A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 60
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However, if the collector reconverts a portion of the R to O, then some ofthe regenerated О will diffuse back ("feed back") to the generator where it adds to the fluxarriving from the bulk. Thus the current at the generator becomes larger than it would bewithout activity at the collector. The feedback effect is also useful for diagnosis and quantification of chemical reactions involving О and R.Generation-collection experiments can be carried out in UME arrays aside from thedouble band.
An obvious extension is to use a triple band so that the middle electrodeserves as a generator and the two flanking electrodes work in parallel as collectors. Amore elaborate approach involves an interdigitated array, which is an extensive series ofparallel bands, the alternate members of which are connected in parallel. One of the setsserves as the generator and the other as the collector.For all of these systems, the dynamics are dependent on the widths of the bands andthe gaps between them. Amatore provides a careful review of theory and application (12).i5.10 REFERENCES1.
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Chem., 54,1727 (1982).41. F. С Anson, Anal. Chem., 38, 54 (1966).52. C. Amatore, B. Fosset, J. Bartelt, M. R. Deakin,and R. M. Wightman, /. Electroanal. Chem.,256,255(1988).42. J. H. Christie, R. A. Osteryoung, and F. C.Anson, /. Electroanal. Chem., 13, 236 (1967).52. I. Fritsch-Faules and L. R. Faulkner, Anal.Chem., 64, 1118, 1127(1992).43. J. H Christie, /. Electroanal. Chem., 13,79 (1967).5.11 PROBLEMS5.1 Fick's law for diffusion to a spherical electrode of radius r 0 is writtendCir, i) = DU2C{r, t) + 2 dCir, t)Solve this expression for C(r, t) with the conditionsC(r, 0) = C*,C(r 0 , 0 = 0(t > 0),andlim C(r, t) = C*Show that the current / follows the expressioni = nFADC* i- +[Hint: By making the substitution v(r,t) = rCir, t) in Fick's equation and in the boundary conditions, the problem becomes essentially the same as that for linear diffusion.]2525.2 Given n = 1, C* = 1.00 mM, A = 0.02 cm , and D = 10" cm /s, calculate the current for diffusion-controlled electrolysis at (a) a planar electrode and (b) a spherical electrode (see Problem 5.1)at t = 0.1, 0.5, 1, 2, 3, 5, and 10 s, and as t —» °°.
Plot both / vs. t curves on the same graph. Howlong can the electrolysis proceed before the current at the spherical electrode exceeds that at the planar electrode by 10%?Integrate the Cottrell equation to obtain the total charge consumed in electrolysis at any time,then calculate the value for t = 10 s. Use Faraday's law to obtain the number of moles reacted bythat time. If the total volume of the solution is 10 mL, what fraction of the sample has been alteredby electrolysis?5.3 Consider a diffusion-controlled electrolysis at a hemispherical mercury electrode protruding from aglass mantle. The radius of the mercury surface is 5 jam, and the diameter of the glass mantle is 5mm.
The electroactive species is 1 mM thianthrene in acetonitrile containing 0.1 M tetra-rc-butylammonium fluoborate, and the electrolysis produces the cation radical. The diffusion coefficient is2.7 X 10" 5 cm2/s. Calculate the current at t = 0.1, 0.2, 0.5, 1, 2, 3, 5, and 10 ms, and also at 0.1,5.11 Problems2230.2, 0.5, 1, 2, 3, 5, and 10 s. Do the same for the system under the approximation that linear diffusion applies. Plot the pairs of curves for the short and long time regimes. How long is the linear approximation valid within 10%?5.4 A disk U M E gives a plateau current of 2.32 nA in the steady-state voltammogram for a speciesknown to react with n = 1 and to have a concentration of 1 vaM and a diffusion coefficient of 1.2 X10~ 5 cm 2 /s.
What is the radius of the electrode?5.5 Derive the sampled-current voltammogram for the reduction of a simple metal ion to a metal thatplates out on the electrode. The electrode reaction isM n + + ne ?± M (solid)Assume that the reaction is reversible, and that the activity of solid M is constant and equal to 1.How does Ej/2 vary with /<j? With the concentration of M n + ?5.6 The following measurements were made at 25°C on the reversible sampled-current voltammogramfor the reduction of a metallic complex ion to metal amalgam (n = 2):Concentration of LigandSalt, NaX (M)Em(volts vs.
SCE)0.10-0.4480.50-0.531LOO-0.566(a) Calculate the number of ligands X~ associated with the metal M 2 + in the complex.(b) Calculate the stability constant of the complex, if EЦ2 for the reversible reduction of the simplemetal ion is +0.081 V vs. SCE. Assume that the D values for the complex ion and the metalatom are equal, and that all activity coefficients are unity.5.7(a) Reductions of many organic substances involve the hydrogen ion. Derive the steady-statevoltammogram for the reversible reactionО + рИ++ ne «± Rwhere both О and R are soluble substances, and only О is initially present in solution at a concentration C Q .(b) What experimental procedure would be useful for determining pi5.8(a) Fill in the derivation of (5.5.37) from Pick's laws for spherical diffusion and the appropriateboundary conditions.(b) Derive (5.5.41) by the method used to reach (5.5.37).(c) Show that the following are special cases of (5.5.41):(1) Equation 5.4.17 for early transients in a reversible system having only species О present inthe bulk.(2) Equation 5.4.54 for steady-state currents in a reversible system having only species О present in the bulk.(3) Equation 5.5.11 for early transients in a quasireversible system having only species О present in the bulk.(4) Equation 5.5.12 for early transients in a quasireversible system having both О and R present in the bulk.(5) Equation 5.5.28 for early transients in a totally irreversible system having only species Оpresent in the bulk.5.9 From (5.5.41) derive the steady-state voltammogram at a hemispherical microelectrode for a reversible system containing both О and R in the bulk.
How does it compare with the result for theanalogous situation in Section 1.4.2(b)?5.10Consider the reversible system О + ne ^ R in which both О and R are present initially.(a) From Fick's laws, derive the current-time curve for a step experiment in which the initial potential is the equilibrium potential and the final potential is any arbitrary value E. Assume that a224 !• Chapter 5.
Basic Potential Step Methodsplanar electrode is used and that semi-infinite linear diffusion applies. Derive the shape of thecurrent-potential curve that would be recorded in a sampled-current experiment performed inthe manner described here. What is the value of Ец{1 Does it depend on concentration?(b) Show that the result of (a) is a special case of (5.5.41).5.11 Derive the Tomes criterion for (a) a reversible sampled-current voltammogram based on semi-infinitelinear diffusion, (b) a totally irreversible sampled-current voltammogram based on semi-infinite lineardiffusion, and (c) a totally irreversible steady-state voltammogram.5.12 Derive (5.7.14) and (5.7.15) from (5.7.1)-(5.7.12).5.13 Derive the shape of the sampled-current voltammogram that would be recorded at a stationary platinum microelectrode immersed in a solution containing only I~. The couple:\~ +2e±± 3I~is reversible.
What is the half-wave potential? Does it depend on the bulk concentration of I"? Isthis situation directly comparable to the case О + ne «=^ R?5.14 Calculate kf for the reduction of Cd 2 + to the amalgam from the data in Figure 5.8.4.5.15 Devise a chronocoulometric experiment for measuring the diffusion coefficient of Tl in mercury.5.16 Consider the data in Figures 5.8.1 to 5.8.3. Calculate the diffusion coefficient of DCB. How well dothe slopes of the two lines in Figure 5.8.3 bear out the expectations for a completely stable, reversible system? These data are typical for a solid planar electrode in nonaqueous media.
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