A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 100
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Even for a one-step, one-electron process, the general case in which k° can adopt any value is very complex. Thereader is referred to the literature for complete discussions (2, 3, 5). Here we examineFigure 10.5.2 An ac polarogramfor3 X l ( T 3 M C d 2 + i n l . 0 MNa2SO4. A£ = 5 mV, W/2TT = 320-0.650-0.550£(V vs. SCE)Hz. [Reprinted with permission fromD. E. Smith, Anal Chem., 35, 1811(1963). Copyright 1963, AmericanChemical Society.]392Chapter 10.
Techniques Based on Concepts of Impedancean important special case in detail, and it will provide us with a good intuitive understanding of the kinetic effects of interest.That special case is the situation in which the dc response from a one-step, one-electron system is effectively nernstian, whereas the ac response is not. This situation is frequently seen in real systems because the time scales of the two aspects can differ sogreatly. That is, k° can be sufficiently large that the mean surface concentrations are keptin the ratio dictated in Edc through the nernstian balance, (10.5.2), even though that rateconstant is not large enough to assure a negligible charge-transfer resistance to the muchfaster ac perturbation.The faradaic impedance in this situation involves both Rct and a, and the magnitudecan be written from (10.2.27):2<ш5л4)(^) ЫТThe parameters Rct and a can both be defined through the assumption of dc reversibility,which allows us to use the same mean surface concentrations as in the previous section.They are defined by (10.5.3) and (10.5.4); thus we can develop a by substitution into(10.5.1).
Rearrangements equivalent to those used in obtaining (10.5.9) then yieldfГC Sh2° ffwhere we have recognized that n = 1.The charge-transfer resistance, Rcb is given by (10.3.2) in terms of the exchange current, /0- Normally we speak of /0 as an equilibrium property defined by bulk concentrations of О and R according to (3.4.6). However, since the mean surface concentrations actlike bulk values for the ac process, we can recognize an effective exchange current for acperturbation that would be given by(io)eff = FAk°[Co(0, 0m] ( 1 ~ a ) [C R (0, t)m]a(10.5.16)anBy determining the mean surface concentrations, Edc controls (z'o)eff d , therefore, Rct.
Amore explicit expression of this dependence is obtained by substitution from (10.5.3),(10.5.4), and (10.5.6), as above:(/0)eff = FAk^e^j^j(Ю.5.17)where j3 = (1 - a). Since Rct = RT/F(io)eff, we haveNow that Rct and a are available, we can write Z f as a function of Edc by substitutioninto (10.5.14). That operation is straightforward, but it yields a rather messy expression.Perhaps more instructive is to examine limiting behavior for high and low frequencies,which can be discerned from (10.5.14).At very low frequencies, Rct is small compared to a/(om; hence the system looks reversible.
This is not surprising; after all we are bringing the time domain of the ac processtoward that of the dc perturbation, which evokes a reversible response. Everything wefound in the previous section about the reversible ac response should also apply to thequasireversible system at the low-frequency limit.As the frequency is elevated, Rct becomes appreciable in comparison to a/ojl/2\ hencereversibility is vitiated. The ac time domain has become shortened enough to strain the10.5 ас Voltammetry3931/2heterogeneous kinetics. Finally, at the high-frequency limit, Rct greatly exceeds o7co ,and Z f approaches Rct itself.
The amplitude of the alternating current is then(10.5.19)This equation describes the shape of the ac polarogram. In general, the response as afunction of dc potential is bell-shaped, much as in the reversible situation. This point isseen by noting the behavior of the factor in parentheses as a becomes large either positively or negatively. However, positive deviations from Ey2 do not evoke the same response as negative deviations; that is, the response is not symmetric and the bell shape isskewed.The peak is easily found by differentiating (10.5.19) with respect to a.
The maximumis reached when ea — /3/a, or=EmRTln-yPa(10.5.20)The peak current amplitude is therefore(10.5.21)These equations, together with those describing the reversible, low-frequency limit,give a good picture of the behavior of the system as со changes. The peak current is at firstlinear with com, but with increasing frequency that dependence is reduced until, at thehigh-frequency limit, / p becomes independent of со. It is easy to see from preceding developments that the frequency dependence reflects the mass transport effects manifested inthe Warburg impedance. The lack of a frequency dependence in (10.5.19) and (10.5.21)comes about because the current is totally controlled at high со by heterogeneous kinetics.Mass transfer plays no role. Not surprisingly, then, / p is proportional to A:0 at high со, and itis totally insensitive to A:0 at low со.
The proportionalities between / p and AE and CQ holdat all frequencies.Note also that since kinetic control of / at high frequencies implies a faradaic impedance that greatly exceeds the Warburg impedance at those frequencies, the current mustbe much smaller than that for a truly reversible system, which shows only the Warburgimpedance at any frequency.
The general reduction in ac response in quasireversible systems is illustrated in Figure 10.5.3. The k° values for all curves shown there are sufficiently great that the assumption of dc reversibility holds. It is easy to see the trend inresponses with decreasing k°; hence one recognizes that there will be a rather small ac response if k° falls below 10~4 to 10~5 cm/s. Systems showing totally irreversible dc polarograms can be almost invisible to the ac experiment.
This fact is useful for analyticalwork (see Section 10.7). n1'The totally irreversible case does yield an ac current, contrary to the impression one might gain from thisline of argument. The current arises from the simple modulation of the dc wave (23, 24). Since the shape ofthat wave is independent of k° (Section 5.5), the ac peak height is also independent of k°. The peak lies nearthe half-wave potential of the dc wave; hence it is shifted substantially from E0' by an amount related to thesize of *°.394 i» Chapter 10. Techniques Based on Concepts of Impedance24.0 -16.08.00.160.080.00Edc ~-0.08-0.16EV2Figure 10.5.3 Calculated acpolarograms for quasireversibleone-step, one-electron systems.Curves (from the top) are for k° —>ооД°= 1Д° = 0.1, and k° = 0.01cra/s.
Other parameters are asfollows: o) = 2500 s" 1 , a = 0.500,D = 9X 10" 6 cm2/s, A = 0.035cm2, Cg = 1.00 X 10~3 M,T =298 K, AE = 5.00 mV. The curvesshow the faradaic current at fmax.[Reprinted from D. E. Smith,Electroanal Chem., 1, 1 (1966),by courtesy of Marcel Dekker,Inc.]The position of the peak is also of interest. Relations derived above show that there isa slight shift with increasing frequency. At low a), the peak comes at E& = ^1/2» J u s t a sfor a reversible system at any frequency. As 00 becomes greater, the peak potential deviates from this value until it reaches the limiting position defined by (10.5.20).
Since a and/3 are generally comparable, we can expect the extent of this shift, (RTIF) In ф/а), to bequite small. In other words, the peak potential for an ac polarogram is always near the formal potential for the couple, provided dc reversibility applies.The phase angle of / a c with respect to Eac is of great interest as a source of kinetic information. This point was suggested in Section 10.3, and it is rooted in equation 10.3.9.We can rewrite that relation asД/2cot ф = 1 + - ^ —(10.5.22)Substitution from (10.5.15) and (10.5.18) and rearrangement gives(10.5.23)The bracketed factor shows that cot ф depends on the dc potential.
Large positive andnegative values of a force cot ф to unity, and hence there must be a maximum in thisquantity near the peak of the polarogram. The precise position is easily found by differentiation, and one ascertains that e~a = fi/a at that point. Thus,^dc - ^ 1 / 2 + ~E^ J(10.5.24)10.5 ас Voltammetry395This maximum point is independent of nearly all experimental variables, for example,A£, CQ, and, most notably, A and со. The difference between £ 1/2 and the potential ofmaximum cot ф provides access to the transfer coefficient a.Actual cot ф data are shown in Figure 10.5.4 for TiCl 4 in oxalic acid solution (25).The electrode reaction is the one-electron reduction of Ti(IV) to Ti(III). Note that the potential of maximum cot ф is independent of frequency, as predicted above.mPlots of cot ф vs. co yield k°, once a is known from the position of [cot ф ] т а х andthe diffusion coefficients are known from other measurements. This point is easily seenfrom (10.5.23), which holds for any value of Edc.
In practice, these plots are usually madefor special values of Edc that give simplified forms of the linear relation.A convenient procedure is to choose Edc = Ey2, for then a = 0, and we have[cot ф]Е= 1+(10.5.25)If one can take DQ = D R = D, then D^D^ = D, and the slope of this particular plot becomes independent of a. Figure 10.5.5 is an example in which the data from Figure 10.5.4at Edc = Em = -0.290 V vs. SCE have been plotted vs. com.Another simplified version of (10.5.23) can be obtained for the potential of maximumcot ф. By substituting e~a = /3/a, we obtain[cot ф]тях = 1 +1)11/2CO(10.5.26)The product of the diffusion coefficients can usually be simplified as above, but a stillmust be known for an evaluation of к0, because of the bracketed factor.Quantitative information about heterogeneous charge-transfer kinetics obtainedfrom ac polarographic data nearly always comes from the behavior of cot ф with po-1020626Figure 10.5.4 Dependence ofthe phase angle on Edc.
Thesystem is 3.36 mM TiCl4 in0.2WMH2C2O4.A£ = 5.00mV, T = 25°C. Points areexperimental; curves arepredicted from experimentalparameters by (10.5.23).[Reprinted with permissionfrom D. E. Smith, Anal. Chem.,35, 610 (1963). Copyright1963, American ChemicalSociety.]396 • Chapter 10. Techniques Based on Concepts of Impedance4.20 3.80 3.40 -|У3.00i 2-60XXXFigure 10.5.5 Plot of cot ф vs.o)m for 3.36 mM TiCl4 in 0.2002.201.80-0.290 V vs. SCE. A£ - 5.001.401 ППXIIIIIIIIIIII81624324048566472808896mV, T = 25°C. [Reprinted withpermission from D. E. Smith,Anal. Chem., 35, 610 (1963).Copyright 1963, AmericanChemical Society.]tential and frequency, rather than from the heights, shapes, or positions of the polarograms.
One reason for this favor toward cot ф is that many experimental variables donot have to be controlled closely or even be known. Among them are CQ, AE, and A.Freedom from knowing A can be a significant advantage. However, the most importantreason for evaluating kinetics through cot ф is that relations (10.5.23) to (10.5.26) holdfor any quasireversible or irreversible system.
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