A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 71
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In turn, m controls the drop time r m a x , because the maximum massthat the surface tension can support (mtm3iX) is a constant defined bymtmaxg= 2тггсу(7.1.14)where g is the gravitational acceleration, r c is the radius of the capillary, and у is the surface tension of the mercury water interface.The first edition covered this topic in greater detail5 and included a demonstrationthat m is inversely proportional to a corrected column height, hC0TT, which is obtainedfrom the actual height by applying two small adjustments.The Ilkovic relation shows that the diffusion-limited current is proportional tom2/3ex^max> which in turn is proportional to h^n h~J^ = h j^u. This square-root dependenceof the limiting current on corrected column height is characteristic of processes that arelimited by the rate of diffusion, and it is used as a diagnostic criterion to distinguish this5First edition, p.
155.270Chapter 7. Polarography and Pulse Voltammetrycase from other kinds of current limitation. For example, the current could be limited bythe amount of space available on an electrode surface for adsorption of a faradaic product,or it might be limited by the rate of production of an electroreactant in a preceding homogeneous chemical reaction.Column height is not of interest in work with an SMDE, because mercury does notflow at the time when currents are sampled.
The test for diffusion control at an SMDE isto vary the sampling time and to examine whether the limiting current varies as the inverse square-root of sampling time. This procedure, like testing for dependence of limiting current on the square-root of column height at a DME, has its origin in the timedependence of the diffusion-layer thickness (Section 5.2.1).7.1.5 Residual CurrentIn the absence of an electroactive substance of interest, and between the anodic and cathodic background limits, a residual current flows (3, 6, 14, 26).
It is composed of a current due to double-layer charging and a current caused by low-level oxidation orreduction of components in the system. The residual faradaic currents arise from (a)trace impurities, (such as heavy metals, electroactive organics, or oxygen), (b) from theelectrode material (which often undergoes slow, potential-dependent faradaic reactions),or (c) from the solvent and supporting electrolyte (which can produce small currents overa wide potential span via reactions that, at more extreme potential, become greatly accelerated and determine the background limits).The nonfaradaic current (often called the charging or capacitive current) can makethe residual current rather large at a DME, even in highly purified systems where thefaradaic component is small.
Because the DME is always expanding, new surface appearscontinuously. It must be charged to reflect the potential of the electrode as a whole; therefore a charging current, iC9 is always required.An expression for it can be obtained as follows. The charge on the double layer isgiven byq=-CiA(E-Ez)(7.1.15)where Q is the integral capacitance of the double layer (Section 13.2.2) and A is theelectrode area. The difference E - Ez is the potential of the electrode relative to thepotential where the excess charge on the electrode is zero. That point, Ez, is calledthe potential of zero charge (the PZC; see Section 13.2.2). One can think of the expansion at the DME as creating new surface in an uncharged state, which then requires charging from the PZC to the working potential. By differentiating (7.1.15),one obtains/c = ^ = C i ( £ z - £ ) f(7.1.16)since Q and E are both effectively constant during a drop's lifetime.
From (7.1.3), dA/dtcan be obtained, and one finds that/c = 0.00567Ci(£z -E)m2/3t~l/3(7.1.17)where ic is in fxA if Q is given in ^F/cm 2 . Typically, Q is 10 to 20 fiF/cm2. Several important conclusions can be drawn from (7.1.17):1. The average charging current over a drop's lifetime is about the same magnitudeas the average faradaic current for an electroactive substance present at the 10~57.1 Behavior at Polarographic Electrodes271M level; thus we understand why the ratio of limiting current to residual currentdegrades badly at a DME in this concentration range.
Charging current, morethan any other factor, limits detection in dc polarography at a DME to concentra6tions above 5 X 10" M or so (Section 7.3.1).2.Note that if C\ and tmax are not strongly varying functions of potential, /c is linear with E. As shown in Figure 7.1.5, experimental residual current curves actually are fairly linear over wide ranges, and this behavior provides thejustification for the common practice of measuring /d for a polarographic waveby extrapolating the baseline residual current as shown in Figure 7.1.6. Notealso that the capacitive current vanishes and changes sign at E = Ez (see alsoFigure 7.3.3).3.Another important contrast between /c and /d is in their time dependencies.
Aswe have seen, /d increases monotonically and reaches its maximum value at r m a x .Х1ЪThe charging current decreases monotonically as Г[see (7.1.17)], because therate of area increase slows as the drop ages. Thus, the charging current is at itsminimal value at £max. This contrast underlies some approaches to increasing polarographic sensitivity by discriminating against ic in favor of /d.
We will discussthem in Section 7.3.Since an SMDE has a fixed area as the current is sampled, dA/dt = 0, and the charging current due to drop expansion is zero. Thus, equation 7.1.17 does not apply to an-0.4-0.4-0.8E {V vs. SCE)-1.2Figure 7.1.5 Residual currentcurve for 0.1 M HC1. The sharplyincreasing currents at potentialsmore positive than 0 V and morenegative than - 1 . 1 V arise fromoxidation of mercury and reductionof H + , respectively.
The currentbetween 0 V and - 1 . 1 V is largelycapacitive. The PZC is near-0.6 V vs. SCE. (From L. Meites,"Polarographic Techniques,"2nd ed., Wiley-Interscience,New York, 1965, p. 101, withpermission.)272Chapter 7. Polarography and Pulse VoltammetryFigure 7.1.6 Method for obtaining /d from a wavesuperimposed on a sloping baseline of residual current.SMDE. In most experiments with an SMDE, the residual current is almost entirely offaradaic origin and is often controlled by the purity of the solvent-supporting electrolytesystem (Section 7.3.2).7.2 POLAROGRAPHIC WAVES7.2.1Reversible SystemsIn Section 7.1.2, we saw that, to a first approximation, current flow at the DME canbe treated as a linear diffusion problem. The time dependence of the area is takeninto account directly in terms of т 2 / 3 г~ 1 / 6 , and a multiplicative factor, (7/3) 1/2 , accounts for increased mass transport due to the "stretching" of the diffusion layer.These concepts apply equally well to i(t) at a potential on the rising portion of thewave, as expressed in (5.4.17).
Consequently the wave shape found for sampledcurrent voltammetry, (5.4.22), applies also to polarography at the DME, which is inessence a sampled-current voltammetric experiment if the rate of potential sweep issufficiently slow that E is virtually constant during a drop's lifetime (27, 28) [seealso Section 5.4.2(b)].Similarly, the surface concentrations are described by (5.4.29) and (5.4.30); hence(5.4.31) and (5.4.32) are valid for the DME. In this case, however, the maximum diffusion current is given by (7.1.7), and the analogues to (5.4.65) and (5.4.66) are(Omax = 708«Z)i / 2 m 2 / 3 C x [CS - C o (0, 01/2(Omax = 7 0 8 ^ r n2/3^xCR(0, 0(7.2.1)(7.2.2)Obviously these relations still follow the forms( 0 m a x = nFAmo[C% - C o (0, t)]( 0 m a x = nFAmR[CR(0, i) - C*](7.2.3)(7.2.4)where m0 is now [(7/3)£>о/тПтах]1/2 and m R is defined analogously.The SMDE adheres in detail to the treatment of Sections 5.4.1 and 5.4.3, providedthat the sampling time т is short enough for linear diffusion to apply, as is true in normalpractice.Section 5.4.4, dealing with applications of reversible sampled-current voltammograms, applies very generally to dc polarography at the DME or to normal pulse polarography at the SMDE.7.2.2Irreversible SystemsThis section concerns special characteristics of irreversible waves at a DME (28, 29, 30).Polarography at an SMDE adheres to the results of Section 5.5.1, provided that linear diffusion effectively applies, as is normally true.7.2 Polarographic Waves273Koutecky treated the totally irreversible system at the DME and expressed the resultas (29, 30)(7.2.5)where x = (l2/l)l/2k{t^JD^and F2(x) is a numeric function computed from a powerseries.
Table 7.2.1 gives some representative values. One can analyze an irreversible polarogram by rinding i/id at various points on the wave, then, from the Koutecky function,finding the corresponding values of x- From these, one obtains &f as a function of E,which can be further distilled to k° and a, if the mechanism is understood well enough tomake these parameters meaningful (e.g., there is a one-step, one-electron reaction or theinitial step in an «-electron process is an irreversible rate-determining electron transfer;see Section 3.5.4).A simplified method for treating totally irreversible polarographic waves was proposed by Meites and Israel (28, 31). From the definition of x and kf,(7.2.6)where к® is the value of kf at E = 0 on the potential scale in use.
Its value is k° exp(afE° ). One can take logarithms of (7.2.6) and rearrange the result to2303RT,aFX(7.2.7)From Koutecky's values of F2(x)> Meites and Israel found that the equationЫт,V-0.130 + 0.9163 log -г-1—.(7.2.8)is valid for 0.1 < (i/id) < 0.94. Substitution into (7.2.7) gives, at 25°C,(7.2.9)TABLE 7.2.1 Shape Function of aTotally Irreversible WaveaXi/idXi/id0.050.10.20.30.40.50.60.70.80.91.01.20.04280.08280.15510.21890.27490.32450.36880.40860.44400.47610.50500.55521.41.61.82.02.53.04.05.010.020.050.00.59700.63260.66230.68790.73910.7730.8250.85770.92680.96290.98511a00Originally reported in references 29 and 30.274Chapter 7. Polarography and Pulse Voltammetrywith(7.2.10)A plot of E vs. log [(/d — /)//] for an irreversible system should be linear with a slope of54.2/a mV at 25°C.
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