Laws Butler-Volmer equation

Since the proton is transferred from a position right in front of the electrode, the assumptions made in the phenomenological derivation of the Butler-Volmer equation may not be valid furthermore, a proton can tunnel through a potential energy barrier in the reaction path. Nevertheless, an empirical law of the form ... 

But before dying to understand the behavior of electrochemical systems, or cells, it was considered useful to disassemble, or analyze, them conceptually into two isolated electrode/electrolyte interfaces and then to study single interfaces. This has been done. The whole treatment so far has concerned itself with a single electrode/ solution interface98 and with the current-potential laws that govern its behavior. The Butler-Volmer equation is the key equation for a single interface. The behavior of an electrochemical system, or cell, must be conceptually synthesized from the behavior of the individual interfaces that combine to form a cell... 

However, the form of (9.38) is not that of the first (cathodic) term in the familiar Butler-Volmer equation (7.24), which itself does indeed give the experimentally required Tafel law at 1) > RT/F. 

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,... 

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] ... 

Charge transport is modeled by Ohm s law (Equation (3.10)) and the charge conservation equation (Equation (3.68)), while the current density distribution at the electrode/electrolyte interface is modeled through the Butler-Volmer equation (Equation (3.102)). It should be noted that, contrarily to Section 3.7, Equation (3.102) is here derived from Equation (3.37) rather than Equation (3.39), because the former allows for a better agreement between experimental and simulated results. Equations (3.40)-(3.42) are used to model, the exchange current density, the activation overpotential, and the ideal potential drop at the electrode/electrolyte interface, respectively. Heat transfer is modeled through Equation (3.6), and the appropriate heat terms for each domain. 

In the numerical model calibration phase, the unknown parameters are those contained in Fick s law and in the Butler-Volmer equation, i.e. the diffusion coefficients representing the porous micro-structural characteristics (e and r), and the electrochemical kinetics parameter (A and Ea). It should be noted that the calibration pro-... 

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 Butler-Volmer equation, together with Equation (3.19), allows the prediction of the current-voltage characteristics of a galvanic or an electro chemical cell. In applications where the current density is high (or the over potential is high), the exponential law of the Butler-Volmer equation implies that 77 can be approximated by a constant value independent of j. Relation (3.19) can then be written in its simplified form (Fig. 3.5) ... 

The wave shapes observed for electrochemically irreversible or quasi-reversible voltammograms are governed by the Tick s law of diffusion (Eq. II. 1.6) and the Butler-Volmer expression (Eq. II. 1.16). By rewriting the Butler-Volmer equation for the case of a reduction A h- n e" B (Eq. n.1.19), it can be shown that, for the limit of extremely fast electron transfer kinetics, kg oo, theNemst law (Eq. n.1.7) is obtained as anticipated. 

This linear approximation to the Butler-Volmer equation is shown in Fig. 3.3. The charge transfer resistance is defined by rearranging Equation (3.22) by analogy with Ohm s law ... 

Figure 4-1 Model for the electrode process includes diffusion (Pick s laws), electrode kinetics (the Butler-Volmer equation), and chemical reaction kinetics.

When modeling fuel cells and gas diffusion electrodes, simplifications are almost always inevitable. These have to be handled with care, since they may easily introduce physical inconsistencies that lead to ill-posed problems or inaccuracy in the mathematical model. If the Tafel equations are used as a simplification to the Butler-Volmer equations, then it has to be established that all part of the electrodes are far from equilibrium. If not, then substantial errors may be introduced by such simplifications (Fig. 18.6). The concentration overpotential has to be accurately introduced in the model, since starved part of the electrodes may have small activation overpotentials and therefore be close to equilibrium for the local gas and electrolyte composition. The use of the Tafel equations in combination with a poor description of the concentration overpotential is a common source of inaccuracy in fuel cell modeling. This may also lead to convergence problems, since the model may not follow the conservation laws and therefore become ill-posed. 

The model of water-filled nanopores, presented in the section ORR in Water-Filled Nanopores Electrostatic Effects in Chapter 3, was adopted to calculate the agglomerate effectiveness factor. As a reminder, this model establishes the relation between metal-phase potential and faradaic current density at pore walls using Poisson-Nernst-Planck theory. Pick s law of diffusion, and Butler-Volmer equation... 

R. de Levie [80] wrote Erdey-Gniz and Volmer were the first to do this, in 1930, when they derived the corresponding rate expression in a paper on the kinetics of the hydrogen electrode. Recently, this basic law of electrode kinetics has become known as the Butler-Volmer equation. Butler was a leading British electrochemist, who had indeed attempted to find an answer to this question. Butler did not find it. In fact, in his 1940 book on electrocapillarity, Butler specifically refers to Erdey-Gruz and Volmer in this respect. The first time that the name of Erdey-Gruz was replaced by that of Butler appears to be in the 1970 textbook by Bockris and Reddy [81], but it may have an earlier origin. At any rate, subsequent textbook authors simply copied it... . 

Simplification of the Butler-Volmer equation can lead to Ohm s law or the Tafel equation. 

Neither the original Butler-Volmer nor the Tafel equations allow for accurate identification of the voltage for open circuit voltage (either tending to infinity, or becoming undefined, depending). The Butler-Volmer equation is used to determine the value of the coefficient k, whereas the coefficient b is determined by taking into account the diffusion laws. 

The first attempt to describe the dynamics of dissociative electron transfer started with the derivation from existing thermochemical data of the standard potential for the dissociative electron transfer reaction, rx r.+x-,12 14 with application of the Butler-Volmer law for electrochemical reactions12 and of the Marcus quadratic equation for a series of homogeneous reactions.1314 Application of the Marcus-Hush model to dissociative electron transfers had little basis in electron transfer theory (the same is true for applications to proton transfer or SN2 reactions). Thus, there was no real justification for the application of the Marcus equation and the contribution of bond breaking to the intrinsic barrier was not established. 

The principle of this method is quite simple The electrode is kept at the equilibrium potential at times t < 0 at t = 0 a potential step of magnitude r) is applied with the aid of a potentiostat (a device that keeps the potential constant at a preset value), and the current transient is recorded. Since the surface concentrations of the reactants change as the reaction proceeds, the current varies with time, and will generally decrease. Transport to and from the electrode is by diffusion. In the case of a simple redox reaction obeying the Butler-Volmer law, the diffusion equation can be solved explicitly, and the transient of the current density j(t) is (see Fig. 13.1) ... 

In this notation, anodic current is positive, while cathodic current is negative. As the later section on oxygen reduction will show, the Tafel slope can change with overpotential. This is because the Butler-Volmer law only applies to outer-sphere reactions. Although it can describe electrode reactions, the equation does not account for repulsive interactions of the adsorbates or changes in the reaction mechanism as potential is changed. 

The dimensionless time (t), potential ( ), and current (i/0 are all as defined in equations (1.4). The exact characteristics of the voltammograms depend on the rate law. In the case of Butler-Volmer kinetics,... 

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