A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 94
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ED as ED is scanned from+0.4 t o - 0 . 6 V.9.5 The ring voltammograms for an RRDE in 5 mM K 3 Fe(CN) 6 and 0.1 M KCl are shown in Figure9.10.2. From the information in these curves, calculate N for this electrode and D for Fe(CN>6~.112What is the predicted slope of iD vs. со ! What are the predicted values of the limiting disk currentanc20D,/,C) * ™g current (/R>/)C with /D = 0 and /D = /D,/,c) at 5000 rpm? Assume v = 0.01 cm /s.9.6 Experiments are performed at an RRDE with the following dimensions: rx = 0.20 cm, r 2 = 0.22cm, r 3 = 0.32 cm. A disk voltammogram (/D vs.
ED) is to be recorded for a rotation rate of 2000rpm. What maximum potential sweep rate should be employed to prevent non-steady-state effectsfrom occurring? What is the transit time with this electrode?9.7 The diffusion coefficient of an electroactive species can be obtained from a limiting-current measurement at an RDE and a transient measurement (e.g., a potential step measurement) at the sameelectrode (at со = 0) under identical conditions. It is not necessary to know the electrode area, n, orC*. Explain how this is accomplished and discuss the possible errors in this procedure.9.8 S. Bruckenstein and P.
R. Gifford [Anal. Chem., 51, 250 (1979)] proposed that ring electrodeshielding measurements at the RRDE could be employed for the analysis of micromolar solutionsby using the equationA/R/ = 0.62rcF7rr 2 D 2 / Vl/6(ol/2NC*where A/R / represents the change in the limiting ring current for /D = 0 and /D = /D,/- (a) Derive thisequation, (b) A plot of /Rj/ vs. ED is given in Figure 9.10.3 for the reduction of Bi(III) to Bi(0) in 0.1 MHNO3. Mass-transfer limited reduction of Bi(III) occurs at -0.25 V.
From the data on the curve duringthe forward scan (ED varied from +1.0 to —0.2 V), calculate N for this RRDE. Explain the large ringcurrent transient observed during the reverse scan (ED from —0.2 to +1.0 V).9.9 The effective time scale for kinetic measurements at a rotating disk electrode is ~ l/co. What rangeof effective times is available for the usual range of rotation rates of an RDE? UMEs provide an al-1200 -5_800s400 -II-400~+0.4+0.20ER (V vs. SCE)-0.2-0.4Figure 9.10.2 Ringvoltammograms (/R vs.
ER) with(l)iD = 0 and (2) i D at itslimiting value, 302 jxA, for anRRDE with r2 = 0.188 cm andr 3 = 0.325 cm, rotated at 48.6rps in a solution of 5.0 mMK 3 Fe(CN) 6 and 0.10 M KCl.9.10 Problems+1.0-0.2367Figure 9.10.3 / R / c vs. £ D for reductionof Bi(III) to Bi(0) for a solutioncontaining 4.86 X 10" 7 M Bi(III) and 0.1M HN0 3 . The ring electrode was held at-0.25 V, and the disk potential wasscanned from +1.0 V at 200 mV/s. Forthis electrode, the slope of /D,/,c vs. Св 1 ( Ш )is 0.934 fiA/fjuM. [Reprinted withpermission from S. Bruckenstein, and P.R. Gifford, Anal.
Chem., 51, 250 (1979).Copyright 1979, American ChemicalSociety.]ternative steady-state electrode system for electrochemical studies. What range of UME radii yieldsthe same effective time range as that calculated for the RDE? Can stationary UMEs be extended toeven shorter times than an RDE at its maximum useful rotation rate? Explain.9.10 Show how eq. (9.3.40) leads to (9.3.39).CHAPTER10TECHNIQUES BASEDON CONCEPTS OFIMPEDANCE10.1 INTRODUCTIONIn previous chapters we have discussed ways of studying electrode reactions throughlarge perturbations on the system.
By imposing potential sweeps, potential steps, or current steps, we typically drive the electrode to a condition far from equilibrium, and we observe the response, which is usually a transient signal. Another approach is to perturb thecell with an alternating signal of small magnitude and to observe the way in which thesystem follows the perturbation at steady state. Many advantages accrue to these techniques. Among the most important are (a) an experimental ability to make high-precisionmeasurements because the response may be indefinitely steady and can therefore be averaged over a long term, (b) an ability to treat the response theoretically by linearized (orotherwise simplified) current-potential characteristics, and (c) measurement over a widetime (or frequency) range (10 4 to 10~6 s or 10~4 to 106 Hz). Since one usually worksclose to equilibrium, one often does not require detailed knowledge about the behavior ofthe i-E response curve over great ranges of overpotential.
This advantage leads to important simplifications in treating kinetics and diffusion.In deriving the theory below, we will rely frequently on analogies between the electrochemical cell and networks of resistors and capacitors that are thought to behave likethe cell. This feature may seem at times to disembody the interpretation from the chemical system, so let us emphasize beforehand that the ideas and the mathematics used in theinterpretation are basically simple. We will do our best to tie them to the chemistry atevery possible point, and we hope readers will avoid letting the details of interpretationobscure their view of the great power and beauty of these methods.10.1.1Types of Techniques (1-12)The prototypical experiment is the faradaic impedance measurement, in which the cellcontains a solution with both forms of a redox couple, so that the potential of the working electrode is fixed.
For example, one might use 1 mM E u 2 + and 1 mM E u 3 + in 1 MNaClO4. A mercury drop of fixed area might be employed as the working electrode,and it might be paired with a nonpolarizable reference such as an SCE, which would actalso as the counter electrode. It is probably easiest to understand the measurement ofimpedance by considering the classical approach with an impedance bridge. The cell isinserted as the unknown impedance into one arm of an impedance bridge, and thebridge is balanced by adjusting R and С in the opposite arm of the bridge, as shown inFigure 10.1.1.368ЮЛ Introduction369Potentiometerto null dc cellvoltageFigure 10.1.1 A bridge circuitfor measurements ofelectrochemical impedance.This operation determines the values of R and С that, in series, behave as the celldoes at the measurement frequency. The impedance is measured as a function of the frequency of the ac source. The technique where the cell or electrode impedance is plottedvs.
frequency is called electrochemical impedance spectroscopy (EIS). In modern practice, the impedance is usually measured with lock-in amplifiers or frequency-response analyzers, which are faster and more convenient than impedance bridges. Such approachesare introduced in Section 10.8. The job of theory is to interpret the equivalent resistanceand capacitance values in terms of interfacial phenomena.
The mean potential of theworking electrode (the "dc potential") is simply the equilibrium potential determined bythe ratio of oxidized and reduced forms of the couple. Measurements can be made at otherpotentials by preparing additional solutions with different concentration ratios. Thefaradaic impedance method, including EIS, is capable of high precision and is frequentlyused for the evaluation of heterogeneous charge-transfer parameters and for studies ofdouble-layer structure.A variation on the faradaic impedance method is ac voltammetry (or, with a DME, acpolarography). In these experiments, a three-electrode cell is used in the conventionalmanner, and the potential program imposed on the working electrode is a dc mean value,Edc, which is scanned slowly with time, plus a sinusoidal component, Eac, of perhaps 5mV peak-to-peak amplitude.
The measured responses are the magnitude of the ac component of the current at the frequency of Eac and its phase angle with respect to Еас.г Аtypical experimental arrangement is shown schematically in Figure 10.1.2. As we will seepresently, this measurement is equivalent to determining the faradaic impedance. The roleof the dc potential is to set the mean surface concentrations of О and R. In general, thispotential differs from the true equilibrium value; hence CQ(0, t) and CR(0, 0 differ fromCQ and CR, and a diffusion layer exists. Note, however, that since E^c is effectivelysteady, this layer soon becomes so thick that its dimensions greatly exceed those of thediffusion zone affected by the rapid perturbations from £ ac .
Thus, the mean surface con-1Alternatively, one could measure the current components in phase with Eac and 90° out of phase with E.dC. Theyprovide equivalent information.370Chapter 10. Techniques Based on Concepts of ImpedancePotentiostatCellHE converterLock-in amp.1ac component + фLow-passfilterdc componentFigure 10.1.2 Schematic diagram of apparatus for an ac voltammetric experiment.centrations C o (0, t) and CR(0, t) look like bulk concentrations to the ac part of the experiment. This same effect is exploited in DPP (see Section 7.3.4).
One usually starts with asolution containing only one redox form, for example Eu 3 + , and obtains continuous plotsof the ac current amplitude and the phase angle vs. Edc. In effect, these plots represent thefaradaic impedance at continuous ratios of C o (0, t) and CR(0, t), all recorded withoutchanging the solution. The amplitude plot is also useful for analytical measurements ofconcentration.EIS and ac voltammetry normally involve excitation signals, £ a c , of very lowamplitude, and they depend essentially on the fact that current-overpotential relations are virtually linear at low overpotentials. In a linear system, excitation at frequency (o provides a current also of frequency со (and only of frequency со). On theother hand, a nonlinear i-E relation gives a distorted response that is not purely sinusoidal; but it is still periodic and can be represented as a superposition (Fourier synthesis) of signals at frequencies <o, 2u>, Ъоо, .
. . The current-overpotential functionfor an electrode reaction is nonlinear over moderate ranges of overpotential, and theeffects of this nonlinearity can be observed and put to use. For example, considersecond (and higher) harmonic ac voltammetry, which is nearly the same as the firstharmonic ac experiment described above. It differs in that one detects an ac component current at 2o>, 3o>, . . .
, etc., instead of the component at the excitation frequency со. Faradaic rectification features excitation with a purely sinusoidal sourceand measurement of the dc component of current flow. Intermodulation voltammetrydepends on the mixing properties of a nonlinear characteristic. One excites with twosuperimposed signals at frequencies (x)\ and o)2 and observes the current at the combination frequencies (sidebands or beat frequencies) a)\ + o)2 and o)\ — a)2.
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