A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 63
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While / p varies with v1/2 for linear diffusion, /c varies with u, so that ic becomes relatively more important at faster scan rates. From (6.2.19) and (6.2.25)I'd _Cdvl/2(l0~5)T " 2.69(6226)or for D o = 10~5cm2/s and Cd= 20 /xF/cm2,\(2.4 XThus at high v and low CQ values, severe distortion of the LSV wave occurs.
This effectoften sets the limits of maximum useful scan rate and minimum useful concentration.In general, a potentiostat controls E + iRw rather than the true potential of the workingelectrode (Sections 1.3.4 and 15.6.1). Since / varies with time as the peak is traversed, theerror in potential varies correspondingly. If /p/?u is appreciable compared to the accuracy ofmeasurement (e.g., a few mV), the sweep will not be truly linear and the condition given by(6.2.1) does not hold. Moreover, the time required for the current to rise to the level given in(6.2.25) depends upon the electrode time constant, RuCd, as shown in (1.2.15). The practicaleffect of Ru is to flatten the wave and to shift the reduction peak toward more negative potentials. Since the current increases with i>1/2, the larger the scan rate, the more Ep will beshifted, so that appreciable Rn causes £ p to be a function of v.
It moves systematically in anegative direction with increasing v (for a reduction). Uncompensated resistance can thushave the insidious effect of mimicking the response found with heterogeneous kinetic limitations (as discussed in Sections 6.3 and 6.4).By using a UME, one can extend the useful range of sweep rates to the 106 V/s region. Because the measured currents at the UME are small, the iRu drop does not perturbthe response or the applied excitation to the same extent as with larger electrodes. Themuch smaller RuCd at the UME also leads to less distortion in the voltammogram. However, even with the UME (6.2.27) applies, so the faradaic wave lies on top of a large capacitive current.
To extract the desired information from the voltammogram, the totalresponse (capacitive plus faradaic) can be simulated (9) or perturbations caused by Cd andRu can be subtracted (10). Alternatively, positive feedback circuitry with a fast responsecan be used to compensate for distortions otherwise caused by Ru (11).234Chapter 6.
Potential Sweep Methods(d)(c)(b)(a)Figure 6.2.3 Effect of double-layercharging at different sweep rates on a linearpotential sweep voltammogram. Curves areplotted with the assumption that Сд isindependent of E. The magnitudes of thecharging current, /c, and the faradaic peakcurrent, /p, are shown. Note that the currentscale in (c) is 10X and in (d) is 100X thatin (a) and (b).An important practical limitation of very fast voltammetry [other than instrumentaland Ru and C& considerations (11)], is the importance of adsorption of even small amountsof electroactive substance or faradaic changes involving the electrode surface (e.g., formation of an oxide layer).
As shown in Section 14.3, for surface processes like doublelayer charging, the current response varies directly with v. Therefore, surface effects ofminor importance at small v can dominate at high scan rates.6.3 TOTALLY IRREVERSIBLE SYSTEMS6.3.1The Boundary Value ProblemFor a totally irreversible one-step, one-electron reaction (О + еboundary condition, (6.2.2), is replaced by (see Section 5.5)R) the nernstian(6.3.1)6.3 Totally Irreversible Systems235wherekf(t) = k° exp{ -af[E(t) - E0']}(6.3.2)Introducing E(t) from (6.2.1) into (6.3.2) yieldskf (0C o (0, 0 = £fiCo(0, t)ebt(6.3.3)- E0)](6.3.4)where b = afu andkfi = k° exp[-af(E{The solution follows in an analogous manner to that described in Section 6.2.1 and againrequires a numerical solution of an integral equation (3, 5).
The current is given byi = FAC%(TTDob)112 xibt)(6.3.5)(6.3.6)(where x(bt) is a function [different from xi*7*)] tabulated in Table 6.3.1. Again, / at anypoint on the wave varies with vl/2 and CQ.For spherical electrodes, a procedure analogous to that employed at planar electrodeshas been proposed. Table 6.3.1 contains values of the spherical correction factor, ф(Ы)employed in the equation/ = /(plane) +TABLE 6.3.1Current Functions for Irreversiblek Charge TransferDimensionlessPotential^Potential6"mV at 25°CTr x(bt)6.235.454.674.283.893.503.112.722.341.951.561.361.170.970.78160140120110100908070605040353025200.0030.0080.0160.0240.0350.0500.0730.1040.1450.1990.2640.3000.3370.3720.406al/2(6.3.7)orф(Ы)DimensionlessPotentialPotential6mV at 25°Cтгтх№Ф№0.0040.0100.0210.0420.0830.1150.1540.1990.2530.580.390.190.00-0.19-0.21-0.39-0.58-0.78-0.97-1.17-1.36-1.56-1.95-2.72151050-5-5.34-10-15-20-25-30-35-40-50-700.4370.4620.4800.4920.4960.49580.4930.4850.4720.4570.4410.4230.4060.3740.3230.3230.3960.4820.6000.6850.6940.7550.8230.8950.9520.9921.000To calculate the current:1.
/ = /(plane) + /(spherical correction).2.i = FADl£c%bm<irll2x(f,t)+ FADoC%(\lrQ)<t>{bt)3. i = 602ADH 2 C^a 1/2 v l/2 {7r l/2 x(bt)+ 0.160[^^/2/(г0о;1/21;1/2)]ф(^0}. Units for step 3 are the same as in Table 6.2.1.^Dimensionless potential is (aF/RT)(E - E0') + In [(7rDob)l/2/k°].Potential scale in mV for 25°C is a(E - E°) + (59.1) In [(ттОоЬ)ш/к0].236Chapter 6. Potential Sweep Methods6.3.2Peak C u r r e n t and PotentialThe function x(bt) goes through a maximum at rrl/2x(bt) = 0.4958. Introduction of thisvalue into (6.3.6) yields the following for the peak current:/p = (2.99 X\05)amAC%D\?vm(6.3.8)where the units are as for (6.2.19). From Table 6.3.1, the peak potential, £ p , is given bya(Ep - E°) + ^In= -0.21 Щ- = -5.34 mV at 25°C°Qor1D 2р- * .
- Ц;к-,V2I0.780 + In1.S51RTafo)k°)47.7mVat 25°Ca(6.3.9)(6.3.10)(6.3.11)where £ р д is the potential where the current is at half the peak value. For a totally irreversible wave, £ p is a function of scan rate, shifting (for a reduction) in a negative directionby an amount !A5RT/aF (or 30/a mV at 25°C) for each tenfold increase in v. Also, ED occurs beyond E (i.e., more negative for a reduction) by an activation overpotential relatedto k°. An alternative expression for / p in terms of Ev can be obtained by combining (6.3.10)with (6.3.6), so that the result contains the value of x(bt) at the peak.
After rearrangementand evaluation of the constants, the following equation is obtained (3, 6):ip = 0.227 FAC$tP exp[-a/(£ p - E0)](6.3.12)A plot of In / p vs. Ep — E0' (assuming E0' can be obtained) determined at different scanrates should have a slope of —af and an intercept proportional to &°.For an irreversible process more complicated than the one-step, one-electron reaction, it is usually not feasible to derive equations describing the current-potential relationship. In the general case, the more practical approach is to compare experimentalbehavior with the predictions from simulations (see Appendix В and Chapter 12). Analytical equations may be achievable for a few of the simpler possibilities (see Section 3.5.4).The most important is an overall и-electron process having an irreversible heterogeneousone-electron transfer as the rate-controlling first step.
In that case, all equations describingcurrents in this section [(6.3.5)-(6.3.8), (6.3.12), and in the notes to Table 6.3.1] apply,but with the right hand side multiplied by n. The equations describing potentials[(6.3.9)—(6.3.11)] apply without alteration.6.4 QUASIREVERSIBLE SYSTEMSMatsuda and Ayabe (5) coined the term quasireversible for reactions that show electrontransfer kinetic limitations where the reverse reaction has to be considered, and they provided the first treatment of such systems. For the one-step, one-electron case,кO + e;e±R(6.4.1)bthe corresponding boundary condition is [from (5.5.3)]_x=0(6 42)6.4 Quasireversible Systems *4 237The shape of the peak and the various peak parameters were shown to be functions of aand a parameter Л, defined as(6.4.3)(6.4.4)The current is given by(6.4.5)where Ф(Е) is shown in Figure 6.4.1.
Note that when Л > 10, the behavior approachesthat of a reversible system.The values of /p, Я р , and Ep/2 depend on Л and a. The peak current is given byip = /p(rev)#(A, a)(6.4.6)where /p(rev) is the reversible /p value (equation 6.2.18), and the function K(A, a) is shown inFigure 6.4.2. Note that for a quasireversible reaction, /p is not proportional to vl/2.The peak potential is- fii/2 = -B(A, otily-J = -26H(A, a) mV at 25°C0.5a = 0.70.4//a0.30.20.10(6.4.7)v/ /Ун Уш1r0.5a = 0.50.40.3jl к /тJ0.20.1//00.5a = 0.30.40.3f0.20.10-128^-—Xfv/•128257-(£-£ 1/2 ),mV(25°C)385Figure 6.4.1 Variation ofquasireversible current function,ЩЕ), for different values of a(as indicated on each graph) andthe following values of Л: (I) Л= 10;(П)Л = 1; (III) Л = 0.1;(IV) Л = 10~2.
Dashed curve isfor a reversible reaction. ЩЕ)= UFAC%D\? (nFIRT)mvmand Л = k°/[D1/2(F/RT)l/2vl/2](for DO = DR = D). [From H.Matsuda and Y. Ayabe, Z.Elektrochem., 59, 494 (1955),with permission. Abscissa labeladapted for this text.]238Chapter 6. Potential Sweep MethodsFigure 6.4.2 Variation of K(A, a) with Л fordifferent values of a. Dashed lines showfunctions for a totally irreversible reaction.K(K, a) - /p//p(rev). [From H. Matsuda andY. Ayabe, Z. Elektrochern., 59, 494 (1955),with permission.]where Н(Л, a) is shown in Figure 6.4.3. For the half-peak potential, we have= Д(Л, а)( Щ-) = 26Д(Л, a) mV at 25°CEp/2 -Ev(6.4.8)where Д(Л, a) is given in Figure 6.4.4.The parameters K(A, а), Н(Л, a), and Д(Л, a) attain limiting values characteristic ofreversible or totally irreversible processes as Л varies.
For example, consider Д(Л, a). ForЛ > 10, Д(Л, a) « 2.2, yielding the Ep - Ep/2 value characteristic of a reversible wave,(6.2.22). For Л < 1(Г 2 and a = 0.5, Д(Л, a) « 3.7, yielding the totally irreversible charac0.1 0.2 0.320•I*11 1f1 /' /1I118 —1 1I /I /160.4 0.5SJSI1 1 1I A 0.6II/1 I /I I II I /0.7/ ///' I / / 0.81 1 1 /1 1 I 1I I I /! 1 1 //1// //// 0.9I14t12//i1 1 1 / 1f/ I// /'//'/< 1.0/ / •AI I I / /1 1 1 / / J' /10' /'/ // // /1 ' / / / / /t S /t t1 1 1 /I/ /11//I/.///w864 -/s2 I1+2+110-11-211Figure 6.4.3 Variation of Н(Л, a) with Лfor different values of a.
Dashed lines showfunctions for a totally irreversible reaction.Н(Л, a) = - ( £ p - Em)F/RT. [From H.Matsuda and Y. Ayabe, Z. Elektrochem.,59, 494 (1955), with permission.]6.5 Cyclic Voltammetry239Figure 6.4.4 Variation of Д(Л, a) with Лand a. Dashed lines show functions for atotally irreversible reaction. Д(Л, а) =(£ p / 2 - Ep)F/RT. [From H. Matsuda andY.
Ayabe, Z. Elektrochem., 59, 494 (1955),with permission.]teristic, (6.3.11). Thus a system may show nernstian, quasireversible, or totally irreversiblebehavior, depending on Л, or experimentally, on the scan rate employed. The appearance ofkinetic effects depends on the time window of the experiment, which is essentially the timeneeded to traverse the LSV wave (see Chapter 12). At small v (or long times), systems mayyield reversible waves, while at large v (or short times), irreversible behavior is observed.
InSection 5.5, we reached the same conclusions in the context of potential step experiments.Matsuda and Ayabe (5) suggested the following zone boundaries for LSV:6Reversible (nernstian)QuasireversibleTotally irreversibleЛ > 15; k° > 03vm cm/s15 > Л > КГ2(1+«>; 0.3u1/2 > ^ ° > 2 X 10~5 vl/2 cm/sЛ < 10~ 2 ( 1 + a ) ; i l ° < 2 X 10~5 vm cm/s6.5 CYCLIC VOLTAMMETRYThe reversal experiment in linear scan voltammetry is carried out by switching the direction of the scan at a certain time, t = A (or at the switching potential, Ex). Thus the potential is given at any time by(0<f<A)(t > A)E = E{-vtE = E{- 2vX + vt(6.5.1)(6.5.2)While it is possible to use a different scan rate (i/) on reversal (12), this is rarely done, andonly the case of a symmetrical triangular wave is considered here.
The theoretical treatmentfollows that of Section 6.2, except that (6.5.2) is used in the concentration-potential equation,rather than (6.2.1), for t > A. This sweep reversal method, called cyclic voltammetry, is extremely powerful and is among the most widely practiced of all electrochemical methods.6.5.1Nernstian SystemsApplication of (6.5.2) in the equation for a nernstian system, (5.4.6), yields (6.2.3), whereS(t)is now given by(t > A)6S(t) = eat~2(Tk(6.5.3)The k° values are based on n = 1, a = 0.5, T = 25°C, and D = 1(Г 5 cm2/s. With v in V/s, Л = k°/Q9Dv)['2.240 >; Chapter 6. Potential Sweep MethodsThe derivation then proceeds as in Section 6.2. The shape of the curve on reversal depends onthe switching potential, £A, or how far beyond the cathodic peak the scan is allowed to proceed before reversal.
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