A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 30
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If the electrode kinetics are fairly facile, thesystem will approach the mass-transfer-limited current by the time such an extreme overpotential is established. Tafel relationships cannot be observed for such cases, becausethey require the absence of mass-transfer effects on the current. When electrode kineticsare sluggish and significant activation overpotentials are required, good Tafel relationships can be seen.
This point underscores the fact that Tafel behavior is an indicator of totally irreversible kinetics. Systems in that category allow no significant current flowexcept at high overpotentials, where the faradaic process is effectively unidirectional and,therefore, chemically irreversible.(d) Tafel Plots (8-11,32)A plot of log / vs. 7], known as a Tafel plot, is a useful device for evaluating kinetic parameters. In general, there is an anodic branch with slope (1 — a)F/23RT and a cathodicbranch with slope —aF/23RT. As shown in Figure 3.4.4, both linear segments extrapolate to an intercept of log i0. The plots deviate sharply from linear behavior as 77 approaches zero, because the back reactions can no longer be regarded as negligible.
Thelog I i I-3.5Slope = - a F-4.5-5.520015010050_l_-50I_l_-100-150I-200Л, mVFigure 3.4.4 Tafel plots for anodic and cathodic branches of the current-overpotential curve forО + e *± R with a = 0.5, T = 298 K, andy0 = Ю" 6 A/cm2.9Note that for a = 0.5, b = 0.118 V, a value that is sometimes quoted as a "typical" Tafel slope.104Chapter 3. Kinetics of Electrode Reactions-2E, V vs.
NHEFigure 3.4.5 Tafel plots for the reduction of Mn(IV) to Mn(III) at Pt in 7.5 M H2SO4 at 298 K. Thedashed line corresponds to a = 0.24. [From K. J. Vetter and G. Manecke, Z. Physik. Chem.(Leipzig), 195, 337 (1950), with permission.]transfer coefficient, a, and the exchange current, /0, are obviously readily accessible fromthis kind of presentation, when it can be applied.Some real Tafel plots are shown in Figure 3.4.5 for the Mn(IV)/Mn(III) system inconcentrated acid (33). The negative deviations from linearity at very large overpotentialscome from limitations imposed by mass transfer. The region of very low overpotentialsshows sharp falloffs for the reasons outlined just above.Allen and Hickling (34) suggested an alternative method allowing the use of data obtained at low overpotentials.
Equation 3.4.11 can be rewritten asi - ioe-°fn (1 - ef7))(3.4.18)orlog1 -= l°g *0 ~ 23RT(3.4.19)so that a plot of log [//(1 - eft)] vs. rj yields an intercept of log /0 and a slope of-aF/23RT. This approach has the advantage of being applicable to electrode reactionsthat are not totally irreversible, that is, those in which both anodic and cathodic processescontribute significantly to the currents measured in the overpotential range where masstransfer effects are not important.
Such systems are often termed quasireversible, becausethe opposing charge-transfer reactions must both be considered, yet a noticeable activation overpotential is required to drive a given net current through the interface.3.4 Implications of the Butler-Volmer Model for the One-Step, One-Electron Process3.4.4105Exchange Current Plots (8-14)From equation 3.4.4, we recognize that the exchange current can be restated aslog i0 = log FAk° + log Cg + j^fE0'- 2~£e q(3.4.20)Therefore, a plot of log /0 vs.
Ещ at constant CQ should be linear with a slope of-aF/23RT. The equilibrium potential Eeq can be varied experimentally by changing thebulk concentration of species R, while that of species О is held constant. This kind of plotis useful for obtaining a from experiments in which /0 is measured essentially directly(e.g., see Chapters 8 and 10).Another means for determining a is suggested by rewriting (3.4.6) aslog i 0 = log FAk° + (1 - a) log Cg + a log C$(3.4.21)Thusд log L \601= l - alog C5MId log io ,\d log ^ -Ci)cand= a(3.4.22)An alternative equation, which does not require holding either C% or Cf constant, isd log(3.4.23)(CR/CQ)The last relation is easily derived from (3.4.6).3.4.5Very Facile Kinetics and Reversible BehaviorTo this point, we have discussed in detail only those systems for which appreciableactivation overpotential is observed.
Another very important limit is the case inwhich the electrode kinetics require a negligible driving force. As we noted above,that case corresponds to a very large exchange current, which in turn reflects a bigstandard rate constant k°. Let us rewrite the current-overpotential equation (3.4.10)as follows:[loCRCo~a)fri(3.4.24)and consider its behavior when /0 becomes very large compared to any current of interest.The ratio ///Q then approaches zero, and we can rearrange the limiting form of equation3.4.24 toefipjp-'-*,3.4.25,and, by substitution from the Nernst equation in form (3.4.2), we obtainCo ( Q ' ° = ef(EQq-E°') ef(E-Eeq)CR(0,026)' - }(3 4KorГ"т- ')=f(E-E«)e( 3 A 2 7 )106Chapter 3.
Kinetics of Electrode ReactionsThis equation can be rearranged to the very important result:(3.4.28)Thus we see that the electrode potential and the surface concentrations of О and R arelinked by an equation of the Nernst form, regardless of the current flow.No kinetic parameters are present because the kinetics are so facile that no experimental manifestations can be seen. In effect, the potential and the surface concentrationsare always kept in equilibrium with each other by the fast charge-transfer processes, andthe thermodynamic equation, (3.4.28), characteristic of equilibrium, always holds. Netcurrent flows because the surface concentrations are not at equilibrium with the bulk, andmass transfer continuously moves material to the surface, where it must be reconciled tothe potential by electrochemical change.We have already seen that a system that is always at equilibrium is termed a reversible system; thus it is logical that an electrochemical system in which the chargetransfer interface is always at equilibrium be called a reversible (or, alternatively, anernstian) system.
These terms simply refer to cases in which the interfacial redox kinetics are so fast that activation effects cannot be seen. Many such systems exist in electrochemistry, and we will consider this case frequently under different sets of experimentalcircumstances. We will also see that any given system may appear reversible, quasireversible, or totally irreversible, depending on the demands we make on the charge-transferkinetics.3.4.6Effects of Mass TransferA more complete i-j] relation can be obtained from (3.4.10) by substituting forC o (0, 0/CQa n d C R (0, 0/CR according to (1.4.10) and (1.4.19):JL=i - ±)е-Ф-i - ±(3.4.29)This equation can be rearranged easily to give / as an explicit function of rj over thewhole range of 77.
In Figure 3.4.6, one can see i-r\ curves for several ratios of 1ф\, whereFor small overpotentials, a linearized relation can be used. The complete Taylor expansion (Section A.2) of (3.4.24) gives, for a/77 < < 1>/ _CQ(0,t)C R (0, t)FT)(3.4.30)which can be substituted as above and rearranged to giveVl Fv'okcW(3.4.31)In terms of the charge- and mass-transfer pseudoresistances defined in equations 1.4.28and 3.4.13, this equation is(3.4.32)Here we see very clearly that when /0 is much greater than the limiting currents,Rct «RmtiC + i?mt,a a n d the overpotential, even near £ e q , is a concentration over-3.5 Multistep Mechanisms-200-300107-400, mVFigure 3.4.6 Relationship between the activation overpotential and net current demandrelative to the exchange current.
The reaction is О + e ^ R with a = 0.5, T = 298 K, and// c = -//a = //. Numbers by curves show /0///.potential. On the other hand, if IQ is much less than the limiting currents, then /?mt,c +^mt,a < < ^ct» a n c * the overpotential near Ещ is due to activation of charge transfer. Thisargument is simply another way of looking at the points made earlier in Section 3.4.3(a).In the Tafel regions, other useful forms of (3.4.29) can be obtained. For the cathodicbranch at high r\ values, the anodic contribution is insignificant, and (3.4.29) becomes=h-JL(3.4.33)orRT, *o , RT, (*'/,c ~ 0(3.4.34)7] = —~ In — + —~ InatiicaFThis equation can be useful for obtaining kinetic parameters for systems in which the normal Tafel plots are complicated by mass-transfer effects.3.5 MULTISTEP MECHANISMS (11, 13, 14, 25, 26, 35)The foregoing sections have concentrated on the potential dependences of the forward andreverse rate constants governing the simple one-step, one-electron electrode reaction.
Byrestricting our view in this way, we have achieved a qualitative and quantitative understanding of the major features of electrode kinetics. Also, we have developed a set of relations that we can expect to fit a number of real chemical systems, for example,Fe(CN)^" + eFe(CN)^~(3.5.1)(3.5.2)Anthracene + e ^ Anthracene"(3.5.3)108Chapter 3. Kinetics of Electrode ReactionsBut we must now recognize that most electrode processes are mechanisms of severalsteps. For example, the important reaction2H + + 2е±±Щ(3.5.4)clearly must involve several elementary reactions. The hydrogen nuclei are separated inthe oxidized form, but are combined by reduction. Somehow, during reduction, theremust be a pair of charge transfers and some chemical means for linking the two nuclei.Consider also the reductionSn 4 + + 2e *± Sn 2 +(3.5.5)Is it realistic to regard two electrons as tunneling simultaneously through the interface? Ormust we consider the reduction and oxidation sequences as two one-electron processesproceeding through the ephemeral intermediate Sn 3 + ? Another case that looks simple atfirst glance is the deposition of silver from aqueous potassium nitrate:Ag + + e ^ A g(3.5.6)However, there is evidence that this reduction involves at least a charge-transfer step, creating an adsorbed silver atom (adatom), and a crystallization step, in which the adatommigrates across the surface until it finds a vacant lattice site.
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