Resistance Faradaic

Failure of materials, hydrogen coadsorption, 1340 Faradaic resistance, 1175 Faiadaic current density. 1250, 1404. 1414 Faraday, Michael, 1050, 1346... 

This total resistivity, RF, will be called the faradaic resistance of the interface. 

In fact, one can imagine (Fig. 7.70) that the electrodic reaction is like a resistor and the faradaic resistance of the overall reaction is a series combination of resistors in an electrical circuit. Then the overall conductance of the circuit is approximately given by the smallest conductance or largest resistance as long as one of the resistors is significantly—say, 10 times—larger than any of the other resistors. 

Now, it has already been seen in Chapter 7 that one may write an equivalent circuit for a simple electrode/solution interface as given in Fig. 7.52, where Rsoln is the resistance of the solution RF is the Faradaic resistance and DL the double-layer capacitance then the relaxation time6 is given by ... 

Fig. 1.6 Equivalent circuit for a two-electrode cell. A single interface is usually represented by the elements in the dashed rectangle. Cdh RP, and Rs denote the double-layer capacitance, the Faradaic resistance, and the solution resistance, respectively...

The above-described situation is but an exception rather than the rule. Generally, the diamond electrode capacitance is frequency-dependent. In Fig. 12 we show a typical complex-plane plot of impedance for a single-crystal diamond electrode [69], At lower frequencies, the plot turns curved (Fig. 12a), due to a finite faradaic resistance Rp in the electrode s equivalent circuit (Fig. 10). And at an anodic or cathodic polarization, where Rf falls down, the curvature is still enhanced. At higher frequencies (1 to 100 kHz), the plot is a non-vertical line not crossing the origin (Fig. 12b). Complex-plane plots of this shape were often obtained with diamond electrodes [70-73],... 

Typical values of o and a, as well as the faradaic resistance Rf, series resistance Rs (and the film resistivity p calculated thereby, assuming that the electrolyte resistance can be neglected) are presented in Table 3 for polycrystalline and single-crystal diamond electrodes and a DLC electrode [69-77]. By comparing the impedance... 

The rather high values of the faradaic resistance RF are due to the above-mentioned low background currents and negligible corrosion rate of diamond in aqueous solutions. Relatively high a values, as compared with other semiconductor electrodes [6], reflect the higher doping level of diamond films the acceptor concentration Na would be no less than 1018 cm-3. 

Recall that the faradaic resistance can be determined as a low-frequency cut-off at the complex-plane plot of impedance spectrum (compare the equivalent circuit in Fig. 10b). Such plots measured in the Fc(CN)63 /4 solutions of different concentrations are given in Fig. 23a [104] (similar results were obtained in [111]). The plots are (somewhat depressed) semicircles, whose radii decreased with increasing redox couple concentration. Figure 23b shows the line plotted by using the data in Fig. 23a, in accord with Eq. (6). We notice that all three methods yielded similar results. 

Fig. 25. Dependence of faradaic resistance measured at the equilibrium redox potential on the polycrystalline film resistivity for (1) Fe(CN)63, 4 and (2) quinone/hydroquinone systems. Reprinted from [110], Copyright (1997), with permission from Elsevier Science.

Reynolds number (characteristic solution velocity x characteristic length/kinematic viscosity) faradaic resistance for an electrochemical reaction electrolyte resistance between a reference electrode and a working electrode Schmidt number (= v/D)... 

Fig. 10.4. Equivalent circuit for the electrode with hydrodynamic modulation R. denotes the solution resistance, C0/. the double layer capacitance, RF the faradaic resistance and Z the impedance contributed by diffusional transport of the reactants to the surface. Hydrodynamic modulation is represented by the modulated current, JD and the resultant current in the external circuit represented by I,.

In the circuit, Rs is the electrolyte resistance, CPE indicates the double-layer capacitance, Rc, is the methanol oxidation charge-transfer resistance, while R1 and Cl are the mass transfer related resistance and capacitance (mainly due to methanol adsorption or CO coverage). The physical expression of these parameters can be deduced from the reaction kinetics. In the methanol oxidation reaction, the overall charge transfer rate is the sum of each charge-transfer step (rct). The Faradaic resistance (Rj) equals the inverse of the DC polarization curve slope ... 

According to the equivalent circuit, at zero frequency the Faradaic resistance equals the sum of the charge-transfer resistance and the mass transfer resistance ... 

Figure 6.61 shows the four parameters (Rf, Rct, Rc. and f) with potential. In the region of 370-600 mV versus RHE, the inverse of the Faradaic resistance and charge-transfer resistance versus electrode potential gives the Tafel slope. The changing features in these plots indicate a change in the mechanism of methanol electrooxidation. 

The faradaic resistance is also increased by formation of the complex, which reduces the concentration of free copper ions in solution, hence reducing the rate of their reduction, but here we choose to discuss only the effect of the diffuse double layer. 

Fig. IIG General equivalent circuit with an unspecified element Z in the faradaic resistance branch, showing that the double layer capacitance can be obtained by extrapolation to sufficiently high frequency.

If diffusion limitation is considered, the overpotential decays more slowly, as shown in Fig. 7K. This should be evident, since the Warburg impedance -W- is added in series with the faradaic resistance / p. In this case the plot of logq versus t is not linear and a much more complex mathematical treatment, taking into account the diffusion equations, must be applied to calculate the kinetic parameters. 

How should one calculate the faradaic resistance from such a plot Surely one cannot just take the distance between the points A and B on the ReZ axis as being equal to R, since this is not the diameter of the semicircle. The distance between the points A and B is equal to the diameter of the semicircle, but this line does not lie on the ReZ axis. What is the physical meaning of the angle of depression ... 

In the absence of mass transport limitations, the local current density at a given potential is determined by the sum of two resistors in series the faradaic resistance and the solution resistance. For values of Wa much less than unity the solution resistance is dominant and the current distributions depends primarily on geometry. This is the realm of primary current distribution. For Wa much greater than unity the faradaic resistance is predominant and secondary current distribution is... 

The correct expression is obtained if we replace the differential faradaic resistance (ar)/ai) in the definition of the Wagner number (Eq. 59M) by the integral resistance (T)/i). The modified Wagner number, f/i is then given by... 

To illustrate the importance of using the integral faradaic resistance, consider two reactions taking place under identical conditions, one having an exchange current density of i = 10 A/cm and the other a... 

Fig. 26M The integral faradaic resistance and the modified Wagner number, as a function of the current density. Parameters used i = 5x10 Alcm p = JO Q. cm L - 2 cm b = b = 0.118 V.

If the interphase is ideally polarizable, the faradaic resistance approaches infinity, and the equivalent circuit shown in Fig. lG(a) can be simplified to that shown in Fig. 2G(b). If it is ideally nonpolari-zable, the faradaic resistance tends to zero, and the equivalent circuit shown in Fig. lG(c) results. Real systems never behave ideally, of course they may approach one extreme behavior or the other, or be... 

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