A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 65
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Then one continues the scan and measures V from the potential axis as a baseline (Figure 6.6.3). The application of this method requires convection-free conditions(quiet, vibration-free solutions, and shielded electrodes oriented to prevent convectionfrom density gradients; see Section 8.3.5), because the waiting time for the current decaymust be 20 to 50 times the time needed to traverse a peak.For the stepwise reduction of a single substance O, that is, О + ще —> Rj (£?),Ri + n2e —» R2 (Е?), the situation is similar to the two-component case, but morecomplicated.
If E\ and E\ are well separated, with E\ > E2 (i.e., О reduces before R^,then two separate waves are observed. The first wave corresponds to reduction of О toRi in an ri\-electron reaction, with Rj diffusing into the solution as the wave is traversed.At the second wave, О continues to be reduced, either directly at the electrode or by reaction with R2 diffusing away from the electrode (Oj + R2 —> 2Rj), in an overall {ri\ +/22)-electron reaction, and R^ diffuses back toward the electrode to be reduced in an n2electron reaction. The voltammogram for this case resembles that of Figure 6.6.1.In general the nature of the i-E curve depends on Д£° (= E2 ~ E\), the reversibilityof each step, and щ and n2. Calculated cyclic voltammograms for different values of AE°in a system with two one-electron steps are shown in Figure 6.6.4. When AE° is between0 and -100 mV, the individual waves are merged into a broad wave whose Ep is independent of scan rate.
When AE° = 0, a single peak with a peak current intermediate betweenthose of single-step le and 2e reactions is found, and Ep - Ep/2 = 21 mV. For AE° > 180mV (i.e., the second step is easier than the first), a single wave characteristic of a reversible 2e reduction (O + 2e ±± R 2 ) is observed (i.e., Д £ р = 23RT/2F).~ 20 to 50 t100-100-200E-300-400Figure 6.6.3 Method ofallowing current of first wave todecay before scanning secondwave.
Upper curve: potentialprogram. Lower curve: resultingvoltammogram. System as inFigure 6.6.1.;246 <Chapter 6. Potential Sweep Methods1.00.50.0-0.51d1.0III(b)IIII(d)Figure 6.6.4 Cyclic voltammogramsfor a reversible two-step system at25°C. Current function is analogous toX(z) defined in (6.2.16). n2lnx = 1.0.(a) A£° = - 1 8 0 mV. (b) AE° = - 9 0mV. (c) AE° = 0 mV. (d) AE° = 180mV. [Reprinted with permission fromD.
S. Polcyn and I. Shain, Anal. Chem.,38, 370 (1966), Copyright 1966,American Chemical Society.]0.50.0-0.5JI-200200A case of particular interest occurs when AE° = -(2RT/F) In 2 = -35.6 mV (25°C).This AE° occurs when there is no interaction between the reducible groups on O, and theadditional difficulty in adding the second electron arises purely from statistical (entropic)factors (17). Under these conditions the observed wave has all of the shape characteristicsof a one-electron transfer even though it is actually the result of two merged one-electrontransfers. This same concept can be extended to the reduction of molecules containing кequivalent, noninteracting, reducible centers (e.g., reducible polymers).
For this case theA£° between the first and kth electron transfers is given byoFo_(2RT(6.6.1)and again the reduction wave, now involving к merged waves, appears like a single oneelectron wave with respect to shape, even though the height corresponds to a ^-electronprocess (18). From these considerations it is clear that for two one-electron transfer reactions, AE° values more positive than -(2RT/F) In 2 represent positive interactions(i.e., the second electron transfer is assisted by the first), while A£° values more negative than —(2RT/F) In 2 represent negative interactions. Step wise electron transfers(EE-reactions) are discussed in more detail in Sections 12.3.6 and 12.3.7; see also Polcyn and Shain (16).Linear sweep and cyclic voltammetric methods have been employed for numerousbasic studies of electrochemical systems and for analytical purposes.
For example, thetechnique can be used for in vivo monitoring of substances in the kidney or brain (19); atypical example that employed a miniature carbon paste electrode to study ascorbic acidin a rat brain is illustrated in Figure 6.6.5. These techniques are especially powerful tools2476.7 Convolutive or Semi-Integral Techniquesvvv—w|__гГXRef1Л> Stainless steelOE(t)E vs. AQ/AQCI+0.61+0.4""Skull^lAg/AgCIМММII^t- 5><^-0.2+0.2ipe resп+ .^^\Teflon1Ns.\N ^ X Y 3rd Scan47 2nd Scan4A 1st ScanCarbon paste- 10ECOAnodic(oxidation)current1- 15(b)(a)Figure 6.6.5 Application of cyclic voltammetry to in vivo analysis in brain tissue, (a) Carbonpaste working electrode, stainless steel auxiliary electrode (18-gauge cannula), Ag/AgCl referenceelectrode, and other apparatus for voltammetric measurements, (b) Cyclic voltammogram for ascorbicacid oxidation at C-paste electrode positioned in the caudate nucleus of an anesthetizedrat.
[From P. T. Kissinger, J. B. Hart, and R. N. Adams, Brain Res., 55,20 (1973), with permission.]in the study of electrode reaction mechanisms (Chapter 12) and of adsorbed species(Chapter 14).6.7 CONVOLUTIVE OR SEMI-INTEGRAL TECHNIQUES6.7.1 Principles and DefinitionsBy proper treatment of the linear potential sweep data, the voltammetric i-E (or i-i) curvescan be transformed into forms, closely resembling the steady-state voltammetric curves,which are frequently more convenient for further data processing.
This transformationmakes use of the convolution principle, (A.1.21), and has been facilitated by the availability of digital computers for the processing and acquisition of data. The solution of the diffusion equation for semi-infinite linear diffusion conditions and for species О initiallypresent at a concentration CQ yields, for any electrochemical technique, the following expression (see equations 6.2.4 to 6.2.6):(6.7.1)If the term in brackets, which represents a particular (convolutive) transformation of theexperimental /(0 data, is defined as /(r), then equation 6.7.1 becomes (20)C o (0, t) =(6.7.2)where(6.7.3)248;Chapter 6. Potential Sweep MethodsFollowing the generalized definition of the Riemann-Liouville operators, this integralcan be considered as the semi-integral of /(0, generated by the operator d~ll2ldt~112, sothat (21, 22)л-1/2^—-1 / 2 i(t) = m{f) = I(t)dt(6.7.4)Both m(t) and I(f), which represent the integral in equation 6.7.3, have been used in presentations of this transformation technique; clearly the convolutive (20) and semi-integral (21, 22) approaches are equivalent.The transformed current data can be used directly, by (6.7.2), to obtain C o (0, t).Under conditions where CQ(0, t) = 0 (i.e., under purely diffusion-controlled conditions),/(0 reaches its limiting or maximum value, // [or, in semi-integral notation, m(t)max],wherez^(6.7.5)orNote the similarity between this expression for the transformed current and that for thesteady-state concentration in terms of the actual current, (1.4.11).
Similarly for species R,assumed absent initially, the corresponding expression resulting from (6.2.9) isThese equations hold for any form of signal excitation in any electrochemical technique applied under conditions in which semi-infinite diffusion is the only form of masstransfer controlling the current. No assumptions have been made concerning the reversibility of the charge-transfer reaction or even the form of the dependence of C o (0, 0and C R (0, 0 on E. Thus, with the application of any excitation signal that eventually drives C o (0, 0 to zero, the transformed current /(0 will attain a limiting value, //, that canbe used to determine CQ by equation 6.7.5 (22).If the electron-transfer reaction is nernstian, the application of equations 6.7.6 and6.7.7 yieldswhere Ещ = £°' + (RT/nF) In (DR/Z)0)1/2, as usual.
Note that this expression is identicalin form with those for the steady-state or sampled-current i-E curves (equations 1.4.16and 5.4.22). Transformation of a linear potential sweep i-E response thus converts thepeaked i-E curve to an S-shaped one resembling a polarogram (Figure 6.7.1).6.7.2Transformation of the Current-Evaluation of I(t)Although analog circuits that approximate I(t) have been proposed (23), the function isusually evaluated by a numerical integration technique on a computer.
Several differentalgorithms have been proposed for the evaluation (24, 25). The i-t data are usually dividedinto N equally spaced time intervals between t = 0 and t = tf, indexed by j ; then I(t) becomes I(kAt), where Д/ = tf/N and к varies between 0 and N, representing t = 0 and6.7 Convolutive or Semi-Integral Techniques249\fl)(d)1.0+5c0 (o, QCo,//-/(00.54\Figure 6.7.1 Variation of /, /,C o (0, 0/C*, and ln[(7/ - /)// with£.
[Adapted from J. C. Imbeauxand J.-M. Saveant, /. Electroanal.Chem., 44, 169 (1973), withpermission.]t = tf as in Figure 6.7.2). One convenient algorithm, which follows directly from the definition of /(0, is (24)/(/) = 1(Ш) =Vтг1/2 j^VkAt-jAt+ ^At(6.7.9)which is obtained from (6.7.3) by using t = kAt and и = jAt, and measuring / at the midpoint of each interval. This can be simplified tourn = -^ §i(jAt -\At) At112(6.7.10)Figure 6.7.2 Division ofexperimental i(f) vs. t [i(t) or vs. E(t)]curve for digital evaluation of 7(0-250 P Chapter 6. Potential Sweep MethodsAnother algorithm, which is especially convenient for digital processing, is77(6.7.1y = l V* 7^-where T(x) is the Gamma function of*, where Г(1/2) = тг1/2, Г(3/2) = |тт 1/2 , Г(5/2) =(|)(|)тг 1/2 , etc.
Other algorithms based on standard methods of numerical evaluation ofdefinite integrals also have been used (20-22, 25, 26).6.7.3Irreversible and Quasireversible ReactionsThe convolutive form for a totally irreversible one-step, one-electron reaction follows directly from the i-E expression with no back reaction, equation 3.3.11:i = FAk°Co(0, t)e~af{E~EO)(6.7.12)and the expression for C o (0, t), equation 6.7.6.
Thus (20),°l/2a^E-E°f)(6.7.13)or. Л4)E-go' + RTbjL + m.biLl™(6 7aFpV2 aFi(t)For a quasireversible one-step, one-electron reaction, the full equation 3.3.11 is employed, along with equations 6.7.6 and 6.7.7, to yieldE-E°^Щ/(0 = k°Dol/2e-afiE-E0f){/, - /(0[l + m}rv RT lfiRT h ~ /(0[l1 + £0]E = E° +Ц\п^-^Ц\п^Ц- —aFV2aFi(t)D(6.7.15)(6.7.16)(6.7.17)where £ = (Do/DR)l/2 and в = exp[/(£ - E0)].In deriving (6.7.14) and (6.7.17), we assumed that Butler-Volmer kinetics apply, asexpressed in the i-E characteristic, (3.3.11). Indeed, this assumption (or the adoption ofsome other model) is necessary before equations can be derived for most electrochemicalapproaches.
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