Potential planar electrodes

Cyclic voltammetry provides a simple method for investigating the reversibility of an electrode reaction (table Bl.28.1). The reversibility of a reaction closely depends upon the rate of electron transfer being sufficiently high to maintain the surface concentrations close to those demanded by the electrode potential through the Nemst equation. Therefore, when the scan rate is increased, a reversible reaction may be transfomied to an irreversible one if the rate of electron transfer is slow. For a reversible reaction at a planar electrode, the peak current density, fp, is given by... 

Eig. 1. Current flow (—) and electrical potential distribution (—) between two planar electrodes separated by an iasulated channel. 

To be specific we consider a planar electrode in contact with a solution of a z — z electrolyte (i.e., cations of charge number z and anions of charge number -z). We choose our coordinate system such that the electrode surface is situated in the plane at x = 0. The inner potential (x) obeys Poisson s equation ... 

This being stated, applying Laplace s transform one obtains from Fick s second law that the maximum current (i.e. the current at the potential corresponding to the maximum of the peak) for a planar electrode is expressed by ... 

Assume that the reaction ox + c <=> red at the planar electrode is diffusion controlled. Sketch and correlate the concentration profiles Cox =f(x), where x is the distance from the electrode surface to the bulk of the solution, with the shape of the current-potential curve for electrolysis carried out at (a) a stationary disk electrode and (b) a rotating disk electrode. Support your explanation by the equations. (Skompska)... 

The important concept in these dynamic electrochemical methods is diffusion-controlled oxidation or reduction. Consider a planar electrode that is immersed in a quiescent solution containing O as the only electroactive species. This situation is illustrated in Figure 3.1 A, where the vertical axis represents concentration and the horizontal axis represents distance from the electrodesolution interface. This interface or boundary between electrode and solution is indicated by the vertical line. The dashed line is the initial concentration of O, which is homogeneous in the solution the initial concentration of R is zero. The excitation function that is impressed across the electrode-solution interface consists of a potential step from an initial value E , at which there is no current due to a redox process, to a second potential Es, as shown in Figure 3.2. The value of this second potential is such that essentially all of O at the electrode surface is instantly reduced to R as in the generalized system of Reaction 3.1 ... 

Figure 3.1 Concentration-distance profiles during diffusion-controlled reduction of O to R at a planar electrode. D0 = DR. (A) Initial conditions prior to potential step. Cq = 1, CR = 0 mM. (B) Profiles for O (dashed line) and R (solid line) at 1,4, and 10 ms after potential step to Es. R is soluble in solution. (C) Profiles for O and R at 1,4, and 10 ms after potential step to Es. R is soluble in the electrode. (D) Potential stepped from Es to Ef at t = 10 ms so that oxidation of R to O is now diffusion-controlled. Profiles are for 11, 15, and 20 ms after step to Es. R is soluble in the electrode.

The Cottrell equation states that the product it,/2 should be a constant K for a diffusion-controlled reaction at a planar electrode. Deviation from this constancy can be caused by a number of situations, including nonplanar diffusion, convection in the cell, slow charging of the electrode during the potential step, and coupled chemical reactions. For each of these cases, the variation of it1/2 when plotted against t is somewhat characteristic. 

It follows from Equation 6.12 that the current depends on the surface concentrations of O and R, i.e. on the potential of the working electrode, but the current is, for obvious reasons, also dependent on the transport of O and R to and from the electrode surface. It is intuitively understood that the transport of a substrate to the electrode surface, and of intermediates and products away from the electrode surface, has to be effective in order to achieve a high rate of conversion. In this sense, an electrochemical reaction is similar to any other chemical surface process. In a typical laboratory electrolysis cell, the necessary transport is accomplished by magnetic stirring. How exactly the fluid flow achieved by stirring and the diffusion in and out of the stationary layer close to the electrode surface may be described in mathematical terms is usually of no concern the mass transport just has to be effective. The situation is quite different when an electrochemical method is to be used for kinetics and mechanism studies. Kinetics and mechanism studies are, as a rule, based on the comparison of experimental results with theoretical predictions based on a given set of rate laws and, for this reason, it is of the utmost importance that the mass transport is well defined and calculable. Since the intention here is simply to introduce the different contributions to mass transport in electrochemistry, rather than to present a full mathematical account of the transport phenomena met in various electrochemical methods, we shall consider transport in only one dimension, the x-coordinate, normal to a planar electrode surface (see also Chapter 5). 

Fig. 2.1 Concentration profiles of species O at a planar electrode calculated from Eq. (2.19) for the application of a potential pulse for different values of...

Fig. 2.3 Experimental current-time curve (a) and logarithmic curves (b) for the application of a constant potential to a graphite disc electrode of radius 0.5 mm (planar electrode) for the reduction of Fe(CN)g. ...

Fig. 2.15 (Solid line) Current-time curves for the application of a constant potential to a spherical electrode calculated from Eq. (2.142). D0 = Dr = 10-5 cm2 s 1, co = cr = 1 mM, rs = 0.001 cm, (E — E ) = -0.2 V, 7=298 K. (Dashed line) Current-time curves for the application of a constant potential to a planar electrode of the same area as the spherical one calculated from Eq. (2.28). (Dotted line) Steady-state limiting current for a spherical electrode calculated from Eq. (2.148). The inner figure corresponds to the plot of the current of the spherical electrode versus j ft...

Fig. 2.21 Evolution of the half-wave potential with the electrode size for spherical white dots) and cylindrical black dots) electrodes. The value of r 2 for a planar electrode has been included for comparison dashed line), ro = rs for a spherical electrode and ro = rc for a cylindrical one.

Fig. 3.1 Real concentration profiles (solid lines) and linear concentration profiles (dashed lines) of the oxidized species at a planar electrode for the application of a potential step, calculated from...

The characteristics of the current-potential curves (polarograms) are similar to that discussed in Sect. 3.2.1.2.2 for planar electrodes. 

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. 

Fig. 3.14 Single potential step chronoamperometry at large overpotentials, (a) Variation of the limiting current with time (b) concentration profiles at the end of the pulse. Planar electrode. / jrn (/)//ii , (t p) /d o"e (/)//d>1c"e ( p) f°r the two kinetic models considered. k° = 10-4 cm/s. Reproduced with permission of reference [30]. In this Figure X = A.

Even in the simplest situation for which a = a2 = 0.5, the global behavior of the response depends upon three parameters, the difference between the formal potentials AEf, and the rate constants of both steps k(j and k. Thus, the observed current-potential curves are the result of the interaction of thermodynamic and kinetic effects so the appearance of two or one waves would not be due solely to thermodynamic stability or instability of the intermediate species but also to a kinetic stabilization or destabilization of the same [4, 31]. This can be seen in Fig. 3.19 in which the current-potential curves of an EE process with AE = 0 mV taking place at a planar electrode with a reversible first step... 

Fig. 4.1 Current density-time curves when both species are soluble in the electrolytic solution and only species O is initially present. Three electrode sizes are considered planar electrode (solid lines), spherical electrode with rs = 10 5 cm (dotted lines), and spherical ultramicroelectrode with rs = 10-5 cm (dashed lines), and three y values y = 0.5 (green curves), y = 1.0 (black curves), and y = 2.0 (red curves). The applied potential sequences are Ei -Ef -> -oo, E2 - E — +oo. n = T2 = 1 s, Cq = 1 mM, cR = 0, D0 = 10-5 cm2 s 1. Taken from [20] with permission...

It is interesting to highlight the case in which D0 = DR = D (i.e., y = 1) since, surprisingly, in this situation the current density corresponding to the second potential pulse remains unaltered when the electrode radius varies from rs —> oo (planar electrode) up to rs —> 0 (ultramicroelectrodes). This can be easily... 

Eq. (4.61) and Table 2.3 of Sect. 2.6), whereas for the second potential pulse the amount of converted charge is much smaller than that obtained at a planar electrode (macroelectrode). Indeed, when the electrode radius becomes small enough the converted charge for the second potential pulse is constant and coincides with (for example, from Eq. (4.62) in the limit rs current-time curves. 

In Fig. 4.5 it can be seen the influence of t2 on the normalized RPV curves calculated from planar, spherical, and disc electrodes from Eqs. (4.67) and (4.36). From these curves, it can be observed that the decrease of t2 causes an increase of the anodic limiting current (with this increase being more noticeable in the case of planar electrodes), whereas it has no effect on the half-wave potential of the responses (marked as a vertical dotted line). 

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