Frequency dependence of the impedance, and

To show the frequency dependence of the impedance and of the phase angle, we can substitute Equation 2-46 into 2-47 and 2-48, aivinz... 

Finally, the basic equivalence of the two measuring techniques should be appreciated. Although there are many ways to approach such a comparison, the following simplified explanation will, we hope, give a more intuitive feeling for the relationship between EIS and PR measurements. As stated above, both techniques rely on the frequency dependence of the impedance of the double-layer capacitance in order to determine the polarization resistance. EIS uses low frequencies to force the capacitor to act like an open circuit. PR measurements use a slow scan rate to do the same thing. To make comparisons, the idea of equivalent scan rate is useful. Suppose that a particular electrochemical system requires EIS measurements to be made down to 1 mHz in order to force 99% of the current through Rp. What would the equivalent scan rate be for PR measurements A frequency of 1 mHz corresponds to a period of 1000 s. If the sine wave is... 

A Bode plot is an alternative representation of the impedance. There are two types of Bode diagram, log z log (or z log ) and 9 log , describing the frequency dependencies of the modulus and phase, respectively. A Bode plot is normally depicted logarithmically over the measured frequency range because the same number of points is collected at each decade. Both plots usually start at a high frequency and end at a low frequency, which enables the initial resistor to be found more quickly. 

In the 1920s, impedance was applied to biological systems, including the resistance and capacitance of cells of vegetables and the dielectric response of blood suspensions. ° Impedance was also applied to muscle fibers, skin tissues, and other biological membranes. " The capacitance of the cell membranes was found to be a function of frequency, and Fricke observed a relationship between the frequency exponent of the impedance and the observed constant phase angle. In 1941, brothers Cole and Cole showed that the frequency-dependent complex... 

Through this equation the Q-factor is also defined, which, in addition, can be used to specify the losses or quality (Q) of the capacitor. In Figure 1.49 the parallel and series equivalent circuits of a capacitor are shown, along with the frequency dependence of the impedance Z. Both the parallel and series circuit can be used synonymously, that is they describe an identical behavior. (The series circuit is normally preferred for high frequencies.) ESR stands for equivalent series resistance . The total impedance Z for the parallel circuit is ... 

Equation 10.3.5 shows that one can, in principle, evaluate /q from data taken at a single frequency. However, doing so is not really wise, because one has no experimental assurance that the equivalent circuit actually mirrors the performance of the system. The best way to check for agreement is to examine the frequency dependence of the impedance. For example, (10.2.25) and (10.2.26) predict that and 1/mCg should both be linear with and should have a common slope, (7, which is quantitatively predictable from the constants of the experiment that is. 

E.C. Dutoit, RLv Melrhaeghe, F. Cardon, W.P. Gomes, Investigation on the frequency-dependence of the impedance of the nearly ideally polarizable semiconductor electrodes CdSe, CdS and Ti02. Berichte der Bunsengellschaft fur physikalische Chemie 79, 1206-1213... 

Investigation of the temperature and frequency dependences of the real and imaginary parts of the complex impedance of ceramic samples was performed between 4.2 and 300 K in the range of 20-10 Hz for Sn24Pi9.3Brxl8-x for a variable Br/I ratio [66]. This study showed that at room temperature and down to about 75 K the electrical resistivity is intrinsic for all compositions. At lower temperatures a substantial contribution to the complex impedance from the grain boundaries appears. 

Useful information about the adsorption kinetics, mobility of the adsorbed polynucleotide segments, and mechanism of electrode processes can be obtained by measurement of the frequency dependence of the impedance of the electrode double layer (EIS) [31, 88, 207-209]. If the adsorption/desorption process is slow with respect to the period of the a.c. potential used for the impedance measurement, the measured capacitance values decrease with increasing frequency (dispersion of the capacity). The frequency effect is most remarkable around the potentials of adsorption/desorption peaks. With more flexible ss polynucleotides, the frequency effect is larger than with the more rigid ds ones [210]. 

From the frequency dependence of the impedance of the electrode double layer represented in a complex impedance plot (the imaginary component Z" is plotted against the real component Zf Cole-Cole, or Nyquist plot), the electric equivalent circuit of the electrode covered with an adsorbed layer can be determined. From such a circuit, the physical parameters of the layer, such as the effective thickness and the degree of molecular order of the layer, can also be evaluated [213-221]. 

F. Cardon, and W. P. Gomes [1975] Investigation on the Frequency-Dependence of the Impedance of the Nearly Ideally Polarizable Semiconductor... 

Figure 14.10 Bode plot for the determination of the frequency dependence on the impedance and phase angle for the immunosensor. From Ref [30] with permission from Elsevier...

Maslii, A.I., Pirogov, V.Y., and Medvedev, V.Z. (1986) Kinetics of electrodeposition of copper from sulfate solutions. Analysis of the frequency dependence of the impedance for model of step-wise discharge without crystallization limitations. Elektrochimiya, 22 (2), 164-169. 

The KirchhofFs equations for the circuit in Figure 7.14 and the material constants obtained for resonator material and castor oil allowed us to calculate the frequency dependencies of the real and imaginary parts of the electrical impedance of the resonator loaded by the film under study for given values C(-, and L°. Here L° is the thickness of oil layer. Then by changing these parameters with the help of the least-squares method, we found the minimum of the criterion function for which the theoretical and experimental data were closest. Figures 7.12 and 7.13 show by the... 

It was shown in [39, 40, 42], by use of the Delahay and Senda scheme, that the frequency dependence d the impedance and phase angle components make it possible to establish the nature of the electrode processes and to calculate their kinetic and adsorption parameters. In the absence of adsorption of the reaction components cot0 increases linearly with wV2 [29] ... 

Unlike the first two peaks, the third peak is practically independent of f re quency and c or-responds to des orption of dihy-droriboflavinfrom the electrode surface. Similar results are also obtainedfrom chronopotentio-metric measurements. The pseudocapacity peak of the reaction splits even at w = 500 rad/sec, which indicates that the protonated semiquinone is very stable in neutral solution. This complication makes it more difficult to determine the adsorption parameters, but the fact that cot0 < 1 and the frequency dependence of the impedance components is similar to that at pH 0 means that the reduction of riboflavin in neutral solution, as in acidic solution, is a reversible process involving adsorption of both reaction components. 

Figure 12 shows the potential dependence of capacity and conductivity of a mercury electrode in a 2.2 10" mole/liter solution of juglone in phosphate buffer solution at pH 6.8. The capacity curve has three peaks two peaks correspond to the half-wave potentials of the prewave and of the main wave on the polarogram, and the third peak is at E —0,8 V. The frequency dependence of the pseudocapacity peaks of the reaction and their corresponding conductivity peaks is similar to the frequency dependence of the impedance components of the electrode in a solution of riboflavin. The... 

In an analysis of an electrode process, it is useful to obtain the impedance spectrum —the dependence of the impedance on the frequency in the complex plane, or the dependence of Z" on Z, and to analyse it by using suitable equivalent circuits for the given electrode system and electrode process. Figure 5.21 depicts four basic types of impedance spectra and the corresponding equivalent circuits for the capacity of the electrical double layer alone (A), for the capacity of the electrical double layer when the electrolytic cell has an ohmic resistance RB (B), for an electrode with a double-layer capacity CD and simultaneous electrode reaction with polarization resistance Rp(C) and for the same case as C where the ohmic resistance of the cell RB is also included (D). It is obvious from the diagram that the impedance for case A is... 

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