A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 52
Текст из файла (страница 52)
Thus it is important to control the analytical conditions,e.g. by employing a medium of controlled pH, buffer strength, and complexing characteristics. The analytical application of sampled current voltammetry is discussed more fullyin Sections 7.1.3 and 7.3.6.(d) Information from Change in Diffusion CurrentFor many processes D for MX p is not very different from that of M, so that /d, as given in(5.4.76) is about the same as that before complexation, as in the example in Figure 5.4.2.However if the ligand, X, is very large, as might occur when X is DNA, a protein, or apolymer, then the size of the species after complexation will be much larger than that of M,and there will be a significant decrease in D and in /d. Under these conditions the change in/d with addition of X can be used to obtain information about Kc and p. An investigation ofthis type was based on the interaction of Со(рпеп)з+ with double-strand DNA (31), wherephen is 1,10-phenanthroline.
The diffusion coefficient decreased from 3.7 X 10~6 cm2/sfor the free Co species to 2.6 X 10~7 cm2/s upon binding to DNA.5.5 Sampled-Current Voltammetry for Quasireversible and Irreversible Electrode Reactions191г 5.5 SAMPLED-CURRENT VOLTAMMETRY FORQUASIREVERSIBLE AND IRREVERSIBLEELECTRODE REACTIONSIn this section, we will treat the one-step, one-electron reaction О + e ^ R using the general (quasireversible) i-E characteristic. In contrast with the reversible cases just examined, the interfacial electron-transfer kinetics in the systems considered here are not sofast as to be transparent. Thus kinetic parameters such as &f, &ь, к® and a influence the responses to potential steps and, as a consequence, can often be evaluated from those responses.
The focus in this section is on ways to determine such kinetic information fromstep experiments, including sampled-current voltammetry. As in the treatment of reversible cases, the discussion will be developed first for early transients, then it will be redeveloped for the steady-state.5.5.1Responses Based on Linear Diffusion at a Planar Electrode(a) Current-Time BehaviorThe treatment of semi-infinite linear diffusion for the case where the current is governed by both mass transfer and charge-transfer kinetics begins according to the patternused in Section 5.4.1.
The diffusion equations for О and R are needed, as are the initialconditions, the semi-infinite conditions, and the flux balance. As we noted there, theselead toCo(x, s) = ^+ А{$)е-«°°)ШхCK(x, s) = -£А(з)е-^'Пх(5.5.1)(5.5.2)where f = {DoIDK)m.For the quasireversible one-step, one-electron case, we can evaluate A(s) by applyingthe condition:JAwhereandvti±f=F/RT.The transform of (5.5.3) isDolJXJ)= k f C o (0, s) - £bCR(0, s)(5.5.6)and, by substitution from (5.5.1) and (5.5.2),A(s) =~^—г-(5.5.7)192Chapter 5. Basic Potential Step MethodswhereЬ1 2/къn(5.5.8)1/2Then,C*(5.5.9)From (5.5.3)(5.5.10)or, taking the inverse transform,2i(t) = FAkf C% ехр(Я 0merfc(Ht )(5.5.11)For the case when R is initially present at CR, equation 5.5.11 becomes/(0 =- къехр(Я2Г)erfc(Htl/2)(5.5.12)At a given step potential, k{, къ, and H are constants. The product exp(x2)erfc(x) is unityfor x = 0, but falls monotonically toward zero as x becomes large; thus the current-timecurve has the shape shown in Figure 5.5.1.
Note that the kinetics limit the current att = 0 to a finite value proportional to kf (with R initially absent). In principle, Iq can beevaluated from the faradaic current at t = 0. Since a charging current also exists in themoments after the step is applied, the faradaic component at t = 0 typically would be determined by extrapolation from data taken after the charging current has decayed [seeSections 1.4.2 and 7.2.3(c)].(b) Alternate Expression in Terms ofrjIf both О and R are present in the bulk, so that an equilibrium potential exists, one can describe the effect of potential on the current-time curve in terms of the overpotential, r\.
Analternate expression for (5.5.12) can be given by noting that%khCt = к0[ф-^Е-Е0)- С&1-°№-Е°'>\(5.5ЛЗ)or, by substituting for k° in terms of i 0 by (3.4.11),(5.5.14)FAkfC*QFigure 5.5.1 Current decayafter the application of a step to apotential where species О isreduced with quasireversiblekinetics.5.5 Sampled-Current Voltammetry for Quasireversible and Irreversible Electrode Reactions193Therefore, (5.5.12) may be written/ = iQ \e~afl] -{Хе~аЩехр(Я 2 0 erfc(№ 1/2 )By similar substitutions into the expression for Я, one hasя=^й^~^р](5.5.15)(5 5л6)-Note that the form of (5.5.12) and (5.5.15) is/ = [/ in the absence of mass-transfer effects] X [f(H, t)]where / ( # , t) accounts for the effects of mass transfer.(c) Linearized Current-Time CurveFor small values of № 1 / 2 , the factor exp(#2f)erfc(№1/2) can be linearized:(5.5.17)Then, (5.5.11) becomes(5.5.18)In a system for which R is initially absent, one can apply a step to the potential region atthe foot of the wave (where k{, hence # , is still small), then plot i vs.
t112 and extrapolatethe linear plot to t = 0 to obtain kf from the intercept.Likewise, (5.5.15) can be written= i0 [е-'*' - e <'-<Wl - Щ£\(5.5.19)This relation applies only to a system containing both О and R initially, so that Ещ isdefined. Stepping from Eeq to another potential involves a step of magnitude 77; thus a plotof / vs.
t1^2 has as its intercept the kinetically controlled current free of mass-transfer effects. A plot of it=Q vs. 7] can then be used to obtain IQ.For small values of 77, the linearized i-rj characteristic, (3.4.12), can be used, so that(5.5.15) becomesi = —exp (H2t) erfc(#f1/2)(5.5.20)Then for small 17 and small Ht112 one has a "completely linearized" form:Firm/RT \?Ht\^1/2 Jpo.z(d) Sampled-Current VoltammetryIn preparation for deriving the shape of a sampled-current voltammogram, let us return to(5.5.11), which is the full current-time expression for the case where only species О ispresent in the bulk.
Recognizing that к\>1к$ = в = exp[f(E — E® )], we find that^ + & )(5.5.22)194 • Chapter 5. Basic Potential Step Methodsand that (5.5.11) can be rephrased asI =1Л ^Since semi-infinite linear diffusion applies, the diffusion-limited current is the Cottrellcurrent, which is easily recognized in the factor preceding the brackets. Thus, we can simplify (5.5.23) to'd(1+#)^i(A)!(5.5.24)where(A) = 77-1/2Aexp(A2)erfc(A)(5.5.25)and(5.5.26)Equation 5.5.24 is a very compact representation of the way in which the current ina step experiment depends on potential and time, and it holds for all kinetic regimes: reversible, quasireversible, and totally irreversible. The function F\(X) manifests the kinetic effects on the current in terms of the dimensionless parameter A, which can bereadily shown to compare the maximum current supportable by the reductive kineticprocess at a given step potential (FAkfC% vs.
the maximum current supportable by diffusion at that potential [/d/(l+£0)]. Thus a small value of A implies a strong kinetic influence on the current, and a large value of A corresponds to a situation where the kineticsare facile and the response is controlled by diffusion. The function FX(A) rises monotonically from a value of zero at A = 0 toward an asymptote of unity as A becomes large(Figure 5.5.2).Simpler forms of (5.5.24) are used for the reversible and totally irreversible limits.For example, consider (5.4.17), which we derived as a description of the current-timecurve following an arbitrary step potential in a reversible system.
That same relationshipis available from (5.5.24) simply by recognizing that with reversible kinetics A is verylarge, so that Fi(\) is always unity. The totally irreversible limit will be considered separately in Section 5.5.l(e).1.21.00.8Ц. 0.60.40.20.010.110Figure 5.5.2 General kinetic functionfor chronoamperometry and sampledcurrent voltammetry.5.5 Sampled-Current Voltammetry for Quasireversible and Irreversible Electrode Reactions1951.21.0Quasireversible -0.8 -^0.6 -Reversible - - ^0.4 -IV0.2 -300if 7 ik Li//гЖ100tAi^T-100IrreversibleFigure 5.5.3 Sampled-currentvoltammograms for various kineticregimes.
Curves are calculated from(5.5.24) assuming Butler-Volmerkinetics with a = 0.5 and т = I s ,Г<$^-3001-50051-700-900(£-£°')/mV2Do = DR= 1 X 1(Г cm /s. Fromleft to right the values of k° are 10,1 X 10~3, 1 X 10" 5 , and 1 X 10~7cm/s.So far, it has been most convenient to think of (5.2.24) as describing the current-timeresponse following a potential step; however it also describes the current-potential curvein sampled-current voltammetry, just as we understood (5.4.17) to do for reversible systems.
At a fixed sampling time т, Л becomes (&fT1/2/Do/2)(l f £6), which is a functiononly of potential among the variables that change during a voltammetric run. At very positive potentials relative to E° , в is very large, and i = 0. At very negative potentials, в - * 0but kf becomes very large; thus F^A) approaches unity, and / ~ /d. From these simpleconsiderations, we expect the sampled-current voltammogram to have a sigmoidal shapegenerally similar to that found in the reversible case. Figure 5.5.3, which contains severalvoltammograms corresponding to different kinetic regimes, bears out this expectation.For very facile kinetics, corresponding to large к0, the wave has the reversible shape,and the half-wave potential is near E°'. (In Figure 5.5.3, where Do = £>R, Ец2 = E0'exactly.) For smaller values of k°, the kinetics must be driven, and the wave is displaced toward more extreme potentials (i.e., in the negative direction if the wave is for a reductionand in the positive direction for an oxidative wave).
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
