A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 89
Текст из файла (страница 89)
Note that at the limiting current, dildE approaches zero, so that a uniform current distribution is always obtained inthis circumstance.9.3.6Considerations in Experimental Application of the RDEThe equations derived for the RDE will not apply at very small or very large values ofo). When со is small, the hydrodynamic boundary layer [y^ ~ 3(v/(o)1^2] becomes large,and when it approaches the disk radius r\, the approximations break down. Thus, theA. 10.0MHCIO4B.
1.0MHCIO4С1.0 MKCID. 0.1 MHCIO 4E. 0.1 MNaOHF. 0.1 MKCIG. 10mMHCIO 4H. 10mMNaOHJ. 10mMKCIK. 1 mM KCI— +1log ( O1-7I-6I-5кII-4J H G FE D СI I I I II I I II-3-2-1Nonuniform currentdistributionВАIII+102--3Лгл (cm)0.54--5Uniform currentdistribution--6J0.1--7ISFigure 9.3.11 Diagnostic plotfor the uniformity of currentdistribution at an RDE.Conductivities of some typicalbackground electrolytes at 25 °C inaqueous solution are marked onlog (/Coo) scale in the figure.
Note:1di/dr] is in units of И " ; Коо isbulk electrolyte conductivity inП ^ с п Г 1 . [From W. J. Alberyand M. L. Hitchman, "Ring-DiscElectrodes," Clarendon, Oxford,1971, Chap. 4, by permission ofOxford University Press.]9.3 Rotating Disk Electrode < 347lower limit for со is obtained from the condition v\ > 3(*V<w) , that is, со > lOv/rf. For21v = 0.01 cm /s and r\ = 0.1 cm, со should be larger than 10 s" . Another problem occurs when recording i-E curves at the RDE at low values of со.
The derivation involvedan assumed steady-state concentration at the electrode surface (i.e., (дСо/dt — 0). Thus,the rate at which the electrode potential is scanned (V/s) must be small with respect toсо to allow the steady-state concentrations to be achieved. If the scan rate is too large fora given со, the i-E curves will not have the S-shape predicted by, for example, (9.3.34),but will instead show a peak, as in linear scan voltammetry at a stationary electrode.The question of transients and the time response at the RDE is covered further in Section 9.5.1.The upper limit for со is governed by the onset of turbulent flow. This occurs at theRDE at a critical Reynolds number, Re c r , larger than about 2 X 105 (7, 13).
In this system,i>ch is the velocity of the edge of the disk согъ and the characteristic distance / is r± itself.Thus, from (9.2.10),vchlcor\(9.3.51)and the condition for nonturbulent flow is со < 2 X \05v/r^. For the r\ and v values assumed, со should be less than 2 X 105 s" 1 . The transition to turbulent flow can occur atmuch lower values of со when the surface of the disk is not perfectly polished, when thereare small bends or eccentricities in the RDE shaft, or when the cell walls are too close to theelectrode surface. Also, at very high rotation rates, there can be excessive splashing or theformation of a vortex around the electrode.
In practice, the maximum rotation rates are frequently at 10,000 rpm or at со ~ 1000 s" 1 . Thus, the ranges of со and/are limited in mostRDE studies to 10 s" 1 < со < 1000 s" 1 or 100 rpm < / < 10,000 rpm.The theory developed for the RDE also assumes that the disk is precisely centeredon the axis of rotation. If the disk is located off the axis of rotation, either because ofthe way it is constructed (Figure 9.3.12) or because the shaft connecting the disk electrode to the rotator is bent, the observed currents will be larger than those for the centered disk.
This result comes about because the disk continually sweeps into a widerarea of solution and gains an additional mass-transfer contribution from radial diffusionin a manner analogous to that found with the rotating ring electrode (Section 9.4.1). Thetheory for this situation has been developed (16) and yields a limiting current density,7iim»o fjlim=l№l(iu/A)sl/3(9.3.52)where //c is given by (9.3.22), and s is an eccentricity factor (s = R/r), as shown in Figure 9.3. i2.Figure 9.3.12 A rotating disk electrode where theelectrode, with radius r, is offset from the axis ofrotation by a distance R.Chapter 9.
Methods Involving Forced Convection—Hydrodynamic MethodsIn standard RDE theory, one also assumes a negligible radial diffusion contributionto the current at the edges of the disk. This is true if the disk radius is sufficiently largethat diffusion to it can be treated as simple linear diffusion. When the disk electrode radius falls in the ultramicroelectrode regime, this is no longer the case, so different equations will apply to the rotating microdisk, as discussed in Section 9.7.9.4 ROTATING RING AND RING-DISK ELECTRODESReversal techniques are obviously not available with the RDE, since the product of theelectrode reaction is continuously swept away from the surface of the disk. At the RDE,reversal of the direction of the potential sweep under conditions where the scan rate is sufficiently slow compared to со (i.e., when no peak is seen during the forward scan) will justretrace the i-E curve of the forward scan.
Information equivalent to that available from reversal techniques at a stationary electrode is obtained by the addition of an independentring electrode surrounding the disk (Figure 9.4.1). By measuring the current at the ringelectrode, some knowledge about what is occurring at the disk electrode surface can beobtained. For example, if the potential of the ring is held at the foot of the wave for О +ne —> R, product R, formed at the disk, will be swept over to the ring by the radial flow,where it will be oxidized back to О (or be collected).The ring can also be used alone as an electrode (the rotating ring electrode), such aswhen the disk is disconnected.
Mass transfer to a ring electrode is larger than to a disk at aElectrode material(e.g., platinum)Insulator(e.g., Teflon)Shaft and ring material(e.g., brass)LJFigure 9.4.1Ring-disk electrode.9.4 Rotating Ring and Ring-Disk Electrodes349given A and a), because flow of fresh solution occurs radially from the area inside the ring,as well as normally from the bulk solution.The theoretical treatment of ring electrodes is more complicated than that of theRDE, since the radial mass-transfer term must be included in the convective-diffusionequation.
While the mathematics sometimes become difficult in these problems, the results are still easy to understand and to apply. The problem will only be outlined here; details of the mathematical treatment are given in references 7 and 15.9.4.1Rotating Ring ElectrodeConsider a ring electrode with an inner radius r 2 and outer radius r 3 [AT = rr(r2 — r2)].This could be, for example, the RRDE illustrated in Figure 9.4.1 with the disk electrode atopen circuit. When this electrode is rotated at an angular velocity, со, the solution flow velocity profile is that discussed in Section 9.3.1. The steady-state convective-diffusionequation that must be solved isd2Co\^ )( 9 A I )This is obtained from (9.3.12).
As with the RDE, symmetry considerations require thatthe concentrations be independent of </>, so that the derivatives in ф vanish. In addition,the mass transfer by diffusion in the radial direction, represented by the termsDo[(d2Co/dr2) + (l/r)(dCo/dr)], is, at usual flow rates, small compared to convection inthe radial direction, (vrdCo/dr); hence the diffusive terms are neglected. The boundaryconditions for the limiting ring current are:CO = C$fory^ooCo = Qaty = 0forг2<г<гъ-^=0aty = 0forr<r2dyWhen the values of vT and vy are introduced [see (9.3.9) and (9.3.10)], we obtain=Dow) bf)(dC0\where B' = 0.51СОЪ12У~112./£o\l/<?2Co\(dC0\(9A2),o...The current at the ring electrode is given by3iR = nFD027r] [-£)Jrdr(9.4.4)r2 \ °У /y=03The area of an infinitesimal section of ring of thickness Sr at a radius r is 7r(r + 8r)2 — тгг2 ~ 2ш(8г).
Thecurrent through this section isnFAnF2irr8r°\ dy ) y = 0The total ring current is the summation of (/R)S,lR *which yields, as 8r —>• 0, eq. (9.4.4).2Jr=r27r 2r=r2(dCo/dy)y=0r8r350Chapter 9. Methods Involving Forced Convection—Hydrodynamic MethodsThe solution to these equations yields the limiting ring current (17):fRi/tC = 0.62nFir{rl -(9.4.5)or, in general,- C0(y = O)]/CS)«R =(9.4.6)This can be written in terms of the disk current (9.3.30), which would be observed underidentical conditions for a disk of radius rl to yieldlR =or'D '(9.4.7)Л(9.4.8)Notice that for given reaction conditions {C% and со), a ring electrode will produce alarger current than a disk electrode of the same area. Thus, the analytical sensitivity of aring electrode (i.e., the current caused by a mass-transfer-controlled reaction of an electroactive species divided by the residual current) is better than that of a disk electrode, andthis is especially true of a thin ring electrode.
However, it is usually more difficult to construct a rotating ring electrode than an RDE.9A 2The Rotating Ring-Disk ElectrodeIn a rotating ring-disk electrode (RRDE), the current-potential characteristics of the diskare unaffected by the presence of the ring, and the properties of the disk are as describedin Section 9.3. (In fact, if the disk current is found to change upon variation of the ring potential or current, one should suspect a defective RRDE or an undesirable coupling of thering and disk through solution uncompensated resistance).
Since RRDE experiments involve the examination of two potentials (that of the disk, £ D , and that of the ring, ER) andtwo currents (zrj and /R), the representation of the results involves more dimensions thanfor experimental results involving a single working electrode. RRDE experiments areusually carried out with a bipotentiostat (Section 15.4.4), which allows separate adjustment of £rj> and £ R (Figure 9.4.2a). However, since most RRDE measurements involvesteady-state conditions, it is possible to use an ordinary potentiostat to control the ring circuit and a simple floating power supply in the disk circuit (Figure 9A.2b).CounterCountertDiskDisk(a)(b)Figure 9.4.2 Block diagram of RRDE apparatus, (a) Bipotentiostat.
(b) Ordinary (threeelectrode) potentiostat and voltage divider.9.4 Rotating Ring and Ring-Disk Electrodes351Several different types of experiments are possible at the RRDE, The most commonare collection experiments, where the disk generated species is observed at the ring, andshielding experiments, where the flow of bulk electroactive species to the ring is perturbed because of the disk reaction.(a) Collection ExperimentsConsider the experiment in which the disk is held at a potential £ D , where the reaction О+ ne —> R produces a cathodic current /D, while the ring is maintained at a sufficiently positive potential, ER, that any R reaching the ring is rapidly oxidized (R —> О + ne), so thatthe concentration of R at the ring surface is essentially zero.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
