Impedance expression

Here is more impedance study the simplest cell. In a real-life experiment, one can only work with a complete circuit, which consists of at least two electrodes. Now, to test our newly acquired impedance knowledge on a real-life problem, let s consider a circuit consisting of two identical electrodes. Draw its equivalent circuit and make a try at its impedance expression. Try harder to imagine its Cole-C ole plot You may also use a computer to simulate the situation by using reasonable parameters. To make the situation less complicated, we assume the interface is ideally polarizable. (Kang)... 

Impedance spectroscopy a single interface. Draw the equivalent circuits for the following electrode/electrolyte interfaces, then derive their impedance expression and explain what their Cole-Cole plot will look like (a) An ideally polarizable interface between electrode and electrolyte, (b) An ideally nonpolarizable interface between electrode and electrolyte, (c) A real-life electrode/... 

So, the term [A0a0 + AR aR ] co-1/2 (1 — i) resembles the Warburg impedance corresponding to diffusional mass transport of A, O and R, with a mobile equilibrium between A and 0, i.e. kQ -> °°, whereupon the term in g = kQ /co would vanish. If, however, kQ has a finite value, the faradaic impedance is enlarged by the Gerischer impedance expressed by the term containing g. 

It is, of course, impossible to provide an exhaustive catalog containing the EHD impedance expressions with all possible mechanisms. We will, only, in the next section illustrate the general procedure to follow, by some examples. 

The impedance expression is calculated with the following assumptions ... 

EHD impedances have been measured on the diffusion plateau at 0.7 V/SCE. The mass transport time constant of the redox couple in solution, which is one of the terms implied in the impedance expression is independent of the interface nature. The Schmidt number Sc was first determined on a bare electrode, and this value of 1540 is further used as a fixed parameter in the analysis of the diagrams obtained on pECBZ films at different Q (Fig. 6-14). The different diagrams are analyzed in the light of the theoretical model predicted by expression (6-34). 

Several quantities characterize the behavior of power lines as far as transient response is concerned. One important quantity is the characteristic impedance, expressed as ... 

The last of these is the impedance which has been considered throughout this chapter. We now consider forced convection. For low frequencies the diffusion layer thickness due to the a.c. perturbation is similar to that of the d.c. diffusion layer in these cases convection effects will be apparent in the impedance expressions. For the rotating disc electrode these frequencies are lower than 40 Hz33. For higher frequencies where the two diffusion layers are of quite different thicknesses, the advantage of hydrodynamic electrodes is that transport is well defined with time, as occurs with linear sweep voltammetry. 

Example 4.4 Impedance Expression for a Nested Circuit Derive an expression for the impedance response of the circuit shown in Figure 4.5(a), and give expressions for the dc current and the current at infinite frequency for an applied potential V. 

It is easy to show that (B + ARt) always has a positive value. The easiest way to determine whether the low-frequency loop is inductive or capacitive is to calculate (Z — Rp ) at zero frequency. If the value is positive, an inductive loop is present if the value is negative, a capacitive loop appears. Thus the same impedance expression (10.77) can yield two completely different equivalent circuits according to the potential and the constant parcimeter values. 

Example 16.3 Evaluation of Double-Layer Capacitance Find the meaning of the effective capacitance obtained using equation (16.41) for the convective-diffusion impedance expressed as equation (11.20), i.e.,... 

The measurement of the cell characteristics in a bridge yields values of R and Cb that in series are equivalent to the whole cell impedance, including the contributions from Rq and Q, which are often not of interest in studies focused on the faradaic process. In general, one desires to separate the faradaic impedance from Rq and Q. It is possible to do so by considering the frequency dependencies of R and Cb, or by evaluating Rq and from separate experiments in the absence of the electroactive couple. Techniques for making such determinations are considered in Section 10.4. For the moment, let us assume that the faradaic impedance, expressed as the series combination R and Cg, is evaluable from the total impedance (see Figure 10.1.14). 

Eliminating the angular frequency [Pg.183]

To describe the corrosion processes occurring in the system NiP- 0.5M NaCl solution the equivalent electrical circuit shown in Fig. 21 was designed with the resulting impedance expressed by (6). 

The diffusion impedance expression due the formation of a thin oxide film is define by [31-32]... 

In previous studies, a theoretical analysis of the faradaic impedance has been performed provided that the final product is also soluble [67-69]. Next, the relationships obtained in Ref. [66] have been extended for the case when an insoluble final product (e.g., metal deposit) is formed [69]. No EC was proposed in the previous investigations [67, 68] until it was found [70] that the general impedance expression corresponds to an EC consisting of five elements that, according to the authors, have no sensible physical meaning. At the same time, we proposed another EC [71], which rigorously followed from analytical expressions and contained less subcircuits. So, two different chemical step + electrochemical step (CE) have been proposed for description ofthe same mechanism (Figure 8.12). To avoid confusion in this situation, we made an attempt to analyze both EC so as to clarify their possible interrelation. 

Similarly to Equation 3.6, the solution of the linearized form of (3.32) and (3.33) yields the impedance expression... 

Thus, the diffusion impedance expressions depend on the electrode separations d at low frequencies. One way to identify the finite Warburg impedance is to use measurements at various values of the electrodes separation d. When LpCO / D 3 (at oo °o), the tank term approaches imity, the diffusion length is negligible compared to the whole region avaUable for diffusion d, and Zpjj-j-approaches infinite length Warburg Zy ... 

If the electrodes are non-blocking, then the Cai is now shunted in parallel by a charge-transfer resistance, as shown in Fig. 1.4(b) and discussed in Section 1.4.2. Evaluation of the impedance expression for this equivalent circuit produces two semicircles. The high-frequency semicircle is related to the bulk electrolyte and the low-frequency semicircle, which is more distant from the origin, arises from interfacial prcx esses. The bulk resistance is the Z value at the high-frequency end of the interface semicircle. 

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