Warburg equation

Analogously, the generalized Warburg equation, representative of the response of constant phase elements in electrochemical impedance spectroscopy experiments, becomes (Nyikos and Pajkossy, 1990 Dassas and Duby, 1995) ... 

Previous discussions and the data in Figure 5-10 can be further generalized for the spatially restricted finite-diffusion situations that were discussed in Section 5-6. The semi-infinite diffusion Warburg equation is replaced by appropriate finite-diffusion expressions, such as Eqs. 5-46 and 5-49 [1, p. 76]. The resulting Nyquist and Bode plots are presented in Figure 5-11 [28]. 

Returning to the fundamental ac harmonic in Fig. 3.42, we wish to establish the relationship between I and the faradaic impedance Z( instead of considering a combination of a series resistance Rs and a pseudo-capacity C8, the alternative is to separate a pure resistance of charge transfer Rct and a kind of resistance to mass transfer Zw, the Warburg impedance the derivation of the polarogram39 then (for AEtc < 8/remV) leads to the equation... 

At high frequencies diffusion of the reactants to and from the electrode is not so important, because the currents are small and change sign continuously. Diffusion does, however, contribute significantly at lower frequencies solving the diffusion equation with appropriate boundary conditions shows that the resulting impedance takes the form of the Warburg impedance ... 

In systems where diffusion phenomena are of significance, the mechanistic study is facilitated by using the general expression for Impedance Z (26). This equation shows for instance how the Warburg coefficient can be evaluated by conducting impedance studies at very low frequencies. These coefficients in turn enable the evaluation of diffusion coefficients for the diffusing species. 

Warburg (Wted. Ann. XLi. 1, 1890) observed that the surface tension of the interface between mercury and dilute acid docixiascs as the amount of the corresponding mercury salt present in the solution increases. He therefore concluded that the salt is positively adsorbed in accordance with Gibbs adsorption equation. The adsorption by mercury of its salts from aqueous solution has been directly observed by McLewis jPA /s. Ghem. Lxxvil. 129,... 

Lippmann s equation may thus be regarded as identical with that of Gibbs expressed in the form given to it by Warburg. 

According to Warburg and Christian the total amount of nucleic acids and protein is calculated using the following equation (the factors F and T are given in Table 1.10 in conjunction with the ratio A280/A260) ... 

This is a form of Sand s equation and dates from the remarkably early time of 1901. It reminds us that there was much progress made in the early days of electrochemistry with diffusion-related situations one might call it the macrophase of the subject. (Hie Warburg impedance was derived even earlier it deals with impedance due to diffusion. See Section 7.5.13.5.) Consideration on a microscale (the things that happen during charge transfer over 5 A at the interface) had another 50 years before it became a funded research topic. 

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). 

The more accurate relationships obtained from the underlying microscopic current equations are discussed below (see Section III.3.iii). There we will see that the impedance that describes the stoichiometry polarization is composed of a so-called Warburg behavior (linear increase with a slope of 45° followed by a semicircular behavior, see Figure 39). Figure 37 shows an experimental example that comes close to the ideal situation. The detailed behavior around the maximum is also in accord with the precise treatment (see Section III.3.iii). 

For a constant phase process, as a diffusion process, the plot is represented as a straight line with one slope (see Figure 8.22) [75], This is evident from Equation 8.86, because when Z, is plotted versus Zr, the Warburg component is represented as a straight line with a unitary slope. 

In the plot in Figure 8.23, it is evident that the ohmic factors are independent of frequency, after this, the ideal activation processes display a semicircular conduct with a frequency which is typical of the corresponding relaxation processes (see Equation 8.88 and Figures 8.20 and 8.21) finally, the concentration processes exhibit a diagonal conduct characteristic of diffusion processes (see Figure 8.22) often referred to as the Warburg behavior [124,129,130] (to see a real Nyquist plot related to an EIS test of a battery, see Section 8.9.1). 

From Equations 3.92 and 3.93, we can see that the frequency dependence at low frequencies comes from Warburg impedance terms hence, the linear correlation of Zm versus Zre features a diffusion-controlled electrode process. As the frequency rises, the presupposition for Equation 3.94 will not exist. The linear correlation will deviate with increasing frequencies. 

The Warburg impedance is only valid if the diffusion layer has an infinite thickness. If the diffusion layer is bounded, the impedance at lower frequencies no longer obeys Equation 4.32. Instead, the bounded Warburg element (BW) should be used to replace the Warburg. The impedance of the series connection between the resistance and the BW, shown in Figure 4.9a, can be calculated by adding their impedances ... 

Equivalent circuits for the catalyst layer are similar to those for porous electrodes, where charge-transfer resistance, capacitance, and Warburg resistance should be considered. The catalyst layer can be conceived of as a whole uniform unit or as a non-uniform circuit. In the case of a uniform unit, the equivalent circuits are similar to the modified ones discussed in Section 4.2.2 2, and the equations in that section apply. In many cases, such as in the presence of adsorbents, the surface is covered by the adsorbed species. For example, in direct methanol fuel cells and in H2/air fuel cells, CO adsorption should be considered. One example is illustrated in Ciureanu s work [7], as shown in Figure 4.31. 

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. 

Turning now to the case in which diffusion control must be considered, we have already seen that the equivalent circuit takes the form shown in Fig. 2K, in which the symbol -W- represents the so-called Warburg impedance which accounts for diffusion limitation. The diffusion equations have been solved for the case of a low amplitude sine wave applied to the interphase. The Warburg impedance is given by... 

Consider the equivalent circuit shown in Fig. 2K, ignoring, for the moment, the Warburg impedance. When a galvanostatic pulse is applied to such a circuit, the response is that shown in Fig. 5M. The equation describing the change of overpotential with time during the transient is... 

Example 2.6 Foundation for Warburg Impedance The following differential equation appears in the treatment of impedance associated with diffusion ... 

As an exercise, the reader can verify that equation (2.73) satisfies both real and imaginary parts of equation (2.70). This development represents the starting point for both the Warburg impedance associated with diffusion in a stationary medium of infinite depth and the diffusion impedance associated with a stationary medium of finite depth. 

Equation (11.52) is known as the Warburg impedance The expression of the cell impedance is obtained by inserting equation (11.52) into equation (11.20). 

Remember 11.2 The Warburg impedance, equation (11.52), applies for diffusion in an infinite stagnant domain. This expression applies as a high-frequency limit for diffusion in a finite domain. 

Equation (11.71) is the Warburg impedance. The impedance response for a stagnant layer is often imposed incorrectly for situations where convective diffusion takes place. The more correct approach is to accovmt explicitly for the role of convection. 

The concentrations follow the diffusion equation. The derivation of this effect leads to an additional impedance (Warburg impedance) as given by... 

The diffusion coefficient of lithium ions in the intercalation electrodes also can be measured, using EIS, by the analysis of Warburg impedance representing diffusion through the electrode with I.C. in Equation (5.28), impermeable B.C. in Equation (5.29), and sinusoidal oscillation B.C. 

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