A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 61
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Offer atleast two possible explanations for the slight inequalities in the magnitudes of the slopes and intercepts in Figure 5.8.3.5.17 An ultramicroelectrode has a "recessed disk" shape as shown in Figure 5.11.1. Assume the orificediameter, d0, is 1 jam and the Pt hemisphere diameter is 10 /АШ, with С = 20 jLtm.
Assume the spacewithin the recess fills with the bulk solution in which this tip is immersed (e.g., 0.01 M Ru(NH 3 )g +inO.lMKCl).(a) What magnitude of steady-state (long time) diffusion current, /d, would be found?(b) In using this electrode to study heterogeneous electron transfer kinetics from the steady statewave shape, what value is appropriate for r$lPlatinumGlassFigure 5.11.1 Recessed working electrodecommunicating with the solution through a smallorifice.5.18. The one-electron reduction of a species, O, is carried out at an ultramicroelectrode having a hemispherical shape with r 0 = 5.0 fim. The steady-state voltammogram in a solution containing 10 mMО and supporting electrolyte yields Д£з/4 = Ец2 ~ £3/4 = 35.0 mV, Д £ 1 / 4 = Ещ — Ещ = 31.5mV, and /d = 15 nA. Assume Do = DR and T = 298 K.(a) Find£>o(b) Using the method in reference 33, estimate k° and a.5.11 Problems < 2255.19 G.
Denault, M. Mirkin, and A. J. Bard [/. Electroanal. Chem., 308, 27 (1991)] suggested that bynormalizing the diffusion-limited transient current, /, obtained at an ultramicroelectrode at shorttimes, by the steady-state current, /ss, one can determine the diffusion coefficient, D, without knowledge of the number electrons involved in the electrode reaction, n, or the bulk concentration of thereactant, C*.(a) Consider a disk ultramicroelectrode and derive the appropriate equation for //zss.(b) Why would this procedure not be suitable for a large electrode?(c) A microdisk electrode of radius 13.1 /xm is used to measure D for Ru(bpy)3 inside a polymer]/2+film. The slope of ///ss vs.
Гfor the one-electron oxidation of Ru(bpy>3 is found to be 0.2381/2s (with an intercept of 0.780). Calculate D.2+(d) In the experiment in part (c), / s s = 16.0 nA. What is the concentration of Ru(bpy) in the film?CHA/PTER6POTENTIAL SWEEPMETHODS6.1 INTRODUCTIONThe complete electrochemical behavior of a system can be obtained through a series ofsteps to different potentials with recording of the current-time curves, as described in Sections 5.4 and 5.5, to yield a three-dimensional i-t-E surface (Figure 6.1.1a). However, theaccumulation and analysis of these data can be tedious especially when a stationary electrode is used. Also, it is not easy to recognize the presence of different species (i.e., to observe waves) from the recorded i-t curves alone, and potential steps that are very closelyspaced (e.g., 1 mV apart) are needed for the derivation of well-resolved i-E curves.
Moreinformation can be gained in a single experiment by sweeping the potential with time andrecording the i-E curve directly. This amounts, in a qualitative way, to traversing thethree-dimensional i-t-E realm (Figure 6.1.1b). Usually the potential is varied linearly withtime (i.e., the applied signal is a voltage ramp) with sweep rates v ranging from 10 mV/s( I V traversed in 100 s) to about 1000 V/s with conventional electrodes and up to 106 V/swith UMEs. In this experiment, it is customary to record the current as a function of potential, which is obviously equivalent to recording current versus time.
The formal namefor the method is linear potential sweep chronoamperometry, but most workers refer to itas linear sweep voltammetry (LSV).1(*)(b)Figure 6.1.1 (a) A portion of the i-t-E surface for a nernstian reaction. Potential axis is in units of60//I mV. (b) Linear potential sweep across this surface.
[Reprinted with permission from W. H.Reinmuth, Anal. Chem., 32, 1509 (1960). Copyright 1960, American Chemical Society.]lrThis method has also been called stationary electrode polarography, however, we will adhere to therecommended practice of reserving the term polarography for voltammetric measurements at the DME.2266.1 Introduction -I 227t0(a)EtE°E(ort)(b)(c)Figure 6.1.2 (a) Linear potential sweep or ramp starting at EL (b) Resulting i-E curve,(c) Concentration profiles of A and A1 for potentials beyond the peak.A typical LSV response curve for the anthracene system considered in Section 5.1 isshown in Figure 6.1.2Z?.
If the scan is begun at a potential well positive of E0' for the reduction, only nonfaradaic currents flow for a while. When the electrode potential reachesthe vicinity of E° the reduction begins and current starts to flow. As the potential continues to grow more negative, the surface concentration of anthracene must drop; hence theflux to the surface (and the current) increases. As the potential moves past E° , the surfaceconcentration drops nearly to zero, mass transfer of anthracene to the surface reaches amaximum rate, and then it declines as the depletion effect sets in. The observation istherefore a peaked current-potential curve like that depicted.At this point, the concentration profiles near the electrode are like those shown inFigure 6.1.2c.
Let us consider what happens if we reverse the potential scan (see Figure6.1.3). Suddenly the potential is sweeping in a positive direction, and in the electrode'svicinity there is a large concentration of the oxidizable anion radical of anthracene. As thepotential approaches, then passes, E°', the electrochemical balance at the surface growsmore and more favorable toward the neutral anthracene species. Thus the anion radicalbecomes reoxidized and an anodic current flows. This reversal current has a shape muchlike that of the forward peak for essentially the same reasons.This experiment, which is called cyclic voltammetry (CV), is a reversal technique andis the potential-scan equivalent of double potential step chronoamperometry (Section 5.7).Cyclic voltammetry has become a very popular technique for initial electrochemical studies of new systems and has proven very useful in obtaining information about fairly complicated electrode reactions.
These will be discussed in more detail in Chapter 12.In the next sections, we describe the solution of the diffusion equations with the appropriate boundary conditions for electrode reactions with heterogeneous rate constantsspanning a wide range, and we discuss the observed responses. An analytical approachbased on an integral equation is used here, because it has been widely applied to thesetypes of problems and shows directly how the current is affected by different experimenA + e - » Ae(a)Figure 6.1.3(b)(a) Cyclic potential sweep, (b) Resulting cyclic voltammogram.228Chapter 6. Potential Sweep Methodstal variables (e.g., scan rate and concentration). However, in most cases, particularlywhen the overall reactions are complicated by coupled homogeneous reactions (Chapter12), digital simulation methods (Appendix B) are used to calculate voltammograms.• 6.2 NERNSTIAN (REVERSIBLE) SYSTEMS6.2.1Solution of the Boundary Value ProblemWe consider again the reaction О + ne «^ R, assuming semi-infinite linear diffusion and asolution initially containing only species O, with the electrode held initially at a potential£j, where no electrode reaction occurs.
These initial conditions are identical to those inSection 5.4.1. The potential is swept linearly at v (V/s) so that the potential at any time isE(t) =E{-vt(6.2.1)If we can assume that the rate of electron transfer is rapid at the electrode surface, so thatspecies О and R immediately adjust to the ratio dictated by the Nernst equation, then theequations of Section 5.4, that is, (5.4.2)-(5.4.6), still apply.
However, (5.4.6) must be recognized as having a time-dependent form:The time dependence is significant, because the Laplace transformation of (6.2.2) cannotbe obtained as it could in deriving (5.4.13),2 and the mathematics for sweep experimentsare greatly complicated as a consequence. The problem was first considered by Randies(1) and Sevcik (2); the treatment and notation here follow the later work of Nicholson andShain (3).
The boundary condition (6.2.2) can be written(6.2.3)^Mwhere S(t) = e~at, в = exp[(nfIRT){E{ - £ 0 ')], and a = (nF/RT)v. As before (see Section 5.4.1), Laplace transformation of the diffusion equations and application of the initialand semi-infinite conditions leads to [see (5.4.7)]Co(x, s) = ~^ + A(s) exp| - ( -±- )x|(6.2.4)The transform of the current is given by [see (5.2.9)].«л(6.2.5)Combining this with (6.2.4) and inverting, by making use of the convolution theorem (seeAppendix A), we obtain3C o (0, t) = C% - [nFA(7rDo)l/2rlf' Kr)(t - тУ1/2 dr(6.2.6)2The Laplace transform of C o (0, 0 = 0CR(O, t) is C o (0, s) = в C R (0, s) only when в is not a function of time; itis only under this condition that в can be removed from the Laplace integral.3This derivation is left as an exercise for the reader (see Problem 6.1).
Equation 6.2.6 is often a useful startingpoint in other electrochemical treatments involving semi-infinite linear diffusion, т in the integral is a dummyvariable that is lost when the definite integral is evaluated.6.2 Nemstian (Reversible) Systems229By letting(6.2.7)nFA(6.2.6) can be writtenCo(0, 0 = Cg - (тг£>оГ1/2 Гдт)(Г •'о(6.2.8)Similarly from (5.4.12) an expression for CR(0, ?)can be obtained (assuming R is initiallyabsent):C R (0, f) =(6.2.9)The derivation of (6.2.8) and (6.2.9) employed only the linear diffusion equations,initial conditions, semi-infinite conditions, and the flux balance.
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