Electron transfer Butler-Volmer equation

As tire reaction leading to tire complex involves electron transfer it is clear that tire activation energy AG" for complex fonnation can be lowered or raised by an applied potential (A). Of course, botlr tire forward (oxidation) and well as tire reverse (reduction) reaction are influenced by A4>. If one expresses tire reaction rate as a current flow (/ ), tire above equation C2.8.11 can be expressed in tenns of tire Butler-Volmer equation (for a more detailed... 

The two-step charge transfer [cf. Eqs. (7) and (8)] with formation of a significant amount of monovalent aluminum ion is indicated by experimental evidence. As early as 1857, Wholer and Buff discovered that aluminum dissolves with a current efficiency larger than 100% if calculated on the basis of three electrons per atom.22 The anomalous overall valency (between 1 and 3) is likely to result from some monovalent ions going away from the M/O interface, before they are further oxidized electrochemically, and reacting chemically with water further away in the oxide or at the O/S interface.23,24 If such a mechanism was operative with activation-controlled kinetics,25 the current-potential relationship should be given by the Butler-Volmer equation... 

Inner-sphere electron-transfer reactions are not expected to obey the Butler-Volmer equation. In these reactions the breaking or formation of a bond, or an adsorption step, may be rate determining. When the reactant is adsorbed on the metal surface, the electrostatic potential that it experiences must change appreciably when the electrode potential is varied. 

On application of an overpotential rj, the Gibbs energy of the electron-transfer step changes by eo[r) — Afa rj), where Afa(rj) is the corresponding change in the potential fa at the reaction site. Consequently, rj must be replaced by [rj — Afa r )] in the Butler-Volmer equation (5.13). 

These arguments are similar to those employed in the derivation of the Butler-Volmer equation for electron-transfer reactions in Chapter 5. However, here the reaction coordinate corresponds to the motion of the ion, while for electron transfer it describes the reorganization of the solvent. For ion transfer the Gibbs energy curves are less symmetric, and the transfer coefficient need not be close to 1/2 it may also vary somewhat with temperature since the structure of the solution changes. 

We assume that k and /c i are independent of the coverage and the electrode potential. We further assume that the rate of the electron-transfer step obeys a Butler-Volmer equation of the form ... 

This current-potential relationship, also known as the Butler-Volmer equation, governs all the (fast and single step) heterogeneous electron transfers. 

It is now time to define some terms. The exchange current (/o) is best thought of as the rate constant of electron transfer at zero overpotential. This current is commonly expressed as a form of current density, Iq/A (cf. equation (1.1)), in which case it is called the exchange current density, io- (Incidentally, this also explains why the Butler-Volmer equation does not include an area term. This follows since both / et and /q are functions of area, thus causing the two area terms to cancel out.)... 

Occasionally, the analyst is required to determine the rate of electron transfer, ket, and can then use the Butler-Volmer equation (equation (7.16)) to determine 7o, from which ket is readily calculated by using equation (7.17). The preferred method of obtaining the exchange currents in such cases is under conditions of infinite rotation speed i.e. via a Koutecky-Levich plot. 

If this is the case, the interest in the phenomenology of /—r) in terms of the Butler Volmer equation (no diffusion control, all electron transfer at the interface) is lessened. It will be acceptable to use equilibrium concepts at the interface for many purposes, and concentrate on the rate-determining transport process outside the interfacial region. 

The Butler-Volmer equation has yielded much that is essentia] to the first appreciation of electrode kinetics. It has not, however, been mined out. One has to dig deeper, and after electron transfer at one interface has been understood in a more general way, electrochemical systems or cells with two electrode/electrolyte interfaces must be tackled. It is the theoretical descriptions of these systems that provide the basis... 

This treatment remains valid for two other possible reaction sequences these are sequences in which there are (a) chemical, i.e., noncharge-transfer, steps before and after a charge-transfer rds and (b) charge-transfer steps before and after a chemical rds. In the latter case, where no charge transfer occurs in the rds, the number of electrons transferred after the rds will be n — y. There will be no effect of potential on the rate of the rds except that arising from previous charge-transfer steps thus, the Butler-Volmer equation for a chemical rds is given as... 

Equation (7.144) is the most general form of the Butler-Volmer equation it is valid for a multistep overall electrodic reaction in which there may be electron transfers in steps other than the rds and in which the rds may have to occur V times per occurrence of the overall reaction. This generalized equation is seen to be of the same form as the simple Butler-Volmer equation for a one-step, single-electron transfer reaction ... 

Fick s first and second laws (Equations 6.15 and 6.18), together with Equation 6.17, the Nernst equation (Equation 6.7) and the Butler-Volmer equation (Equation 6.12), constitute the basis for the mathematical description of a simple electron transfer process, such as that in Equation 6.6, under conditions where the mass transport is limited to linear semi-infinite diffusion, i.e. diffusion to and from a planar working electrode. The term semi-infinite indicates that the electrode is considered to be a non-permeable boundary and that the distance between the electrode surface and the wall of the cell is larger than the thickness, 5, of the diffusion layer defined as Equation 6.19 [1, 33] ... 

The first exponential term in both equations is independent of the applied potential and is designated as k and A(L for the forward and backward processes, respectively. These represent the rate constants for the reaction at equilibrium, e.g. for a monolayer containing equal concentrations of both oxidized and reduced forms. However, the system is at equilibrium at E0/ and the products of the rate constant and the bulk concentration are equal for the forward and backward reactions, i.e. k must equal Therefore, the standard heterogeneous electron transfer rate constant is designated simply as k°. Substitution into Equations (2.19) and (2.20) then yields the Butler-Volmer equations as follows ... 

If the adsorption step itself is rate-limiting, one must have available rate expressions for the adsorption and the desorption steps. The flux in (2.108) is then split into two opposing components. Using the notation of Delahay and Mohilner [201,403], there is a forward flux vj, adding to the adsorbate s surface concentration and backward flux tadsorbed substance. These obey rate equations rather analogous to those for electron transfer, the Butler-Volmer equation, in the sense that there are rate constants that are potential dependent. For the forward and backward rates, we have... 

Corrosion current density — Anodic metal dissolution is compensated electronically by a cathodic process, like cathodic hydrogen evolution or oxygen reduction. These processes follow the exponential current density-potential relationship of the - Butler-Volmer equation in case of their charge transfer control or they may be transport controlled (- diffusion or - migration). At the -> rest potential Er both - current densities have the same value with opposite sign and compensate each other with a zero current density in the outer electronic circuit. In this case the rest potential is a -> mixed potential. This metal dissolution is related to the corro-... 

An alternative theory to the Butler-Volmer theory for electron transfer is provided by the Marcus-Hush theory (Marcus, 1968 Hush, 1968) which assumes a potential-dependent a. Since in most cases a is essentially independent of potential, use of the simpler Butler-Volmer equation is usually adequate. 

That is, for an irreversible electron-transfer process, the rate-limiting step over a wide range of potentials is the electron-transfer step rather than diffusion. The constant is related to the electrode potential and the standard rate constant, ko, by the Butler-Volmer equation described above. Use of the Butler-Volmer equation and Fick s laws of diffusion enables the voltammetric response of an irreversible process to be understood. 

The rate of electron transfer and its potential dependence can be described by the Butler-Volmer equation (20) (see Section 2). An electron transfer often initiates a cascade of homogeneous chemical reactions by producing a reactive radical anion/cation. The mechanism can be described mathematically by a rate equation for each species these form part of the electrochemical model. The rate law of the overall sequence is probed by the voltammetric experiment. 

The dependence of the rate of elementary electron transfer reactions on the applied potential, , is governed by the Butler-Volmer equation. For irreversibly adsorbed redox active species, this rate can be expressed, without loss of generality, in terms of the surface concentration of the reduced form of the species, Tred, as follows ... 

It is fair to say that the effect of ultrasound upon the fundamental electron transfer processes at an electrode have been less widely studied than the effects upon mass transport phenomena. Electrode kinetics is defined by the Butler—Volmer equation, which by a series of practical assumptions reduces to the Tafel equation [44],... 

The electron transfer process across the electrode/electrolyte interface is a heterogeneous reaction. The rate at which electron transfer takes place across that interface is described in terms of a heterogeneous electron transfer rate constant. The kinetics can be described via the Butler-Volmer equation ... 

In this equation, and represent the surface concentrations of the oxidized and reduced forms of the electroactive species, respectively k° is the standard rate constant for the heterogeneous electron transfer process at the standard potential (cm/sec) and oc is the symmetry factor, a parameter characterizing the symmetry of the energy barrier that has to be surpassed during charge transfer. In Equation (1.2), E represents the applied potential and E° is the formal electrode potential, usually close to the standard electrode potential. The difference E-E° represents the overvoltage, a measure of the extra energy imparted to the electrode beyond the equilibrium potential for the reaction. Note that the Butler-Volmer equation reduces to the Nernst equation when the current is equal to zero (i.e., under equilibrium conditions) and when the reaction is very fast (i.e., when k° tends to approach oo). The latter is the condition of reversibility (Oldham and Myland, 1994 Rolison, 1995). 

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