A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 101
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We have derived them for the situationin which dc reversibility applies; however, they hold regardless of that condition.Demonstrations of this point are available in the literature (3, 5). Their unconditionalvalidity is a big asset, for it frees the experimenter from having to achieve special limiting conditions.As in the previous section, we have assumed semi-infinite linear diffusion to a planarelectrode throughout the mathematical discussion here.
With a reversible dc process, theeffects of sphericity and drop growth at the DME are exactly as discussed in Section10.5.1. In general, the sphericity has a negligible impact and drop growth can be accommodated by using an explicit expression for A as a function of time. If dc reversibilitydoes not apply, these factors influence the ac response in more complex ways (3, 5, 23).The reader is referred to the literature for details.10.5.3 Linear Sweep ac Voltammetry at Stationary Electrodes (26,27)The previous two sections have dealt generally with ac voltammetry as recorded by theapplication of Edc in successive steps and with a renewal of the diffusion layer betweeneach step. The DME permits the most straightforward application of that technique, butother electrodes can be used if there is a means for stirring the solution between steps sothat the diffusion layer is renewed.
On the other hand, this requirement for periodic renewal is inconvenient when one wishes to use stationary electrodes, such as metal or carbon disks, or a hanging mercury drop. Then one prefers to apply Edc as a ramp and torenew the diffusion layer only between scans. In this section, we will examine the expected ac voltammograms for reversible and quasireversible systems when Edc is imposedas a linear sweep and we will compare them with the results obtained above for effectively constant Edc.10.5 ас Voltammetry < 397The strategy is exactly that used before.
The time domains associated with variationsin Edc and Eac are assumed to differ greatly, so that the diffusional aspects of the two partsof the experiment can be uncoupled. This assumption will hold as long as the scan rate,u, is not too large compared to the ac frequency (28). More precisely, dE&Jdt = v shouldbe much smaller than the amplitude of dE/dt, which is &Ea>.
Then, we can take the meansurface concentrations enforced by Е$с as effective bulk values for the ac perturbation,just as we did earlier. The current amplitude and phase angle then follow easily from theimpedance properties.(a) Reversible SystemsLet us consider a completely nernstian system О + ne ^ R in which R is initially absent.The starting potential for the linear sweep is rather positive with respect to E°\ and thescan direction is negative. Semi-infinite linear diffusion is assumed. The mean surfaceconcentrations, C o (0, t)m and CR(0, t)m, are exactly those obtained in the analogous linearsweep experiment without superposed ac excitation, and they adhere always to the nernstian relation (10.5.2).The arguments leading to equation 5.4.26 show that it applies without reference tothe kinetic properties of the electrode reaction or the nature of the excitation waveform.For the present purpose, we can rewrite it asЯо 2 О)(0, 0 m + ^ / 2 C R ( 0 , t)m = C%D}I2(10.5.27)Substitution from (10.5.2) then reveals that the mean surface concentrations are exactly asgiven in (10.5.3) and (10.5.4).
In other words, those relations, which were derived earlierexpressly for step excitation, have been shown here to apply regardless of the manner bywhich Edc is attained.12This conclusion is very important because it implies that all relations and all qualitative conclusions presented in Section 10.5.1 also hold for linear sweep ac voltammetry ofreversible systems at a stationary electrode.(b) Quasireversible SystemsAn important special case of quasireversibility is the situation in which a one-step, oneelectron process is sufficiently facile to maintain a reversible dc response, but not facileenough to show a negligible charge-transfer resistance, Rcb to the ac perturbation.If the dc response is nernstian, (10.5.2) and (10.5.27) hold, and the mean surface concentrations are given by (10.5.3) and (10.5.4), which are the same relations used in thetreatment of Section 10.5.2.
Thus, all of the equations and qualitative conclusions reachedthere for quasireversible ac polarograms also apply to the corresponding linear sweep acvoltammograms.These precise parallels between linear sweep voltammetry and ac polarography nolonger persist when there is a lack of dc reversibility. Treating such a case is more complex than the situations we have examined above because the mean surface concentrationsare affected by the concentration profiles throughout the diffusion layer, and the surfacevalues applicable at any potential generally depend on the waveform used to attain thatpotential (26, 27).Linear sweep ac voltammetry allows precise, rapid kinetic measurements at solidelectrodes. It can therefore be used to characterize these electrodes themselves, which12Equation 10.5.27 is based on semi-infinite linear diffusion; hence this conclusion applies strictly only toplanar electrodes.
Work at an SMDE can be affected by sphericity (27).398 • Chapter 10. Techniques Based on Concepts of Impedancemay be of considerable interest, or to study electrode processes operating outside theworking range of mercury or taking place in controlled environments where the DME orSMDE may be inconvenient.10,5*4 Cyclic ac Voltammetry (26,27)Cyclic ac voltammetry is a simple extension of the linear sweep technique; one simplyadds the reversal scan in E$c.
This technique retains the best features of two powerful,complementary methodologies. Conventional cyclic voltammetry is especially informative about the qualitative aspects of an electrode process. However, the response waveforms lend themselves poorly to quantitative evaluations of parameters. Cyclic acvoltammetry retains the diagnostic utility of conventional cyclic measurements, but itdoes so with an improved response function that permits quantitative evaluations as precise as those obtainable with the usual ac approaches. Although this technique is notwidely employed, it can be a useful adjunct to dc cyclic voltammetry.Treatments of cyclic ac voltammetry follow the familiar pattern. The ac and dc timescales are independently variable, but are assumed to differ markedly.
Then a treatment ofthe dc aspect yields mean surface concentrations, which are used to calculate faradaic impedances that define the ac response by amplitude and phase angle. The electrode is assumed to be stationary and the solution is regarded as quiescent for the duration of the dccycle.(a) Reversible SystemsCyclic ac voltammograms for completely nernstian systems are easy to predict on thebasis of results from the previous section. The mean surface concentrations, Q)(0, t)m andCR(0, t)m, adhere to (10.5.3) and (10.5.4) unconditionally; hence at any potential they arethe same for both the forward and reverse scans. The cyclic ac voltammogram shouldtherefore show superimposed forward and reverse traces of ac current amplitude vs.
Edc.We expect a peak-shaped voltammogram that adheres in every way to the conclusionsreached in Section 10.5.1 about the general ac voltammetric response to a reversible system at a planar electrode.Figure 10.5.6 contrasts the responses from the ac and dc versions of cyclic voltammetry for the purely nernstian case. Kinetic reversibility is shown in the dc experiment by aFaradaicbaselinefor reversescanSuperimposedforward andreverse scansReversescandc voltammetryac voltammetryFigure 10.5.6 Comparison of response wave forms for cyclic dc and cyclic ac voltammetry for areversible system.10.5 ас Voltammetry -Щ 399peak separation near 60/'n mV (25°C), regardless of scan rate.
In the ac experiment, it isshown by identical forward and reverse peak potentials and by peak widths of 90/n mV(25°C), again regardless of scan rate. Chemical stability of the reduced form is demonstrated in the dc experiment by a peak current ratio, |/p?r//P;f|, of unity. Given chargetransfer reversibility, the same thing is shown by the ratio of peak ac current amplitudes,|/ p r // p f |. The advantage in the ac experiment is that the reversal response has an obviousbaseline for quantitative measurements, whereas the baseline for reversal currents in thedc response is more difficult to fix.(b) Quasireversible SystemsIt continues to be helpful to consider two separate cases of quasireversibility in a onestep, one-electron reaction. In both, a significant polarization resistance is manifested inthe ac response, but in one instance the dc aspect appears reversible, whereas more generally it is not.When dc reversibility obtains, a theoretical description is straightforward, becausethe mean surface concentrations still adhere to (10.5.3) and (10.5.4), regardless of themanner in which the dc potential determining them was established.
Thus the forwardand the reverse traces again overlap precisely. The shape of the peak and its position adhere to the relations derived in Section 10.5.2, where this kinetic case was considered indetail.If dc reversibility does not hold, then the situation becomes quite complex. Themean surface concentrations at a given dc potential tend to depend on the way inwhich that potential is reached. In general, the surface concentrations at any Edc willdiffer for forward and reverse scans; therefore we can expect the corresponding tracesto differ in the voltammogram. In the dc cyclic voltammogram, increasingly sluggishelectron transfer causes greater splitting of the forward and reverse peaks, becauselarger activation overpotentials are needed to motivate charge transfer. This splittingalso manifests the fact that the surface concentrations undergo the transition fromnearly pure О to virtually pure R in different potential regions for the two scan directions.
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