A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 54
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The details are left to Problem 5.8(c).5.5.3Steady-state Voltammetry at a UMEOur concern now is with steady-state responses in systems with quasireversible or irreversible electron-transfer kinetics. We can limit (5.5.37) to the steady state simply by imposing the condition that 8 «1, which implies that у —> 1 andKs) =1ros(5.5.43)Since nothing in the brackets depends on s, the current is readily obtained by inversion.With rearrangement the result is:(5.5.44)which describes the steady-state current for any kinetic regime at a sphere or hemisphere.At very negative potentials relative to E°, в approaches zero and к becomes verylarge, so that the limiting current is given byFADOC%(5.5.45)as we have already seen. By dividing (5.5.44) with (5.5.45), one has1 + #c(l + ?0)(5.5.46)which compactly describes all steady-state voltammetric waves at spherical electrodes.One can easily see that as the potential is changed from values far positive of E tovalues far negative, i/id goes from zero to unity.
Thus a sigmoidal curve is found generallyin steady-state voltammetry. Figure 5.5.4 is a display of voltammograms corresponding tothe three kinetic regimes. For the reversible case, к —» °° at all potentials, and (5.5.46)200Chapter 5. Basic Potential Step Methods1.2fid/гт~~1.0 0.8 :-"~р-— Quasireversible-/гSS 0-6 "0.4 -0.2 -200Reversible IP/ —"—~~~£- \ ^ TotallyIrreversible/АУ У100-1009-20011-300-400-500(£-£°')/mVFigure 5.5.4 Steady-statevoltammograms at a spherical orhemispherical electrode for variouskinetic regimes.
Curves arecalculated from (5.5.46) assumingButler-Volmer kinetics with a = 0.5,and ro = 5 /mi, DQ = Z)R =1 X 10" 5 cm2/s. From left to rightthe values of £° are 2, 2 X 1(Г 2 ,2 X 10~3, and 2 X 10~4 cm/s.collapses to (5.4.54), represented as the leftmost curve in Figure 5.5.4. Smaller values ofk° cause a broadening and a displacement of the wave toward more extreme potentials,just as for waves based on sampled transients.(a) Total IrreversibilityIn Section 5.5.l(e), we determined that the criterion for total irreversibility is that в ~ 0over all points on the wave that are measurably above the baseline.
Thus, the limitingform of (5.5.46) is(5.5.47)1 +кOne can substitute for k{ and rearrange (5.5.47) to the formaF(5.5.48)which has a half-wave potential given byE\p, ~ERT(5.5.49)so that a standard plot of E vs. log[(/d - /)//] is expected to be linear with a slope of2303RT/aF (i.e., 59.I/a mV at 25°C) and an intercept of £ 1/2 . From the slope and intercept, one can obtain a and k° straightforwardly.(b) Kinetic RegimesWe can define the boundaries between the kinetic regimes in steady-state voltammetry inessentially the manner used in Section 5.5.l(f), but the focus now must be placed on thevalue of the parameter к at E° , which is Г^/DQ. This quantity, designated as к 0 , has asignificance for steady-state voltammetry essentially the same as that of A0 for voltammetry based on semi-infinite linear diffusion.
Even though currents are small at UMEs, current densities can be extremely high because of the very high mass transfer rates that canapply at UMEs. This is the aspect of their nature that provides access to heterogeneousrate constants of the most facile known reactions.5.5 Sampled-Current Voltammetry for Quasireversible and Irreversible Electrode Reactions201If a system is to appear reversible, then к at potentials in the neighborhood of E°must be large enough that (5.5.46) converges to (5.4.54). This will be true within the limits of experimental precision if к0 > 10.As argued in Section 5.5.l(f), total irreversibility applies when the wave is displacednegatively to such a degree that в ~ 0 across the whole wave.
If Ец2 — E0' is at least asnegative as -4.6RT/F, that condition will be satisfied. Thus the second term in (5.5.49)must be more negative than -4.6RT/F, implying that log к 0 < - 2 a .The quasireversible regime lies between these boundaries, in the range 10~ 2 Q ! <ic° < 10.(c) Other Electrode ShapesThe foregoing discussion is rigorous only for spherical and hemispherical electrodes,which are called uniformly accessible because there are no differences in mass transferover the electrode surface. Steady-state voltammetry can be carried out readily at otherUMEs, but for quasireversible and totally irreversible systems, the results are affected bythe nonuniformity in the flux at different points on the electrode surface.
At a disk UME,for example, mass transfer can support a flux to points near the edge that is much higherthan to points near the center; thus the kinetics must be activated more strongly to supportthe diffusion-limited current at the edge than in the center. The recorded voltammogramwould represent an average of behavior, with contributions from different points weightedby their diffusion-limited fluxes.There is a significant contrast here with Section 5.4.2(e), where we found that the results for reversible systems observed at spherical electrodes could be extended generallyto electrodes of other shapes. This is true for a reversible system because the potentialcontrols the surface concentration of the electroactive species directly and keeps it uniform across the surface.
Mass transfer to each point, and hence the current, is consequently driven in a uniform way over the electrode surface. For quasireversible andirreversible systems, the potential controls rate constants, rather than surface concentrations, uniformly across the surface. The concentrations become defined indirectly by thelocal balance of interfacial electron-transfer rates and mass-transfer rates.
When the electrode surface is not uniformly accessible, this balance varies over the surface in a way thatis idiosyncratic to the geometry. This is a complicated situation that can be handled in ageneral way (i.e., for an arbitrary shape) by simulation. For UME disks, however, the geometric problem can be simplified by symmetry, and results exist in the literature to facilitate the quantitative analysis of voltammograms (12).5.5.4Applications of Irreversible i-E Curves(a) Information from the Wave HeightExactly as in the reversible case, the plateau of an irreversible or quasireversible wave iscontrolled entirely by diffusion and can be used to determine any variable that contributesto /jj.
The most important applications involve the evaluation of C*, but it is sometimesuseful to determine n, A, D, or TQ from i&. Section 5.4.4(a), which covers these ideas, iswholly applicable to irreversible and quasireversible systems.(b) Information from the Wave Shape and PositionWhen the wave is not reversible, the half-wave potential is not a good estimate of the formal potential and cannot be used directly to determine thermodynamic quantities in themanner discussed in Section 5.4.4. In the case of a totally irreversible system, the waveshape and position can furnish only kinetic information, but quasireversible waves can202 P Chapter 5. Basic Potential Step Methodssometimes provide approximate values of E0' in addition to kinetic parameters.
Becausethe interpretation and information content of a wave's shape and position depends on thekinetic regime, it is essential to be able to diagnose the regime confidently.Wave shape is a useful indicator toward that end, especially if the «-value is known.One can characterize reversibility either by the slope of a plot of E vs. log[(/d - /)//] (the"wave slope") or by the difference \Ещ ~ Ещ\ (the Tomes criterion). Table 5.5.1 provides a summary of expectations for sampled-current voltammetry based either on earlytransients or on the steady state in all three kinetic regimes.
For reversible systems, thesefigures of merit are near 60/n mV at ambient temperatures. Significantly larger figuresoften signal a degree of irreversibility. For example, if the one-step, one-electron mechanism applies and a is between 0.3 and 0.7 (commonly true), then a totally irreversible system would show |£з/4 — Ещ\ between 65 and 150 mV.
Except when a is toward theupper end of the range, such behavior would represent a clear departure from reversibility. Similar effects are seen in wave slopes; however it is not always easy to analyze themprecisely, because wave-slope plots are slightly nonlinear for quasireversible voltammograms and for totally irreversible voltammograms based on early transients. The advantage of the Tomes criterion is that it is always applicable.If the electrode process is more complex than the one-step, one-electron model (e.g.,n > 1 with a rate-determining heterogeneous electron transfer), then the wave shape canbecome extremely difficult to analyze.
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