Koutecky equation

Fig. 4. Left side of Koutecky equation plotted versus the bovine superoxide dismutase concentration. Sodium tetraborate 0.025 Af triphenylphosphineoxide 0.025% (wt.) pH = 9.76 tg = 2.90sec temperature = 25°C. (Reprinted with permission from A. Rigo etal., icc(roanai. Chem. Inlerfac. Electrochem.,57,291-296, 1974.)...

Sheng et al. [34] recently compared the rotating disk electrode (RDE) studies of the HOR in acid and alkaline and those for the carbon-supported Pt (Pt/C) for the first time. RDE studies are used to separate the current into its kinetic- and diffusion-based constituents, as shown by the Levich-Koutecky equation ... 

Although, in theory, the Koutecky-Levich equation can be applied to estimate n y and k at any part of the voltammogram (provided that the conditions stated above are satisfied), for practical reasons only limiting (plateau) currents can be acquired with adequate reproducibility to yield suitable Koutecky-Levich plots. 

The above 1-dimensional model may be extended to 3-dimensions, in a straightforward manner, and yields a substrate whose surface is completely covered by adatoms. The TBA again leads to a difference equation and boundary conditions which can be solved directly (Grimley 1960). We do not intend to discuss the 3-dimensional case here and, instead, direct the reader to the loc. sit. articles. However, in passing, we note that, even when there is no direct interaction between the atoms in the adlayer, an important indirect interaction occurs between them via the substrate by a delocalization of the bonding electrons in directions parallel to the surface (Koutecky 1957, Grimley 1960). This topic is discussed in Chapter 8. 

It is interesting to note that there is no complete symmetry between the role of substrate diffusion and electron transport in their combination with the catalytic reaction, as can be seen in the structures compared in the equations and also in the fact that linear Koutecky-Levich plots are not obtained in all cases, as noted in Table 4.1. 

It can be shown by rearranging the Levich equation and inserting via the Butler-Volmer equation that the Koutecky-Levich equation takes the following form ... 

Occasionally, the analyst is required to determine the rate of electron transfer, ket, and can then use the Butler-Volmer equation (equation (7.16)) to determine 7o, from which ket is readily calculated by using equation (7.17). The preferred method of obtaining the exchange currents in such cases is under conditions of infinite rotation speed i.e. via a Koutecky-Levich plot. 

We therefore advise that the reader should consult a recent series of papers published by Galvez et al. [171, 172] encompassing all the mechanisms mentioned in Sect. 7.1, elaborated for both d.c. and pulse polarography. The principles of the Galvez method are clearly outlined in the first part of the series [171]. It is similar to the dimensionless parameter method of Koutecky [161], which enables the series solutions for the auxiliary concentration functions cP and cQ exp (kt) and

combined directly with the partial differential equations of the type of eqn. (203). In some of the treatments, the sphericity of the DME is also accounted for. The results are usually visualized by means of predicted polarograms, some examples of which are reproduced in Fig. 38. Naturally, the numerical description of the surface concentrations at fixed potential are also immediately available, in terms of the postulated power series, and the recurrent relationships obtained for the coefficients of these series. 

Koutecky obtained the numerical solution for the catalytic current at a DME by rigorous solution of the convective-diffusion equations by the expanding-plane model with =0 [199]. Subsequently, an approximate analytical solution was obtained which holds over the whole range of k j with an error of not more than 1% [188]. The equations are... 

Note that the finite electrode volume has not been considered to deduce Eq. (2.137), i.e., we have used the so-called Koutecky approximation (see Eq. (2.134) and [52]). Therefore, when amalgamation takes place, these equations with the lower sign cannot be used for very small spherical electrodes for which numerical treatments considering null flux at the center of the electrode are needed. 

In Eq. (4.39), the upper sign refers to solution soluble product and the lower one to amalgam formation. When species R is amalgamated inside the electrode, the applicability of this analytical equation is limited by Koutecky approximation, which considers semi-infinite diffusion inside the electrode, neglecting its finite size and simplifying the calculations. Due to the limitations of this approximation, the analytical and numerical results coincide only for < 1 with a relative difference < 1.7% [20]. For higher values of c,. a numerical solution obtained with the condition (3cR/3r)r=0 = 0 should be used. 

The Dimensionless Parameter is a mathematical method to solve linear differential equations. It has been used in Electrochemistry in the resolution of Fick s second law differential equation. This method is based on the use of functional series in dimensionless variables—which are related both to the form of the differential equation and to its boundary conditions—to transform a partial differential equation into a series of total differential equations in terms of only one independent dimensionless variable. This method was extensively used by Koutecky and later by other authors [1-9], and has proven to be the most powerful to obtain explicit analytical solutions. In this appendix, this method will be applied to the study of a charge transfer reaction at spherical electrodes when the diffusion coefficients of both species are not equal. In this situation, the use of this procedure will lead us to a series of homogeneous total differential equations depending on the variable, v given in Eq. (A.l). In other more complex cases, this method leads to nonhomogeneous total differential equations (for example, the case of a reversible process in Normal Pulse Polarography at the DME or the solutions of several electrochemical processes in double pulse techniques). In these last situations, explicit analytical solutions have also been obtained, although they will not be treated here for the sake of simplicity. 

To solve the problem corresponding to the second potential step, the Koutecky s dimensionless parameter method [3] has been applied by assuming that the solutions of the differential equation system (G.29) are functional series of the dimensionless variables Xi and p ... 

Due to the formation of an intermediate complex, this type of reaction mechanism was described as being analogous to Michaelis-Menten kinetics [39]. A common error made when examining the behaviour of systems of this type is to use the Koutecky-Levich equation to analyse the rotation speed-dependence of the current. This is incorrect because the Koutecky-Levich analysis is only applicable to surface reactions obeying strictly first-order kinetics. Applying the Koutecky-Levich analysis to situations where the surface kinetics are non-linear, as in this case, leads to erroneous values for the rate constants. Below, we present the correct treatment for this problem based on an extension of a model originally developed by Albery et al. [42]... 

As an alternative to EHD measurements, the presence of a surface film of constant thickness can be detected using Koutecky-Levich analysis. From equation (10.32),... 

Vandeputte et al. [122] used both of these equations to derive more accurate values of the constants than could be obtained from the Koutecky-Levich analysis alone, and hence derived rate constants for the predissociation of the thiosulfate complex involved in the electrodeposition of silver from thiosulfate solutions. 

The elimination of transport effects is not so readily achieved. One relatively simple procedure is to measure currents (/m) as a function of electrode angular velocity (co) using a rotating disc electrode. Currents free of diffusive transport effects (/k) can then be obtained by application of the Koutecky-Levich equation,... 

Figure 9 Use of the Koutecky-Levich equation to correct for diffusive transport effects on the anodic dissolution of Cu in 1 mol dm-3 NaCl recorded on a rotating disk electrode.

The fit to the Koutecky-Levich equation, Fig. 9, demonstrates that the anodic dissolution of Cu occurs under mass-transport control, and extrapolation of these fits to co 1/2 = 0 yields kinetically controlled currents, 7k, free from transport effects and appropriately used in Tafel plots. 

Oct. 14, 1922, Kromeriz, then Czechoslovakia - Aug. 10, 2005, Berlin, Germany) Koutecky was a theoretical electrochemist, quantum chemist, solid state physicist (surfaces and chemisorption), and expert in the theory of clusters. He received his PhD in theoretical physics, was later a co-worker of -> Brdicka in Prague, and since 1967 professor of physical chemistry at Charles University, Prague. Since 1973 he was professor of physical chemistry at Freie Universitat, Berlin, Germany. Koutecky solved differential equations relevant to spherical -> diffusion, slow electrode reaction, - kinetic currents, -> catalytic currents, to currents controlled by nonlinear chemical reactions, and to combinations of these [i-v]. For a comprehensive review of his work on spherical diffusion and kinetic currents see [vi]. See also Koutecky-Levich plot. 

Koutecky-Levich plot — The diffusion-limited current fiim> diff at a -> rotating disk electrode is given by the -> Levich equation based totally on mass-transfer-limited conditions. The disk current in the absence of diffusion control, i.e., in case of electron transfer control, would be... 

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