J. Phys. Chem. B 1998, 102, 1515-1521
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Heterogeneous ECE Processes at Channel Electrodes: Analytical Theory. Distinguishing Hetero- and Homogeneous ECE Reactions W. J. Aixill, J. A. Alden, F. Prieto, G. A. Waller, and R. G. Compton* Physical and Theoretical Chemistry Laboratory, Oxford UniVersity, South Parks Road, Oxford OX1 3QZ, U.K.
M. Rueda Department of Physical Chemistry, Faculty of Pharmacy, UniVersity of SeVilla, 41012 SeVilla, Spain ReceiVed: June 25, 1997; In Final Form: December 16, 1997
Analytical theory is developed for heterogeneous ECE reactions occurring at channel electrodes. Simple expressions are presented that allow the ready mechanistic interpretation of experimental data and the deduction of corresponding rate constants. These are shown to be in excellent agreement with numerical simulations and consistent with experiments conducted on the reduction of nitromethane in buffered aqueous solution at Hg/Cu electrodes deposited on a platinum substrate thought to proceed via a heterogeneous ECEEE mechanism. The experimental resolution between heterogeneous and homogeneous ECE processes in general is discussed.
Introduction
SCHEME 1
Many electrode processes proceed via a succession of coupled electron transfer events and chemical transformations of intermediates. Such reactions are often readily classified by the Testa and Reinmuth notation1 in which the symbols E and C are used to represent the sequence of such steps so that, for example, ECE denotes the following mechanism:
E step:
A ( e- f B
C step:
BfC
E step:
C ( e- f products
However it has been recognized and elegantly argued by Laviron2 that identification of the C step as being authentically homogeneous or heterogeneous is not necessarily straightforward, especially if the mechanistic interrogation relies exclusively on voltammetric methods. In particular, recourse to spectroelectrochemical procedures may be essential in some situations. In the following we first develop the voltammetric theory for heterogeneous ECE processes at channel electrodes and compare the mass transport and cell geometry dependence of the limiting current response with that for the analogous homogeneous process for which the theory is well established3 so as to establish the circumstances under which channel electrode voltammetry may or may not discriminate between the two limiting possibilities. The methodology is then applied to the characterization of the reduction of nitromethane at mercury electrodes in aqueous solution which is thought to involve the uptake of four electrons and four protons in addition to the cleavage of a N-O bond to give methylhydroxylamine.4-8 Evidently a large number of mechanistic possibilities exist in this case! Nevertheless, impedance voltammetry9-12 has been successfully deployed to unravel some of the key details of the
mechanism in the case of the reduction of nitromethane in aqueous solution. Specifically in basic solution9,10,12 it wasestablished that the electron transfers were Nernstian on the time scales studied, and the mechanism starts with an elementary one-electron transfer followed by a fast heterogeneous chemical step, C1, as shown in Scheme 1. The radical thus formed, possibly •CH2-N(O-)OH, can then undergo another chemical reaction (loss of hydroxide), C2 in the scheme, to form a species which is immediately transformed to methylhydroxylamine in a second reduction involving the uptake of three electrons and three protons. On pure mercury electrodes the second chemical reaction, C2, was found to be relatively slow so that the inferred radical intermediate, •CH2-N(O-)OH, had time to diffuse from the interface, thus partially preventing the second reduction. It is therefore apparent that the process can be represented as an ECEEE mechanism of either the heterogeneous or homogeneous type depending on whether step C1 or C2 is rate determining. In the following we will explore the ability of channel electrode voltammetry to identify the different mechanisms.
S1089-5647(97)02080-4 CCC: $15.00 © 1998 American Chemical Society Published on Web 02/06/1998
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Aixill et al.
Figure 1. (a) Practical channel flow cell for mechanistic electrochemical studies. (b) Schematic diagram which defines the coordinate system adopted in the text.
Theory We consider a heterogeneous ECE mechanism where the chemical transformation
BfC occurs exclusively on the electrode surface. The convectivediffusion equations for a channel electrode describing the A and B distributions in time (t) and space (x,y) are 2
∂ [A] ∂[A] ∂[A] ) DA 2 - Vx ∂t ∂x ∂y
(1)
∂[B] ∂2[B] ∂[B] ) DB 2 - Vx ∂t ∂x ∂y
(2)
where DL is the diffusion coefficient of species L () A or B) and the Cartesian coordinates x and y can be understood with reference to Figure 1. Vx is the solution velocity in the x-direction; the components in the y- and z-directions are zero. Given laminar flow conditions and that a sufficiently long leadin length exists upstream of the electrodes so as to allow the full development of Poiseuille flow, then Vx is parabolic
(
Vx ) Vo 1 -
)
(h - y)2 h2
(3)
where h is the half-height of the cell and Vo is the solution velocity at the center of the channel. Equations 1-3 assume that axial diffusion effects may be neglected; this is valid provided the electrodes considered are not of microelectrode dimensions.13-15 The boundary conditions relevant to the case of the transportlimited electrolysis of A are as follows
all y, x < 0, [A] ) [A]bulk, [B] ) 0, [C] ) 0 (4) y ) 0, 0 < x < xe, [A] ) 0, ∂[A] ∂[B] DA ) -DB + khet[B] (5) ∂y ∂y y ) 0, 0 < x < xe, [C] ) 0 y ) 2h, all x, DA
∂[A] ∂[B] ) DB )0 ∂y ∂y
(6) (7)
The set of mass transport eqs 1 and 2 can be solved under the above-specified boundary conditions by direct application of an implicit finite-difference method previously optimized for the solution of mass transport problems in the channel electrode geometry.16,17 References 14 and 15 contain a fully comprehensive account of the computation of the concentration profiles within a channel electrode; the interested reader is directed to those sources for further detail. In the present work such simulations are used for comparison with the analytical results developed below; these were performed using programs written in FORTRAN 77, available on request from the authors, and executed on a Silicon graphics Indigo.2 In typical computations 1000 grid points were used over the electrode length (xe) and 1000 over that fraction (φ) of the total channel depth (2h) which corresponds to no less than the maximum size of the diffusion layer thickness ( -1)
(25)
1
n!
sin[2/3(n + 1)π] (31/3x)n
(20)
The use of eqs 24 and 25 then permits the deduction of the following low-K approximation:
neff ) 1 - 1.2377Khet + 1.6982Khet2 - 2.6085Khet3 + and is the solution to the following differential equation
d2Ai(x) dx2
4.3867Khet4 (26) Alternatively, for high-Khet values
) x (Ai(x))
(21)
neff ) 2 - 0.7369Khet-1 + 0.4797Khet-2 - 0.2583Khet-3 + 0.0952Khet-4 - 0.01856Khet-5 (27)
To characterize the ECE behavior we seek the effective number of electrons transferred as a function of Khet. Equation 8 indicates that
neff )
L-1
∫01(∂L[A]/∂ξ)ξ)0 dχ + L-1∫01KhetL[B]ξ)0 dχ 1 L-1∫0 (∂L[A]/∂ξ)ξ)0 dχ
(22)
where L-1 signifies the inverse Laplace transform operation. neff takes values between 1 and 2. The lower value corresponds to slow rate constants (khet) or conditions of fast mass transport,
Theoretical Results and Discussion Figure 2 shows a plot of neff against Khet for an ECE process as deduced by numerical simulation of eqs 1 and 2. Also shown is the behavior calculated from the series in eqs 26 and 27 using all the terms given. Comparison of the analytical and numerical predictions shows that the low-K approximation holds to within 0.1% for log Khet < -0.35 while the high-K equation describes the simulated data to within 0.1% for log Khet > 0.45. In the range -0.35 < log Khet < 0.45 the simulated curve was found to be described by the equation
neff ) 1.5602 + 0.5439 log Khet
(28)
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Figure 3. Variation of neff with Vo1/3 for an authentic heterogeneous ECE process with khet/cm s-1 ) 0.0004 and 0.0014 for a conventional channel flow cell of the geometry specified in the text.
Figure 5. The data of Figure 3 analyzed in terms of a homogeneous ECE reaction.
Figure 4. Variation of neff with Vo1/3 for an authentic heterogeneous ECE process with khet/cm s-1 ) 0.014 and 0.0550 for a fast flow channel electrode cell of the geometry specified in the text.
to within 0.85%. It may be concluded that eqs 26-28 are adequate to describe the experimental behavior for almost all experimental purposes. Equations 26-28 define the channel electrode voltammetric response for a heterogeneous ECE process; analogous equations have been established and verified for the corresponding homogeneous process.3 It is therefore interesting to ask, can the two mechanisms be confidently distinguished on the basis of limiting current flow rate data alone? To pursue this question we note that such experiments may be conducted either using a conventional channel electrode18,19 characterized by a typical geometry of 2h ) 0.1 cm, w ) 0.3 cm, d ) 0.6 cm, and xe ) 0.3 cm and a volume flow rate, Vf, range of 4 × 10-4 to 0.25 cm3 s-1 or by a “fast flow” channel electrode23 where the corresponding parameters are 2h ) 0.0125 cm, w ) 0.15 cm, d ) 0.2 cm, xe ) 0.0025 cm, and 0.1 < Vf/cm3 s-1 < 2.5. To examine the possibility of mechanistic resolution, neff versus flow rate data was generated for authentic heterogeneous ECE processes with D ) 2 × 10-5 cm2 s-1 and rate constants of khet ) 0.0004 and 0.0014 cm s-1 in the case of the conventional channel and of khet ) 0.014 and 0.055 cm s-1 for the fast flow situation. Figures 3 and 4 show the variation of neff with Vo (cm s-1) in each case. These data were then analyzed according to the homogeneous ECE theory which predicts neff is a unique function of the dimensionless rate constant,
Khomo ) khomo(4h4xe2/9Vf2D)1/3
(29)
where Vf ) 4Vohd/3. The known3 working curves permit the inference of Khomo as a function of flow rate for the “synthetic” heterogeneous ECE data. If the latter were indistinguishable from a homogeneous ECE process, then a plot of Khomo found in this way against (flow rate)-2/3 would be a straight line passing through the origin. Figures 5 and 6 show that this is
Figure 6. The data of Figure 4 analyzed in terms of a homogeneous ECE reaction.
not the case but rather that significantly curved plots are obtained. This observation suggests that provided neff data can be found which is sufficiently deviated from limiting values of 1 or 2susing either a conventional or fast flow channel electrodesthen variable mass transport measurements should permit a clear resolution of homogeneous and heterogeneous ECE reactions. Finally it is illuminating to use the simulation data to visualize the concentration profiles of the species A, B, and products in a channel electrode for a heterogeneous ECE process. Simulations were carried out for a geometry of 2h ) 0.0125 cm, xe 0.0025 cm, w ) 0.15 cm, d ) 0.2 cm, and a diffusion coefficient of 2 × 10-5 cm2 s-1. Figure 7 shows three sets of concentration profiles, each set showing the steady-state spatial distributions of A, B, and the product species for different flow rates (Vf) and heterogeneous rate constants (khet). Comparison of parts a-c with d-f of Figure 7 show the effect of increasing the kinetics at a fixed flow rate; as expected an increased formation of product at the expense of species B is evident. Comparison parts d-f with g-i of Figure 7 show the effect of decreasing the flow rate for fixed kinetics; again this causes increased product formation at the expense of, now, both A and B. Experimental Section The fast flow channel electrode system has been described previously.23 In essence a pressurized system is used to force solutions at flow rates of up to ca. 5 cm3 s-1 through the channel cell which is fabricated in silica (Optiglass Ltd, Hainault, Essex, U.K.). The cross-sectional area of the flow cell is approximately 2 mm × 0.1 mm. Platinum microband electrodes were made
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J. Phys. Chem. B, Vol. 102, No. 9, 1998 1519
Figure 7. Concentration profiles for a heterogeneous ECE process in a channel electrode of the geometry specified in the text. The different profiles relate to the following values of volume flow rate (Vf/cm3 s-1) and heterogeneous rate constant (khet/cm s-1): (a) profile of A for Vf ) 2.9 and khet ) 0.25; (b) profile of B for Vf ) 2.9 and khet ) 0.25; (c) profile of product species for Vf ) 2.9 and khet ) 0.25; (d) profile of A for Vf ) 2.9 and khet ) 0.055; (e) profile of B for Vf ) 2.9 and khet ) 0.055; (f) profile of product species for Vf ) 2.9 and khet ) 0.055; (g) profile of A for Vf ) 0.1 and khet ) 0.055; (h) profile of B for Vf ) 0.1 and khet ) 0.055; (i) profile of product species for Vf ) 0.1 and khet ) 0.055. In all cases a fraction 0.02 of the channel depth (2h) is shown in the y-direction.
by sealing a platinum foil strip (Goodfellow Metals, Cambridge, U.K.) between two glass rods and lapping the unit to a smooth finish on one side using standard glass-working methods. The glass was chosen to be soda-glass rather than Pyrex, since the linear expansion coefficient of the former is more compatible with that of platinum24 and its lower softening temperature facilitates construction.24 The face forming the electrode was lapped to a flat surface and polished using a sequence of silicon carbide and alumina abrasives. Atomic force microscopy was used to monitor the lapping and polishing procedure to ensure that this did not cause undercutting of the electrode. The precise dimensions of the electrodes were measured using either a scanning electron microscope or a Topometrix AFM. To produce the Hg/Cu working electrodes the platinum band electrodes were electroplated first with copper using an aqueous 5 mM cupric nitrate/ 0.1 M KCl/20 mM NH4OH solution and second with mercury from a 0.12 M KCN/0.15 M Hg(NO3)2 solution following etching of the surface using 3 M KCN/1 M KOH to slightly recess the electrode so that a flat surface was obtained after the plating procedure. All solutions were made up using Elgastat (High Wycombe, U.K.) UHQ grade water of resistivity (18 MΩ cm). Solutions of nitromethane (puriss, absolute, Fluka) were prepared im-
mediately prior to the experiment and were buffered using buffer solutions based on the following: pH 1.8 0.2 M phosphoric acid/0.08 M KOH; pH 3-5 0.1 M citric acid/0.1 M KH2PO4; pH 7 0.2 M KH2PO4; pH 8-9 0.2 M boric acid. In each case the buffers were adjusted to the desired pH by addition of KOH solution and additionally contained 1.0 M KCl. Solutions were thoroughly purged of oxygen before use with nitrogen which had been presaturated by prepassage through a solution identical to that being degassed. Experimental Results and Discussion Experiments conducted using conventional channel flow cells on the reduction of nitromethane in aqueous solution (pH 8.3) using Au/Hg electrodes had previously indicated25 currents consistent with the passage of nearly four electrons and a diffusion coefficient of 2.0 × 10-5 cm2 s-1 for nitromethane. Spectroelectrochemical data suggested the operation of an ECEEE mechanism. Accordingly in the present work experiments were conducted using a “fast flow” channel cell so as to bring the effective number of electrons transferred into a range, 1 < neff < 4, amenable to investigation by variation of the mass transport. Experiments were conducted using Hg/Cu electrodes prepared as described above by plating a 25 µm (xe) microband
1520 J. Phys. Chem. B, Vol. 102, No. 9, 1998
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Aixill et al.
b
c
d
e
f
Figure 8. The variation of the transport-limited current with the electrolyte flow rate as measured ([) in buffered solutions of pH (a) 1.8, (b) 3.0, (c) 4.0, (d) 5.0, (e) 7.0, and (f) 9.0. The open symbols (0) show the behavior simulated for an ECEEE mechanism using the values of khet plotted in Figure 9.
electrode and solution flow rates in the range 0.024-3.0 cm3 s-1. Solutions of nitromethane (ca 1.5 mM) and pHs ranging from 1.8 to 9.0 were examined and displayed voltammograms with half-wave potentials of ca. -1.3 ( 0.1 V (vs Pt pseudoreference electrode) with no systematic variation with pH. In all cases the transport-limited current was measured as a function of flow rate. Typical plots are shown in Figure 8. In all cases there is a steady increase in an approximately linear fashion, and the neff values lie in the ranges 1.04-1.61 (pH 1.8); 1.391.88 (pH 3); 1.62-1.93 (pH 4); 2.34-2.92 (pH 5); 2.00-2.92 (pH 7); 2.32-2.87 (pH 9). The approximate cube-root dependence in Figure 8 arises from the relatively weak dependence of neff on flow rate recognized above for electrode processes in which electron-transfer steps are separated by heterogeneous (but not homogeneous) chemical steps. The possiblity for mistaking any individual plot in Figure 8 for simple Levich-
type behavior should be apparent. Moreover the near linearity of the plots in Figure 8 provides a clear indication that the multielectron process of interest does not contain interleaved kinetically significant coupled chemical steps of a homogeneous nature. Rather the deviation of the neff values below 4 must result from heterogeneous chemistry where, as derived above, the sensitivity of neff toward the flow rate is considerably weaker. Accordingly data such as that in Figure 8 was analyzed in terms of the working curve in Figure 2, adapted for an ECEEE process,25 and best fit values of khet found in each case. The corresponding theoretical plots are shown in Figure 8. Good agreement between theory and experiment can therefore be seen across the entire flow rate range indicating consistency of the data with a heterogeneous ECEEE mechanism. Figure 9 shows the variation of the best fit value of khet with pH, suggesting the step C1 shows a very weak increase with pH. We note
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J. Phys. Chem. B, Vol. 102, No. 9, 1998 1521 References and Notes
Figure 9. The variation of khet with pH.
that for mercury12 and for Au/Hg electrodes25 a value khet ) 0.06 cm s-1 at a pH of 8.3 was deduced, suggesting that the electrode surfaces prepared for the present studies showed a little more electrocatalytic activity in respect of the step C1. Conclusions Variable flow rate channel electrode voltammetry can successfully distinguish the character of the chemical step in ECEtype mechanisms as hetero- or homogeneous. In particular, in the former case the effective number of electrons transferred is only weakly sensitive to solution flow rate so that the transportlimited current approximately follows a cube-root (Levich) behavior as expected in the absence of any coupled kinetics. Acknowledgment. We thank the British Council and the Spanish Ministry of Education and Science for support through the Acciones Integradas scheme, the EC for a research training Grant for F.P. under the Fourth Framework Program (contract no. ERB FMB ICT95 0219), the EPSRC for studentships for J.A.A. and W.J.A., and Keble College for a Senior Scholarship for J.A.A.
(1) Testa, A. C.; Reinmuth, W. Anal. Chem. 1961, 33, 1320. (2) Laviron, E. J. Electroanal. Chem. 1995, 391, 187. (3) Leslie, W.; Alden, J. A.; Compton, R. G.; Silk, T. J. Phys. Chem. 1996, 100, 14130. (4) Petru, F. Collect. Czech. Chem. Commum. 1947, 12, 620. (5) Suzuki, M.; Elving, P. J. Collect. Czech. Chem. Commun. 1960, 25, 3202. (6) Gavioli, G. B.; Grandi, G.; Andreolli, R. Collect. Czech. Chem. Commum. 1971, 36, 730. (7) Wawzonek, S.; Tsung-Yuan, S. J. Electrochem. Soc. 1973, 120, 745. (8) Guidelli, R.; Foresti, M. L. J. Electroanal. Chem. 1978, 88, 65. (9) Rueda, M.; Sluyters-Rehbach, M.; Sluyters, J. H. J. Electroanal. Chem. 1989, 261, 23. (10) Prieto, F.; Rueda, M.; Navarro, I.; Sluyters-Rehbach, M.; Sluyters, J. H. J. Electroanal.Chem. 1992, 327, 1. (11) Prieto, F.; Rueda, M.; Navarro, I.; Sluyters-Rehbach, M.; Sluyters, J. H. J. Electroanal. Chem. 1996, 405, 1. (12) Prieto, F.; Navarro, I.; Rueda, M. J. Phys. Chem. 1996, 100, 16346. (13) Compton, R. G.; Fisher, A. C.; Wellington, R. G.; Dobson, P. J.; Leigh, P. A. J. Phys. Chem. 1991, 95, 1991. (14) Alden, J. A.; Compton, R. G. J. Electroanal. Chem. 1996, 404, 27. (15) Alden, J. A.; Compton, R. G. J. Phys. Chem. B 1997, 101, 8941. (16) Compton, R. G.; Pilkington, M. B. G.; Stearn, G. M.; J. Chem. Soc. Faraday Trans. I, 1988, 84, 2155. (17) Fisher, A. C.; Compton, R. G. J. Phys. Chem. 1991, 95, 7538. (18) Compton, R. G.; Dryfe, R. A. W. Progr. React. Kinet. 1995, 20, 245. (19) Compton, R. G.; Dryfe, R. A. W.; Eklund, J. C. Res. Chem. Kinet. 1994, 1, 239. (20) Le´veˆque, M. A. Ann. Mines. Mem. Ser. 1928, 12/13, 201. (21) Miles, J. Integral Transforms in Applied Mathematics; Cambridge University Press: London, 1971. (22) Abramowitz, M.; Stegun, I. A. Handbook of Mathematical Functions; Dover: New York, 1970. (23) Rees, N. V.; Dryfe, R. A. W.; Cooper, J. A.; Coles, B. A.; Compton, R. G. J. Phys. Chem. 1995, 99, 7096. (24) Moore, J.; Davis, C. C.; Coplan, M. A.; Greer, S. C. Building Scientific Apparatus, 2nd ed.; Addison-Wesley: Redwood City, CA, 1989; Chapter 2. (25) Prieto, F.; Webster, R. D.; Alden, J. A.; Aixill, W. J.; Waller, G. A.; Compton, R. G.; Rueda, M. J. Electroanal. Chem. 1997, 437, 183.
