A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 99
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In suchcases Nyquist plots can be obtained and compared to theoretical models based on the appropriate equations representing the rates of the various processes and their contributionsto the current. It may be useful in these cases to represent the system by an equivalent circuit involving different components (resistors, capacitors, inductors). However, suchequivalent circuits are not unique and one cannot easily guess the form or structure of theequivalent circuit from the processes involved in the reaction scheme (19). Electrode surface roughness and heterogeneity can also be significant factors in the ac response in EIS.Indeed, even for simple electron-transfer reactions, measurements that can be made usefully with a smooth and homogeneous mercury electrode are often not possible with solidelectrodes.EIS has been applied to a variety of electrochemical systems, including those involved in corrosion, electrodeposition, polymer films, and semiconductor electrodes.
Representative studies can be found in EIS symposia proceedings (20-22).10.5 AC VOLTAMMETRYWe noted in Section 10.1 that ac voltammetry is basically a faradaic impedance technique in which the mean potential, £ dc , is imposed potentiostatically at arbitrary valuesthat usually differ from the equilibrium value. Ordinarily, Edc is varied systematically(e.g., linearly) on a long time scale compared to that of the superimposed ac variation,£ a c (10 Hz to 100 kHz). The output is a plot of the magnitude of the ac component ofthe current vs.
Edc. The phase angle between the alternating current and Eac is also ofinterest.Treatments of this problem are greatly simplified by uncoupling the long-term diffusion due to £ d c from the rapid diffusional fluctuations due to Eac. We do that by recognizing that Edc sets up mean surface concentrations that look like bulk values to the acperturbation because of the difference in time scale. In Section 10.3, we defined thefaradaic impedance in terms of bulk concentrations; thus the current response in acvoltammetry as a function of Edc is readily obtained by substituting the surface concentrations imposed by Edc directly into these impedance relations.
Since this strategy is simpleand intuitive, we will pursue it. More rigorous treatments are available in the literature forthe interested reader (2, 3, 5). The results are the same by either approach.The mean surface concentrations enforced by £ d c depend on many factors: (a) theway in which Edc is varied; (b) whether or not there is periodic renewal of the diffusionlayer; (c) the applicable current-potential characteristic; and (d) homogeneous or heterogeneous chemical complications associated with the overall electrode reaction. For example, one could vary £ d c in a sequential potentiostatic manner with periodic renewal of thediffusion layer, as in sampled-current voltammetry. This is the technique that is actuallyused in ac polarography, which features a DME and effectively constant 2sdc during thelifetime of each drop.
Alternatively one could use a stationary electrode and a fairly fastsweep without renewal of the diffusion layer. Both techniques have been developed andare considered below. The effects of different kinds of charge-transfer kinetics will alsobe examined here, but the effects of homogeneous complications are deferred to Chapter12. Throughout the discussion, one should keep in mind that the chief strength of ac10.5 ас Voltammetry < 389voltammetry is the access it gives to exceptionally precise quantitative information aboutelectrode processes. Diagnostic aspects certainly exist, but they are more subtle than withother methods.10.5.1ac Polarography in a Reversible SystemLet us consider the ac response at a renewable stationary mercury drop electrodeimmersed in a solution containing initially only species О in the nernstian processО + ne <=^ R.
The dc potential starts at a value considerably more positive than E0' and isscanned slowly in a negative direction. During the lifetime of a single drop, Edc is effectively constant; hence the dc part of the experiment is conventional polarography and istreated as a series of individual step experiments (see Sections 7.1 and 7.2).Since the charge-transfer resistance is completely negligible, (10.3.10) always applies where'-JB-Гn2F2A\/2 |_£>o/2Co(0, t)mand the mean concentrations Co(0, t)m and CR(0, 0 mtion:[i1Z>k/2CR(0, 0 m Jaredetermined by the nernstian rela—1nFA'— (£dc-Eu)(10.5.2)The arguments that led to (5.4.29) and (5.4.30) apply equally to the dc part of this experiment; hence we writeCo(0, 0 m = CZ{ j i ^ r(Ю.5.3))C R ( 0 , t)m = Сwhere £ is DQ2/DJ[2, as usual. Thus the faradaic impedance is obtained by substitutioninto (10.5.1) and then into (10.3.8):W(A)(Ю.5.5,Let us note now that £0m can be written# m = ea(10.5.6)whereg(10-5.7)and Ец2 is the reversible half-wave potential defined in (5.4.21):*i/2 = £°'+gln|p(Ю.5.8)By substitution from (10.5.6), we find that the term in parentheses in (10.5.5) is e~a +2 + ea, which is also 4 cosh2(a/2).
Thus we haveZf = . , 4 f jw. cosh2ff)(10.5.9)390 - Chapter 10. Techniques Based on Concepts of ImpedanceIn Section 10.3 we saw that the faradaic current for a reversible system leads Eac byexactly 45°. If £ a c = AE sin cot, thenif(10.5.10)Sinand the amplitude of this current, which is the chief observable, is simplyД£ _ n F Aco Doz2fCOART cosh (a/2)lit(10.5.11)Figure 10.5.1 is a display of the ac polarogram defined by this equation. The bellshape derives from the factor cosh~2(a/2), and it reflects the potential dependence of theimpedance, Zf. The maximum in the current occurs at all = 0 or at Edc = Ey2, which isnear E° .
As one moves away from that potential, either positively or negatively, the impedance rises sharply and the current falls off. The physical basis for this behavior wasoutlined in Section 10.3. In effect, the current is controlled by the limiting reagent, that is,the smaller of the two surface concentrations. At potentials far from E°\ where only smallamounts of one reagent can exist at the surface, only small currents can flow.The peak current at Edc = Ey2 comes easily from (10.5.11). Since cosh(0) = 1,ART(10.5.12)From this relation and (10.5.11) one can show straightforwardly that the shape of the acpolarogram adheres to(10.5.13)(See Problem 10.1.)The same results hold for the DME, where one must account for the effect of dropgrowth on the polarogram. The use of linear diffusion relations for the dc part of the15010050-50-100-150Figure 10.5.1 Shape of a reversible acvoltammetric peak for n - 1.10.5 ас Voltammetry «I 391experiment has already been justified (see Section 7.1.2), and that justification is evenmore valid for the ac part because of its shorter time scale.
Thus the peculiarities of theexpanding sphere are felt only in the changing area A with time, and that factor is directly accountable by substitution of (7.1.3) into (10.5.11). Since A grows as t2/3> as thedrop ages, the current also shows the same dependence. Thus we can expect the current to oscillate as successive drops grow and fall. Maxima should be observed at theend of each drop's life. Experimental results in Figure 10.5.2 bear out these expectations. Measurements carried out on the envelope of the ac polarogram can be treatedby all relations derived above, provided that A is defined as the area just before dropfall.A number of important properties of the reversible ac voltammogram can be deducedfrom (10.5.11)—(10.5.13).
Among them are the direct proportionalities between / p and n2,(om, and CQ. There is also a proportionality to A£; however, this relation is a limited one,because the linearized i-E characteristic underlying the derivation of Zf becomes invalid ifA£ is too large. For linearity within a few percent, A£ must be less than about 10/n mV.Not surprisingly, the width of the peak at half height also depends on AE if large valuesare used. If it is kept below 10/n mV, there is a constant width of 90A/n mV at 25°C. Atlarger AE the peaks are broader.10.5.2 Voltammetric ac Response to Quasireversibleand Irreversible SystemsWhen heterogeneous kinetics become sluggish enough to be visible, one requires a moreelaborate theory to predict the voltammetric ac response.
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