A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 44
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(a) Step waveforms applied in a series ofexperiments, (b) Current-time curves observed in response to the steps, (c) Sampled-currentvoltammogram.5.1 Overview of Step Experiments\159A+тtС)(b)Figure 5.1.4Double potential stepchronoamperometry.(a) Typical waveform.(b) Current response.ticular methods. Chapter 7 covers the details of many forms of voltammetry based onstep waveforms, including normal pulse voltammetry and its historical predecessorsand successors.Now consider the effect of the potential program displayed in Figure 5.1.4 a. The forward step, that is, the transition from E\ to E2 at t = 0, is exactly the chronoamperometricexperiment that we discussed above.
For a period r, it causes a buildup of the reductionproduct (e.g., anthracene anion radical) in the region near the electrode. However, in thesecond phase of the experiment, after t = т, the potential returns to E\9 where only the oxidized form (e.g., anthracene) is stable at the electrode. The anion radical cannot coexistthere; hence a large anodic current flows as it begins to reoxidize, then the current declines in magnitude (Figure 5.1.4/?) as the depletion effect sets in.This experiment, called double potential step chronoamperometry, is our first example of a reversal technique. Such methods comprise a large class of approaches, all featuring an initial generation of an electrolytic product, then a reversal of electrolysis so thatthe first product is examined electrolytically in a direct fashion.
Reversal methods makeup a powerful arsenal for studies of complex electrode reactions, and we will have muchto say about them.5.1.2DetectionThe usual observables in controlled-potential experiments are currents as functions oftime or potential. In some experiments, it is useful to record the integral of the currentversus time. Since the integral is the amount of charge passed, these methods are coulometric approaches. The most prominent examples are chronocoulometry and double potential step chronocoulometry, which are the integral analogs of the correspondingchronoamperometric approaches. Figure 5.1.5 is a display of the coulometric response tothe double-step program of Figure 5.1.4a.
One can easily see the linkage, through the integral, between Figures 5.1.4b and 5.1.5. Charge that is injected by reduction in the forward step is withdrawn by oxidation in the reversal.Of course, one could also record the derivative of the current vs. time or potential, butderivative techniques are rarely used because they intrinsically enhance noise on the signal (Chapter 15).Several more sophisticated detection modes involving convolution (or semiintegration), semidifferentiation, or other transformations of the current function alsofind useful applications.
Since they tend to rest on fairly subtle mathematics, we deferdiscussions of them until Section 6.7.160Chapter 5. Basic Potential Step MethodsQt5.1.3Figure 5.1.5 Response curve for doublepotential step chronocoulometry. Stepwaveform is similar to that in Figure 5.1 Aa.Applicable Current-Potential CharacteristicsWith only a qualitative understanding of the experiments described in Section 5.1.1, wesaw that we could predict the general shapes of the responses. However, we are ultimately interested in obtaining quantitative information about electrode processes fromthese current-time or current-potential curves, and doing so requires the creation of a theory that can predict, quantitatively, the response functions in terms of the experimentalparameters of time, potential, concentration, mass-transfer coefficients, kinetic parameters, and so on. In general, a controlled-potential experiment carried out for the electrodereactionkf(5.1.2)can be treated by invoking the current-potential characteristic:i = FAk° [C o (0, t)e-^-E°r)~ C R (0, , ) e a - a № * 0 ' ) ](5.1.3)in conjunction with Fick's laws, which can give the time-dependent surface concentrations CQ(0, t) and C R (0, f).
This approach is nearly always difficult, and it sometimes failsto yield closed-form solutions. The problem is even more difficult when a multistepmechanism applies (see Section 3.5). One is often forced to numerical solutions or approximations.The usual alternative in science is to design experiments so that simpler theory can beused. Several special cases are easily identified:(a) Large-Amplitude Potential StepIf the potential is stepped to the mass-transfer controlled region, the concentration of theelectroactive species is nearly zero at the electrode surface, and the current is totally controlled by mass transfer and, perhaps, by the kinetics of reactions in solution away fromthe electrode. Electrode kinetics no longer influence the current, hence the general i-Echaracteristic is not needed at all.
For this case, / is independent of E. In Sections 5.2 and5.3, we will be concentrating on this situation.(b) Small-Amplitude Potential ChangesIf a perturbation in potential is small in size and both redox forms of a couple are present(so that an equilibrium potential exists), then current and potential are linked by a linearized i-r] relation. For the one-step, one-electron reaction (5.1.2), it is (3.4.12),i = ~kfv(5.1.4)5.2 Potential Step Under Diffusion Control < 161(c) Reversible (Nernstian) Electrode ProcessFor very rapid electrode kinetics, we have seen that the i-E relation collapses generally toa relation of the Nernst form (Sections 3.4.5 and 3.5.3):E = E° + ^ l n ° , ' ;(5.1.5)nF CR(0, t)Again the kinetic parameters k° and a are not involved, and mathematical treatments arenearly always greatly simplified.(d) Totally Irreversible Electron TransferWhen the electrode kinetics are very sluggish (k° is very small), the anodic and cathodicterms of (5.1.3) are never simultaneously significant.
That is, when an appreciable net cathodic current is flowing, the second term in (5.1.3) has a negligibly small effect, and viceversa. To observe the net current, the forward process must be so strongly activated (byapplication of an overpotential) that the back reaction is virtually totally inhibited. In suchcases, observations are always made in the "Tafel region," hence one of the terms in(5.1.3) can be neglected (see also Sections 3.4.3 and 3.5.4).(e) Quasireversible SystemsUnfortunately, electrode processes are not always facile or very sluggish, and we sometimes must consider the whole i-E characteristic. In such quasireversible (or quasi-nernstian cases), we recognize that the net current involves appreciable activated componentsfrom the forward and reverse charge transfers.In delineating these special situations, we are mostly concentrating on electrodeprocesses that are chemically reversible; however the mechanism of an electrode processoften involves an irreversible chemical transformation, such as the decay of the electrontransfer product by a following homogeneous reaction.
A good specific example featuresanthracene in DMF, which we have already considered previously. If a proton donor,such as water, is present in the solvent, the anthracene anion radical is protonated irreversibly and several other steps follow, eventually yielding 9,10-dihydroanthracene.Treating any case in which irreversible chemical steps are linked to heterogeneous electron transfer is much more complicated than dealing with the heterogeneous electrontransfer alone.
One of the simplified cases, given in (a)-(d) earlier might apply to theelectron-transfer step, but the homogeneous kinetics must also be added into the picture.Even in the absence of coupled-solution chemistry, chemically reversible electrodeprocesses can be complicated by multistep heterogeneous electron transfer to a singlespecies. For example, the two-electron reduction of Sn 4+ to Sn 2+ can be treated and understood as a sequence of one-electron transfers.
In Chapter 12, we will see how morecomplicated electrode reactions like these can be handled.5.2 POTENTIAL STEP UNDER DIFFUSION CONTROL5.2.1 A Planar ElectrodePreviously, we considered an experiment involving an instantaneous change in potentialfrom a value where no electrolysis occurs to a value in the mass-transfer-controlled regionfor reduction of anthracene, and we were able to grasp the current-time response qualitatively. Here we will develop a quantitative treatment of such an experiment.
A planarelectrode (e.g., a platinum disk) and an unstirred solution are presumed. In place of the162Chapter 5. Basic Potential Step Methodsanthracene example, we can consider the general reaction О + ne —» R. Regardless ofwhether the kinetics of this process are basically facile or sluggish, they can be activatedby a sufficiently negative potential (unless the solvent or supporting electrolyte is reducedfirst), so that the surface concentration of О becomes effectively zero. This condition willthen hold at any more extreme potential.
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