A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 55
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An exception is the case where the initial step isthe rate-determining electron transfer [Section 3.5.4(b)], in which case all that has beendiscussed for totally irreversible systems also applies, but with the current multiplied consistently by n.Although a large wave slope is a clear indicator that a system is not showing clean reversible behavior, it does not necessarily imply that one has an electrode process controlled by the kinetics of electron transfer.
Electrode reactions frequently include purelychemical processes away from the electrode surface. A system involving "chemical complications" of this kind can show a wave shape essentially identical with that expected fora simple electron transfer in the totally irreversible regime. For example, the reduction ofnitrobenzene in aqueous solutions can lead, depending on the pH, to phenylhydroxylamine (32):HPhNO 2 + 4H + + 4e -> PhNOH + H 2 O(5.5.50)However, the first electron-transfer stepPhNO 2 + e -» PhNO 2 T(5.5.51)is intrinsically quite rapid, as found from measurements in nonaqueous solvents, suchas DMF (32). The irreversibility observed in aqueous solutions arises because of theseries of protonations and electron transfers following the first electron addition. If oneTABLE 5.5.1 Wave Shape Characteristics at 25°C in Sampled-Current VoltammetryLinear DiffusionKinetic RegimeWave Slope/mVReversible (n > 1)Linear, 59.
l/nQuasireversibleNonlinear(n= 1)Irreversible in == 1) NonlinearSteady State|£ 3 / 4 - Si/4|/mVWave Slope/mV\E3/4 -56.4//IBetween 56.4 and45.0/a45.0/aLinear, 59A/nNonlinear56A/nBetween 56.4 and56.4/a56.4/aLinear, 59.1 /aEm\/mV5.5 Sampled-Current Voltammetry for Quasireversible and Irreversible Electrode Reactions203treated the observed voltammetric curve of nitrobenzene using the totally irreversibleelectron-transfer model, kinetic parameters for the electron transfer might be obtained,but they would be of no significance.
Treatment of such complex systems requires amore complete elucidation of the electrode reaction mechanism as discussed in Chapter 12.Before one uses wave shape parameters to diagnose the kinetic regime, one mustbe sure of the basic chemistry of the electrode process. It is easier to establish confidence on this point by investigating the system with a method, such as cyclic voltammetry, that can provide direct observations in the forward and reverse directions(Chapter 6).If the system shows totally irreversible behavior based on the kinetics of interfacialelectron transfer, then kinetic parameters can be obtained in any of several ways:1.Point-by-point evaluation of kf.
From a recorded voltammogram, one can measure ///jj at various potentials in the rising portion of the wave and find the corresponding values of kf by a procedure that depends slightly on the voltammetricmode: (a) If the voltammetry is based on linear diffusion, then one uses a tableor plot of Fi(A), such as Figure 5.5.2, to determine A for each i/i^. Given т andDQ, a value of kf can then be calculated from each value of A. (b) In the case ofsteady-state voltammetry, one uses the array of ili& and (5.5.47) to obtain corresponding values of к, which in turn will yield values of kf if DQITQ is known.This approach involves no assumption that the kinetics follow a particularmodel.
If a model is subsequently assumed, then the set of kf values can be analyzed to obtain other parameters. It is common to assume the Butler-Volmermodel, which implies that a plot of log kf vs. E — E0' will provide a from theslope and k° from the intercept. Of course, this procedure requires knowledge ofEOf by some other means (e.g., potentiometry), because it cannot be determinedfrom the wave position.2.Wave-slope plot. Totally irreversible steady-state voltammograms give linearplots of E vs.
log [(/d — /)//] in accord with (5.5.48). The slope provides a andthe intercept at Eo> yields k° if Do/r0 is known. This approach involves the assumption that Butler-Volmer kinetics apply. For a totally irreversible wavebased on early transients, the wave-slope plot is predicted to be slightly curved;consequently it does not have quantitative utility.3.Tomes criterion and half-wave potential. As one can see from Table 5.5.1,|£з/4 — Ещ\ for a totally irreversible system provides a directly.
That figure canthen be used in conjunction with (5.5.30) [for early transients] or 5.5.49 [forsteady-state voltammetry] to obtain k°. Butler-Volmer kinetics are implicit andE0' must be known.4.Curve-fitting. The most general approach to the evaluation of parameters is toemploy a nonlinear least-squares algorithm to fit a whole digitized voltammogram to a theoretical function. For a totally irreversible wave, one could develop a fitting function from (5.5.28) [for early transients] or (5.5.47) [forsteady-state currents] by using a specific kinetic model to describe the potentialdependence of kf in terms of adjustable parameters.
If the Butler-Volmer modelis assumed, the appropriate substitution is (5.5.4), and the adjustable parametersare a and k°. (The latter might be carried in the fitting process as A0 or к0.) Thealgorithm then determines the values of the parameters that best describe theexperimental results.Chapter 5. Basic Potential Step MethodsIf the voltammetry is quasireversible, one cannot use simplified descriptions of the waveshape, but must analyze results according to the appropriate general expression, either(5.5.24) or (5.5.44). The most useful approaches are:1. Method ofMirkin and Bard (33). If the voltammetry is based on steady-state currents, one can analyze a quasireversible wave very conveniently in terms of twodifferences, \Ещ — Ey2\ and \Ey4 - Ец2\.
Mirkin and Bard have published tables correlating these differences with corresponding sets of k° and a; hence onecan evaluate the kinetic parameters by a look-up process. Reference 33 containsa table for uniformly accessible electrodes, which applies to a spherical or hemispherical UME. A second table is given for voltammetry at a disk UME, whichis not a uniformly accessible electrode.2.Curve fitting. This method applies to voltammetry based on either transient orsteady-state currents and proceeds essentially exactly as described for totally irreversible systems, except that the fitting function must be developed from(5.5.24) or (5.5.44).For a quasireversible wave, Ец2 is not far removed from Eo> and is sometimes usedas a rough estimate of the formal potential.
Better estimates can be made from fundamental equations after the kinetic parameters have been evaluated from the wave shape. Thetables published by Mirkin and Bard for their method actually provide п(Ец2 - Е°) withsets of£° and a (33).Whenever one is concerned with the evaluation of kinetic parameters, it is importantto remember that the kinetic regime is defined partly by experimental conditions and thatit can change if those conditions are altered. The most important experimental variable affecting the kinetic regime in voltammetry based on linear diffusion is the sampling time т.For steady-state voltammetry, it is the radius of the electrode TQ.
See Sections 5.5.l(f) and5.5.3(b) for more detailed discussion. In estimating kinetic parameters, the actual shape ofthe electrode can be important. For example in making small electrodes (sub-/xm radius),the metal disk is sometimes recessed inside the insulating sheath and has access to the solution only through a small aperture (Problem 5.17). Such an electrode will show a limiting current characteristic of the aperture radius, but the heterogeneous kinetics will begoverned by the radius of the recessed disk (34, 35).5.6 MULTICOMPONENT SYSTEMS AND MULTISTEP CHARGETRANSFERSConsider the case in which two reducible substances, О and O', are present in the samesolution, so that the consecutive electrode reactions О + ne —> R and O' + n'e —> R/can occur.
Suppose the first process takes place at less extreme potentials than the second and that the second does not commence until the mass-transfer-limited region hasbeen reached for the first. The reduction of species О can then be studied without interference from O\ but one must observe the current from O' superimposed on that causedby the mass-transfer-limited flux of O. An example is the successive reduction of Cd(II)and Zn(II) in aqueous KC1, where Cd(II) is reduced with an Ец2 near —0.6 V vs.SCE, but the Zn(II) remains inactive until the potential becomes more negative thanabout -0.9 V.In the potential region where both processes are limited by the rates of mass transfer[i.e., C o (0, t) = CO'(0> t) = 0], the total current is simply the sum of the individual diffu-5.6 Multicomponent Systems and Multistep Charge Transfers ^ 205sion currents.
For chronoamperometry or sampled-current voltammetry based on lineardiffusion, one hasl2('d)total = -J^ji (nD d C* + n' DtfCp)(5.6.1)where t is either a sampling time or a variable time following the potential step. For sampled-current voltammetry based on the steady-state at an ultramicroelectrode('d)totai = FA(nmoC% + n'mo> C%>)(5.6.2)fwhere mo and mo are defined by Table 5.3.1 for the particular geometry of the UME.In making measurements by sampled-current voltammetry, one would obtain traceslike those in Figure 5.6.1.
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