A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 75
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For a stationary planar electrode, the relationships worked outabove apply directly. For an SMDE, they apply to the extent that /d,DC * s ш е Cottrell current for an electrolysis of duration r and is not disturbed by the convection associated withthe establishment of the drop. For a DME, the picture is complicated by the steady expansion of area, but it turns out (47, 48) that (7.3.8) is still a good approximation if / d D C isunderstood as the Ilkovic current for time r [(7.3.1) or (7.3.4)] and the pulse width isshort compared to the preelectrolysis time [i.e., (т - т')/ т' < 0.05].For a reversible system, the shape of the RPV wave can be derived from the generaldouble-step response given in (5.7.14). We confine our view to the situation where the forward electrolysis always takes place in the diffusion limited region, so that 0' = exip[nf(Eb —E°)] ~ 0. Then we have for the current sampled in a reverse pulse to any value of E:nFAD}?C* lГ/![{Vl\ll^J(where 0" = ехр[и/(£ - E0')].
Of the three terms in (7.3.9), the first and the third togetherare -/d,Rp, as defined in (7.3.7), and the second is ~/d,Np/(l + £#"); thus*=4+^ + £0"(7.3.10)286Chapter 7. Polarography and Pulse VoltammetrySubstitution for /^NP according to (7.3.8) and rearrangement gives(9*RPtez(7.3.11)d,RPBy taking the natural logarithm and defining Emobtain= E0' + (RT/nF) In}Q / 2 ),we(7.3.12)which is identical to the shape function for a reversible composite wave worked out inSection 1.4.2(b).
We have now established, as one suspects by a glance at Figure 7.3.8Z?that the half-wave potential, the total height, and the wave slope of the RPV wave are allexactly as for the corresponding NPV wave. These results were derived here for a systemwhere simple semi-infinite linear diffusion applies; however Osteryoung and Kirowa-Eisner (47) show that they apply at a DME, too.The principal use of RPV is to characterize the product of an electrode reaction, especially with respect to stability. It is obvious that if species R decays appreciably during theperiod of the experiment, particularly on the time scale of the pulse, it cannot be fullyavailable to be reoxidized during the pulse.
Consequently, the anodic plateau current mustbe smaller in magnitude than expected from (7.3.7). If the decay is very fast, R will becompletely unavailable and the anodic plateau current will be zero. The ratio of plateauheights in RPV and NPV quantifies the stability, and with proper theory, one can obtainthe rate constant for the following chemistry. Chapter 12 covers this kind of issue formany different mechanisms and methods.As in the application just discussed, the focus in RPV is often on the magnitude ofthe wave heights, rather than wave shapes and positions.
One can think of RPV as a wayto present double potential step chronoamperometric data conveniently on a potentialaxis, because the features of interest are rooted in chronoamperometric theory, as we havealready seen for the derivations done in this section. Thus one can make direct and confident use of the extensive published results for double-step chronoamperometry to treatdata from RPV in various chemical situations.7.3.4Differential Pulse Voltammetry(a) General Polarographic ContextSensitivities even better than those of normal pulse voltammetry can be obtained withthe small-amplitude pulse scheme shown in Figures 7.3.9 and 7.3.10, which show thebasis for differential pulse voltammetry (DPV) (6, 35, 41-45).
The figures focus on theЛ у Drop fall5-100 msec —>j10 to 100 mV-0.5 to 4 sec-•П—•Drop fallTimeDrop fallFigure 7.3.9 Potentialprogram for several dropsin a differential pulsepolarographic experiment.The 10-100 mV potentialchange late in the droplifetime is the pulse height,7.3 Pulse Voltammetry < 287Second current sampleFigure 7.3.10Events for a single dropof a differential pulsepolarographic experiment.Timespecial case of differential pulse polarography (DPP), but the waveform and measurement strategy are general for the method in its broader sense. DPP resembles normalpulse polarography, but several major differences are evident: (a) The base potential applied during most of a drop's lifetime is not constant from drop to drop, but instead ischanged steadily in small increments, (b) The pulse height is only 10 to 100 mV and ismaintained at a constant level with respect to the base potential, (c) Two current samples are taken during each drop's lifetime.
One is at time r\ immediately before thepulse, and the second is at time r, late in the pulse and just before the drop is dislodged,(d) The record of the experiment is a plot of the current difference, 8i = i(r) - i(r'),versus the base potential. Obviously the name of the method is derived from its relianceon this differential current measurement. The pulse width (—50 ms) and the waiting period for drop growth (0.5 to 4 s) are both similar to the analogous periods in the normalpulse method.Figure 7.3.11 is a block diagram of the experimental system and Figure 7.3.12a is anactual polarogram for 10~~6 M Cd 2 + in 0.01 M HC1. For comparison, the normal pulse response from the same system is given in Figure 7.3.12Z?.Note that the differential measurement gives a peaked output, rather than the wavelike response to which we have grown accustomed.
The underlying reason is easily understood qualitatively. Early in the experiment, when the base potential is much morepositive than E0' for Cd 2 + , no faradaic current flows during the time before the pulse,г111PotentialprogrammerсPulsesequencerPotentiostat1HEConverterDropknocker1— Sample/Holdi(T)г—Sample/Hold("0i | |iJ—Differenceamplifier—RecorderFigure 7.3.11 Schematicexperimental arrangement fordifferential pulse polarography.For clarity, this diagram showsthe functions as they would beorganized in separate electronicstages in a free-standing DPP unit.In contemporary instruments, acomputer performs many of theroles delineated here, includingsequencing, recording of data,calculation of difference signals,and display of results.288Chapter 7.
Polarography and Pulse Voltammetry-0.4-0.6E (V vs. SCE)-0.8(a)Baseline-0.4-0.6-0.8E{Vvs. SCE)(b)Figure 7.3.12 Polarograms at a DME for10" 6 M C d 2 + in 0.01 M HC1. (a) Differentialpulse mode, AE = —50 mV. (b) Normal pulsemode.and the change in potential manifested in the pulse is too small to stimulate the faradaicprocess. Thus /(r) - Z(r') is virtually zero, at least for the faradaic component.
Late inthe experiment, when the base potential is in the diffusion-limited-current region, Cd 2 +is reduced during the waiting period at the maximum possible rate. The pulse cannot increase the rate further; hence the difference i(r) - i(r') is again small. Only in the regionof E0' (for this reversible system) is an appreciable faradaic difference current observed.There the base potential is such that C d 2 + is reduced during the waiting period at somerate less than the maximum, since the surface concentration CQ(0, t) is not zero. Application of the pulse drives C o (0, t) to a lower value; hence the flux of О to the surface andthe faradaic current are both enhanced.
Only in potential regions where a small potentialdifference can make a sizable difference in current flow does the differential pulse technique show a response.7.3 Pulse Voltammetry289The shape of the response function and the height of the peak can be treated quantitatively in a straightforward manner. Note that the events during each drop's lifetime actually comprise a double-step experiment. From the birth of the drop at t = 0 until theapplication of the pulse at t = r', the base potential E is enforced.
At later times, the potential is E + AE, where AE is the pulse height. Each drop is born into a solution of thebulk composition, but generally electrolysis occurs during the period before r' and thepulse operates on the concentration profiles that prior electrolysis creates. This situation isanalogous to that considered in Section 5.7, and it can be treated by the techniques developed there. Even so, we will not take that approach, because the essential simplicity of theproblem is obscured.Instead, we begin by noting that the preelectrolysis period r' is typically 20-100times longer than the pulse duration т — т'.
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