Butler-Volmer equation exchange current density

The exchange current density for common redox couples (at room temperature) can range from 10-6 pAcm-2 to A cm"2. Equation (1-24) can be written in terms of the exchange current to give the Butler-Volmer equation ... 

Figure 3a is an illustration of the effect of surface overpotential on the limiting-current plateau, in the case of copper deposition from an acidified solution at a rotating-disk electrode. The solid curves are calculated limiting currents for various values of the exchange current density, expressed as ratios to the limiting-current density. Here the surface overpotential is related to the current density by the Erdey Gruz-Volmer-Butler equation (V4) ... 

We assume that both reactions obey the Butler-Volmer equation, and denote the corresponding transfer coefficients by a and 2, the exchange current densities by jo,i and jo,2> and the equilibrium potentials by total current density is zero we have ... 

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.)... 

Figure 5. Measurement and analysis of steady-state i— V characteristics, (a) Following subtraction of ohmic losses (determined from impedance or current-interrupt measurements), the electrode overpotential rj is plotted vs ln(i). For systems governed by classic electrochemical kinetics, the slope at high overpotential yields anodic and cathodic transfer coefficients (Ua and aj while the intercept yields the exchange current density (i o). These parameters can be used in an empirical rate expression for the kinetics (Butler—Volmer equation) or related to more specific parameters associated with individual reaction steps.(b) Example of Mn(IV) reduction to Mn(III) at a Pt electrode in 7.5 M H2SO4 solution at 25 Below limiting current the system obeys Tafel kinetics with Ua 1/4. Data are from ref 363. (Reprinted with permission from ref 362. Copyright 2001 John Wiley Sons.)...

The real case, a partly polarizable (and hence partly nonpolarizable) electrode, can be described in terms of the exchange current density i0. From the linearized Butler-Volmer equation [Eq. (7.25)], then ... 

Calculating Exchange Current Densities and Rate Constants from Impedance Plots. If one takes the Butler-Volmer equation (7.24) under the reversible condition, i.e that in which the overpotential, rj, tends to zero, then,... 

The procedure for correcting forthe departure from equilibrium to nonequilibrium surface coverage consists in (1) writing down the actual concentration in the Butler-Volmer equation or its relevant special case and (2) transforming this expression into one involving the equilibrium exchange-current density Iq, which contains the bulk concentration. 

For a given overpotential, the effective current density depends on the magnitude of the charge-transfer coefficient a as well as on the exchange current density iu. If the overpotential is high enough—that is, if either —(a Frj/RT) or (a Fr/)/RT > 1—then one of the partial current densities in the Butler-Volmer equation overrules the other ... 

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 other version of the Butler-Volmer equation the potential term is expressed by using the - overpotential (q) and the - exchange current density (ja) instead of E, Ef, and, respectively ... 

Exchange current density — When an electrode reaction is in equilibrium, the reaction rate in the anodic direction is equal to that in the cathodic direction. Even though the net current is zero at equilibrium, we still envisage that there is the anodic current component (If) balanced with the cathodic one (Ic). The current value /() = Ja = IC is called the exchange current . The corresponding value of current density jo = Io/A (A, the electrode area) is called the exchange current density . If the rate constants for an electrode reaction obey the Butler-Volmer equation, jo is given by... 

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 + ... 

Physically, the reason for the dramatic difference between performances of cathode and anode active layers is the exchange current density ia at the anode the latter is 10 orders of magnitude higher than at the cathode [6]. Due to the large ia, the electrode potential r]a is small. The anode of PEFC, hence, operates in the linear regime, when both exponential terms in the Butler-Volmer equation can be expanded [178]. This leads to exponential variation of rja across the catalyst layer with the characteristic length (in the exponent)... 

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... 

Additional parameters specified in the numerical model include the electrode exchange current densities and several gap electrical contact resistances. These quantities were determined empirically by comparing FLUENT predictions with stack performance data. The FLUENT model uses the electrode exchange current densities to quantify the magnitude of the activation overpotentials via a Butler-Volmer equation [1], A radiation heat transfer boundary condition was applied around the periphery of the model to simulate the thermal conditions of our experimental stack, situated in a high-temperature electrically heated radiant furnace. The edges ofthe numerical model are treated as a small surface in a large enclosure with an effective emissivity of 1.0, subjected to a radiant temperature of 1 103 K, equal to the gas-inlet temperatures. 

C Exchange current density in the Butler-Volmer equation (A/cm )... 

Reiss solved the equation of continuity for both minority and majority carriers and used a modified Butler-Volmer equation to describe the net nonequilibrium rate for charge transfer with exchange current densities of minority and majority carriers as... 

These equations are straight lines in a plot of E versus log i, such as that in Fig. 2. This plot of versus log i is called an Evans diagram. Actually, the Butler-Volmer equation is described better by the curve in Fig. 5. The net current at the reversible potential is zero because the forward and reverse current, each equal to the exchange current density, balance each other. The log of the current density approaches negative infinity at the reversible potential at which the net current density goes to zero, and the polarization curve points down at the reversible potential when plotted on semilogarithmic axes. 

In order to explain the water vapor influence, one has to refer to the following equations. According to Geyer et al. [1997], the dependence of the anodic charge transfer resistance (Ret,a in the equivalent circuit used) on the mole fraction of water vapour (X) in the gas can be explained by assuming a Butler-Volmer kinetic where the exchange current density (to) is proportional to powers of the concentrations of the reactants (hydrogen and water vapour) ... 

Figure 4.10 Polarization curves (linear scale) calculated using the Butler-Volmer equation, with n = 2 and a = 0.5, for different exchange current densities.

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