Equivalent circuit electrode with charge transfer

Figure 4.10 (a) Equivalent circuit of an electrode with charge transfer. R electrolyte resistance,... 

Complexation of alkali metal ions by 15-crown-5 and 12-crown-4 terminated SAMs was studied by EIS [78,80]. The impedance of SAM-covered electrodes in the presence of a redox probe (usually Ru(NH3)g ) was described by the Randles equivalent circuit, and the charge-transfer resistance Rct changed systematically with the metalion concentration. Binding of Na" " ions... 

F ure 3.44 Equivalent circuit of metal-elecirolyte interface for (a) an electrode with charge transfer reaction, showing the double layer capacitance and the charge transfer resistance and for (b) an ideally polarizable electrode with infinite charge transfer resistance. 

When the polymer flhn is oxidized, its electronic conductivity can exceed the ionic conductivity due to mobile counterions. Then, the film behaves as a porous metal with pores of limited diameter and depth. This can be represented by an equivalent circuit via modified Randles circuits such as those shown in Figure 8.4. One Warburg element, representative of linear finite restricted diffusion of dopants across the film, is also included. The model circuit includes a charge transfer resistance, associated with the electrode/fllm interface, and a constant phase element representing the charge accumulation that forms the interfacial double... 

FIG. 7 Simplified equivalent circuit for charge-transfer processes at externally biased ITIES. The parallel arrangement of double layer capacitance (Cdi), impedance of base electrolyte transfer (Zj,) and electron-transfer impedance (Zf) is coupled in series with the uncompensated resistance (R ) between the reference electrodes. (Reprinted from Ref. 74 with permission from Elsevier Science.)... 

Very often, the electrode-solution interface can be represented by an equivalent circuit, as shown in Fig. 5.10, where Rs denotes the ohmic resistance of the electrolyte solution, Cdl, the double layer capacitance, Rct the charge (or electron) transfer resistance that exists if a redox probe is present in the electrolyte solution, and Zw the Warburg impedance arising from the diffusion of redox probe ions from the bulk electrolyte to the electrode interface. Note that both Rs and Zw represent bulk properties and are not expected to be affected by an immunocomplex structure on an electrode surface. On the other hand, Cdl and Rct depend on the dielectric and insulating properties of the electrode-electrolyte solution interface. For example, for an electrode surface immobilized with an immunocomplex, the double layer capacitance would consist of a constant capacitance of the bare electrode (Cbare) and a variable capacitance arising from the immunocomplex structure (Cimmun), expressed as in Eq. (4). 

Figure 7.1 (A) Typical controlled-potential circuit and cell OA1, the control amplifier OA2, the voltage follower (Vr = Er) OA3, the current-to-voltage converter. (B) Equivalent circuit of cell Rc, solution resistance between auxiliary and working electrodes Ru, solution resistance between reference and working electrodes, Rs = Rc + Ru and Cdl, capacitance of interface between solution and working electrode. (C) Equivalent circuit with the addition of faradaic impedance Zf due to charge transfer. Potentials are relative to circuit common, and working electrode is effectively held at circuit common (Ew = 0) by OA3.

The equivalent circuit corresponding to this interface is shown in Fig. 6.1b. The charge-transfer resistances for the exchange of sodium and chloride ions are very low, but the charge-transfer resistance for the polyanion is infinitely high. There is no direct sensing application for this type of interface. However, it is relevant for the entire electrochemical cell and to many practical potentiometric measurements. Thus if we want to measure the activity of an ion with the ion-selective electrode it must be placed in the same compartment as the reference electrode. Otherwise, the Donnan potential across the membrane will appear in the cell voltage and will distort the overall result. 

The simplest and most common model of an electrochemical interface is a Randles circuit. The equivalent circuit and Nyquist and Bode plots for a Randles cell are all shown in Figure 2.39. The circuit includes an electrolyte resistance (sometimes solution resistance), a double-layer capacitance, and a charge-transfer resistance. As seen in Figure 2.39a, Rct is the charge-transfer resistance of the electrode process, Cdl is the capacitance of the double layer, and Rd is the resistance of the electrolyte. The double-layer capacitance is in parallel with the charge-transfer resistance. 

Figure 3.1 shows a typical equivalent circuit of an electrochemical cell. Rel represents the electrolyte resistance between the working electrode surface and the point of reference electrode Cd is a pure capacitor of the capacity associated with the double layer of the electrode/electrolyte interface and Zf refers to the Faradaic impedance, which corresponds to the impedance of the charge transfer at the electrode/electrolyte interface. The connection of X, and Cd in Figure 3.1 is in parallel. The impedance X, can be subdivided in two equivalent ways, as seen in Figure 3.1 b ... 

Faradic impedance (//) is directly related to the rates of charge transfer reactions at and near the electrode/electrode interface. As shown in Figure 3.1, the Faradaic impedance acts in parallel with the double-layer capacitance Cd, and this combination is in series with the electrolyte resistance Rei The parameters Rei and Cd in the equivalent circuit are similar to the idea of electrical elements. However, X/ is different from those normal electrical elements because Faradaic impedance is not purely resistive. It contains a capacitive contribution, and changes with frequency. Faradaic impedance includes both the finite rate of electron transfer and the transport rate of the electroactive reagent to the electrode surface. It is helpful to subdivide Zj into Rs and Cs, and then seek their frequency dependencies in order to obtain useful information on the electrochemical reaction. 

The simplest model is the connection of resistor and capacitor in either series or parallel. Figure 4.2a shows the connection of a resistor and a capacitor in series. This equivalent circuit is the simplest model for an ideal polarizable electrode, with the assumption that neither the charge transfer on the electrode surface nor the diffusion limitations are present. 

Figure 5.17c shows the equivalent circuit of the PEM fuel cell symmetrical gas supply arrangement. Fitting the spectra of the symmetrical cells for both the anode and the cathode, as seen in Figure 5.17c, Rctanode and Rct,catho

An ideally polarizable electrode behaves as an ideal capacitor because there is no charge transfer across the solution/electrode boundary. In this case, the equivalent electrical model consists of the solution resistance, R, in series with the double-layer capacitance, Cdi. An analysis of such a circuit was presented in Section I.2(i). 

These laws, found empirically by Faraday over half a century prior to the discovery of the electron, can now be shown to be simple consequences of the electrical nature of matter. In any electrolysis, a reduction must occur at the cathode to remove electrons flowing from the external circuit into the electrode and an oxidation must occur at the anode to supply the electrons that leave the electrolytic cell at this electrode. By the principle of continuity of current, electrons must be discharged at the cathode at exactly the same rate at which they are supplied to the anode. By definition of the equivalent mass for oxidation-reduction reactions (that fraction of the molar mass associated with the transfer of one mole of electrons), the number of equivalents of electrode reaction must be proportional to the amount of charge transported into or out of the electrolytic cell and must, indeed, be equal to the number of moles of electrons transported in the circuit. The Faraday constant ) is equal to the charge of one mole of electrons ... 

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