A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 90
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We are interested in the magnitude of the ring current, /R, under these conditions; that is, we want to know how much ofthe disk-generated R is collected at the ring. The approach is again to solve the steady-statering convective-diffusion equation, (9.4.3), this time for species R:д2СЛ)The boundary conditions are more complex because of the structure of the system:1.W 4 9 )At the disk (0 < r < rx), the flux of R is related to that of О by the usual conservation equation:(dCr( 9 41From the results in Section 9.3.2,dy Jy=02.(9.4.1nFADRIn the insulating gap region (rj ^ r < r 2 ), no current flows, so that(9.4.12)3.At the ring (r 2 < r < r 3 ), under limiting current conditions,CR(y = 0) = 0(9.4.13)We assume that R is initially absent from the bulk solution (lim CR = 0) and that thebulk concentration of О is CQ. AS in (9.4.4), the ring current is given byrdr(9.4.14)This problem can be solved in terms of dimensionless variables using the Laplace transform method and results have been given in terms of Airy functions (18, 19).
It turnsout that the ring current is related to the disk current by a quantity N, the collection efficiency,N=(9.4.15)'Dthat depends only on r b r 2 , and r 3 and is independent of w, CQ, DO, DR, etc. The collection efficiency can be calculated fromN = 1 - F(a/I3) + /32/3[l - F(a)] - ( 1 + a + £)2/3{l - F[(a//3)(1 + a + 0)]} (9.4.16)352Chapter 9. Methods Involving Forced Convection—Hydrodynamic Methodswhere a = ir-Jr^ ~ 1, >3 is given by (9.4.8), and the F values are defined byf d + <?I/3)313( 9 A 1 7 )The function F(6) and values of N for different ratios r2/ri and гъ1г2 are tabulated in reference 18.
One can also determine N experimentally for a given electrode, by measuring -/RA'D f° r a system where R is stable. Once N is determined, it is a known constantfor that RRDE. For example, for an RRDE with rx = 0.187 cm, r 2 = 0.200 cm, andr 3 = 0.332 cm, N = 0.555; that is, 55.5% of the product generated at the disk is collected at the ring. Qualitatively, N becomes larger as the gap thickness (r 2 - rx) decreases and as the ring size (r^ — r 2 ) increases. The concentration profiles of R in thevicinity of the RRDE surface are shown in Figure 9.4.3.In a typical collection experiment, one plots /^ and /R as functions of ED (at a constant £R) (Figure 9.4.4a).
Stability of the product is assured if N is independent of i& anda). If R decomposes at a rate sufficiently high that some is lost in its passage from disk toring, the collection efficiency will be smaller and will be a function of со, /r> or C o . Information about the rate and mechanism of decay of R can thus be obtained from RRDE collection experiments (see Chapter 12). Information about the reversibility of the electrodereaction can be obtained by plotting the ring voltammogram (/R VS. £ R ) at a constant valueof £ D , and comparing the £ 1 / 2 with that of the disk voltammogram (Figure 9.4.4/?).(b) Shielding ExperimentsThe current at the ring electrode for the reduction of О to R when the disk is at open circuit is given by (9.4.5) to (9.4.8).
The limiting current at the ring with /D = 0, denoted /R his given by (9.4.8), which is rewritten as(9.4.18)_ o2/3:~ P lwhere /D / is the limiting current that could be achieved at the disk electrode if it wereactive.If the disk current is changed to a finite value, /D, the flux of О to the ring will be decreased. The extent of this decrease will be the same as the flux of stable product R to thering in a collection experiment—Мр. Hence the limiting ring current /R / is given by(9.4.19)DiscGapRingFigure 9.4.3 Concentrationprofiles of species R at an RRDE.Concentrations increase fromcurve 1 to curve 6.
For the disk (0< r < п), дСк/дг = 0; in the gap(П < r < r2), (дСк/Эу)у=0 = 0;and at ring surface (r 2 < r < r 3 ),CR(y = 0) = 0. [From W. J. Alberyand M. L. Hitchman, "Ring-DiscElectrodes," Clarendon, Oxford,1971, Chap. 3, by permission ofOxford University Press.]9.5 Transients at the RDE and RRDE - 353z(cath)M• О + ne -» RR -> О + ne(anod)(a)• О + ne - > RNiD, lim;0to, limto, lim - to, lim ~J=f>2/3;PlD, limfFigure 9.4.4 (a) Diskvoltammogram. (7) /D vs. £ D and(2) iR vs. ED with ER = El.(b)Ring voltammograms. (3) /R vs.ER, I'D = 0 (£ D = £ 0 and(b)(4) iR vs.
ER, iD = iDMc (ED = E2).(This equation holds for any value of /D, including /D = 0 and / D = /D,/.) Using (9.4.18),we have for the special case / D = / D / :— "^Г/R~2/3^(9.4.20)Thus, when the disk current is at its limiting value, the ring current is decreased bythe factor (1 - Nfi~2/3). This factor, always less than unity, is called the shieldingfactor. These relations are easier to understand when the complete 1-Е curves are considered (Figure 9.4.4b). One sees that the effect of switching /D from 0 to zD / is toshift the entire ring voltammogram (/R vs. £ R ), which is assumed to be reversible, bythe amount M D ,/.Other dual electrode systems that operate at steady state and show similar shieldingand collection effects include microelectrode arrays and the scanning electrochemical microscope (SECM).
With microelectrode arrays (Section 5.9.3), one monitors diffusion between two neighboring electrodes. In a similar way, one can use the SECM (Section 16.4)to study diffusion between an ultramicroelectrode tip and substrate electrode. In both ofthese systems, convective effects are absent and the time for interelectrode transit is governed by the distance between the electrodes.9.5 TRANSIENTS AT THE RDE AND RRDEAlthough a major advantage of rotating disk electrode techniques, compared to stationary electrode methods, is the ability to make measurements at steady state without the need to consider the time of electrolysis, the observation of currenttransients at the disk or ring following a potential step can sometimes be of use inunderstanding an electrochemical system.
For example the adsorption of a component,354Chapter 9. Methods Involving Forced Convection—Hydrodynamic MethodsA, on the disk electrode can be studied by noticing the transient shielding of thering current for the electrolysis of A upon stepping the disk potential to a valuewhere A is adsorbed.9.5.1Transients at the RDEThe treatment of the non-steady-state problem at the RDE requires solution of the usual diskconvective-diffusion equation, (9.3.14), but with inclusion of the дС/dt term, that is,(9.5.1)where В' = 05\o?l2v 1 / 2 .
This has been accomplished by approximation methods (20,21) and by digital simulation (22). For a potential step to the limiting current region of thei-E curve, the instantaneous value of //, denoted //(*), is given approximately by (20). iff) _л _ ^__J-m2ir2Dot)- 1 + 2 2 exp(9.5.2)m=\where ij{ss) is the value of // as t —» °°, and 8O is given in (9.3.25). An implicit approximateequation for R(t) obtained by the "method of moments" has also been proposed (21):1.8049Ы1 3- R(t)]H[i-arctan2R(t) + 1——V V3(9.5.3)Both of these results are in good agreement with the digital simulation (22); a typicaldisk transient is shown in Figure 9.5.1. At short times, when the diffusion layer thicknessis much thinner than 80, the potential step transient follows that for a stationary electrode [equation 5.2.11].
The time required for the current to attain its steady-state valuecan be obtained from the curve in Figure 9.5.1. The current is within 1% of i[(ss) at atime r when<OT(D/J/) 1 / 3 (0.51) 2 / 3 > 1.3(9.5.4)1/3or, taking (D/v) — 0.1, when cor > 20. Thus for со = 100 s (or a rotation rate of about1000 rpm), т « 0.2 s.0.51.01.52.0a>f(ZVv)1/3(0.51)2/32.5Figure 9.5.1 Disk-currenttransient for potential step at thedisk: Curve is simulated; pointsare from theoretical equationsgiven by Bruckenstein and Prager(O) and Siver (•). [From К. В.Prater and A.
J. Bard,/. Electrochem. Soc, 117, 207(1970), with permission of thepublisher, The ElectrochemicalSociety, Inc.]9.5 Transients at the RDE and RRDE9.5.2355Transients at the RRDEConsider the experiment in which the ring of the RUDE is maintained at a potential whereoxidation of species R to О can occur, and the disk is at open circuit or at a potentialwhere no R is produced. If R is then generated at the disk by a potential step to an appropriate value or by a constant current step, a certain time will be required for R to transitthe gap from the outside of the disk to the inside edge of the ring (the transit time, t'). Anadditional time will be required until the disk current attains its steady-state value.
Therigorous solution for the ring current transient, /R(0» involves solving the non-steady-stateform of (9.4.9):(9>5-5)This rather difficult problem is discussed by Albery and Hitchman (23), and several approaches to the solution and approximate equations are given. These may also be obtained by the digital simulation method (see Appendix B.5) (22), and Figure 9.5.2 is adisplay of typical simulated ring-current transients for both a current step [to ivj(ss)] anda potential step to the limiting-current region at the disk electrode. Note that the ring current rises more rapidly when a potential step is applied.
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