A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 81
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A planar workingelectrode and an unstirred solution are assumed, with only species О initially present at aconcentration CQ. These conditions are the same as those in Section 5.4.1, so that the diffusion equations and general boundary conditions, (5.4.2) to (5.4.5), apply:дСо(х, i(8.2.1)'These latter methods are employed infrequently and are discussed in more depth in the first edition, Sections7.4.3 and 7.6.308Chapter 8. Controlled-Current Techniques—R '= £>JR2'(8.2.2)At t = 0 (for all JC) "and\C0(x, t) = Cgas x —> oo (for all i)CR(x, r) = 0(8.2.3)<8 2 4>»=0-Since the applied current i(t) is presumed known, the flux at the electrode surface is alsoknown at any time, by the equation [see (4.4.29)]:U=oThis boundary condition involving the concentration gradient allows the diffusion problem to be solved without reference to the rate of the electron-transfer reaction, in contrastwith the concentration-potential boundary conditions required for controlled-potentialmethods.
Although in many controlled-current experiments the applied current is constant, the more general case for any arbitrarily applied current, i(f)9 can be solved readilyand includes the constant-current case, as well as reversal experiments and several othersof interest.For species O, application of the Laplace transform method to (8.2.1) and (8.2.3) yieldsCo(x, s) = -^- + A(s) exp| - ( ^ - )x|(8.2.6)The transform of (8.2.5) is'dC0(x,s)\=x=onFABy taking the indicated derivative of (8.2.6) and substituting in (8.2.7) to eliminate A(s),we have(8.2.8)By substitution of the known function, i(s), and employing the inverse transform, CQ(X, t)can be obtained. Similarly, the following expression for CR(x, s) can be derived:Note that direct inversion of (8.2.8) and (8.2.9) using the convolution property leads to(6.2.8) and (6.2.9).
These integral forms are also convenient for solving controlled-current problems.8.2.2 Constant-Current Electrolysis—The Sand EquationIf i(t) is constant, then i(s) = i/s and (8.2.8) becomesCo(x, *) = % s/2Wv^ exP "f if-У *inFAD^s312}L \Do)J8.2 General Theory of Controlled-Current Methods309The inverse transform of this equation yields the expression for CQ(X, t):Typical concentration profiles at various times during a constant-current electrolysis aregiven in Figure 8.2.1.
Note that C o (0, 0 decreases continuously, yet [dCo(x, t)/dx]x=0 isconstant at all times after the onset of electrolysis.An expression for CQ(0, 0 can be obtained by setting x = 0 in (8.2.11), or directly byinverse transform of (8.2.8) with x = 0:C o ( 0 , s) = - ^ -(8.2.12)to yieldC o (0, t) ==C%C% -lit1'2(8.2.13)Figure 8.2.1 Concentration profiles of О and R (in dimensionless form) at various values of tlrindicated on the curves.310 P Chapter 8. Controlled-Current TechniquesAt the transition time, r, CQ(0, t) drops to zero and (8.2.13) becomes(8.2.14)This equation, known as the Sand equation, was first derived by H. J.
S. Sand (1).As discussed in Section 8.1.2, the flux of О to the electrode surface beyond the transition time is not large enough to satisfy the applied current, and the potential shifts to amore extreme value where another electrode process can occur (Figure 8.2.2). The actualshape of the E-t curve is discussed in the next sections.The measured value of r at known / (or better, the values of ir1^2 obtained at various currents) can be used to determine n, A, CQ or DQ.
For a well-behaved system, thetransition time constant, irl^2/CQ, is independent of / or CQ. A lack of constancy in thisparameter indicates complications to the electrode reaction from coupled homogeneouschemical reactions (Chapter 12), adsorption (Chapter 14), or measurement artifacts,such as double-layer charging or the onset of convection (Section 8.3.5).Note that (8.2.11) can be written in a convenient form with dimensionless groupingsC0(x, t)IC%, t/т, and xo = x/[2(Dot)l/2] for (0 < t < r):(8.2.16)In a similar way, the following equations hold for CR(x, t) when (0 < t < r):CR(*,Q/Д1/2= f ( r I [exp(^R) - TTL/ZXR erfc(^R)]»where дгк = */[2(£>R01/2] and f = (Do/DR)l/2.(8.2.17)Accordingly,(8.2.18)See Figure 8.2.1.-0.24-0.12+0.120.5tin1.0Figure 8.2.2 Theoreticalchronopotentiogram for a nernstianelectrode process.8.3 Potential-Time Curves in Constant-Current Electrolysis8.2.3311Programmed Current ChronopotentiometryIt is possible to use currents that are programmed to vary with time in a special way,rather than remaining constant (2, 3).
For example, a current that increases linearly withtime could be used:(8.2.19)i(t) = ptThe treatment follows that for a constant-current electrolysis, hi this case the transform isJ3i(s) = 4:sz(8.2.20)so that (8.2.8) becomes, at x = 0,C o (0, t) = Cg(8.2.22)~zwhere Г(5/2) is the mathematical gamma function, equal with this argument to 1.33. Thissame treatment can be employed with any power function of time.A particularly interesting applied current is one varying with the square root of time:i(0 = ptm(8.2.23)i(s) = ^—rjr2s3/2(8.2.24)~с°(0'!> = ^-ШБ$?<8 2 25)C o (0,0 = C S -P-'.-(8.2.26)Again, defining the transition time т as that time when CQ(0, 0 = 0> an expression equivalent to the constant-current Sand equation results, but with т (rather than r1^2) proportionalto CQ and p:^colll= 2nFATr- D\?(8.2.27)Because this current excitation function is fairly difficult to generate, the techniquehas not been used very much.
Nevertheless, it would be advantageous for step wiseelectron-transfer reactions and multicomponent systems (see Section 8.5).8.3 POTENTIAL-TIME CURVES IN CONSTANTCURRENT ELECTROLYSIS8.3.1Reversible (Nernstian) WavesFor rapid electron transfer, a nernstian relationship links the potential with the surfaceconcentrations of О and R (Sections 3.4.5 and 3.5.3). Substitution of the expressions forC o (0, 0 and C R (0, 0, equations 8.2.16 and 8.2.18, into (3.5.21) yields (4)(8.3.1)312 • Chapter 8. Controlled-Current Techniqueswhere Ет/4, the quarter-wave potential, isThus Ет/4 is the chronopotentiometric equivalent of the voltammetric Ец2 value (Figure8.2.2).
The test for reversibility of an E-t curve is a linear plot of E vs. log [(r 1 / 2 tl/2)/tl/2] having a slope of 59/n mV, or a value of \Ет/4 - Е3т/4\ = 47.9/л mV (at 25°C).8.3.2Totally Irreversible WavesFor a totally irreversible one-step, one electron reactionO + e5>R(8.3.3)the current is related to the potential by either of the following equations (5):i = nFAk°fCo(0, i) exp ^pi = nFAk°Co(0, t) е х р Г " а / 7 ( ^ г " £ ) 1(8.3.4a)(8.3.4b)When the expression for CQ(0, t), (8.2.16), is substituted into these equations, the following expressions result:Equivalent expressions can be obtained by using the Sand equation and substituting forT l/2.(8.3.6b)Thus, for a totally irreversible reduction wave, the whole E-t wave shifts toward morenegative potentials with increasing current, with a tenfold increase in current causing ashift of 23RT/aF (or 59/a mV at 25°C). Note that uncompensated resistance between thereference and working electrodes will also cause the E-t curve to shift with increasing /.For a totally irreversible wave, \ET/4 - E3r/4\ = 33.8/a mV at 25°C.8.3.3Quasireversible WavesFor the quasireversible one-step, one-electron process,kfO + e^±R(8.3.7)kba general E-t relationship is found from combining the current-overpotential equation,(3.4.10), with the equations for C o (0, 0, (8.2.16), and C R (0, 0, (8.2.18).
A bulk concen-8.3 Potential-Time Curves in Constant-Current Electrolysis313tration of R, CR, is required, so that a starting equilibrium potential can be defined (6,7). This requirement adds the term CR to (8.2.18). The overall result is(8.3.8)Alternative forms can be written in terms of the current density, j , and the heterogeneousrate constants,or, when C R = 0,(8.3.9b)where kf and kb are defined in (3.3.9) and (3.3.10).Usually, the study of the kinetics of quasireversible electrode reactions by constantcurrent techniques (generally called the galvanostatic or current step method) involvessmall current perturbations, and the potential change from the equilibrium position is alsosmall. When both О and R are initially present, the linearized current-potential-concentration characteristic, (3.5.33), can be employed.
Combination with equations 8.2.13 and8.2.18 (with the latter modified by an added term, CR) yields(8.3.10)The same result can be obtained by linearization of (8.3.8). Thus, a plot of 77 vs. tl/2,for small values of 77, will be linear, and /0 can be obtained from the intercept. Thismethod is the constant-current analog of the potential step method discussed in Section8.3.4General Effects of Double-Layer CapacityBecause the potential is changing during the application of the current step, there is always a nonfaradaic current that contributes to charging of the double-layer capacitance. IfdA/dt = 0, then ic is given byic = -ACd(dV/dt)= -ACd(dE/dt)(8.3.11)Thus, of the total applied constant current, /, only a portion, /f, goes to the faradaic reaction:if= i - ic(8.3.12)Since dE/dt is a function of time, ic and if also vary with time, even when / is constant.This situation can be treated as a case of programmed current chronopotentiometry if anexplicit form of dE/dt or drj/dt is known.For the case where the one-step, one-electron process applies and the general r\-t expression can be linearized, then (8.3.10) can be adapted as follows (8):(8.3.13)314Chapter 8.
Controlled-Current TechniqueswhereN = ~-( J1/2+ J\(8.3.14)The intercept of the j]-t^2 plot allows the determination of 1/z'o only when this term is appreciable compared to (RT/F)AC&N2 (8). To overcome this limitation for fast electrontransfer reactions, where /Q is large, the galvanostatic double-pulse method has beenproposed (Section 8.6).8.3.5Practical Issues in the Measurement of Transition TimeAs discussed in Section 8.3.4, the presence of a finite double-layer capacity results in acharging current contribution proportional to dE/dt (equation 8.3.11) and causes /f to differ from the total applied current, /. This effect, which is largest immediately after application of the current and near the transition (where dE/dt is relatively large), affectsthe overall shape of the E-t curve and makes measurement of т difficult and inaccurate.A number of authors have examined this problem and have proposed techniques for measuring т from distorted E-t curves or for correcting values obtained in the presence of significant double-layer effects.In the simplest approach, ic is assumed to be constant for 0 < t < r.
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