Butler-Volmer equation partial reactions

Large Cathodic Current We have seen from Figure 6.7 that for the large negative values of overpotential r], the partial cathodic current density i approaches i, i i. For these conditions the Butler-Volmer equation (6.45) can be simplified. Analysis of Eq. (6.45) shows that when rj becomes more negative, the first exponential term in the equation (corresponding to the anodic partial current) decreases, whereas the second exponential term (corresponding to the cathodic partial reaction) increases. Thus, under these conditions. 

It is an experimental fact that whenever mass transfer limitations are excluded, the rate of charge transfer for a given electrochemical reaction varies exponentially with the so-called overpotential rj, which is the potential difference between the equilibrium potential F0 and the actual electrode potential E (t) = E — Ed). Since for the electrode reaction Eq. (1) there exists a forward and back reaction, both of which are changed by the applied overpotential in exponential fashion but in an opposite sense, one obtains as the effective total current density the difference between anodic and cathodic partial current densities according to the generalized Butler-Volmer equation ... 

If only the charge-transfer reaction is hindered and not any of the other preceding or following partial reactions, the experimentally measurable total overpotentiai at the working electrode of the PEVD system equals the activation overpotentiai and is related to the PEVD current via the Butler-Volmer equation ... 

The relationship between overpotential and current density of a single, activation-controlled electrochemical reaction is the Butler-Volmer equation, Eq. (10). As the equation shows, the rates of anodic and cathodic partial reactions are exponentially dependent on the overpotential. The net current is the sum of the anodic and cathodic partial currents. 

The only quantity in this equation that can be measured directly is the total current density. The partial current densities for each of the three reactions are obtained by weighing the deposit and analyzing its composition. Each of the current densities is calculated from the Butler-Volmer equation... 

It is obvious that this reaction can only lead to metal dissolution, if the metal electrode potential is negative from the hydrogen electrode potential. This is the reason for the classification of metals into noble metals (the equilibrium potential is more positive than the standard hydrogen potential) and non-noble metals (the equilibrium potential is more negative than the standard hydrogen potential). The kinetic of the total process can be described by the Butler-Volmer equation for the two partial reactions. 

Equations (4.91) to (4.93) can be applied to any cathodic partial reaction for which the charge-transfer step obeys the Butler-Volmer equation. In corrosion, oxygen reduction is often under mixed control. Figure 4.25 shows the cathodic polarization curve for oxygen reduction, measured on a platinum electrode [6]. The shape of the curve suggests a relatively low value for the ratio I o/ltil-... 

The theory can be generalized by also taking into account the anodic term of the Butler-Volmer equation. If the anodic partial reaction is controlled by charge transfer, and if the cathodic reaction is under mixed control, the polarization curve is described by equation (4.94). 

Metal dissolution usually follows the Butler-Volmer equation (Eqs. (1-26) and (1-28)) independent from which surface position the transfer of the metal cation may occur. The relation for the exchange current density may contain the concentration of a catalyzing or complexing anion. The catalysis of iron dissolution by OH" ions is a typical example which has been studied in detail (Bonhoeffer and Heusler, 1956). Its mechanism includes a sequence of partial reactions of an electrode process and its in-... 

If charge transfer is rate limiting, the Butler-Volmer equation (35) described the current-potential relationship of each partial reaction. With the overvoltage of the metal dissolution reaction, r j = - reduction reaction, rjg... 

The two successive electron transfer reactions are assumed to obey the Butler-Volmer law with the values of standard potentials, transfer coefficient, and standard rate constants indicated in Scheme 6.1. It is also assumed, matching the examples dealt with in Sections 2.5.2 and 2.6.1, that the reduction product, D, of the intermediate C, is converted rapidly into other products at such a rate that the reduction of B is irreversible. With the same dimensionless variables and parameters as in Section 6.2.4, the following system of partial derivative equations, and initial and boundary conditions, is obtained ... 

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