A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 31
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Electrode processes may alsoinvolve adsorption and desorption kinetics of primary reactants, intermediates, and products.Thus, electrode reactions generally can be expected to show complex behavior, andfor each mechanistic sequence, one would obtain a distinct theoretical linkage betweencurrent and potential.
That relation would have to take into account the potential dependences of all steps and the surface concentrations of all intermediates, in addition to theconcentrations of the primary reactants and products.A great deal of effort has been spent in studying the mechanisms of complex electrode reactions. One general approach is based on steady-state current-potential curves.Theoretical responses are derived on the basis of mechanistic alternatives, then one compares predicted behavior, such as the variation of exchange current with reactant concentration, with the behavior found experimentally.
A number of excellent expositions of thisapproach are available in the literature (8-14, 25, 26, 35). We will not delve into specificcases in this chapter, except in Problems 3.7 and 3.10. More commonly, complex behavior is elucidated by studies of transient responses, such as cyclic voltammetry at differentscan rates. The experimental study of multistep reactions by such techniques is covered inChapter 12.3.5.1Rate-Determining Electron TransferIn the study of chemical kinetics, one can often simplify the prediction and analysis of behavior by recognizing that a single step of a mechanism is much more sluggish than allthe others, so that it controls the rate of the overall reaction.
If the mechanism is an electrode process, this rate-determining step (RDS) can be a heterogeneous electron-transferreaction.A widely held concept in electrochemistry is that truly elementary electron-transferreactions always involve the exchange of one electron, so that an overall process involving a change of n electrons must involve n distinct electron-transfer steps. Of course, itmay also involve other elementary reactions, such as adsorption, desorption, or variouschemical reactions away from the interface. Within this view, a rate-determining electrontransfer is always a one-electron-process, and the results that we derived above for the3.5 Multistep Mechanisms109one-step, one-electron process can be used to describe the RDS, although the concentrations must often be understood as applying to intermediates, rather than to starting speciesor final products.For example, consider an overall process in which О and R are coupled in an overallmultielectron processО + ne *± R(3.5.7)by a mechanism having the following general character:О + n'e +± O'(net result of steps preceding RDS)(3.5.8)O'+e^R'КR' + ri'e <± R(RDS)(3.5.9)(net result of steps following RDS)(3.5.10)Obviously л' +n" + 1 = л .
1 0The current-potential characteristic can be written as[C O '(0, t)e-af(E-E^aшaЕlv t 0 t h e R D S- CR/(0, г ) ^ ( 1 " а У ( Е " ^ ) ](3.5.11)T h i swhere £°ds> > & %& P Prelation is (3.3.11) written for the RDS andmultiplied by n, because each net conversion of O' to R' results in the flow of n electrons,not just one electron, across the interface. The concentrations CQ>(0, t) and C R '(0, i) arecontrolled not only by the interplay between mass transfer and the kinetics of heterogeneous electron transfer, as we found in Section 3.4, but also by the properties of the preceding and following reactions. The situation can become quite complicated, so we willmake no attempt to discuss the general problem.
However, a few important simple casesexist, and we will develop them briefly now. 113.5.2Multistep Processes at EquilibriumIf a true equilibrium exists for the overall process, all steps in the mechanism are individually at equilibrium. Thus, the surface concentrations of O' and R' are the values in equilibrium with the bulk concentrations of О and R, respectively. We designate them as(C(y)eq and (C R ') e q - Recognizing that / = 0, we can proceed through the treatment leadingto (3.4.2) to obtain the analogous relation?ds) = < £ ф(3.5.12)For the mechanism in (3.5.8)-(3.5.10), nernstian relationships define the equilibria for thepre- and postreactions, and they can be written in the following forms:(C )C*п'/(Ещ-Е°рТ&)е10_Q^Y(£eq--is post) _ ^ R ; ^eq^ ^ ^чThe discussions that follow hold if either or both of n' or n" are zero.1^n the first edition and in much of the literature, one finds n.d used as the n value of the rate-determining step.As a consequnce n.d appears in many kinetic expressions.
Since n.d is probably always 1, it is a redundant symboland has been dropped in this edition. The current-potential characteristic for a multistep process has often beenexpressed asi = nFAk0 [C o (0, ^-«"аЯЯ-я 0 ') _ C R ( 0 ? ^(i-«)«a/(£-£ 0 ')]This is rarely, if ever, an accurate form of the i-E characteristic for multistep mechanisms.110Chapter 3. Kinetics of Electrode Reactionswhere E^Q and £p0St apply to (3.5.8) and (3.5.10), respectively. Substitution for the equilibrium concentrations of O' and R' in (3.5.12) givesCRo>Recognizing that n = n' + n" + 1 and that E2.10)for the overall process is (see Problem.(v _ ^rds + "'£pre + ""^postwe can distill (3.5.14) into^/(£ e q-£°') = _2.(3.5.16)CRwhich is the exponential form of the Nernst equation for the overall reaction,(3.5.17)Of course, this is a required result if the kinetic model has any pretense to validity, and itis important that the В V model attains it for the limit of / = 0, not only for the simple onestep, one-electron process, but also in the context of an arbitrary multistep mechanism.The derivation here was carried out for a mechanism in which the prereactions and postreactions involve net charge transfer; however the same outcome can be obtained by a similar method for any reaction sequence, as long as it is chemically reversible and a trueequilibrium can be established.3.5.3Nernstian Multistep ProcessesIf all steps in the mechanism are facile, so that the exchange velocities of all steps arelarge compared to the net reaction rate, the concentrations of all species participating inthem are always essentially at equilibrium in a local context, even though a net currentflows.
The result for the RDS in this nernstian (reversible) limit has already been obtainedas (3.4.27), which we now rewrite in exponential form:°' ' 1 = ef{E~E^(3.5.18)Equilibrium expressions for the pre- and post-reactions link the surface concentrations ofO' and R' to the surface concentrations of О and R. If these processes involve interfacialcharge transfer, as in the mechanism of (3.5.8)—(3.5.10), the expressions are of the Nernstform:n'f(E-E%e)eC=O(°> 0n"f(E-E%st)eQ(00=C R (0,0By steps analogous to those leading from (3.5.12) to (3.5.16), one finds that for the reversible systemС RVU, t)(3 52 0 )3.5 Multistep Mechanisms111which can be rearranged toJ?TCr\(0.
t)(3.5.21)This relationship is a very important general rinding. It says that, for a kineticallyfacile system, the electrode potential and the surface concentrations of the initial reactantand the final product are in local nernstian balance at all times, regardless of the details ofthe mechanism linking these species and regardless of current flow. Like (3.5.17),(3.5.21) was derived for pre- and postreactions that involve net charge transfer, but onecan easily generalize the derivation to include other patterns. The essential requirement is12that all steps be chemically reversible and possess facile kinetics.A great many real systems satisfy these conditions, and electrochemical examinationof them can yield a rich variety of chemical information (see Section 5.4.4).
A good example is the reduction of the ethylenediamine (en) complex of Cd(II) at a mercury electrode:Cd(en)^+ + 2e ^ Cd(Hg) + 3en(3.5.22)3*5.4 Quasireversible and Irreversible Multistep ProcessesIf a multistep process is neither nernstian nor at equilibrium, the details of the kinetics influence its behavior in electrochemical experiments, and one can use the results to diagnose the mechanism and to quantify kinetic parameters.
As in the study of homogenouskinetics, one proceeds by devising a hypothesis about the mechanism, predicting experimental behavior on the basis of the hypothesis, and comparing the predictions against results. In the electrochemical sphere, an important part of predicting behavior isdeveloping the current-potential characteristic in terms of controllable parameters, such asthe concentrations of participating species.If the RDS is a heterogeneous electron-transfer step, then the current-potential characteristic has the form of (3.5.11). For most mechanisms, this equation is of limited direct utility, because O' and R' are intermediates, whose concentration cannot be controlled directly.Still, (3.5.11) can serve as the basis for a more practical current-potential relationship, because one can use the presumed mechanism to reexpress Qy(0, i) and CR/(Q, t) in terms ofthe concentrations of more controllable species, such as О and R (36).Unfortunately, the results can easily become too complex for practical application.
Forexample, consider the simple mechanism in (3.5.8)—(3.5.10), where the pre- and postreactions are assumed to be kinetically facile enough to remain in local equilibrium. The overall nernstian relationships, (3.5.19), connect the surface concentrations of О and R to thoseof O' and R'. Thus, the current-potential characteristic, (3.5.11), can be expressed in termsof the surface concentrations of the initial reactant, O, and the final product, R.i = nFAk°rdsCo(0, t)e-n'f{E-EQv^e-af{E-E^(3.5.23)This relationship can be rewritten asi = nFA[kfCo(0, t) - kbCR(0, t)](3.5.24)12In the reversible limit, it is no longer appropriate to speak of an RDS, because the kinetics are not ratecontrolling. We retain the nomenclature, because we are considering how a mechanism that does have an RDSbegins to behave as the kinetics become more facile.112 P Chapter 3.
Kinetics of Electrode Reactionswhere=kfО^екThe point of these results is to illustrate some of the difficulties in dealing with a multistep mechanism involving an embedded RDS. No longer is the potential dependence ofthe rate constant expressible in two parameters, one of which is interpretable as a measureof intrinsic kinetic facility. Instead, k° becomes obscured by the first exponential factorsin (3.5.25) and (3.5.26), which express thermodynamic relationships in the mechanism.One must have ways to find out the individual values of n\ n", £p re , E®osV and E®ds beforeone can evaluate the kinetics of the RDS in a fully quantitative way. This is normally adifficult requirement.More readily usable results arise from some simpler situations:(a) One-Electron Process Coupled Only to Chemical EquilibriaMany of the complications in the foregoing case arise from the fact that the pre- andpostreactions involve heterogeneous electron transfer, so that their equilibria depend on E.Consider instead a mechanism that involves only chemical equilibria aside from the ratedetermining interfacial electron transfer:0 + Y;0'•f e(net result of steps precedinggRDS)(3 .5.27)(RDS)(3 .5.28)X>kuDI' <3-R+z(net result of steps following RDS)(3 .5.29)where Y and Z are other species (e.g., protons or ligands).
If (3.5.27) and (3.5.29) are sofacile that they are always at equilibrium, then CQ'(0, 0 and CR'(0, 0 in (3.5.11) are calculable from the corresponding equilibrium constants, which may be available from separate experiments.(b) Totally Irreversible Initial StepSuppose the RDS is the first step in the mechanism and is also a totally irreversible heterogeneous electron transfer:O + AR'R' + ri'e -+ R(RDS)(net result of steps following RDS)(3.5.30)(3.5.31)The chemistry following (3.5.30) has no effect on the electrochemical response, except toadd n" electrons per molecule of О that reacts. Thus, the current is n = 1 + n" times bigger than the current arising from step (3.5.30) alone. The overall result is given by the firstterm of (3.3.11) with C O '(0, 0 = C o (0, t),i = nFAk°Co(0,t)e~af(E~E°"ds)(3.5.32)Many examples of this kind of behavior exist in the literature; one is the polarographic reduction of chromate in 0.1 M NaOH:+ 4H 2 O + 3e -> Cr(OH)4 + 4OH"(3.5.33)Despite the obvious mechanistic complexity of this system, it behaves as though it has anirreversible electron transfer as the first step.(c) Rate-Controlling Homogeneous ChemistryA complete electrode reaction may involve homogeneous chemistry, one step of whichcould be the RDS.
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