A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 49
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Here we consider theshape of this curve for a reversible couple and the kinds of information one can obtainfrom it.Equation 5.4.17 really answers the question for us. For a fixed sampling time т,(5.4.18)which can be rewritten as£0 =(5.4.19)and expanded:^1/2A1/2RT,—pinnF(5.4.20)When г(т) = /d(T)/2, the current ratio becomes unity so that the third term vanishes.
Thepotential for which this is so is £ 1/2 , the half-wave potential:(5.4.21)5.4 Sampled-Current Voltammetry for Reversible Electrode Reactions1791-'d\1id(x)/2ла-ы-НТТ I i I i I i I i i i i 1Y i i i 1 i M i l l ' ^1 1 1i ii150100500-50 -100 -150 -200 -250{E-E°')NFigure 5.4.1 Characteristics of a reversiblewave in sampled-current voltammetry.This curve is for n = 1, T = 298 K, andDo = DRP- Because Do Ф DR, El/2 differsslightly from E°\ in this case by about 9 mV.For n > 1, the wave rises more sharply to theplateau (see Figure 5.4.2).and (5.4.20) is often written(5.4.22)These equations describe the voltammogram for a reversible system in sampled-currentvoltammetry as long as semi-infinite linear diffusion holds.
It is interesting to compare(5.4.20) and (5.4.22) with the wave shape equations derived in a naive way for steadystate voltammetry in Section 1.4.2(a). They are identical in form.As shown in Figure 5.4.1, these relations predict a wave that rises from baseline tothe diffusion-controlled limit over a fairly narrow potential region (—200 mV) centeredon Ey2. Since the ratio of diffusion coefficients in (5.4.21) is nearly unity in almost anycase, Ещ is usually a very good approximation to E0' for a reversible couple.Note also that E vs.
log [(/d - i)/i] should be linear with a slope of 2303RT/nF or59Л/п mV at 25°C. This "wave slope" is often computed for experimental data to test forreversibility. A quicker test [the Tomes criterion (25)] is that \Ey4 — Ещ\ = 56A/n mV at25°C. The potentials Ey4 and Ещ are those for which / = 3/d/4 and / = /d/4, respectively.If the wave slope or the Tomes criterion significantly exceeds the expected values, thesystem is not reversible. (See also Section 5.5.4).(c) Concentration ProfilesTaking the inverse transforms of (5.4.14) and (5.4.15) yields the concentration profiles:Co(x, t) = Cg1/2J(5.4.23)(5.4.24)Some other convenient equations relating to concentrations can also be written. Letus solve for^4(s) and B(s)jn (5.4.7) and (5.4.8) in terms of the transformed surface concentrations C o (0, s), and C R (0, s), then substitute into (5.4.10):Co(0, s) - -£•(5.4.25)or, using the inverse transform,,0,0=(5.4.26)180Chapter 5.
Basic Potential Step MethodsThe more general relation for R initially present isDl^2Co(0, t) + /^ / 2 C R (0, t) = C%D\>2 +(5.4.27)For the special case when Do = £)R,C o (0, t) + CR(0, t) = C% + C%(5.4.28)Equations 5.4.26 to 5.4.28 were derived without reference to the sixth boundary conditionin the diffusion problem; hence they do not depend on any particular electrochemical perturbation or i-E function, and they hold for virtually any electrochemical method. Theprincipal assumptions are that semi-infinite linear diffusion applies and that О and R aresoluble, stable species.7Returning now to the step experiments for which (5.4.23) and (5.4.24) apply, we seethat the surface concentrations are(5.4.29)C R (0, 0 = C%(5.4.30)Since (5.4.17) shows that i(t)/id(t) = 1/(1 + £0,.1с o(0, 0сЛ£c*«xo°w(5.4.31)(5.4.32)We will use these relations in Section 5.4.3 to simplify the interpretation of reversiblesampled-current voltammograms in various chemical situations.
The reader interested in aquick view of applications can proceed directly to that point and beyond. However, a fullview of reversible waves needs to include those recorded by sampling steady-state currents, so the next section is devoted to that topic.5.4.2 Steady-State Voltammetry at a UME(a) A Step to an Arbitrary Potential at a Spherical ElectrodeLet us consider again the reaction О + ne ±± R in an experiment involving a step of anymagnitude, but in contrast to the limitations of the previous section, let us allow the experiment to proceed beyond the regime where semi-infinite linear diffusion applies.
For themoment let us also restrict the electrode geometry to a sphere or hemisphere of radius r 0 .Species О is present in the bulk, but R is absent. We begin each experiment at a potentialat which no current flows; and at t = 0, we change E instantaneously to a value anywhereon the reduction wave.The governing equations are<?Co(r, 0 _72/d Co(r, i)2 <?C0(r, 0\(5.4.33)Note also that for the step experiments under discussion, (5.4.23) and (5.4.24) show that C0(x, t) + CR(x, t) =C* at any point along the profiles, when Do = DR.5.4 Sampled-Current Voltammetry for Reversible Electrode Reactions >\ 181дг=/^м+^ \ys,\(5А34)/Co(r, 0) = С%Hm C o (r, 0 - CgCR(r9 0) = 0(5.4.35)lim CR(r, 0 = 0= 0(5.4.36)(5.4.37)By the method addressed in Problem 5.1, one can show that (5.4.33)-(5.4.36) togetheryield the general solutionsCo(r, s) = ^+^e-WV^(5.4.39)(5.4.40)Transformation and application of the flux balance, (5.4.37), in the same manner used inSection 5.4.1 givewhere ^ = (Do/DR)m!f^W61*1 +r0(s/Do)l/21 +ro(s/DR)l/2(5A41)andWe have not yet called upon reversibility; hence (5.4.41) and (5.4.42) hold for any i-Echaracteristic.By assuming reversibility and applying condition (5.4.38), we evaluate A(s) essentially in the same way as in Section 5.4.1.
The result is{тТ^е)^еШо)1Пг°(5А44)so that the transformed profiles are( 5 A 4 6 )The current is obtained from the slope of the concentration profiles at the electrodesurface, for example,i(t) = nFADo(°V\/r=ro(5.4.47)182 • Chapter 5. Basic Potential Step Methodswhich can be transformed on t to give= nFADo/dCa(r, s)(5.4.48)By allowing (5.4.48) to operate on (5.4.45), we obtain the transform of the current-timerelationship,j_rwhich can be usefully reexpressed asEquation 5.4.50 describes current flow at the sphere in all time domains, includingthe early transient and steady-state regimes. Unfortunately, the complete current transform is not readily inverted to produce a closed-form result, because у is a complex function of s. Still, one can develop useful results by considering limiting cases. Our mainconcern here is in distinguishing the early transient and steady-state limits, which can bedone by recognizing the role of r0(s/Do)l/2 in (5.4.43) and (5.4.50).
The transform variable s has units of frequency [e.g., s" 1 ] and is, in fact, an alternate representation of timein the experiment. Thus (D o /s) 1 / 2 has units of length, and r0(s/Do)l/2 relates the radius ofcurvature of the electrode to the diffusion layer thickness.When r0(s/Do)l/2 »1, the diffusion layer is thin compared to r 0 , and the system isin the early transient regime, where linear diffusion applies. Then the parenthesized factorin (5.4.50) collapses to s~ 1 / 2 D o ~ 1 / 2 and у -» l/£, so that(5.4.51)i(s) =which is readily inverted to produce (5.4.16), as required.
Section 5.4.1 fully covers theconsequences of this case.On the other hand, when r0(s/Z)o)1/2 < < 1» the diffusion layer thickness greatly exceeds r 0 and the system is in the steady-state regime. By inspection, one sees that у —> 1and the parenthesized factor in (5.4.50) becomes l/ros, so thati(s) =nFADoCo(5.4.52)which is easily inverted to the steady-state analogue of (5.4.16),(5.4.53)This relation is the general response function for a step experiment in a reversible system when the sampling of current occurs in the steady-state regime.
Thesteady-state limiting current, (5.2.21) or (5.3.2), is the special case for the diffusionlimited region, where в —> 0. Let us represent this limiting current as /^ and rewrite(5.4.53) as(5.4.54)5.4 Sampled-Current Voltammetry for Reversible Electrode Reactions183This result is analogous to (5.4.17) and has essentially the same interpretation. The important difference in behavior at the steady state is that the key relations depend onthe first power of diffusion coefficients, rather than on their square roots.
This effect isseen in the numerator of (5.4.53) vs. that of (5.4.16) and also in the appearance of£ 2 = (DO/DR) in (5.4.53) and (5.4.54) vs. | in the analogous relations (5.4.16) and(5.4.17). The factor 1/(1 + £20) has a value between zero (for very positive potentials relative to E°) and unity (for very negative potentials); thus / has a value between zero and> much like the representation in Figure 5.1.3.(b) Conditions for Recording Steady-State VoltammogramsIn conception, sampled-current voltammetry involves the recording of an i(r)~E curveby the application of a series of steps to different final potentials E. The current issampled at a fixed time r after the step, then /(r) is plotted vs. E.
This defining protocol can be relaxed considerably when sampling occurs in the steady-state regime.Since the current is independent of time, it does not matter when sampling occurs orhow precisely the sampling time is controlled. If the system is chemically reversible, italso does not matter how the steady-state was reached. One need not reinitialize thesystem after each step; thus the potential can be taken directly from step value to stepvalue as long as the system has enough time to establish the new steady state beforesampling occurs.Actually one need not even apply steps.
It is satisfactory to change the potential linearly with time and to record the current continuously, as long as the rate of change issmall compared to the rate of adjustment in the steady state. Section 6.2.3 contains a discussion of the required conditions in more quantitative terms. Virtually all "sampled current voltammetry" at UMEs is carried out experimentally in this linear-sweep form, butthe results are the same as if a normal sampled-current voltammetric protocol were employed, except with respect to the charging-current background [see Sections 6.2.4 and7.3.2(c)].(c) Shape of the WaveBy rearranging (5.4.54), one derives the reversible steady-state voltammogram as(5.4.55)This equation has the familiar form seen in (5.4.22), but the half-wave potential differsfrom that defined in (5.4.21), because the second term contains the first power of the diffusion coefficients, rather than the square root.(5.4.56)Thus, the shape of the reversible steady-state sampled-current voltammogram is identical to that of the reversible early-transient sampled-current voltammogram (Figure5.4.1), and the comments made about wave shape in Section 5.4.1(b) also apply in thesteady-state case.
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