A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 48
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The corresponding expression of Fick's second law(see Table 4.4.2) is:I и t n ( / , I)dt1ик^глТ, I)Щu (-"0Vwhere r describes radial position normal to the axis of symmetry, and z is the positionalong the length. Since we normally assume uniformity along the length of the cylinder,дС/dz = d2C/dz2 = 0, and z drops out of the problem. The boundary conditions are exactlyIn practice, it would be difficult to achieve a diffusion layer as thick as 500 jitm, because convection wouldnormally begin to manifest itself before 60 s.By analogy to the rigorous result for the spherical system, one can estimate the current at the disk as the simplelinear combination of the Cottrell and steady-state terms:This approximation is accurate at the short-time and long-time limits and deviates from the Aoki-Osteryoungresult by only a few percent in the range of Figure 5.3.2b.
The largest error ( — h 7%) is near т = 1, as onewould expect.5.3 Diffusion-Controlled Currents at Ultramicroelectrodes175those used in solving the spherical case (see Section 5.2.2), and the result is available inthe literature (10)A practical approximation, reported by Szabo et al. (24) to be valid within 1.3%, is~2ехр(-0.05тг1/2 т 1/2 )i =7T1/2T1/21ln(5.29450.7493T1/2(5.3.13)where r = 4Dot/rQ. In the short-time limit, when т is small, only the first term of (5.3.13)is important and the exponential approaches unity.
Thus (5.3.13) reduces to the Cottrellequation, (5.2.11), as expected for the situation where the diffusion length is small compared to the curvature of the electrode. In fact, the deviation from the Cottrell current resulting from the cylindrical diffusion field does not become as great as 4% until т reaches—0.01, where the diffusion layer thickness has become about 10% of r 0 .In the long-time limit, when т becomes very large, the first term in (5.3.13) dies awaycompletely, and the logarithmic function in the denominator of the second term approaches In т 1/2 . Thus the current becomes'qss =2nFADoC%rolnr(5.3.14)Because this relationship contains r, the current depends on time; therefore it is not asteady-state limit such as we found for the sphere and the disk.
Even so, time appears onlyas an inverse logarithmic function, so that the current declines rather slowly in the longtime limit. It can still be used experimentally in much the same way that steady-state currents are exploited at disks and spheres. In the literature, this case is sometimes called thequasi-steady state.(d) BandUMEIn the same way that a disk electrode is a two-dimensional diffusion system behavingvery much like the simpler, one-dimensional, hemispherical case, a band electrode is atwo-dimensional system behaving much like the simpler hemicylindrical system.
The coordinate system used to treat diffusion at the band is shown in Figure 5.3.3. At shorttimes, the current converges, as we now expect, to the Cottrell form, (5.2.11). At longtimes, the current-time relationship approaches the limiting form,qsswln(64Dot/w2)(5.3.15)Thus, the band UME also does not provide a true steady-state current at long times.z axis (x = 0)Flux intomantle .a 0Inlaid UME band( / » w)Mantle in z = 0 plane (extendsbeyond diffusion layer boundary)Figure 5.3.3 Diffusionalgeometry at a bandelectrode.
Normally thelength of the electrode isvery much larger than thewidth, and the threedimensional diffusion at theends does not appreciablyviolate the assumption thatdiffusion occurs only alongthe x and z axes.176Chapter 5. Basic Potential Step MethodsTABLE 53.1 Form of mo for UMEs of Different GeometriesCylinder6Band"2D 02TTD02w\n(64Dot/w ] )a5.3.3r lnToDiskHemisphereSphere4D0£>oDorrr0Г°Long-time limit is to a quasi-steady state.S u m m a r y of Behavior at UltramicroelectrodesAlthough there are some important differences in the behavior of UMEs with differentshapes, it is useful here to recollect some common features in the responses to a largeamplitude potential step:First, at short times, where the diffusion-layer thickness is small compared to the critical dimension, the current at any UME follows the Cottrell equation, (5.2.11), and semiinfinite linear diffusion applies.Second, at long times, where the diffusion-layer thickness is large compared to thecritical dimension, the current at any UME approaches a steady state or a quasi-steadystate.
One can write the current in this limit in the manner developed empirically in Section 1.4.2,i s s = nFAmoC%(5.3.16)where m® is a mass-transfer coefficient. The functional form of YHQ depends on geometryas given in Table 5.3.1.In practical experiments with UMEs, one normally tries to control the experimentalconditions so that the electrode is operating either in the short-time regime (called theearly transient regime or the regime of semi-infinite linear diffusion in the remainder ofthis book) or in the long-time limit (called the steady-state regime).
The transition regionbetween these two limiting regimes involves much more complicated theory and offers noadvantage, so we will not be considering it in much detail.У 5.4 SAMPLED-CURRENT VOLTAMMETRY FOR REVERSIBLEELECTRODE REACTIONSThe basic experimental methodology for sampled-current voltammetry is described inSection 5.1.1, especially in the text surrounding Figure 5.1.3. After studying the diffusion-controlled responses to potential steps in Sections 5.2 and 5.3, we now understandthat the result of a sampled-current experiment might depend on whether the sampling occurs in the time regime where a transient current flows or in the later period, when asteady-state could be reached. This idea leads us to consider the two modes separately forreversible chemistry in Sections 5.4.1 and 5.4.2 below.
Applications of reversible voltammograms are then treated in Section 5.4.4.5.4.1 Voltammetry Based on Linear Diffusion at a Planar Electrode(a) A Step to an Arbitrary PotentialConsider again the reaction О + ие «^ R in a Cottrell-like experiment at an electrodewhere semi-infinite linear diffusion applies,5 but this time let us treat potential steps of5It is most natural to think of this experiment as taking place at a planar electrode, but as shown in Sections5.2.2 and 5.3, the required condition is realized with any electrode shape as long as the diffusion layer thicknessremains small compared to the radius of curvature of the electrode.5.4 Sampled-Current Voltammetry for Reversible Electrode Reactions177any magnitude. We begin each experiment at a potential at which no current flows; and att = 0, we change E instantaneously to a value anywhere on the reduction wave.
We assume here that charge-transfer kinetics are very rapid, so thatalways.The equations governing this case are 6dCo(x,t)_2d2Co(x,t))lim Co(x, i) = С о(5.4.3)lirn^ CR(x, t) = 0(5.4.4)and the flux balance is= 0(5.4.5)It is convenient to rewrite (5.4.1) as(5.4.6)In Section 5.2.1, we saw that application of the Laplace transform to (5.4.2) and consideration of conditions (5.4.3) and (5.4.4) would yieldcCo(x, s) = -f- + A(s) e~CR(JC, s) = B(s) e~V^x(5.4.8)Transformation of (5.4.5) givesx s)\/дСт>(х s)\which can be simplified by evaluating the derivatives from (5.4.7) and (5.4.8):-A(s) D\? sm - B(s) D^2 sl/2 = 0l/2Thus, В = —A(s)^, where f = (Do/DR) .tion, (5.4.1); hence our results:(5.4.10)So far we have not invoked the Nernst rela-Co(x, s) = ^- + A(s) e-{slD°)l'2x(5.4.11)x, s) = -A(s)£ e~(s/D^/2x(5.4.12)hold for any i-E characteristic.
We will make use of this fact in Section 5.5.We introduce_the assumption of reversibility to evaluate A(s). Transformation of(5.4.6) shows that C o (0, s) = 0CR(O, s); thusC*6(5.4.13)Clearly, (5.4.3) implies that R is initially absent. The case for CR(x, 0) = CR, follows analogously, and is leftas Problem 5.10.178;Chapter 5. Basic Potential Step Methodsand A(s) = -CQ/S(1+ £0). The transformed profiles are then1/2 vCo(x, s) = —(5.4.14)C R (x, s) =(5.4.15)Equation 5.4.14 differs from (5.2.7) only by the factor 1/(1 + £0) in the second term.Since (1 + £0) is independent of x and t, the current can be obtained exactly as in thetreatment of the Cottrell experiment by evaluating i(s) and then inverting:(5.4.16)This relation is the general response function for a step experiment in a reversiblesystem.
The Cottrell equation, (5.2.11), is a special case for the diffusion-limited region,which requires a very negative E - E°\ so that 0 —> 0. It is convenient to represent theCottrell current as id(t) and to rewrite (5.4.16) asid(t)W = 1 +£0(5.4.17)Now we see that for a reversible couple, every current-time curve has the same shape; butits magnitude is scaled by 1/(1 + £0) according to the potential to which the step is made.For very positive potentials (relative to £ ° ) , this scale factor is zero; thus i(t) has a valuebetween zero and id(t), depending on E, as sketched in Figure 5.1.3.(b) Shape of the Current-Potential CurveIn sampled-current voltammetry, our goal is to obtain an i(r)-E curve by (a) performingseveral step experiments with different final potentials E, (b) sampling the current response at a fixed time т after the step, and (c) plotting /(т) vs. E.
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