Warburg parameter

Evidently, the Warburg parameter contains no information about the rate of charge transfer. The value of kt has to be determined either from -Ret rev or fromp. ... 

In order to obtain a surveyable expression for the faradaic impedance, it is convenient to define the Warburg parameters a0 and oR by... 

Figure 4.5.69. Time elapsed of the Warburg parameter W from the Nernst impedance (Eq. 82) for the ( ) Pt/C and ( ) PtRu/C anodes after evaluation of the time dependent impedance spectra with the equivalent circuit from Figure 4.5.67.

Figure 20. Time dependence of EIS parameters for zinc-coated steel with polyurethane topcoat after exposure to artificial and natural seawater (a)yj, (b) and (c) Warburg coefficient. 

In the Warburg impedance, parameters and C are not constant but depend on frequency according to Eq. (12.28). Figure 12.16ft shows plots of the values of... 

In the case of reactions that are not completely irreversible (or not completely reversible), we must account for both the kinetic factors (e.g., the polarization resistance R and the concentration changes (the Warburg impedance). The simplest equivalent circuit for this case is shown in Fig. 12.15c, while Fig. 12.17c shows the impedance diagram for this circuit (AJS = 10 = 1 Q the other parameters... 

Under this electrochemical configuration, it is commonly accepted that the system can be expressed by the Randles-type equivalent circuit (Fig. 6, inset) [23]. For reactions on the bare Au electrode, mathematical simsulations based on the equivalent circuit satisfactorily reproduced the experimental data. The parameters used for the simulation are as follows solution resistance, = 40 kS2 cm double-layer capacitance, C = 28 /xF cm equivalent resistance of Warburg element, W — R = 1.1 x 10 cm equivalent capacitance of Warburg element, IF—7 =l.lxl0 F cm (

charge-transfer resistance, R = 80 kf2 cm. Note that these equivalent parameters are normalized to the electrode geometrical area. On the other hand, results of the mathematical simulation were unsatisfactory due to the nonideal impedance behavior of the DNA adlayer. This should... 

Warburg impedance is a well-known term in the field of impedance spectroscopy because of the early date at which it was published, the formulation came before the rest of the properties of the interface were known. In fact, for nearly all real situations examined in electrochemistry, the Warburg impedance is relatively small. Thus, for a concentration of 1 mol liter and a frequency of 1 kilocycle s l, and using the normal parameters for room temperature, the resistance is in the milliohm cm-2 range. 

Obviously, the faradaic impedance equals the sum of the two contributions f ct, the charge transfer resistance, and Zw = aco-1/2 (1 — i), the Warburg impedance. Again, the meaning of the parameters Rct and a is still implicit at this stage of the treatment and explicit expressions have to be deduced from an explicit rate equation, e.g. the expressions given in eqns. (51). 

In plant cell cultures, shake flask culture is an indispensable stage of cultivation. Investigations in a shake flask are very essential and critical to bioprocess scale-up and optimization. We have developed a simple and convenient technique based on the principle of the Warburg manometric method to measure 02 uptake rate (OUR) and C02 evolution rate (CER) of suspended cells in a shake flask culture. This technique has been successfully applied to suspension cultures of Panax notoginseng cells, and some important bioprocess parameters, such as OUR, CER, respiratory quotient (RQ), SOUR and specific CER (SCER), were quantitatively obtained [99]. As long as the environment temperature is strictly controlled to within an error of 0.1 °C, the measuring system is accurate and reproducible, is easy to operate, is economical, and is also able to treat many samples simultaneously. 

EIS data analysis is commonly carried out by fitting it to an equivalent electric circuit model. An equivalent circuit model is a combination of resistances, capacitances, and/or inductances, as well as a few specialized electrochemical elements (such as Warburg diffusion elements and constant phase elements), which produces the same response as the electrochemical system does when the same excitation signal is imposed. Equivalent circuit models can be partially or completely empirical. In the model, each circuit component comes from a physical process in the electrochemical cell and has a characteristic impedance behaviour. The shape of the model s impedance spectrum is controlled by the style of electrical elements in the model and the interconnections between them (series or parallel combinations). The size of each feature in the spectrum is controlled by the circuit elements parameters. 

However, although powerful numerical analysis software, e.g., Zview, is available to fit the spectra and give the best values for equivalent circuit parameters, analysis of the impedance data can still be troublesome, because specialized electrochemical processes such as Warburg diffusion or adsorption also contribute to the impedance, further complicating the situation. To set up a suitable model, one requires a basic knowledge of the cell being studied and a fundamental understanding of the behaviour of cell elements. 

Z(a)i,ak) = Zre(a)i,ak) +jZim(cai,ak) is the model function, which can be altered using the adjustable parameters the model function can often be presented by an equivalent circuit, involving such elements as resistance, capacitance, and Warburg in series and/or in parallel ... 

Electronic spectra provide a simple and convenient way to monitor changes induced in the oxidase by various chemical treatments. Indeed, spectral observations were at the core of the pioneering observations of MacMunn (12), Keilin (96), and Warburg (97) and more recently many investigators have examined the spectra of isolated oxidase, mitochondrial particles, and electron transport particles. The spectra of the fully oxidized [oxidase (IV)] (97a) and the fully reduced [oxidase (0)] oxidase have been well characterized (52) (Table V). In Table VI are spectral parameters for ligand complexes of various oxidation states (98-103). Although the spectra of most of these complexes have been... 

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. 

At first glance, it may not be obvious that such an approach should work. It is well known, for example, that the impedance spectrum associated with an electrochemical reaction limited by the rate of diffusion through a stagnant layer (either the Warburg or the finite-layer diffusion impedance) can be approximated by an infinite number of RC circuits in series (the Voigt model). In theory, then, a measurement model based on the Voigt circuit should require an infinite number of parameters to adequately describe the impedance response of any electrochemical system influenced by mass transfer. 

Here, the impedance response is independent of the working point, and the frequency dependence is determined solely by the material parameters of the composite. For / <linear branch appears only at frequencies co > a/Cfr). Doublelayer charging and proton transport dominate the overall electrode response in this regime, whereas Faradaic processes are insignificant due to the high frequencies. An equivalent representation of this system is an RC-transmission line [130], Since no fractality or branching of the network is assumed, the response resembles that of a Warburg impedance with a characteristic proportionality Z a where... 

A graphic illustration of these equations is presented in Fig. 11(b). Although, in simple cases, the process parameters may be obtained graphically, the best way to analyze the impedances is by the complex nonlinear least-squares approximation technique. The following parameters may be obtained from such fits 7 , R, and the Warburg coeffi-... 

Experiments carried out on monocrystalline Au(lll) and Au(lOO) electrodes in the absence of specific adsorption did not show any fre-quency dispersion. Dispersion was observed, however, in the presence of specific adsorption of halide ions. It was attributed to slow adsorption and diffusion of these ions and phase transitions (reconstructions). In their analysis these authors expressed the electrode impedance as = R, + (jco iJ- where is a complex electrode capacitance. In the case of a simple CPE circuit, this parameter is = T(Jcaif. However, an analysis of the ac impedance spectra in the presence of specific adsorption revealed that the complex plane capacitance plots (C t vs. Cjnt) show the formation of deformed semicircles. Consequently, Pajkossy et al. proposed the electrical equivalent model shown in Fig. 29, in which instead of the CPE there is a double-layer capacitance in parallel with a series connection of the adsorption resistance and capacitance, / ad and Cad, and the semi-infinite Warburg impedance coimected with the diffusion of the adsorbing species. A comparison of the measured and calculated capacitances (using the model in Fig. 29) for Au(lll) in 0.1 M HCIO4 in ths presence of 0.15 mM NaBr is shown in Fig. 30. 

The parameter cr is the Warburg coefEcient, from which the diffusion coefficient is calculated. 

As for n, which is the parameter that provides the gap between the actual capacitace and a pure capacitance, it is independent of temperature. The values are close to 0.5 (Warburg diffusion mechanism), sUghtly higher for sintered beta-alumina and sintered zirconia (0.62 and 0.59), slightly lower for glass and the... 

AC impedance spectroscopy also has seen extensive utility in the study of the hole injection or recombination process depicted in Fig. 14. An equivalent circuit for this process is illustrated in Fig. 17 it does resemble the circuit in Fig. 16(a), except for the Warburg component [84]. Early studies [85-88] utilized the recombination resistance parameter, / r, that was extracted from model fits of the measured AC impedance data. This parameter was seen to be inversely related to the hole injection current, thus signifying that it is indeed related to the recombination process. However, the challenge is to differentiate... 

The analysis above has shown that it is possible to derive kinetic parameters from the impedance spectra of a redox system. The charge-transfer resistance is directly related to the exchange current (Eq. 27) and a medium diffusion coefficient for oxidized and reduced species can be calculated from the coefficient a of the Warburg impedance (Eq. 33). 

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