on a Pt/Ru catalyzed electrode at various temperatures. It can be seen that there is a significant increase in the current density with increasing temperature. From the data in this figure, the kinetic parameters of methanol oxidation can be estimated. For an electrochemical reaction controlled purely by electron transfer kinetics, if the reaction overpotential is large enough (>60mV), the Butler-Volmer equation can be simplified to the form of a Tafel equation, which is similar to Eqn (12.6) ... 

Tafel equation is an empirical equation, but development of the electrode kinetics allowed dipper understanding of charge transfer processes at electrified interfaces. If reaction (2) is a single step -electron charge transfer a general expression holds [2] ... 

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