Tafel region/equation

The region of potential that corresponds to rj/p 1 is called the anodic Tafel region. Equation (4.36) then becomes ... 

It is evident from these expressions that since in the Tafel region / (the current density actually determined) must be greater than /(, (the equilibrium exchange current density), the signs of the overpotentials will conform to equations 1.60 and 1.61, i.e. will be negative and will be positive. 

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. 

Between the limiting current plateaux of a voltammogram and the linear region close to Eeq described by (6.38) there is a region of potential for irreversible reactions where j depends exponentially on potential. This is the Tafel region. Considering a system where there is only O in bulk solution, that is there is only reduction, from the equation... 

A "small" perturbation in this context is one for which nrjF/vRT 1 or i/i 1. The linearity of the response allows easier and more rigorous mathematical treatment and is, therefore, often preferred. It is interesting to note that a linear response is also obtained when a small perturbation is applied to a system far away from equilibrium. To show this, we write the usual rate equation for an activation controlled process in the linear Tafel region (cf. Eq. 7F) namely ... 

Whereas the charge-injection method is a small-amplitude perturbation method in which measurement is conducted during open-circuit decay, we now discuss a different open-circuit measurement, in which the initial overpotential is high, in the linear Tafel region. The equations we need to solve are similar to Eqs. 9K and lOK, except that the value of the current in Eq. lOK is that corresponding to the linear Tafel region, namely... 

To obtain the stoichiometric number experimentally, it is necessary to measure i by two independent methods from extrapolation of the linear Tafel region to q = 0 and from micropolarization measurements. Equation 51F can then be used to calculate v, since the value of n is determined independently. Unless one considers a very complex system, both n and v are integers, and it is easy to distinguish experimentally among a small number of possible values of v. 

D is the mean diffusion coefficient of and v kinematic viscosity (see Section 5.3.1), and co = 2nf the angular velocity with/the rotation frequency. In the anodic Tafel region the following equation is obtained ... 

The Stern-Geary equation has been modified to minimize errors under particular conditions. Errors in Eq. (5.8) are especially significant in systems in which the corrosion potential is close to one of the reversible potentials— that is, is outside the Tafel region. Mansfeld and Oldham [10] have developed a set of equations that provides less error than Eq. (5.8) when this situation applies (see Appendix, Section 29.2). 

Equation (4.89) is valid if the electrodeposition process is an under activation control in the Tafel region, the limiting diffusion current density is the same in both the middle and at the edge of the electrode, and / is the cell current corresponding to i. During the current pulses, the amplitude values of current densities and current should be substituted in Eq. (4.89) producing... 

The Tafel equations contain information about both the exchange current density, 7, and the transfer coefficient, olq. Plots of log 171 vs 77 are more com-monly used than the true Tafel plots of 77 vs log 171, simply because 17 is now usually the controlled variable. Fig. 3.4 illustrates plots of this kind, and shows how 7q is obtained from the extrapolation of the data obtained in the limiting Tafel regions at high positive and negative values of 77. The relationships between the slopes of the plots and the value of are given by... 

It is seen that b disappears and Equation 4.80 coineides with Equation 4.129, derived below, for the case of ideal oxygen transport at the small current. Small fo means small fjo or large diffusion coefficient. In both these cases, the oxygen transport does not eontribute to the potential loss. The respective polarization plots cover linear and Tafel regions (Figure 4.15, lower solid curves). 

Using the example of iron corroding in a hydrochloric acid solution, if the iron sample is maintained at the natural corrosion potential of —0.2 V, no current will flow through the auxiliary electrode. The plot of this data point in the study would equate to that of A or C in Figure 1.5. As the potential is raised, the current flow will increase and curve AB will approximate the behavior of the true anodic polarization curve. Alternatively, if the potential were lowered below —0.2 V, current measm-ements would result in the curve CD and approximate the nature of the cathodic polarization curve. By using the straight line portions, or Tafel regions, of these curves, an approximation of the corrosion ciurent can be made. 

When or , the Butler-Volmer equation reduces to the one-term Tafel equation as the first or second exponential term, respectively, may be approximated as zero. Consequently, a plot of In f vs — for the region highlighted in Fig. 2.3 (often called the Tafel region) should yield a straight hne of gradient aF/RT, so allowing measurement of the transfer coefficient (a), a (as shown in Eq. 2.1) is known as a transfer coefficient and is a measure of the position of the transition state between the oxidised and reduced species. Typically it has a value of around 0.5. 

Equation (5.9) indicates that there is a linear relationship between r/and Ini. A part of the polarization curve where (5.9) is valid is called the Tafel region. 

Tafel region — Part of the current density vs. electrode potential relationship which can be described in sufficiently good approximation with the Tafel equation. Part of the current density vs. electrode potential relationship which can be described in sufficiently good approximation with the Tafel equation. 

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. 

Equation (2.15) is valid only for potential values far from the equilibrium potential. Limitations by mass transport should be negligible. If these assumptions are met, the conditions correspond to the so-called Tafel region, where j depends exponentially on the electrode potential. The Tafel region is defined by the Terfel equation, which can be written in a general form (for anodic current density jf) as. 

The region of micro-polarization, where the j/r) plot is linear, can extend to about il/i) < 0.2 whereas the linear Tafel region starts at about r]/b > 1. As a result, the intermediate region of 0.2 < t]/b < 1 would be left unused, as far as the evaluation of kinetic parameters is concerned. For fast reactions, such as the HER on platinum, this represents a loss of crucial data, since it may be difficult to extend the measurements to overpotentials much above rj/h = 1, because of mass-transport limitations. Fortunately, modern microcomputers allow us to make use of this intermediate region. To do this, we write the full equation for an activation-controlled electrode reaction as follows ... 

It is interesting to note that a linear response is also obtained when a small perturbation is applied to a system far away from equilibrium. To prove this, we write the usual rate equation for an activation controlled process in the linear Tafel region... 

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 ... 

This conclusion is in complete agreement with the experimental results. Figure 6.1 shows the S vs n dependence for different solutions, viz. for a surface-inactive electrolyte, and for solutions to which specifically adsorbed anions and cations have been added. It can be seen that the results for different solutions differ considerably. If, however, we assume, in accordance with the usual equations of the theory of slow discharge[1], that the change in overpotential at a constant concentration of H" " ions is exactly equal to the change in the local -potential (the coefficient (1 - a)/a of the ij -poten-tial is equal to unity, since a = 1/2 in the Tafel regions for the investigated solutions), the displacement of one curve with respect to the other by the difference in overpotentials enables us to... 

Initially, the curve conforms to the Tafel equation and curve AB which is referred to as the active region, corresponds with the reaction Fe- Fe (aq). At B there is a departure from linearity that b omes more pronounced ns the potential is increased, and at a potential C the current decreases to a very small value. The current density and potential at which the transition occurs are referred to as the critical current density, and the passivation potential Fpp, respectively. In this connection it should be noted that whereas is determined from the active to passive transition, the Flade potential Ef is determined from the passive to active transition... 

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. 

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