Faradaic reactions impedance

A constant phase element (CPE) rather than the ideal capacitance is normally observed in the impedance of electrodes. In the absence of Faradaic reactions, the impedance spectrum deviates from the purely capacitive behavior of the blocking electrode, whereas in the presence of Faradaic reactions, the shape of the impedance spectrum is a depressed arc. The CPE shows... 

In the equivalent electric scheme of the entire electrochemical cell (Figure 1.5b), we note, starting from the working electrode, the presence of a capacitance, Cd, in parallel with an impedance, Zf, which represents the Faradaic reaction. The presence of the supporting electrolyte in excess indeed induces the formation of an electrical double layer, as sketched in... 

Figure 7. Equivalent circuit for interphase (Raei) resistance of semiconductor (Retec) electrolyte resistance, (Rfar) fara-daic resistance (Csc) space charge capacitance (CDl) double-layer capacitance and (z) parallel impedances associated with surface states, faradaic reactions, etc.

Figure 4.13a shows the most commonly applied model, which represents a polarizable electrode (simple Faradaic reaction) with replacement of the doublelayer capacitance by a CPE. All the parameters in the model have direct physical meanings Rei represents the electrolyte resistance and Rct represents the charge-transfer resistance. The CPE describes the depression of the semicircle, which is often observed in real systems. The total impedance can be obtained as follows ... 

Some electrochemical systems can be described as blocking electrodes for which no Faradaic reaction can occur. At steady state, the current density for such a system must be equal to zero. The transient response of a blocking electrode is due to the charging of the double layer. At short times or high frequency, the interfacial impedance tends toward zero, and the solution adjacent to Ihe electrode can then be considered to be an equipotential surface. The short-time or high-frequency current distribution, therefore, follows the primary distribution described in the... 

The representation of an Ohmic impedance as a complex number represents a departure from standard practice. As will be shown in subsequent sections, the local impedance has inductive features that are not seen in the local interfacial impedance. As the calculations assumed an ideally polarized blocking electrode, the result is not influenced by Faradaic reactions and can be attributed only to the Ohmic contribution of the electrolyte. 

Table 10.1 Some useful relationships for the development of the impedance response associated with Faradaic reactions.

Equations (10.17), (10.19), and (10.21) are extremely useful relationships for the development of the impedance response associated with Faradaic reactions. As they are used repeatedly in this chapter, generalized forms of these equations are summarized in Table 10.1. 

An electrical circuit that yields the impedance response equivalent to equation (10.25) for a single Faradaic reaction is presented in Figure 10.2. Such a circuit may provide a building block for development of circuit models as shown in Chapter 9 for the impedance response of a more complicated system involving, for example, coupled reactions or more complicated 2- or 3-dimensional geometries. 

Remember 10.4 Low-frequency inductive loops in the impedance response can be attributed to Faradaic reactions that involve adsorbed intermediate species. Such systems can be described in terms cf electrical circuits that inwlve inductances. 

The impedance associated with a simple Faradaic reaction without diffusion can be expressed in terms of a CPE as... 

The admittance format is not particularly well suited for analysis of electrochemical and other systems for which identification of Faradaic processes parallel to the capacitance represents the aim of the impedance experiments. When plotted in impedance format, the characteristic time constant is that corresponding to the Faradaic reaction. When plotted in admittance format, the characteristic time constant is that corresponding to the electrol5rte resistance, and that is obtained only approximately when Faradaic reactions are present. 

This technique was the first used to measure double-layer parameters (principally of the dropping mercury electrode) and later to measure electrode impedance in the presence of a faradaic reaction to determine the kinetics of electrode processes. The use of ac bridges provides meas-... 

IV. IMPEDANCE OF A FARADAIC REACTION EWOLVING ADSORPTION OF REACTING SPECIES... 

III.l [see also Eq. (17) and Fig. 2], and that in the presence of a faradaic reaction [Section III. 2, Fig. 4(a)] are found experimentally on liquid electrodes (e.g., mercury, amalgams, and indium-gallium). On solid electrodes, deviations from the ideal behavior are often observed. On ideally polarizable solid electrodes, the electrically equivalent model usually cannot be represented (with the exception of monocrystalline electrodes in the absence of adsorption) as a smies connection of the solution resistance and double-layer capacitance. However, on solid electrodes a frequency dispersion is observed that is, the observed impedances cannot be represented by the connection of simple R-C-L elements. The impedance of such systems may be approximated by an infinite series of parallel R-C circuits, that is, a transmission line [see Section VI, Fig. 41(b), ladder circuit]. The impedances may often be represented by an equation without simple electrical representation, through distributed elements. The Warburg impedance is an example of a distributed element. 

Problems similar to those observed on ideally polarizable solid electrodes also arise in the presence of faradaic reactions at these electrodes. In the next section, various models used to explain solid electrode impedance behavior are presented. 

In the presence of the faradaic reaction, assuming that the faradaic impedance can be expressed as a simple equivalent resistance, the complex plane plots represent a rotated semicircle p ig. 28(b)], instead of a semicircle centered on the Z axis. Similarly, the double-layer capacitance in the presence of the faradaic reaction may be obtained as... 

The frequency dispersion of porous electrodes can be described based on the finding that a transmission line equivalent circuit can simulate the frequency response in a pore. The assumptions of de Levi s model (transmission line model) include cylindrical pore shape, equal radius and length for all pores, electrolyte conductivity, and interfacial impedance, which are not the function of the location in a pore, and no curvature of the equipotential surface in a pore is considered to exist. The latter assumption is not applicable to a rough surface with shallow pores. It has been shown that the impedance of a porous electrode in the absence of faradaic reactions follows the linear line with the phase angle of 45° at high frequency and then... 

Let us consider the double-layer model circuit as shown in Fig. 3.4. This circuit can be modified based on Randles circuit [2], a prevalent circuit in electrochemistry [7]. It consists of an active electrolyte resistance Rg in series with the parallel combination of the double-layer capacitance Cj and an impedance of a faradaic reaction. The faradaic reaction consists of an active charge transfer resistance R and Warburg resistance Rw- Hence, the electrical equivalent circuit can be modified as shown in Fig. 3.5. 

Impedance of the Faradaic Reactions in the Presence of Mass Transfer... 

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