A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 58
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The large body of chronoamperometric theory for systems with coupledchemistry can be used directly to describe chronocoulometric experiments, because thereare no differences in fundamental assumptions. The only differences are that the responseis integrated in chronocoulometry and that the chronocoulometric experiment manifestsmore visibly the contributions from double-layer capacitance and electrode processes ofadsorbates.5.8.3Effects of Heterogeneous KineticsIn the foregoing discussion, we have examined only situations in which electroreactantsarrive at the electrode at the diffusion-limited rate.
At the extreme potentials required toenforce that condition, the heterogeneous rate parameters are experimentally inaccessible.On the other hand, if one wished to evaluate those parameters, it would be useful to obtaina chronocoulometric response governed wholly or partially by the interfacial chargetransfer kinetics. That goal can be reached by using a step potential that is insufficientlyextreme to enforce diffusion-controlled electrolysis throughout the experimental time domain.
In other words, steps must be made to potentials in the rising portion of the sampledcurrent voltammogram corresponding to the time scale of interest, and that time scalemust be sufficiently short that electrode kinetics govern current flow for a significant period.5.8 Chronocoulometry«i 215The appropriate experiment involves a step at t = 0 from an initial potential whereelectrolysis does not occur, to potential E, where it does.
Let us consider the special case(45, 46) in which species О is initially present at concentration CQ and species R is initially absent. In Section 5.5.1, we found that the current transient for quasireversibleelectrode kinetics was given by (5.5.11). Integration from t = 0 provides the chronocoulometric response:(5.8.9)where H = (kf/D}j2) + (£b/£>//2). For Htm > 5, the first term in the brackets is negligiblecompared to the others; hence (5.8.9) takes the limiting form:(5.8.10)A plot of the faradaic charge vs. ft2 should therefore be linear and display a negative intercept on the Q-axis and a positive intercept on the t1^2 axis. The latter involves a shorterextrapolation, as shown in Figure 5.8.4, hence it can be evaluated more precisely. Designating it as t\l2', we find H by the relation:H =7Г 'It}'2(5.8.11)With H in hand, kf is found from the linear slope, 2nFAkfC%l{HTr112).
Note that when E isvery negative, H approaches kf/D}^2, and the slope approaches the Cottrell slope,2nFAD^2CQ/TTl/2. Moreover, H is large, so that the intercept approaches the origin. Thislimiting case is clearly that treated in Section 5.8.1.Equations 5.8.9 and 5.8.10 do not include contributions from adsorbed species ordouble-layer charging. For accurate application of this treatment, one must correct forthose terms or render them negligibly small compared to the diffusive component to thecharge.In practice, it is quite difficult to measure kinetic parameters in this way, so themethod is not widely practiced.
The principal value in considering the problem is in the101520<F, msec172Figure 5«8.4 Chronocoulometric response for10 mM C d 2 + in 1 M Na 2 SO 4 . The working electrodewas a hanging mercury drop with A = 2.30 X 10~ 2cm 2 . The initial potential was -0.470 V vs. SCE,and the step potential was -0.620 V. The slope of theplot is 3.52 ^C/ms 1 7 2 and t\12 = 5.1 ms 1/2 . [From J. H.Christie, G. Lauer, and R. A. Osteryoung, /.Electroanal С hem., 7, 60 (1964), with permission.]216Chapter 5. Basic Potential Step Methodsinsight that it provides to the origin of negative intercepts in chronocoulometry, which arerather common, especially with modified electrodes (Chapter 14). The lesson here is thata rate limitation on the delivery of charge to a diffusing species produces an interceptsmaller than predicted in Sections 5.8.1 and 5.8.2.
A negative intercept clearly indicatessuch a rate limitation. It may be due to sluggish interfacial kinetics, as treated here, but itmay also be from other sources, including slow establishment of the potential because ofuncompensated resistance. Using a more extreme step potential can ameliorate this behavior if it is not itself the object of study.5.9 SPECIAL APPLICATIONS OF ULTRAMICROELECTRODESThe large impact of ultramicroelectrodes is rooted in their ability to support very usefulextensions of electrochemical methodology into previously inaccessible domains of time,medium, and space. That is, UMEs allow one to investigate chemical systems on timescales that could not previously be reached, in media that could not previously be employed, or in microstructures where spatial relationships are important on a distance scalerelevant to molecular events.5.9ACell Time Constants and Fast ElectrochemistryIn Section 2.2 we learned that the establishment and control of a working electrode's potential is carried out operationally by adjusting the charge on the double layer.
In Sections1.2.4 and 5.2.1, we found that changing the double-layer charge, hence changing the potential, involves the cell time constant, /?uCd, where Ru is the uncompensated resistanceand Cd is the double-layer capacitance. It is not meaningful to try to impose a potentialstep on a time scale shorter than the cell time constant. In fact, the full establishment of apotential step requires ~5i? u C d , and the added time for taking data normally implies thatthe step must last at least 10/?uCd, and often more than 100/?uCd. To a large extent, thesize of the electrode controls the cell time constant and, therefore, the lower limit of experimental time scale.For example, let us consider a disk-shaped working electrode operating in an electrolyte solution such that the specific interfacial capacitance (capacitance per unit area, C d )is in the typical range of 10-50 fiF/cm2.
ObviouslyC d = m%C°d(5.9.1)With a radius of 1 mm, Cd is 0.3-1.5 /xF, but for TQ = 1 /im, C d is six orders of magnitudesmaller, only 0.3-1.5 pF.The uncompensated resistance also depends on the electrode size, although in a lesstransparent way. As the current flows in solution between the working electrode and thecounter electrode, one can think of it as passing along paths of roughly equal length, terminated by the faces of the two electrodes.
These paths do not generally involve thewhole of the electrolyte solution, but are largely contained in the portion bounded by theelectrodes and the closed surface representing the locus of minimum-length connectionsbetween points on the perimeters of the electrodes (Figure 5.9.1). Usually the counterelectrode is much larger than the working electrode; hence this solution volume is broadlybased on the end connecting to the counter electrode, but narrowly based at the workingelectrode. The precise value of Ru depends on where the tip of the reference electrode intercepts the current path. Figure 5.9.1 shows the situation for a working electrode havinga radius one tenth that of the counter electrode, but if the working electrode is a UME, itsradius can easily be a thousandth or even a millionth of the counter electrode's radius.
In5.9 Special Applications of Ultramicroelectrodes «i 217Electrolyte SolutionVolume ContainingCurrent PathsCounterElectrodeFigure 5.9.1 Schematic representationof the volume of solution containingcurrent paths between disk-shapedworking and counter electrodes situatedon a common axis. Current paths arelargely, but not strictly, confined to thevolume defined by minimum-lengthconnections between electrodeperimeters.Electrolyte Solutionsuch a case, all of the current must pass through a solution volume of extremely smallcross-sectional area near the working electrode, and it turns out that this is the part of thecurrent path that defines the value of Ru.The resistance offered by any element of solution to uniform current flow is1/(KA), where / is the thickness of the element along the current path, A is the crosssectional area, and к is the conductivity.
Thus the resistance of the disk-shaped volumeof solution adjacent to the working electrode and extending out a distance ro/4 is1/(4тпсго). A similar relation applies for the counter electrode; but its radius is typically 103 to 10 6 bigger than r 0 . One can readily see that the resistance contributed by amacroscopic portion of the current path extending out from the counter electrode isnegligible compared to that developed in the tiny part of the solution less than ro/4from the working electrode.In a system with spherical symmetry, which would apply approximately to any working electrode that is essentially a point with respect to the counter electrode, the uncompensated resistance is given by (47),147ГКГП(5.9.2)where x is the distance from the working electrode to the tip of the reference.
In a UMEsystem, it is not generally practical to place a reference tip so that x is comparable to r 0 ;thus the parenthesized factor approaches unity, and Ru becomes 1/(4тткг0).Note that Ru is inversely proportional to r 0 , so that Ru rises as the electrode is madesmaller. This behavior is rooted in the considerations given above, for as r 0 is reduced, thesolution volume controlling Ru also becomes smaller, but with a length scale that shrinksproportionately with r 0 and a cross-sectional area that shrinks with the square.
The effectof decreasing area overrides that of decreasing thickness.From (5.9.1) and the limiting form of (5.9.2), one can express the cell time constantas(5.9.3)Even though Ru rises inversely with r 0 , C d decreases with the square; hence i ? u Q scaleswith TQ. This is an important result indicating that smaller electrodes can provide access tomuch shorter time domains. Consider, for example, the effect of electrode size in a systemwith C^ = 20 fjiF/cm2 and к = 0.013 П ^ с п Г 1 (characteristic of 0.1 M aqueous KC1 atambient temperature).
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