A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 56
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The diffusion current for the first wave can be subtracted fromthe total current of the composite wave to obtain the current attributable to O' alone. Thatis,4 = OdXotai ~ «d(5-6-3)where i& and i& are the current components due to О and O', respectively.This discussion assumes that the reactions of О and O' are independent and that theproducts of one electrode reaction do not interfere with the other. While this is frequentlythe situation, there are cases where reactions in solution can perturb the diffusion currentsand invalidate (5.6.3) (36). The classic case is the reduction of cadmium ion and iodate ata mercury electrode in an unbuffered medium, where О is Cd 2 + and O' is 10^.
The reduction of IO^~ in the second wave occurs by the reaction IO^~ + 3H 2 O + 6e —»I~ +6OH~. The liberated hydroxide diffuses away from the electrode and reacts with Cd 2 +diffusing toward the electrode, causing precipitation of Cd(OH)2 and thus decreasing thecontribution of the first wave (from reduction of Cd 2 + to the amalgam) at potentialswhere the second wave occurs. The consequence is a second plateau that is much lowerthan that observed if the reactions were independent (or if the solution were buffered).Similar considerations hold for a system in which a single species О is reduced inseveral steps, depending on potential, to more than one product.
That is,О + nxe -* Ri(5.6.4)R! + n2e -> R 2(5.6.5)where the second step occurs at more extreme potentials than the first. A simple exampleis molecular oxygen, which is reduced in two steps in neutral solution. Figure 5.6.2 showsthe behavior of this system in the polarographic form of sampled-current voltammetry.(See Chapter 7 for more on polarography.) In the first reduction step, oxygen goes to hydrogen peroxide with a two-electron change manifested by a wave near —0.1 V vs.
SCE.У tЕ('d)total =1'I:'Figure 5.6.1 Sampled-currentvoltammogram for a two-componentsystem.206Chapter 5. Basic Potential Step Methods10 цА--0.6-0.8-1.0E (V vs. SCE)Figure 5.6.2 Polarographic form of sampled-current voltammetry for air-saturated 0.1M KNO3with Triton X-100 added as a maximum suppressor. The working electrode is a dropping mercuryelectrode, which produces oscillations as individual drops grow and fall.
This curve was recordedin the classical mode using a recorder that was fast enough to follow current changes near the endof drop life, but not at drop fall, when the current goes almost to zero. The top edge of the envelopecan be regarded as a sampled-current voltammogram.A second two-electron step takes hydrogen peroxide to water. At potentials less extremethan about —0.5 V, the second step does not occur to any appreciable extent; hence onesees only a single wave corresponding to a diffusion-limited, two-electron process.
Atstill more negative potentials, the second step begins to occur, and beyond —1.2 V oxygen is reduced completely to water at the diffusion-limited rate.For the entire process [(5.6.4) and (5.6.5)], it is clear that at potentials for which thereduction of О to R 2 is diffusion controlled, the early transient current following a potential step is simply(5.6.6)and the steady-state current at a UME is/ d = FAYHQ CQ(HI+n2)(5.6.7)where MQ is given for the particular geometry in Table 5.3.1.
Equations for currents measured in sampled-current voltammetric experiments can be written analogously. Our focushere is on the limiting current resulting from a multistep electron transfer involving agiven chemical species. There are, in addition, many interesting kinetic and mechanisticaspects to processes involving sequential electron transfers, but we defer them for consideration in Chapter 12.5.7 Chronoamperometric Reversal Techniques <l 2075.7 CHRONOAMPEROMETRIC REVERSAL TECHNIQUESAfter the application of an initial potential step, one might wish to apply an additionalstep, or even a complex sequence of steps. The most common arrangement is the doublestep technique, in which the first step is used to generate a species of interest and thesecond is used to examine it. The latter step might be made to any potential within theworking range, but it usually is employed to reverse the effects of the initial step.
Anexample is shown in Figure 5.7.1. Suppose an electrode is immersed in a solution ofspecies О that is reversibly reduced at E° . If the initial potential, Ev is much morepositive than E°', no electrolysis occurs until, at t = 0, the potential is changedabruptly to £ f , which is far more negative than E°'. Species R is generated electrolytically for a period r, then the second step shifts the electrode to the comparatively positive value Er. Often ET is equal to Ev The reduced form R then can no longer coexistwith the electrode, and it is reoxidized to O.
This approach, like other reversal techniques, is designed to provide a direct observation of R after its electrogeneration.That feature is useful for evaluating R's participation in chemical reactions on a timescale comparable to r.This sort of experiment is not useful in the steady-state regime, because the currentobserved in the reversal step at steady state reflects the bulk concentration of R, ratherthan that generated in the forward step.
Consequently, we treat the experiment describedjust above under the condition that semi-infinite linear diffusion applies.5,7.1 Approaches to the ProblemTo obtain a quantitative description of this experiment, one might consider first the resultof the forward step, then use the concentration profiles applicable at т as initial conditions for the diffusion equation describing events in the reversal step. In the case outlinedabove, the effects of the forward step are well known (see Section 5.4.1), and this directapproach can be followed straightforwardly. More generally, reversal experiments present very complex concentration profiles to the theoretician attempting to describe thesecond phase, and it is often simpler to resort to methods based on the principle of superposition (37, 38).
We will introduce the technique here as a means for solving the presentproblem.The applied potential can be represented as the superposition of two signals: a constant component £ f for all t > 0 and a step component Er - Ef superimposed on the con-Figure 5.7.1 General waveformfor a double potential stepexperiment.208Chapter 5. Basic Potential Step MethodsEEfEr0Component IFigure 5.7.21Component IICompositeA double-step waveform as a superposition of two components.stant perturbation for t > т. Figure 5.7.2 is an expression of this idea, which is embodiedmathematically as(t > 0)E(t) =E{ + ST(t)(ET - Ef)(5.7.1)where the step function 5T(r) is zero for t < r and unity for t > r.
Similarly, the concentrations of О and R can be expressed as a superposition of two concentrations that may beregarded as responsive to the separate potential components:Co(x, t) = Clo(x, t) + ST(0Cg(x, t - т)CR(JC, t) = ClR(x, t) + ST(f)Cg(Jc, t - T)(5.7.2)(5.7.3)Of course, the boundary conditions and initial conditions for this problem are mosteasily formulated in terms of the actual concentrations C0(x, t) and CR(x, t), and we writethe initial situation asСо(х, 0) = CgCK(x9 0) = 0(5.7.4)During the forward step we haveC(C\ t\ — СOv-'j L) — ^ OС (C\ t\ — С^Rv^9 f /R(5.7.5)For reasons discussed below, we will treat only situations where the O/R couple is nernstian; thus(5.7.6)C o — 0 CT>whereв' = cxp[nf(Ef -E0)](5.7.7)The reversal step is defined byC o (0, t) = CoC R (0, t) =(5.7.8)and(5.7.9)c0 =where(5.7.10)ff1 =By relying on (5.7.9) and (5.7.10), we have again confined our treatment to systems inwhich the electron transfer is nernstian. At all times, the semi-infinite conditions:lim Co(x, i) = Cglim CR(x, t) = 0(5.7.11)and the flux balance:/O«U)= "/R(0,0are applicable.(5.7.12)5.7 Chronoamperometric Reversal Techniques ^1 209Note that all of these conditions, as well as the diffusion equations for О and R, are1linear.
An important mathematical consequence is that the component concentrations CQ ,11nCQ , CR , and C R can all be carried through the problem separately. Each makes a separable contribution to every condition. We can therefore solve individually for each component, then combine them through (5.7.2) and (5.7.3) to obtain the real concentrationprofiles, from which we derive the current-time relationship. These steps, which are de10tailed in the first edition, are left for the reader now as Problem 5.12.The method of superposition can succeed when linearity exists and separability of thecomponent concentrations can be assured. Unfortunately, many electrochemical situationsdo not satisfy this requirement, and in such instances other predictive methods, such assimulation, must be applied.Quasireversible electron transfer in a system with chemically stable О and R has beenaddressed, initially on the basis of a special case (39), and subsequently in a general wayyielding a series solution (40) that allows the extraction of kinetic parameters from experimental data under a wide variety of conditions.5.7.2Current-Time ResponsesSince the experiment for 0 < / < т is identical to that treated in Section 5.4.1, the currentis given by (5.4.16), which is restated for the present context as(5.7.13)From the treatment outlined in the previous section, the current during the reversalstep turns out to beф) =~nFAD\?C*0 U!i\Г!111—&- {{r^W ~ T+*r Д ^ П Г ] " ^ T ^ J(5ЛЛ4)A special case of interest involves stepping in the forward phase to a potential on thediffusion plateau of the reduction wave (0' ~ 0, CQ « 0), then reversing to a potential onthe diffusion plateau for reoxidation (0" -> °°, C R ~ 0).
In that instance, (5.7.14) simplifies to the result first obtained by Kambara (37):11 I(5.7.15)Note that this relation could also have been derived under the conditions CQ = 0 andCR = 0 without requiring nernstian behavior. It therefore holds also for irreversible systems, provided large enough potential steps are employed.Figure 5.7.3 shows the kind of current response predicted by (5.7.13) and (5.7.14). Incomparing a real experiment to the prediction, it is inconvenient to deal with absolute currents because they are proportional to AD 0 1 / 2 , which is often difficult to ascertain. Toeliminate this factor, the reversal current, —/r, is usually divided by some particular valueof the forward current.
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