A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 76
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Thus the preelectrolysis establishes a thickdiffusion layer, and the pulse is able to modify only a small part of it. In fact, the experiment can be approximated by assuming that the pulse cannot distinguish the actual finite concentration profiles appearing at its start from those of a semi-infinitehomogeneous solution with bulk concentrations equal to the values of CQ(0, t) andCR(0, t) enforced by potential E.
The role of the preelectrolysis is therefore to set up"apparent bulk concentrations" that vary during successive drops from pure О to pureR (or vice versa), as the scan is made. For a given drop, we take the differential faradaiccurrent as the current that would flow at time т - r' after a potential step from E toE + AE.Now let us restrict our consideration to a nemstian system in which R is initially absent. The results from Section 5.4.1 show that the surface concentrations during preelectrolysis at potential E areCo(0, t) = C S ( Y ^ )C R (0, t) = C*(^^j(7.3.13)where в = exp[nf(E — E° )]. We regard these values as the apparent bulk concentrations(Сд)арр a n d (^R)app f° r m e pulse.
Since the system is nernstian, they are in equilibriumwith potential E. The problem is now simply to find the faradaic current flow after a stepfrom equilibrium to E + AE in a homogeneous medium of bulk concentrations (Co)appand (Cf ) a p p .Through the approach of Section 5.4.1 (see also Problem 5.10), that current isstraightforwardly found to be.
_ nFAD^1~W2[(Cg)app - П ф'а р р]((1 + &')}where Or = exp[nf(E + AE — £° )]. Substitution according to (7.3.13) gives. = nFAD^cZ _(ffl - gfl')зi5)It is convenient (42) to introduce the parameters Рд ап< ^ о", where^\EKl \+ ^-E°')2(7.3.16))\and(7.3.17)290 • Chapter 7.
Polarography and Pulse VoltammetryIn this notation, £0 = PA/a and £0' = PAa\ thusi =nFADWcirРАа)and we take the differential faradaic current Si = /(т) - /(т') asThe bracketed factor describes Si as a function of potential. When E is far more positive than E°\ PA is large and 5/ is virtually zero. When E is much more negative then E° ,PA approaches zero, and so does Si. Through the derivative d(Si)/dPA one can easily show(42) that Si is maximized at PA = 1, which implies that_Fo>AEAESince A£ is small, the potential of maximum current lies close to Ey2.
Also, given thatAE is negative in this experiment, we see that the peak anticipates Еу2 by AE/2.The height of the peak is(7.3.21)where the quotient (1 - cr)/(l + &) decreases monotonically with diminishing \AE\ andreaches zero for AE = 0. When AE is negative Si is positive (or cathodic), and viceversa. The quotient's maximum magnitude, which applies at large pulse amplitudes, isunity. In that limit, (S/) m a x is equal to the faradaic current sampled on top of the normal pulse voltammetric wave obtained under the same timing conditions. As (7.3.3)notes, that current is п¥АО\^С%1тт1п(т - r') 1 / 2 . Under usual conditions, AE is notlarge enough to realize this greatest possible (S/)max- Table 7.3.1 shows the influence of| AE\ on (1 - a)l{\ + <T), which is also the ratio of the peak height to the limiting value.For analysis, a typical AE is 50 mV, which gives a peak current from 45% to 90% ofthe limiting value, depending on n.The width of the peak at half height, W\/2, increases as the pulse height grows larger,because differential behavior can be seen over a greater range of base potential.
NormallyTABLE 7.3.1 Effect of PulseAmplitude on Peak Height"+ a)(1 -, mV-10-50-100-150-200aп =20.09710.4530.7500.8990.9600.1930.7500.9600.995—From E. P. Parry and R. A.Osteryoung, Anal. Chem., 37,1634 (1965).0.2850.8990.995——7.3 Pulse Voltammetry I 291one refrains from increasing \AE\ much past 100 mV, because resolution is degraded unacceptably. The precise form of W\/2 as a function of AE is complicated and is of no realuse.
However, it is of interest to note the limiting width as AE approaches zero. By simplealgebra that turns out to be (42)Wm = 3.52RT/nF(7.3.22)At 25°C, the limiting widths for n = 1,2, and 3 are 90.4, 45.2, and 30.1 mV, respectively.Real peaks are wider, especially if the pulse height is comparable to or larger than the limiting width.Since the faradaic current measured in differential pulse polarography is never largerthan the faradaic wave height found in the corresponding normal pulse experiment, thesensitivity gain in the differential method obviously does not come from enhancedfaradaic response.
Instead, the improvement comes from a reduced contribution frombackground currents. If the background current from interfacial capacitance or from competing faradaic processes does not change much from the first current sample to the second, then the subtractive process producing 8i tends to cancel the backgroundcontribution.(b) Behavior at a DMEBecause dA/dt is never zero at the DME, capacitive currents contribute to the backgroundand require consideration. We assume that the current samples i(r) and i(rr) are taken atconstant potential, so that the charging current arises entirely from dA/dt.
From (7.1.17),we express these contributions as/c(r) = 0.00567Q(£z - E - АЕ)т2/3т~1/321Ъ11Ъ/ (T') = 0.00567Q(£ z - Е)т т'~(7.3.23)(7.3.24)Cthus the contribution to the differential current is8ic =I C (T)-I C (T')= 0.00567Qm 2/3 T- 1/3 \(EZ - E - A£) - ( j j(Ez - E)(7.3.25)where Q has been taken as constant over the range from E to E + Д£. For the usual operating conditions, (т/т') 1/3 is very close to unity; hence the bracketed factor is approximately —AE:8ic « -0.00567CiA£m 2 / V 1 / 3(7.3.26)For a negative scan 8i is positive, and vice versa.
A comparison between (7.3.2) and(7.3.26) shows that the capacitive contribution to differential pulse measurements differsfrom that in tast and normal pulse polarography by the factor AE/(EZ — E). Over mostregions of polarographic operation AE is smaller than Ez — E by an order of magnitudeor more. Note also that the capacitive background in differential pulse polarography isflat, insofar as Q is constant over a potential range. In contrast, normal pulse and tastmeasurements feature a sloping background because of the dependence on (Ez - E).This difference is apparent in Figure 7.3.12, and the greater ease in evaluating the differential faradaic response is obvious.Background currents also arise at a DME from electrolysis of impurities in solution(frequently from O2, even in deaerated solutions) or from slow faradaic reactions of majorsystem components (such as H + ). It is often true that the rates of these processes do notchange greatly as the potential shifts from E to E + AE and with the elapse of time from7' to т within a given measurement cycle; thus the subtraction of current samples does292 • Chapter 7.
Polarography and Pulse Voltammetryhelp to suppress the faradaic background, but normally does not eliminate it altogether, aswe will see in Section 7.3.6. In practical analysis by DPP at the DME, the faradaic background is often the dominant factor limiting sensitivity.The improvements manifested in the differential method yield sensitivities that areoften an order of magnitude better than those of normal pulse polarography. Detectionlimits as low as 10~8 M can be achieved, but doing so requires close attention to selectionof the medium. See Section 7.3.6 for more details.(c) Behavior at an SMDEUnder normal experimental conditions for DPP at an SMDE, there is no appreciable charging current contribution, because dE/dt and dA/dt are both essentially zero at the momentof sampling.
The faradaic current from the process of interest is the same as at a DME ofequal mature drop size and is given by (7.3.19). Faradaic contributions to the backgroundare normally also the same at an SMDE vs. a DME of equal drop size, because thesefaradaic processes are usually not affected by the history of the drop's evolution. Since theSMDE preserves the DME's sensitivity to the sample while eliminating one component ofbackground, there is a sensitivity advantage at the SMDE in any situation where the charging current background at the DME is appreciable.
Otherwise the SMDE and DME willprovide comparable performance with respect to sensitivity.An important additional advantage of the SMDE lies in the rapid formation and stabilization of the drop, which allow the use of preelectrolysis times as short as 500 ms. This timecontrols the duration of the scan; and the use of a short preelectrolysis period can save muchtime in practical analysis, often 80% of the scan time required for DPP at a DME.(d) Behavior at Nonpolarographic ElectrodesDPV can be carried out quite successfully at a stationary electrode, such as a Pt disk or anHMDE, even though such systems do not allow physical renewal of the solution near theelectrode with each measurement cycle. As we have seen above, the DPV method isbased on the concept of using the preelectrolysis at potential E to establish "apparent"bulk concentrations, which are then interrogated with the pulse.
If the system is kinetically reversible, the preelectrolysis can establish those conditions as well as at a renewedelectrode, despite the fact that the effects of prior cycles are not erased from the diffusionlayer. In fact, because the changes in potential from cycle to cycle are small, the cumulative effect of successive cycles is gradually to thicken the diffusion layer in a manner thatsupports the assumptions used in the treatment of wave shape and peak height given inSection 7.3.4(a).At solid electrodes of all kinds, the background is rarely dominated by charging current, but rather by faradaic processes involving the electrode material, solvent, or supporting electrolyte.
DPV allows for moderation of background contributions by taking thedifference between current samples. Even so, the residual background is typically higherthan at mercury, and one cannot usually achieve the sensitivity that can be obtained inDPP.On the other hand, one has the freedom to use shorter preelectrolysis times and pulsewidths at stationary electrodes vs.
the DME or SMDE, because one does not have to waitfor drop formation. This feature is used to advantage in the practice of square wavevoltammetry, which is covered in Section 7.3.5.(e) Peak ShapesIn the course of deriving the peak height for DPP in Section 7.3.4(a), we also derived theshape of the peak for a reversible system in the limit of small Д£, and we discussed the7.3 Pulse Voltammetry293effects of larger A£. The results and conclusions of that section are valid for a reversiblereaction at any type of electrode at which semi-infinite linear diffusion applies.We will not treat the application of differential measurements to irreversible systems.Instead we will note only that, as \AE\ tends toward zero, the response in any differentialscan approaches the derivative of the normal pulse voltammogram. This fact is easilydemonstrated for the reversible system (see Problem 7.7).
If a system shows irreversibility because of slow heterogeneous kinetics, one can still expect to see a differential response, but the peak will be shifted from Eo> toward more extreme potentials by anactivation overpotential (i.e., toward the negative for a cathodic process and toward thepositive for an anodic one). Also, the peak width will be larger than for a reversible system, because the rising portion of an irreversible wave extends over a larger potentialrange. Since the maximum slope on the rising portion is smaller than in the correspondingreversible case, (5/)max wn"l be smaller than the value predicted by (7.3.21). If the irreversibility is caused by following chemistry, the peak will also be broad and low, but willbe less extreme than E0' for reasons discussed in Chapter 12.The range of time scales for the differential pulse experiment is the same as for normal pulse voltammetry, hence a given system ordinarily shows the same degree of reversibility toward either approach.
However, the degree of reversibility toward pulsemethods may differ from that shown toward conventional polarography for reasons discussed in Section 7.3.2.7.3.5 Square Wave VoltammetryExceptional versatility is found in a method called square wave voltammetry (SWV),which was invented by Ramaley and Krause (49), but has been developed extensivelyin recent years by the Osteryoungs and their coworkers (35, 45, 50). One can view it ascombining the best aspects of several pulse voltammetric methods, including the background suppression and sensitivity of differential pulse voltammetry, the diagnosticvalue of normal pulse voltammetry, and the ability to interrogate products directly inmuch the manner of reverse pulse voltammetry.
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