Linear electrode kinetics

Fig. 9.4 Potential difference between the working and reference electrodes (a) and relative error in the determination of WE polarization resistance (b) as functions of misalignment between the WE and CE normalized by the solid-electrolyte thickness (s/d), as calculated by the finite-element analysis assuming linear electrode kinetics [8, 25]. At high s/d ratios, the experimentally measured value stabilizes at a small Nemst potential due to gas-phase polarization of the working electrode,...

The distribution of current (local rate of reaction) on an electrode surface is important in many appHcations. When surface overpotentials can also be neglected, the resulting current distribution is called primary. Primary current distributions depend on geometry only and are often highly nonuniform. If electrode kinetics is also considered, Laplace s equation stiU appHes but is subject to different boundary conditions. The resulting current distribution is called a secondary current distribution. Here, for linear kinetics the current distribution is characterized by the Wagner number, Wa, a dimensionless ratio of kinetic to ohmic resistance. 

Over the years the original Evans diagrams have been modified by various workers who have replaced the linear E-I curves by curves that provide a more fundamental representation of the electrode kinetics of the anodic and cathodic processes constituting a corrosion reaction (see Fig. 1.26). This has been possible partly by the application of electrochemical theory and partly by the development of newer experimental techniques. Thus the cathodic curve is plotted so that it shows whether activation-controlled charge transfer (equation 1.70) or mass transfer (equation 1.74) is rate determining. In addition, the potentiostat (see Section 20.2) has provided... 

The solution of the Laplace equation is not trivial even for relatively simple geometries and analytical solutions are usually not possible. Series solutions have been obtained for simple geometries assuming linear polarisation kinetics "" . More complex electrode kinetics and/or geometries have been dealt with by various numerical methods of solution such as finite differencefinite elementand boundary element. ... 

Linearizing the kinetic term as before, a set of three unknown linear equations is obtained, which is completed by the finite difference expression of the initial and boundary conditions. Inversion of the ensuing matrix allows the calculation of C at each node of the calculation grid and finally, of the current flowing through the electrode, or of the corresponding dimensionless function, by means of its finite difference expression. Calculation inside thin reaction layers may thus be more efficiently carried out than with explicit methods. The combination of the Crank-Nicholson... 

The electrode kinetics of the Zn(II)/ Zn(Hg) system was investigated in PC + DMSO mixtures [73] containing 0.1 M TEAP. It was found that the log A s,corr varies linearly with the Gibbs energy of transfer of the Zn(II) ion. 

Fig. 8.11. A cyclic voltammogram for a reversible charge-transfer reaction. (Reprinted from V. D. Parker, Linear Sweep and Cyclic Voltammetry, in Comprehensive Chemical Kinetics, Electrode Kinetics, Principles and Methodology, C. H. Bamford and R. C. Compton, eds., copyright 1986, p. 148, with permission from Elsevier Science.)...

Define and explain the following terms for electrode kinetics irreversible, quasi-reversible, linear region, and reversible. (Bockris)... 

Linear sweep voltammetry Ep measurements have not been applied extensively for the study of heterogeneous charge transfer kinetics. A serious problem with the use of this method is that Ep in itself is not significant in this respect but rather Ep — Etev is the quantity of interest. While AEP in CV is readily measured, this cannot be said for Etev using only LSV as a measurement technique. Therefore, there does not appear to be any advantage in LSV for the study of electrode kinetics. A more detailed analysis of the LSV wave, by convolution potential sweep or normalized potential sweep voltammetry (both to be discussed later) can provide both a and k°. 

Chapter 1 serves as an introduction to both volumes and is a survey of the fundamental principles of electrode kinetics. Chapter 2 deals with mass transport — how material gets to and from an electrode. Chapter 3 provides a review of linear sweep and cyclic voltammetry which constitutes an extensively used experimental technique in the field. Chapter 4 discusses a.c. and pulse methods which are a rich source of electrochemical information. Finally, Chapter 5 discusses the use of electrodes in which there is forced convection, the so-called hydrodynamic electrodes . 

Fig. 3.12 Variation of the ratio between the linear diffusion layer thickness of slow and fast electrode reactions for spherical electrodes, <5 /5, with the electrode kinetics (through the dimensionless parameter k° /t/D) and the electrode size (through rsj aJnDt).

In Fig. 3.14a, the dimensionless limiting current 7j ne(t)/7j ne(tp) (where lp is the total duration of the potential step) at a planar electrode is plotted versus 1 / ft under the Butler-Volmer (solid line) and Marcus-Hush (dashed lines) treatments for a fully irreversible process with k° = 10 4 cm s 1, where the differences between both models are more apparent according to the above discussion. Regarding the BV model, a unique curve is predicted independently of the electrode kinetics with a slope unity and a null intercept. With respect to the MH model, for typical values of the reorganization energy (X = 0.5 — 1 eV, A 20 — 40 [4]), the variation of the limiting current with time compares well with that predicted by Butler-Volmer kinetics. On the other hand, for small X values (A < 20) and short times, differences between the BV and MH results are observed such that the current expected with the MH model is smaller. In addition, a nonlinear dependence of 7 1 e(fp) with 1 / /l i s predicted, and any attempt at linearization would result in poor correlation coefficient and a slope smaller than unity and non-null intercept. 

An aim of the model is to determine the influence of the various mass transport parameters and show how they influence the polarization behavior of three-dimensional electrodes. In the model we have adopted relatively simple electrode kinetics, i.e., Tafel type, The approach can also be applied to more complicated electrode kinetics which exhibit non-linear dependency of reaction rate (current density) on reactant concentration. 

The electrolysis measurements were conducted at three flow rates i) anolyte 3.4 ml/min and catholyte 4.4 ml/min ii) anolyte 11.8 ml/min and catholyte 11 ml/min) in) anolyte 22 ml/min and catholyte 27 ml/min). The tests were run at ambient temperature and pressure. Linear sweep voltammetry data obtained for the AHA and Nafion 115 membranes indicated very little effect of the flow rate on the electrode kinetics as long as the mass transport limitation is not reached. Apparently, the higher flow rates of reactants passing through the electrodes do not speed up the electrochemical conversion rates in the electrolyser used in this study. 

A variety of outer-sphere reactions were studied on diamond electrodes by Swain, Miller, Ramesham, and others, using potentiodynamic curves taken under the linear potential scan. This method is appropriate for both qualitative and quantitative characterization of the electrode kinetics (for details, see monographs [90, 91]). 

An example of the size of the impurity effects that may arise is shown in Fig. 1, which gives the electrode kinetics for the ferro-ferricyanide reaction on three different zinc oxide single crystals of varying conductivity. Each of the crystals was in excess of 99.999% pure. As can be seen, each crystal gives a linear Tafel plot under cathodic bias. However, the exchange currents, i.e, the extrapolations back to the reversible potential (+. 19 volts), differ by a factor of about 1000 and... 

There is an apparent discrepancy between the treatment of electrode kinetics under Temkin conditions, at intermediate values of the coverage, and the results shown in Fig. 141(b) for the adsorption pseudocapacitance in the same region. For the purpose of calculating the kinetic parameters, we have assumed that 0 is a linear function of potential. This is a valid assumption, as we can see in Fig. 21. Yet such a linear dependence of 6 on should give rise to a constant value... 

It is "common wisdom" in electrode kinetics that the region of micropolarization, where the i7ri plot is linear, can extend to about... 

Disregarding for a moment the electrochemical aspect of this isotherm, we note that 0 is proportional to logC, (as opposed to the Langmuir isotherm, where it is proportional to a linear function of the concentration.) A simitar "logarithmic isotherm" was developed by Temkin. His derivation is much more complex, but in the final analysis it is based on the same physical assumptions. It has, therefore, become common to refer to Eq. 141 as the Temkin isotherm, although Temkin has never used it in this form. It is this approximate form of the Frumkin isotherm which is applied to electrode kinetics, as we shall see below. 

In suitable cases, pulse techniques such as chronocoulometry or rapid linear-sweep voltammetry also can be employed to monitor the electrode kinetics within the precursor state "i.e., to evaluate directly the first-order rate constant, k, [Eq. (a) in 12.3.7.2] rather than k. Such measurements are analogous to the determination of rate parameters for intramolecular electron transfer within homeogeneous binuclear complexes ( 12.2.2.3.2). Evaluation of k is of particular fundamental interest because it yields direct information on the energetics of the elementary electron-transfer step (also see 12.3.7.5). 

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