A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 43
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Electroanal. Chem.,352, 83 (1993).21. H. C. Berg, "Random Walks in Biology," Princeton University, Press, Princeton, NJ, 1983.32. A. A. Moya, J. Castilla, and J. Homo, /. Phys.Chem., 99, 1292 (1995).4.6 PROBLEMS4.1 Consider the electrolysis of a 0.10 M NaOH solution at platinum electrodes, where the reactions(anode) 2OH" -> \O2 + H 2 O + 2e(cathode) 2H2O + 2e -* H 2 + 2OH"Show the balance sheet for the system operating at steady state. Assume 20e are .passed in theexternal circuit per unit time, and use the AQ values in Table 2.3.2 to estimate transference numbers.4.2 Consider the electrolysis of a solution containing 10" l M Fe(ClO 4 ) 3 and 10" 1 M Fe(C10 4 ) 2 at platinum electrodes:(anode) F e 2 + ^ F e 3 + + e(cathode) F e 3 + + e -> F e 2 +Assume that both salts are completely dissociated, that the Л values for F e 3 + , F e 2 + , and СЮ4" areequal, and that lOe are passed in the external circuit per unit time.
Show the balance sheet for thesteady-state operation of this system.4.3 For a given electrochemical system to be described by equations involving semi-infinite boundaryconditions, the cell wall must be at least five "diffusion layer thicknesses" away from the electrode.For a substance with D = 10~5 cm2/s, what distance between the working electrode and the cellwall is required for a 100-s experiment?4.6 Problems1554.4 The mobility, u}, is related to the diffusion coefficient, Dy by equation 4.2.2. (a) From the mobility++data in Table 2.3.2, estimate the diffusion coefficients of H , I~, and Li at 25°C. (b) Write theequation for the estimation of D from the Л value.4.5 Using the procedure of Section 4.4.2, derive Fick's second law for spherical diffusion (equation4.4.18).
[Hint: Because of the different areas through which diffusion occurs at r and at r + dr, it ismore convenient to obtain the change of concentration in dr by considering the number of molesdiffusing per second rather than the flux.]С Н AfcT E R5BASIC POTENTIALSTEP METHODSThe next three chapters are concerned with methods in which the electrode potential isforced to adhere to a known program. The potential may be held constant or may be varied with time in a predetermined manner as the current is measured as a function of timeor potential.
In this chapter, we will consider systems in which the mass transport of electroactive species occurs only by diffusion. Also, we will restrict our view to methods involving only step-functional changes in the working electrode potential. This family oftechniques is the largest single group, and it contains some of the most powerful experimental approaches available to electrochemistry.In the methods covered in this chapter, as well as in Chapters 6 and 7, the electrodearea, A, is small enough, and the solution volume, V, is large enough, that the passage ofcurrent does not alter the bulk concentrations of electroactive species.
Such circumstancesare known as small AIV conditions. It is easy to show on the basis of results below thatelectrodes with dimensions of several millimeters operating in solutions of 10 mL or moredo not consume a significant fraction of a dissolved electroactive species in experimentslasting a few seconds to a few minutes (Problem 5.2). Several decades ago, Laitinen andKolthoff (1, 2) invented the term microelectrode to describe the electrode's role undersmall A/V conditions, which is to probe a system, rather than to effect compositionalchange.1 In Chapter 11, we will explore large AIV conditions, where the electrode is intended to transform the bulk system.5.1 OVERVIEW OF STEP EXPERIMENTS5.1.1Types of TechniquesFigure 5.1.1 is a picture of the basic experimental system.
An instrument known as apotentiostat has control of the voltage across the working electrode-counter electrodepair, and it adjusts this voltage to maintain the potential difference between the workingand reference electrodes (which it senses through a high-impedance feedback loop) inaccord with the program defined by a function generator.
One can view the potentiostatdecent years have seen the rapid development of extremely small working electrodes, of dimensions in themicrometer or nanometer range, which have a set of very useful properties. In much of the literature and incasual conversation, these are also called "microelectrodes," in reference to their dimensions. They alwaysprovide small A/V conditions, so they are indeed microelectrodes within the definition given above, but muchlarger electrodes also belong to the class. To preserve the usefulness of the earlier term, very small electrodeshave been called ultramicroelectrodes (see Section 5.3).
That distinction is respected consistently in theremainder of this book, although it now seems likely that the new usage of the term "microelectrode" will soondisplace the historic one altogether.1565.1 Overview of Step ExperimentsFunctiongenerator157PotentiostatE controlledi(t) measuredFigure 5.1.1Experimental arrangementfor controlled-potentialexperiments.alternatively as an active element whose job is to force through the working electrodewhatever current is required to achieve the desired potential at any time. Since the current and the potential are related functionally, that current is unique.
Chemically, it isthe flow of electrons needed to support the active electrochemical processes at ratesconsistent with the potential. Thus the response from the potentiostat (the current) actually is the experimental observable. For an introduction to the design of such apparatus,see Chapter 15.Figure 5.1.2a is a diagram of the waveform applied in a basic potential step experiment. Let us consider its effect on the interface between a solid electrode and an unstirredsolution containing an electroactive species.
As an example, take anthracene in deoxygenated dimethylformamide (DMF). We know that there generally is a potential regionwhere faradaic processes do not occur; let E\ be in this region. On the other hand, we canalso find a more negative potential at which the kinetics for reduction of anthracene become so rapid that no anthracene can coexist with the electrode, and its surface concentration goes nearly to zero. Consider E2 to be in this "mass-transfer-limited" region. What isthe response of the system to the step perturbation?First, the electrode must reduce the nearby anthracene to the stable anion radical:(5.1.1)An + e —> An*This event requires a very large current, because it occurs instantly.
Current flows subsequently to maintain the fully reduced condition at the electrode surface. The initial reduction has created a concentration gradient that in turn produces a continuing flux ofanthracene to the electrode surface. Since this arriving material cannot coexist with theelectrode at E2, it must be eliminated by reduction. The flux of anthracene, hence the cur--елt3 > t2 > f1 > 0j_0(a)t0x(b)0(c)Figure 5.1.2 (a) Waveform for a step experiment in which species О is electroinactive at E\, butis reduced at a diffusion-limited rate at Ei- (b) Concentration profiles for various times into theexperiment, (c) Current flow vs. time.t158Chapter 5.
Basic Potential Step Methodsrent as well, is proportional to the concentration gradient at the electrode surface. Note,however, that the continued anthracene flux causes the zone of anthracene depletion tothicken; thus the slope of the concentration profile at the surface declines with time, andso does the current. Both of these effects are depicted in Figures 5.1.2b and 5.1.2c.
Thiskind of experiment is called chronoamperometry, because current is recorded as a function of time.Suppose we now consider a series of step experiments in the anthracene solution discussed earlier. Between each experiment the solution is stirred, so that the initial conditions are always the same. Similarly, the initial potential (before the step) is chosen to beat a constant value where no faradaic processes occur. The change from experiment to experiment is in the step potential, as depicted in Figure 5.1.3a. Suppose, further, that experiment 1 involves a step to a potential at which anthracene is not yet electroactive; thatexperiments 2 and 3 involve potentials where anthracene is reduced, but not so effectivelythat its surface concentration is zero; and that experiments 4 and 5 have step potentials inthe mass-transfer-limited region. Obviously experiment 1 yields no faradaic current, andexperiments 4 and 5 yield the same current obtained in the chronoamperometric caseabove.
In both 4 and 5, the surface concentration is zero; hence anthracene arrives as fastas diffusion can bring it, and the current is limited by this factor. Once the electrode potential becomes so extreme that this condition applies, the potential no longer influencesthe electrolytic current. In experiments 2 and 3 the story is different because the reductionprocess is not so dominant that some anthracene cannot coexist with the electrode.
Still,its concentration is less than the bulk value, so anthracene does diffuse to the surfacewhere it must be eliminated by reduction. Since the difference between the bulk and surface concentrations is smaller than in the mass-transfer-limited case, less material arrivesat the surface per unit time, and the currents for corresponding times are smaller than inexperiments 4 and 5. Nonetheless, the depletion effect still applies, which means that thecurrent still decays with time.Now suppose we sample the current at some fixed time т into each of these stepexperiments; then we can plot the sampled current, /(т), vs. the potential to which thestep takes place.
As shown in Figures 5.1.3b and 5.1.3c, the current-potential curve hasa wave shape much like that encountered in earlier considerations of steady-statevoltammetry under convective conditions (Section 1.4.2). This kind of experiment iscalled sampled-current voltammetry, several forms of which are in common practice.The simplest, usually operating exactly as described above, is called normal pulsevoltammetry. In this chapter, we will consider sampled-current voltammetry in a general way, with the aim of establishing concepts that apply across a broad range of par-E\4 5(a)(b)(c)Figure 5.1.3 Sampled-current voltammetry.
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