A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 88
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Thus,since // varies as eo1/2, / at any potential should also vary as co1/2. A deviation of a plot of /vs. a>l/2 from a straight line intersecting the origin suggests that a kinetic limitation is involved in the electron-transfer reaction. For example, for a totally irreversible one-step,one-electron reaction, the disk current isi = FAkf(E)C0(y = 0)(9.3.36)00') From (9.3.31),where kf(E) = k° exp [ - « / ( £ - E ')].(9.3.37)or, with rearrangement and definingiK(9.3.38)=one obtains the Koutecky-Levich equation:(9.3.39)Here, iK represents the current in the absence of any mass-transfer effects, that is, thecurrent that would flow under the kinetic limitation if the mass transfer were efficient enough to keep the concentration at the electrode surface equal to the bulkvalue, regardless of the electrode reaction.
Clearly, i/o)1/2C is a constant only wheniK [or kf (E)] is very large. Otherwise, a plot of / vs. (oxl2 will be curved and tend toward the limit / = iK as o)1/2 —» °° (Figure 9.3.6). A plot of 1// vs. l/col/2 should be/ Levich line (id ~ co1/2)/ independent of co1/2*>1/2Figure 9.3.6 Variation of / with w1/2 at anRDE (at constant ED) for an electrodereaction with slow kinetics.Chapter 9.
Methods Involving Forced Convection—Hydrodynamic Methodslinear and can be extrapolated to eo l^2 = 0 to yield l/z#. Determination of i% at different values of E then allows determination of the kinetic parameters k° and a (Figure 9.3.7). A typical application of this procedure is illustrated in Figure 9.3.8,which shows such plots for the reduction of O2 to HO^~ at a gold electrode in alkaline solution.It is instructive to consider a more general derivation of (9.3.39), since the RDEcan be used to study the kinetics of processes other than electron transfer at modified electrode surfaces (see Section 14.4).
When mass transport and another processoccur in series, the rates of both processes must be the same at steady state. Thus,for the case where electron transfer at the electrode surface is the rate-limitingprocess,% - Co(y = 0)]/8 o = KE)C0(y = 0) = UnFA(9.3.40)It is left as an exercise for the reader (Problem 9.10) to show that solving this equation forCo(y = 0) and use of equations (9.3.23) and (9.3.38) lead to (9.3.39). For another ratelimiting process (e.g., diffusion of an electroreactant through a film coated on the electrode), the term k(E)Co(y = 0) would be replaced by the appropriate expression.
Thiswould yield an equation in the general form of the Koutecky-Levich equation, with theextrapolation to co~l/2 —> 0 allowing the determination of the kinetic parameter for thatprocess [see, for example, Section 14.4.2].For a quasireversible one-step, one-electron reaction, a general current-potential relationship can be derived in a similar manner. The i-rj equation, (3.4.10), can be written1=\со(у = 0)1 _tt _ Гск(у = о)1 Л-а'о L С% JL С* J(9.3.41)where b = ехр(^т?/ДГ). This equation, combined with (9.3.31) and (9.3.32), yieldsba1 - bb \-a(9.3.42)which can be reexpressed as(9.3.43)1/1*r1/2Figure 9.3.7 Koutecky-Levich plots atpotential E\, where the rate of electron transfer issufficiently slow to act as a limiting factor,and at E2, where electron transfer is rapid, forexample, in the limiting-current region.
The slopeof both lines is I3439.3 Rotating Disk Electrodeо0.6Cor recte<j disk ci10.8 —111111•0.41I—4"—/-0.2/010.91111110.70.50.3Disk potential, V vs. H/p - Pd10.1I0.010.020.030.04co"1/2 (rpm)- 1/2(b)Figure 9.3.8 (a) iD vs. E at2500 rpm and (b) KouteckyLevich plots for the reduction ofO 2 to HO^ at a gold electrode inO2-saturated (~1.0 mM) 0.1 MNaOH at an RDE (A = 0.196cm2).
The potential was swept at1 V/min. T = 26°C. (ij representsthe corrected current attributableto O2 reduction.) [From R. W.Zurilla, R. K. Sen, and E. Yeager,/. Electrochem. Soc, 125, 1103(1978). Reprinted by permissionof the publisher, TheElectrochemical Society, Inc.]Thus, \ji vs. a) 1 / 2 at a given value of 17 is predicted to be linear for this case as well, andthe intercept of the plot allows the determination of kinetic parameters.Alternative forms of (9.3.39) and (9.3.43) are sometimes given in the literature andare listed here for convenience. If the more general kinetic relation for the one-step, oneelectron process, equation 3.2.8, is used in the derivation, then the equation for 1// at thedisk becomes11ГРБ2% +Р^\]II1 ici/oП /^O-n~ A/O.
. I/Z|_\JSJLV(JJII(9.3.44)_|If the reverse (e.g., anodic) reaction can be ignored, then (9.3.44) yieldsFAkfC%kf/(0.62v-l/6D%3col/2)FAkfC%1+kf8o/Do(9.3.45)344Chapter 9. Methods Involving Forced Convection—Hydrodynamic Methodswhere 8Q is as defined in (9.3.25). This equation, easily derived from (9.3.40), is useful indefining the conditions for kinetic or mass-transfer control at the RDE.
When ^SQIDQ «1,the current is completely under kinetic (or activation) control. When kf8o/Do »1, themass-transfer-controlled equation results. Thus, if the RDE is to be used for kinetic measurements, kfSo/Do should be small, say less than 0.1; that is, kf < 0ADo/8o.Applications of RDE techniques to electrochemical problems have been reviewed(7-10, 12, 13).9.3.5Current Distribution at the RDEIn the preceding derivations, we assumed that the resistance of the solution was verysmall. With this condition, the current density is expected to be uniform across the diskand independent of the radial distance. Although this is frequently the case in real systems, the actual current distribution depends on the solution resistance, as well as themass- and charge-transfer parameters of the electrode reaction.
This topic has been treatedby Newman (14) and discussed by Albery and Hitchman (15).Consider first the primary current distribution, which represents the distributionwhen the surface overpotentials (activation and concentration) are neglected, and the electrode is taken as an equipotential surface. For a disk electrode of radius r\ embedded in alarge insulating plane with a counter electrode at infinity, the potential distribution undersuch conditions is as shown in Figure 9.3.9.
The current flows in a direction perpendicularFigure 9.3.9 Primary current distribution at an RDE. Solid lines show lines of equal potential atvalues of ф/фо, where ф$ is the potential at the electrode surface; that is, ф represents the potentialof the disk measured against an infinitesimal reference electrode (whose presence does not perturbthe current distribution) located at different indicated points in solution. Dotted lines are lines ofcurrent flow. The number of lines per unit length represents the current density j .
Note that j ishigher toward the edge of the disk than at the center. [From J. Newman, /. Electrochem. Soc, 113,501 (1966). Reprinted with permission of the publisher, The Electrochemical Society, Inc.]9.3 Rotating Disk Electrode345to the equipotential surfaces, and the current density is not uniform across the disk surface, but is instead much larger at the edge (r = r{) than at the center (r = 0). This situation arises because the ionic flux at the edge occurs from the side, as well as from thedirection normal to the disk. The total current flowing to the disk under total resistive control is (4, 14)/ = 4кг 1 (Д£)(9.3.46)where к is the specific conductivity of the bulk solution, and AE is the potential difference in solution between the disk and counter electrodes. Thus, the overall resistance,Яа, isRu = 1/4/crj(9.3.47)When electrode kinetics and mass-transfer effects are included, the current distribution (now called the secondary current distribution) is more nearly uniform than the primary one.
Albery and Hitchman (15) have shown that the current distribution can beconsidered in terms of the dimensionless parameter, p, given byP = -£•(9.3.48)KEwhere RE is the electrode resistance due to both charge transfer and concentration polarization. The secondary current distribution as a function of p is shown in Figure 9.3.10.Note that as p —> °° (i.e., high solution resistance and small RE), the current distribution1.81011819l If JCurve123456789101.6 --1.4 -p0.0790.2040.3930.8991.5712.5943.9275.91815.708oo1.2 --11.0————^40.8/7f/i\\H/If//If/6If IsIn1Ж1//23•/1560.6 -7 _—8^y9_"To0.4_____—-""10.210.4I0.6rlr,I0.81.0Figure 9.3.10 Secondarycurrent distribution at an RDE.[From J.
Newman,/. Electrochem. Soc, 113,1235 (1966) as modifiedby W. J. Albery and M. L.Hitchman, "Ring-DiscElectrodes," Clarendon,Oxford, 1971, Chap. 4, withpermission of the publishers,The Electrochemical Society,Inc., and Oxford UniversityPress.]346 • Chapter 9. Methods Involving Forced Convection—Hydrodynamic Methodsapproaches the primary one. Conversely, for small values of p (highly conductive solutionsand large RE) a fairly uniform current distribution is obtained.
To avoid a nonuniform distribution, the conditions must be such that p < 0.1 (15). By takingRE + Ru = ^j](9.3.49)(where dE/di is the slope of the current-potential curve at a given value of E) and combining with (9.3.47) and (9.3.48), we obtain the condition for a uniform distribution (15):§<036rlK(9.3.50)A plot of the values of dildE that satisfy this condition at different values of rx and /c,taken from Albery and Hitchman, is shown in Figure 9.3.11.
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