Butler-Volmer expression

In addition, the following relationship results from the Butler-Volmer expression of the linearized activation-driving force law ... 

However, as we saw in section 3.3 for platinum on YSZ, the fact that i—rj data fits a Butler—Volmer expression does not necessarily indicate that the electrode is limited by interfacial electrochemical kinetics. Supporting this point is a series of papers published by Svensson et al., who modeled the current—overpotential i—rj) characteristics of porous mixed-conducting electrodes. As shown in Figure 28a, these models take a similar mechanistic approach as the Adler model but consider additional physics (surface adsorption and transport) and forego time dependence (required to predict impedance) in order to solve for the full nonlinear i—rj characteristics at steady state. 

In electrodics, the reaction rate is expressed in terms of current density i (Section 7.2.1). Thus one would expect, by analogy, the electrochemical order of the reaction to be given by an expression similar to (7.145) which should result from the Butler-Volmer expression ... 

Two simplifications of the ButLer-Volmer expression are often encountered. First, at low surface overpotentials equation 22 can be written as... 

Fig. 1.22 Schematic current-potential response for the reduction of species O calculated from the Butler-Volmer expression (Eqs. 1.92 and 1.101, dashed line) and from the general expression which includes mass transport influence (Eq. 1.193, solid line)...

To obtain quantitative expressions for the corrosion current and the corrosion potential, one has to substitute the proper expression for the metal-dissolution- and electronation-current densities. If no oxide films form on the surface of the corroding metal and neither of the current densities is controlled by mass transport, i.e., there is no concentration overpotential, one can insert the Butler-Volmer expression for the deelectronation- and electronation-current densities. Thus,... 

From the Butler-Volmer expressions, the true rate constant is then... 

In real (as opposed to model) electrochemical cells, the net current flowing will often be partly determined by the kinetics of electron transfer between electrode and the electroactive species in solution. This is called heterogeneous kinetics, as it refers to the interface instead of the bulk solution. The current in such cases is obtained from the Butler-Volmer expressions relating current to electrode potential [73,74,83,257,559]. We have at an electrode the process (2.18), with concentrations at the electrode/electrolyte interface cj q and cb,Oj respectively. We take as positive current that going into the electrode, i.e., electrons leaving it, which corresponds to the reaction (2.18) going from left to right, or a reduction. Positive or forward (reduction) current if is then related to the potential E by... 

Although such terms as Butler-Volmer equation or Butler-Volmer expression or Butler-Volmer kinetics or Butler-Volmer model are widely used in the literature, see e.g., Refs, [ii-xii], its definition is ambiguous and even the name is questionable in the light of the historical facts [viii, xiii, xiv]. 

In general, the phenomenological Butler-Volmer expression is assumed for the forward (k ) and backward (kb) rate constants of the ET reaction shown by Equation (2) ... 

Friedrich and Filers [113] have proposed a corrosion model of development and derived electron transfer equations based on the Butler-Volmer expression which can be simplified into three cases. Case 1 is when both the forward and reverse processes of developer oxidation are important. Case 2 is when the net rate is limited by the forward rate of developer oxidation. Case 3 corresponds to a rate which is limited by the kinetics of both developer oxidation and silver halide reduction. 

The Butler-Volmer expression is not described here as it is described in all fundamental textbooks on electrodics, but instead the smaller expression is given here ... 

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. 

The Phenomenological Butler-Volmer Expression Considering symmetric energy surfaces for the initial and final states of the reaction described in Eq. (1), the activation energy will be affected by a fraction obs of the Galvani potential difference... 

If we assume that the concentrations at the reaction planes are equal to that in the bulk, Eq. (14) is simplified to the corresponding Butler-Volmer expression for ET at ITIES,... 

C. How Well Does the Butler-Volmer Expression Approximate the Marcus Formalism 118... 

The voltage dependence of MNA electron transfer described by Eqs. (14) is measurably different from that described by the Butler-Volmer expression [42], where ... 

These relationships allow us to estimate the error in the rate constants that obtain when using the simple Butler-Volmer expression (and assuming that the MNA expression is correct) at any given potential the rate constants deduced using the Butler-Volmer expression will be larger than the MNA values by a factor of exp[0.1459 The general ex-... 

A correlation of the experimentally measured relaxation rate constant, fcm, to meaningful physical parameters was described by Smalley et al. [23]. That approach was based on the Butler-Volmer expression. In this... 

Let a and b denote concentrations of reduced and oxidized species present in the film and at any time t the total site concentration Cx = a + b, whereas at time t = 0, b = 0 and Cx - a. Now charge transfer across the conducting/nonconducting boimdary is given by the Butler-Volmer expression... 

We now examine the situation at x = 0. In this case the charge transfer is again described by the Butler-Volmer expression as follows... 

The solution of these equations gives the potential distribution in the electrodes and in the electrolyte. The reaction terms couple the electrolyte and electrode potentials through the reaction kinetics, which are described by Arrhenius expressions for both forward and backward reactions at one electrode surface for a one-electron charge transfer reaction. These terms become a Butler-Volmer expression by introducing the contribution of the electric potential difference at the electrode surface to the activation energy. This results in the following expression for the local charge transfer current density in the electrode [142] ... 

Fig. 18.6 Dimensionless concentration profiles of oxygen in an agglomerate for different overpotentials with full Butler-Volmer expression accounting for concentration overpotential (a) and Tafel expression (b). The reaction rate close to the middle of the agglomerates in (a) is...

A general anode contamination model was developed by Zhang et al. (2005) and is capable of describing the effect of various contaminant species. St.-Pierre s generalized contamination model that is applicable to both the cathode and anode (St.-Pierre, 2009) has already been discussed briefly in the Section 8.3.1.2. All of the above models use similar reactions to describe tbe kinetics of HOR and the contamination reactions, along with Butler-Volmer expression to determine the PC s overpotential or current density. The reaction network and rate constants for a contamination process involving a general contaminant P is shown below ... 

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