A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 82
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This is notstrictly so, of course, since dE/dt and Сд (a function of E) change throughout the E-t curve(9, 10); however the approximation leads to(8.3.15)i = if+ic1/2UT112LT/rwhere / f r 1 /2 /Co is the "true" chronopotentiometric constant, a, equal to nFAD\^TTXI2/2.In the last term, icr is the total number of coulombs needed to charge an average doublelayer capacitance from the initial potential to the potential at which т is measured (A£),so that icr ~ (Q)avg A£» a n d is represented by a correction factor b. Thus the final equation isiTmIC% = a + Ь/С%тт(8.3.17)Plots based on (8.3.17) can thus be used to extract a and b from the observed data (e.g., aplot of ir vs. r 1 / 2 yields a slope aC% and an intercept b).An equation of this form can also be used to correct for formation of an oxide film(e.g., on a platinum electrode during an electrochemical oxidation) and for electrolysis ofadsorbed material in addition to diffusing species.
Under these conditions, (8.3.15) becomes (10):*' = ' f + ' c + 'ox + l 'ads(8.3.18)where /ox is the current going to formation (or reduction) of the oxide film and / a( j s is thecurrent required for the adsorbed material. A treatment similar to that given above againyields (8.3.17), where b is now an overall correction factor including Qox = / o x r andGads ~ nFY, where Г is the number of moles of adsorbed species per square centimeter (Section 14.3.7). Although these approximations are rough, treatments ofactual experimental data by (8.3.17) yield fairly good results, even at rather low concentrations and short transition times, where these surface effects are mostimportant (11).8.3 Potential-Time Curves in Constant-Current Electrolysis3151.000.750.500.25Figure 8.3.1 Fraction of total currentcontributing to the faradaic process (/f//) vs. timefor a nernstian electrode process.
К values of(1) 5 X 10~4; (2) КГ 3 ; (3) 2 X 10" 3 ; (4) 5 X 10(5) 0.01. [From W. T. de Vries, /. Electroanal.Chem., 17, 31 (1968), with permission.]2.0A more rigorous approach involves only the assumption that Q is independent of E(12-14). In this case, one must solve the diffusion equation, (8.2.1), beginning with theusual boundary conditions, (8.2.3). The flux condition, (8.2.5), is replaced by(dCrdC0 ,-£\+ACdx=00dEdt(8.3.19)In addition, one needs the appropriate i-E characteristic (i.e., for a reversible, totally irreversible, or quasireversible reaction).
The resulting nonlinear integral equation must beevaluated numerically. Alternatively, the problem can be addressed by digital simulationtechniques. Figures 8.3.1 and 8.3.2 illuminate the effects of different relative contributions of double-layer charging on /f (at constant /) and on the E-t curves of a nernstian reaction. The charging contribution is represented there by the dimensionless parameter, K,defined asK=^(8.3.20)400200IItil-200-400-10.5In3 \1.01.5\2.0Figure 8.3.2 Effect of double-layercapacitance on chronopotentiograms fora nernstian electrode reaction.
Values ofК as in Figure 8.3.1. [From W. T. deVries, /. Electroanal. Chem., 17, 31(1968), with permission.]316Chapter 8. Controlled-Current TechniquesFigure 8.3.3 (a) Shielded electrodefor maintaining linear diffusion andsuppressing convection, (b) Tubes to whichshielded electrode is attached to provide:(1) horizontal electrode, diffusion upward;(2) horizontal electrode, diffusion downward;(3) vertical electrode. [Reprinted withpermission from A.
J. Bard, Anal. С hem.,33, 11 (1961). Copyright 1961, AmericanChemical Society.]2(b)The effect of double-layer charging is clearly most important at small т values (seeequation 8.3.20). Problems with distorted E-t curves and the difficulty of obtaining corrected т values have discouraged the use of controlled-current methods as opposed to controlled-potential ones.In both controlled-current and controlled-potential methods, problems develop atlong experimental times because of the onset of convection and the nonlinearity of diffusion. Convective effects, caused by motion of the solution with respect to the electrode, can arise by accidental vibrations transmitted to the cell (e.g., by hood fans,vacuum pumps, passing traffic) or as a result of density gradients building up at theelectrode surface because of differences in density between reactants and products (socalled "natural convection"). Convective effects can be minimized by using electrodeswith glass mantles (shielded electrodes; Figure 8.3.3) and by orienting the electrodehorizontally so that the denser species is always below the less dense one (15, 16).
Vertically oriented electrodes (e.g., foils or wires) often suffer from convection effects evenat not very long times (e.g., 60 to 80 s). The shielded electrode also has the virtue ofconstraining diffusion to lines normal to the electrode surface so that true linear diffusion conditions are approached. An unshielded electrode, such as a platinum diskimbedded in glass can show appreciable "sphericity" effects when the diffusion layerthickness is not negligible with respect to the electrode dimensions; that is, material candiffuse to the unshielded electrode from the sides. This effect causes increases in thetransition time (or anomalously large currents in controlled-potential methods).
Withproperly oriented shielded electrodes, however, linear diffusion conditions can be maintained for 300 s or longer.8.4 REVERSAL TECHNIQUES8.4.1Response Function PrincipleA useful technique for treating reversal methods in chronopotentiometry (and other techniques in electrochemistry) is based on the response function principle (2, 17). Thismethod, which is also used to treat electrical circuits, considers the system's response to aperturbation or excitation signal, as applied in Laplace transform space.
One can write thegeneral equation (2)R(s) =(8.4.1)where ^(s) is the excitation function transform, R(s) is_the response transform, which describes how the system responds to the excitation, and S(s) is the system transform, which8.4 Reversal Techniques317connects the excitation and the response. For example for current excitation we can write,from (8.2.8) at x = 0,Co(0, s) = C%ls - [nFADl£sll2Y%)(8.4.2)Cg - C o (0, s) = [nFADl£sll2Yri{s)(8.4.3)orIn this_case ty(s) = i(s) (the transform of the applied current perturbation), R(s) =CQ - C o (0, s),the transform of the concentration response to the perturbation, andS(s) = [nFAD^s^2]'1, which is characteristic of the system under excitation (semi-infinitelinear diffusion).
For controlled-current problems involving different systems (e.g., spherical or cylindrical diffusion, first-order kinetic complications) other system transformswould be employed.2 We have illustrated how this equation could be employed for constant and programmed current methods, using appropriate i(s) functions. We now extendits use to reversal techniques.8.4.2Current Reversal (18,19)Consider a solution where only О is present initially at a concentration CQ, semi-infinite linear diffusion conditions prevail, and a constant cathodic current / is applied for a time t\(where t\<T\, with r\ being the forward transition time).
At t\ the current is reversed, that is,the direction of the current is changed from cathodic to anodic, so that R formed during theforward step is oxidized to O, and the time r 2 (measured from t{) at which CR at the electrodesurface drops to zero is noted. At r 2 , the reverse transition time, the potential shows a rapidchange toward positive values. We desire an expression for r 2 . This is most easily accomplished using the "zero shift theorem" of Section A.1.7. Since for 0 < t < th i(t) = /, and fort\<f^t\-\- r 2 , /(0 = —/, the expression for the current, using step function notation, is/(0 = i + S,,(0(-20(8.4.4)so that the transform is given byIntroducing this into (8.4.3), we obtainс* —f- ~ C o (0, s) = (|j(l - 2 e-^s)(nFADl£sll2ylfC(8.4.6)The analogous expression for CR(0, s) [see (8.2.9)] isCR(0, s) = U\l- 2 e~^s)(nFAD^2sl/2yl(8.4.7)The inverse transform of (8.4.7) yields an expression for CR(0, t) at any time [recall thatSh(i) = 0 for t < tx\ Sh{t) =\fort>tx\ Section A.1.7]:At t = tx + r 2 , CR(0, 0 = 0, so that {tx + т 2 ) 1 / 2 = 2т] / 2 andт2 = ф2(8.4.9)This approach, and transform methods in general, are useful only for linear problems; hence second-orderreactions or nonlinear complications cannot be treated by this technique.318 • Chapter 8.
Controlled-Current TechniquesFigure 8.4.1 Typical experimentalchronopotentiogram with current reversal. Oxidationof diphenylpicrylhydrazyl (DPPH) followed byreduction of the stable radical cation, DPPH • .Solutionwas acetonitrile containing 1.04 mM DPPH and 0.1 MNaC104. The current was 100 /JLA, and a shieldedplatinum electrode of area 1.2 cm 2 was employed.[Reprinted with permission from E.
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