A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 84
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The t112 term in each case accounts forelectrolytic modification of the surface concentrations. In the situation yielding(8.3.10), 77 was induced by current /; but in this case it is first induced by ix and thensupported by /2The differential double-layer capacitance can also be obtained from these data by theequation:(8.6.3)This relation rests on the idea that the total charge in the first step, *Vlvis purely nonfaradaic in the limit of very short t\.The GDP method does not require knowledge of the diffusion coefficients for reactant and products, or of the Q value, for the calculation of /Q. Measurements using instrumentation developed by Aoyagi and coworkers suggest that rate constants of very rapidelectrode reactions (~1 cm/s) can be determined using this technique.8.7 CHARGE STEP (COULOSTATIC) METHODS8.7.1 PrinciplesIn the charge-step (or coulostatic) method, a very short-duration (e.g., 0.1 to 1 fis) currentpulse is applied to the cell, and the variation of the electrode potential with time after thepulse (i.e., at open circuit) is recorded.
The length of the current pulse is chosen to be sufficiently short that it causes only charging of the electrical double layer, so that even avery fast charge-transfer reaction does not proceed to an appreciable extent during thistime. The pulse then serves only to inject a charge increment, Ag, and, in fact, under theseconditions the method of charge injection or the actual shape of the injecting pulse (thecoulostatic impulse) is unimportant. For example, the charge can be injected by discharging a small capacitor across the electrochemical cell (Figure 8.7.1) or with a pulse generator connected to the cell by a capacitor or switching diodes.Charge injection system_LFast recording systemFigure 8.7.1 Circuit for charge-step orcoulostatic method. In practice, the cellmay be held initially at a potential £ e qby means of a potentiostat that isdisconnected immediately before thecharge injection.8.7 Charge Step (Coulostatic) Methods.
323For the circuit in Figure 8.7.1, when the relay is in position A, the capacitor,Cinj, is charged by the voltage source, Vinj, until the capacitor is charged by anamountA*=Qnj^nj(8.7.1)9For example, for Vinj = 10 V and C in j = 10~ F, Aq = 0.01 fiC. When the relay switchesto position В the charge is delivered to the electrochemical cell. Because the double-layercapacitance, Q , is much larger than Сщ, essentially all of the charge will flow into thecell. The time required for this charge injection will depend on the cell resistance, RQ(Figure 8.7.2), with the time constant for injection being essentially СщЯп (Problem 8.6).This injected charge causes the potential of the electrode to deviate from its original valueEeQ to a value E(t = 0), whereE(t = 0) - £ e q = ф = 0) =-eq-Aq(8.7.2)The charge on Q now discharges through the faradaic impedance (i.e., the heterogeneouselectron-transfer process), and the open circuit potential moves back toward Eeq as 17(0decreases to zero.
Since the total external current / is zero, we have from (8.3.11) and(8.3.12),drl(8.7.3)or7,(0 =V(t = 0) + jr(8.7.4)Solution of (8.7.4) with the appropriate expression for if yields the desired expression forthe variation of E (or 17) with t. Note that if no faradaic reaction is possible at E(t = 0) (i.e.,at an ideally polarized electrode), Q remains charged and the potential will not decay [i.e.,for if = 0,Е = Ещ + ф = 0) at all t].We will now examine the E-t behavior following a coulostatic impulse for severalcases of interest. Details of the theoretical treatments have been given by Delahay (29, 30)and Reinmuth (31, 32) and their coworkers, who first described the application of thistechnique.OAuxЩ<— RefzfWk(b)Figure 8.7.2 Equivalent circuit of cell with(a) /?Q, the solution resistance, Cd, the doublelayer capacitance, and Zf, the faradaicimpedance. The faradaic impedance representsthe effect of the heterogeneous electron-transferprocess.
Often Z f is broken down into thecomponents shown in (b), where the chargetransfer resistance Rct manifests the kinetics ofheterogeneous charge transfer, and thecomponents of the Warburg impedance,/?w and Cw, manifest diffusional mass transfer(see Section 10.1.3).324 • Chapter 8. Controlled-Current Techniques8.7.2Small-Signal AnalysisWhen a chemically reversible, but kinetically sluggish, system is being investigated andthe potential excursion is sufficiently small, that is, when r)(t = 0) < < RT/nF and masstransfer effects are absent, one can use the linearized i-rj relation, (3.5.49),-7] =RT(8.7.5)nFi0Гr){t)dtRTC,(8.7.6)1to describe /f in (8.7.4).
Thus,(t) = r]{t = 0) -VThis equation can be solved readily by the Laplace transform method (Problem 8.7) toyieldr\{t) — 7]{t =RTQГ сnFi00) expMj(8.7.7)•ад,(8.7.8)Thus under these conditions, the potential relaxes exponentially toward Eeq with a timeconstant тс, governed by the rate of the charge-transfer reaction (Figure 8.7.3). Thisresult can also be obtained from the equivalent circuit in Figure 8.7.2Z? by noting thatRw and C w are negligible, so that Cd discharges only through the charge-transfer resistance Rcb given by (3.5.50), with a time constant CdRct. When (8.7.7) holds, a plot| [ < — Pulse width01020300102030Figure 8.7.3 Typical coulostatic relaxation curves for atotally irreversible reaction.8.7 Charge Step (Coulostatic) Methods325of In ITJI VS. t is linear with an intercept \r\(t = 0)| [which can be used to determine Qby (8.7.2)] and a slope — l/r c , which yields the charge-transfer resistance and the exchange current.On the other hand, when Rct is negligible compared to the mass-transfer impedance,which is the case for a nernstian system, the following expression applies:(8.7.9)RTCA(8.7.10)The general small-signal expression for the case where both charge and mass-transferterms are significant is (32)V(t) =erfc(yr1/2)][J exp(/32?) erfc(/8r1/2) - /3(8.7.11)(8.7.12)(where the + is associated with /3 and the - with y).
Note that /3 + у = Т£)/2/тс and£ y = 1/TC.Clearly the analysis of experimental data for the determination of /0 is easiest when(8.7.7) applies; this requires that r c > > T D . Detailed discussions of the analysis of coulostatic data and relaxation curves have appeared (33, 34).8.7.3 Large Steps—Coulostatic AnalysisConsider the application of a charge step sufficiently large that the potentialchanges from £eq to a value, E(t = 0), corresponding to the diffusion plateau of thevoltammetric wave. We assume that the double-layer capacity, Q, is independent ofpotential in this region. The faradaic current that flows under these conditions at aplanar electrode is given by (5.2.11).
Introduction of this expression into (8.7.4)yields,-1/2AE = E(t) - E(t = 0) =dt2nFAD)$-C%txl1(8.7.13)(8.7.14)The sign of AE is positive, since the electrode relaxes from a more negative initial potential toward more positive values. A plot of AE vs. t1^2 is linear with a zero interceptand a slope 2nFAD^2CQ/(7Tl/2Cd), which is proportional to the solution concentration(Figure 8.7.4).This method has been suggested for the determination of small concentrations ofelectroactive materials (35, 36), but it has not been widely applied, probably because it requires recording of the E-t curve and is less readily automated than, for example, pulsevoltammetry.326Chapter 8.
Controlled-Current Techniquesr 1/20.5f-\1.0-—U10~—.60—5x10"7 M20 - i40 -—7\4O- 6\\\\io- 5I \ I |\\ 2 x 10- 610"6Figure 8.7.4 A£ vs. t curves for a planarelectrode for several values of C*, with n = 2,D = l(T 5 cm 2 /s, and Q = 20 ju,F/cm2. [Reprintedwith permission from P. Delahay, Anal.
С hem.,\\I0 0.01 0.05 0.1 0.2r(s)I0.5\1.034, 1267 (1962). Copyright 1962, AmericanChemical Society. ]The technique can also be extended to cover large potential excursions to the risingportion of the voltammetric wave. For example, for a reversible process, the appropriateequation for the faradaic current would be (5.4.16), as long as measurements are made atsmall A£ values so that E remains close to E(t = 0) and the surface concentrations donot change appreciably during the measurement. A plot of the slope of the Д£ vs.
t1'curve at any potential vs. E then yields a charge-step voltammogram that resembles anordinary voltammogram (37). Because the coulostatic method requires the determinationof a slope at each data point, incremental changes in Aq to cause changes in potential,and renewal of initial conditions before each step, it requires digital computer controland data acquisition.8.7.4Application of Charge-Step MethodsThe charge-step or coulostatic methods described here have some advantages in the studyof electrode reactions. Since the measurement is made at open circuit with no net externalcurrent flow, the ohmic drop is not of importance and measurements can be made in highlyresistive media. Moreover, because the relaxation occurs by discharge of the double-layercapacitance, the usual competition between faradaic and charging current is replaced by anequality of ic and if, and Q no longer interferes in the measurement.
However, some fundamental limitations still exist in charge-step measurements. High values of RQ, increase thetime required to deliver charge to the cell. Also, the high voltage, Ущ, appears across thecell at the instant of charge injection and tends to overload the measuring amplifier, whichmust be adjusted to a high sensitivity to determine the small changes in Д£.
An amplifiercapable of rapid recovery from overloads is required, and parasitic oscillations caused bystray capacitance and unmatched impedances generally occur following the pulse. Thus, ingeneral, measurements cannot be made for a time interval of about 0.5 /JLS following thepulse. The technique appears to be useful for the determination of /0 values up to about 0.1A/cm2 or k° values up to 0.4 cm/s. A discussion of the experimental conditions and reviews of applications of charge-step methods have appeared (34, 38).8.7.5 Coulostatic Perturbation by Temperature JumpThe potential of an electrode can be perturbed in a manner directly analogous to thecoulostatic approach described above by an abrupt change in a variable other than charge,8.7 Charge Step (Coulostatic) Methods < 327such as the pressure or solution composition, that will shift the electrode from equilibrium. Probably the most straightforward way of doing this is by changing the electrodetemperature (i.e., by a temperature jump) (39-42).
This is conveniently effected by usinga thin (e.g., 1-25 fxm) metal film on a dielectric like glass and irradiating the film from thebackside (through the dielectric) with a pulsed laser (41, 42). The film is sufficiently thickthat no light penetrates the film and all of the absorbed light is converted to heat. Underthese conditions photoemission of electrons into the solution (39, 40), as discussed in Section 18.3.1, does not occur. A schematic diagram of the apparatus used is shown in Figure8.7.5. By using a fast laser and thin metal film, measurements in the nanosecond domainare possible.The rapid temperature change of the electrode perturbs the equilibrium at the electrode-solution interface and causes a change in the potential of the electrode measuredwith respect to a reference electrode.
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