Butler-Volmer equation current-potential dependence

Under current flow, the kinetics of the electrode reactions comes into effect. The Butler-Volmer equation describes the dependency of the current passing through the electrode interface (current density per unit geometric area) on a small voltage excursion (called overvoltage q) from the respective equilibrium potential E . ... 

Hydrogen evolution, the other reaction studied, is a classical reaction for electrochemical kinetic studies. It was this reaction that led Tafel (24) to formulate his semi-logarithmic relation between potential and current which is named for him and that later resulted in the derivation of the equation that today is called "Butler-Volmer-equation" (25,26). The influence of the electrode potential is considered to modify the activation barrier for the charge transfer step of the reaction at the interface. This results in an exponential dependence of the reaction rate on the electrode potential, the extent of which is given by the transfer coefficient, a. 

In writing out the Butler-Volmer equation, it has been assumed that, apart from factors concerning the potential-energy barrier, the current density depends only on the concentrations of reactants on the solution side of the interface. The metal surface was always considered empty, i.e., not blocked with any species, intermediate radical, or products. 

In the case of sufficiently high mass transport, i.e., the absence of concentration polarization phenomena, and when the electrochemical process is controlled by the charge transfer rate at one electrode, the dependency of the current on the electrode potential, V, , is given by the Butler-Volmer equation (Eq. (10)),... 

The potential dependence of both terms Aads and ifdes is described by the well-known Butler-Volmer equation [3.308, 3.309]. Both terms involve the exchange current density, s/j g + ... 

Therefore, the current density depends on the exchange current density ( o), transfer coefficient ( p), overpotential r ), and temperature (r). Fig. 7 represents typical current-overpotential curves based on Eq. (39). The net current is the result of the combined effects of the forward (anodic) and reverse (cathodic) currents. Although the Butler-Volmer equation for an electrochemical reaction in PEMFC is valid over the full potential range, simpler approximate equations may often be used for limited conditions. Thus, for the common value dp = 1/2, Eq. (39) becomes... 

E25.24(b) The potential needed to sustain a given current depends on the activities of the reactants, but the over potential does not. The Butler-Volmer equation says... 

Figure 6.2 Current-potential dependence for a redox system following the Butler-Volmer equation...

The reason why so much attention is devoted to the Butler-Volmer equation and to the exponential dependence of current on potential is interesting to note. First of all, many electrochemical experiments, particularly the fundamental ones, are carried out in the region where the influence of transport control from the solution side is purposely avoided. One simply calculates what the limiting current will be (see below) and then makes one s experiments in the situation when the currents examined are much less than the limiting current, so that the Butler-Volmer equation is applicable. 

If the electrocrystalli2ation is controlled by the formation of two or three-dimensional isolated nuclei, the current-potential relationship has a stronger overpotential dependence than predicted by the Butler-Volmer equation [58],... 

Equation (3.21) is known as the Butler-Volmer equation, and it forms the basis for the theoretical description of electrode processes. The terms in the square brackets represent the anodic (positive) and cathodic (negative) contributions to the net current, and /q is a scaling factor that depends on the values of, Co and Cr (Equation (3.18)). The symmetry of the current-potential... 

The steady-state potential of this sensing electrode (mixed potential) and the corresponding EMF of the sensor are established when the rates of the two electrochemical reactions are equal [13]. In order to estimate the mixed potential, one should consider the absolute values of the cathodic and anodic currents, expressed by the Butler-Volmer equation, taking also into account the rate from the catalytic reaction which determines the amounts of adsorbed species at the three-phase boundaries. A detailed analysis [8] concluded either a logarithmic or linear dependence for the concentration of reacting gas on mixed potential. The former... 

Often, the exponential dependence of the dark current at semiconductor-electrolyte contacts is interpreted as Tafel behavior [49], since the Tafel approximation of the Butler-Volmer equation [50] also shows an exponential increase of the current with applied potential. One should, however, be aware of the fundamental differences of the situation at the metal-electrolyte versus the semiconductor-electrolyte contact. In the former, applied potentials result in an energetic change of the activated complex [51] that resides between the metal surface and the outer Helmholtz plane. The supply of electrons from the Fermi level of the metal is not the limiting factor rather, the exponential behavior results from the Arrhenius-type voltage dependence of the reaction rate that contains the Gibbs free energy in the expraient It is therefore somewhat misleading to refer to Tafel behavior at semiconductor-electrolyte contacts. 

Equation (10.25) evolves to different possible forms of the Butler-Volmer equation, which predicts the dependence on the potential of the density current, 0 and R concentrations at the electrode being supposed not to change in dependence of the flux of current, i.e., of charge. A first one makes use of the standard heterogeneous kinetic constant, i.e., the kinetic constant of both forward and backward reaction at E = E° ... 

O) refers to a vanishingly short time when the current potential dependence is described by the Butler-Volmer equation with no diffusion control. f(0) is obtained by the extrapolation of the i - -Ji plot to = 0 according to Equation 1.131c. 

The velocity and pressure fields for the gas mixtures are solved first in the coupled gas channel-gas diffuser domains disregarding the changes in composition of the gas mixtures. This enables one to solve the flow and pressure fields for the gas mixtures first, and once these fields are found, the equations for the other dependent variables may be solved. The gas species concentrations are dependent on the transfer current densities therefore, the transport equations for the gas components are solved iteratively, together with the Butler-Volmer equations for anode and cathode catalyst layers. After convergence is achieved, one proceeds to solve for the transport equations related to the liquid water flow, for the membrane phase potential and current densities. Because the source terms of the energy equations are functions of the current density, a new level of iterations is needed, except for the velocity and pressure fields of the gas mixtures. 

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,... 

The current density at the pore wall, j, depends of the local overvoltage, t], according to some Butler-Volmer kinetics, which are not given here explicitly. The first boundary condition [Eq. (28.72)] is equivalent to the definition of (< —

electrode pore, so it defines the electric potential there. The second boundary condition [Eq. (28.73)] demands that no charge flux exits the electrolyte at the top of the pore. This differential equation can be solved in combination with a reaction rate expression, for example, Butier-Volmer kinetics. 

Following the seminal paper of Butler (37) in 1924 on the kinetic basis of Nernst equilibrium potentials, an electrochemical rate equation was written by Erdey-Gruz and Volmer (14), for a net current-density i, in terms of components of i for the forward and backward directions of the process. They recognized that only some fraction (denoted by a or 3) of the electrical energy change riF associated with change of electrode potential, would exponentially modify the current, giving a potential-dependent rate-equation of the form ... 

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