Migration and Diffusion Coupled with a Fast Preceding Reaction

voltammograms are compared with experimental volta- mmograms for the reduction of strong and weak acids at. Pt microelectrodes with excess of and with...
2 downloads 0 Views 108KB Size
Download PDF
Anal. Chem. 1999, 71, 167-173

Migration and Diffusion Coupled with a Fast Preceding Reaction. Voltammetry at a Microelectrode Aleksander Jaworski, Mikolaj Donten, and Zbigniew Stojek*

Department of Chemistry, University of Warsaw, ul. Pasteura 1, 02-093 Warsaw, Poland Janet G. Osteryoung*

Department of Chemistry, North Carolina State University, Raleigh, North Carolina 27695-8204

A mathematical model implemented by simulation is presented for voltammetry of a reversible couple that involves a fast preceding chemical reaction and mixed diffusional and migrational transport. The hydrogen couple, H+/H2, fulfills the above criteria. For strong acids there is no preceding reaction, whereas for weak acids the preceding reaction is HA ) H+ + A-. The computed voltammograms are compared with experimental voltammograms for the reduction of strong and weak acids at Pt microelectrodes with excess of and without supporting electrolyte. The key assumption in the calculations is that the flux of hydrogen ion is independent of the anion. This assumption is supported by the experimental fact that the wave heights in the absence of supporting electrolyte of several strong acids of equal concentration and with anions of various size are identical. Microelectrodes make possible accurate voltammetry in solutions of low ionic strength.1 This extension of an experimentally accessible regime has triggered work on theoretical description of the processes involving migrational transport. Mathematical models for simple electrode processes, that is, with 1:1 electrode reaction stoichiometry, equal diffusion coefficients of reactant and product, steady state, and only a few kinds of ions present, are satisfactory at present.2-7 There has also been substantial progress in modeling ECE, comproportionation, and successive electrontransfer reactions, in which the interplay between migration and homogeneous reaction is taken into account.8-11 However, there (1) Wightman, R. M.; Wipf, D. O. Voltammetry at Ultramicroelectrodes. In Electroanalytical Chemistry; Bard A. J., Ed.; Marcel Dekker: New York, 1988; Vol. 15. (2) Amatore, C.; Deakin, M. R.; Wightman, R. M. J. Electroanal. Chem. 1987, 255, 49. (3) Amatore, C.; Bartelt, J.; Deakin, M. R.; Wightman, R. M. J. Electroanal. Chem. 1988, 256, 255. (4) Oldham, K. B. J. Electroanal. Chem. 1988, 250, 1. (5) Baker, D. R.; Verbrugge, M. W.; Newman, J. J. Electroanal. Chem. 1991, 314, 23. (6) Oldham, K. B., J. Electroanal. Chem. 1993, 337, 91. (7) Myland, J, C.; Oldham, K. B. J. Electroanal. Chem. 1993, 347, 49. (8) Norton, J. G.; Benson, W. E.; White, H. S. Anal. Chem. 1991, 63, 1909. (9) Norton, J. G.; White, H. S. J. Electroanal. Chem. 1992, 325, 341. (10) Norton, J. G.; Anderson, S. A.; White, H. S. J. Phys. Chem. 1992, 94, 3. 10.1021/ac980425s CCC: $18.00 Published on Web 12/03/1998

© 1998 American Chemical Society

does not exist a conceptually adequate theoretical treatment of the problem of coupled homogeneous preceding reaction, diffusion, and migration that is consistent with the experimental evidence available. Reduction of weak acids is an experimental system that falls into this category and is both attractive and demanding as a model because hydrogen ion and the acid anion usually differ greatly in their diffusion coefficients. Voltammetric or amperometric monitoring of acid concentration under conditions near steady state is analytically important. Examples in which this approach could be superior to potentiometric measurements employing the glass electrode include measurements in media containing either charged colloidal particles12 or polyelectrolytes,13-15 probing hydrogen ion concentration in a small zone (e.g., near a large electrode), or any situation in which one cannot wait for the potential of a pH electrode to stabilize. Other approaches have also recognized the need for nonpotentiometric pH sensors.16 It has been known for a long time that the reduction wave of a weak acid, HA, is preceded by its dissociation to form the electroactive species, hydrogen ion. That is, the reaction proceeds by a CE mechanism. A series of papers have appeared on the reduction of weak acids under conditions of homogeneous equilibrium, the case where the preceding dissociation is sufficiently fast that the homogeneous reaction is at equilibrium throughout the depletion layer.17-22 Most of this work has been (11) Amatore, C.; Paulson, S. C.; White, H. S. J. Electroanal. Chem. 1997, 439, 173. (12) Morris, S. E.; Osteryoung, J. In Electrochemistry in Colloids and Dispersions; Mackay, R., Texter, J., Eds.; VCH: New York, 1992. (13) Morris, S. E.; Ciszkowska, M.; Osteryoung, J. G. J. Phys. Chem. 1993, 94, 10453. (14) Ciszkowska, M.; Osteryoung, J. G. J. Phys. Chem. 1994, 98, 3194. (15) Roberts, J. M.; Linse, P.; Osteryoung, J. G. Langmuir 1998, 14, 204. (16) Hickman, J. J.; Ofer, D.; Laibinis, P. E.; Whitesides, G. M.; Wrighton, M. S. Science 1991, 252, 688. (17) Ciszkowska, M.; Stojek, Z.; Morris, S. E.; Osteryoung, J. G. Anal. Chem. 1992, 64, 2372. (18) Stojek, Z.; Ciszkowska, M.; Osteryoung, J. G. Anal. Chem. 1994, 66, 1507. (19) Jaworski, A.; Stojek, Z.; Osteryoung, J. G. Anal. Chem. 1995, 67, 3349. (20) Ciszkowska, M.; Stojek, Z.; Osteryoung, J. G. J. Electroanal. Chem. 1995, 389, 49. (21) Daniele, S.; Lavagnini, I.; Baldo, M. A.; Magno, F. J. Electroanal. Chem. 1996, 404, 105. (22) Daniele, S.; Baldo, M. A.; Simonetto, F. Anal. Chim. Acta 1996, 331, 117.

Analytical Chemistry, Vol. 71, No. 1, January 1, 1999 167

done at microelectrodes, which minimize the formation of H2 bubbles (due to much more efficient transport in the spherical field) and permit experiments without added electrolyte. The weak acids of p(Ka/Ca) below approximately 3 display, at a microelectrode, steady-state, transport-limited currents that are proportional, over a relatively wide range, to the analytical concentration of acid, Ca ) C*HA + C*H+, where the asterisk denotes equilibrium concentration in bulk solution, and for which the values without added electrolyte are twice the values observed in excess supporting electrolyte. This factor of 2 has been obtained for a variety of acids and over a range of analytical concentrations.18 An acceptable theoretical prediction of these observations has proven difficult to come by. Kharkats and Sokirko have discussed the case of preceding reaction without supporting electrolyte.23-25 They considered reduction of the cation in partially dissociated electrolytes by assuming either equilibrium23 or slow dissociation/recombination.24,25 However, an application of their approach to the reduction of acids in aqueous media leads to predictions that are completely at odds with the experimental results. In the paper by Oldham, only the case of common anion was considered.26 Hicks and Fedkiw elaborated on the influence of both common ion and inert electrolyte.27 The wave heights calculated for reduction of acids in the absence of supporting electrolyte, according to these models, are at odds with experimental results. Our previous efforts to describe theoretically the preceding reaction case for weak acids used two different concepts.19,28 In the first effort,19 all species participating in the acid dissociation equilibrium, except for H2, were assigned the apparent diffusion coefficient, Da, defined by DaCa ) DH+C*H+ + DHAC*HA, where DH+ and DHA are the known diffusion coefficients of the species indicated. It had been found earlier that for weak acids the diffusion-limited current is proportional to DaCa.18 Simulations based on this assumption yield chronoamperometric currents without added electrolyte higher by a factor of 2 than those with excess electrolyte. In the second successful model, that employed by Xie et al.,28 the apparent diffusion coefficient is assigned to the hydrogen ion only (Da ) DH+ + DHA(C*HA/C*H+)). In ref 28 the value of Da is much larger than that of DH+, as required to include the flux of hydrogen ion arising from the dissociation of the acid. In this treatment, the diffusion coefficient of the acid anion does not appear. The papers mentioned above are focused only on predicting the transport-limited current; none of them deals with the dependence of current on potential. Other theoretical results reported in the literature include simulations of voltammograms for weak acids, including those that dissociate slowly, but with excess supporting electrolyte to avoid the complication of migration.21,29 A paper dealing with voltammetric investigation of complexation equilibria in the presence of a low level of supporting electrolyte, which is a problem of some similarity to that of weak acids, has appeared recently.30 However, the complexes examined are assumed to be inert. In passing, it should be mentioned that (23) Kharkats, Yu. I. Sov. Electrochem. 1988, 24, 503. (24) Kharkats, Yu. I., Sokirko, A. V. J. Electroanal. Chem. 1991, 303, 17. (25) Sokirko, A. V.; Kharkats, Yu. I. Soviet Electrochem. 1989, 25, 287. (26) Oldham, K. B. Anal. Chem. 1996, 68, 4173. (27) Hicks, M. T.; Fedkiw, P. S. J. Electroanal. Chem. 1997, 424, 75 (28) Xie, Y.; Liu, T. Z.; Osteryoung, J. G. Anal. Chem. 1996, 68, 4124. (29) Fleischmann, M.; Lasserre, F.; Robinson, J.; Swan, D. J. Electroanal. Chem. 1984, 177, 97.

168

Analytical Chemistry, Vol. 71, No. 1, January 1, 1999

the treatment by Vetter31 of complex ion equilibria is formally equivalent to that of Hicks and Fedkiw.27 The goal of this paper was to develop a rigorous model that can be used to generate complete current-potential curves for reduction of weak acids without and with excess of supporting electrolyte. Assignment of appropriate values for the diffusion coefficients of the ionic species is of key importance here, since the real potential at the electrode surface, in the absence of supporting electrolyte, depends strongly on the resistance between the electrodes. Simulation techniques were chosen as a tool, since they make possible exploration of variables without the restricting assumptions necessary to solve explicitly problems of this complexity. Finally, this work is not limited to steady-state conditions. Predictions of the model are tested against experimental data for acetic acid. EXPERIMENTAL SECTION All reagents were of analytical reagent purity and were used as received. The acids (hydrochloric, nitric, glacial acetic, trichloroacetic, p-toluenesulfonic) were from either Baker or Fluka. Lithium perchlorate (Baker) and sodium perchlorate (Fluka) were used as supporting electrolytes. Ultrapure water (MilliQ, Millipore Corp.) was employed in all solutions and for rinsing. Linear sweep and cyclic voltammetries were performed using a model 273A potentiostat (EG&G PARC) controlled via software on a 486 DX2 66-MHz personal computer (Protech). All measurements were made at 22 °C. Solutions were deoxygenated before voltammetric scans and then blanketed with a stream of argon. A platinum wire counter electrode and an SCE reference electrode were used. A Pt quasireference electrode was used in preliminary experiments with strong acids. The reference junction was isolated from the solutions without added electrolyte with an auxiliary Vycor-tipped bridge containing only deionized water. Working electrodes were 0.8-mm-radius Pt disk electrode (Bioanalytical Systems) and 11-µm-radius Pt disk microelectrodes (Project Ltd., Warsaw, Poland). The electrodes were initially polished to mirror finish with 1- and 0.05-µm alumina (Buehler) on a polishing cloth. Subsequently, the surface was renewed before voltammetric scans by polishing briefly with 0.05-µm alumina. The electrode was rinsed to remove alumina with a direct stream of water from a wash bottle impinging on the electrode surface. Water was wicked from the surface of the electrode using a dry tissue. The condition of the electrode surface was examined optically using a metallurgical microscope PME 3 (Olympus). After the polishing procedure, the electrode was cleaned and activated electrochemically in a separate cell containing 0.1 M HClO4, while the solution was constantly deoxygenated with a stream of argon. Just for this activation, the platinum electrode was used in a three-electrode system with auxiliary and quasireference electrodes of platinum wires. The potential was cycled from the oxidation peaks of adsorbed hydrogen to the beginning of oxidation of water with a scan rate of 50 mV s-1. The scanning was stopped at the potential responding to the most negative reduction peak of adsorbed hydrogen. Usually it was necessary to do about 50 cycles to get well-shaped signals for a large (30) Palys, M.; Stojek, Z.; Bos, M.; van der Linden, W. E. Anal. Chim. Acta 1997, 337, 5. (31) Vetter, K. J. Electrochemical Kinetics; Plenum Press: New York, 1967; 177 ff.

electrode. For microelectrodes, because of difficulties in evaluation of the shapes of peaks due to very small currents, the 50-cycle treatment was applied routinely. The electrode was taken from the solution and rinsed with a stream of water from a wash bottle. Water was wicked away from the insulator leaving the electrode surface wet. Then the electrode was put into another cell containing a solution of the acid to be studied. For a well-prepared surface, initial scans exhibit a steeply rising wave with a welldefined plateau. The plateau level and the slope decrease slightly on subsequent scans, apparently because hydrogen is not completely oxidized and removed from the platinum electrode. Simulations were performed by several different implementations of the finite difference method using unequal space intervals with ∆ri ) 1.01∆ri-1. Crank-Nicolson and Hopscotch methods were used also to make sure the simulated voltammograms are as free as possible of computational artifacts. For the purpose of comparison with the experimental voltammograms obtained for acetic acid, the following values were used in the calculations: DHA ) 9.75 × 10-6 cm2 s-1 (for HOAc), DH2 ) 5.85 × 10-5 cm2 s-1, DH+ ) 9.3 × 10-5 cm2 s-1, r0 ) (4/2π)(11 µm or 0.8 mm), and KHOAc ) 1.8 × 10-5 M, where r0 is the electrode radius. The hemispherical geometry of the electrode was chosen for the model to simplify the calculations and thus to avoid the lower accuracy and longer calculation time in a two-dimensional simulation. The experimental results were obtained using disk electrodes. To compare these results quantitatively with each other, the radii for hemispherical and disk electrodes were chosen in such a way that the steady-state currents were equal (rh ) (4/ 2π)rd). The 11-µm-radius disk is equivalent to a hemisphere of radius 7 µm. RESULTS AND DISCUSSION Preliminary Experiments with Strong Acids. Four strong acids were chosen for voltammetric examination: trichloroacetic, hydrochloric, nitric, and p-toluenesulfonic. These acids were selected for their differing size of anion, to see if they give identical wave heights for equal concentrations in the absence of supporting electrolyte. It was found earlier for perchloric acid17 that the values of wave heights without added electrolyte are twice the values observed in excess supporting electrolyte. Examples of voltammograms obtained for 1.00 mM solutions of these acids are shown in Figure 1. The voltammetric waves, obtained at a Pt microelectrode, were indeed of identical height. The value of hydrogen ion diffusion coefficient was calculated from the limiting currents obtained, after correcting these currents to the real steady-state value according to the procedure given by Sinru et al.32 The value obtained, for ionic strength of 1.00 mM, was (8.5 ( 0.2) × 10-5 cm2 s-1 or, corrected33 to an ionic strength of 0.1 M, (7.9 ( 0.2) × 10-5 cm2 s-1. This value may be compared with reported values at 0.1 M ionic strength of (7.91 ( 0.04) × 10-5,21 8.5 × 10-5,34a and 7.9 × 10-5 cm2 s-1.34b This is the result expected absent electrolyte and, in dilute solution, provided the Nernst-Einstein (32) Sinru, L.; Osteryoung, J. G.; O’Dea, J.; Osteryoung, R. A. Anal. Chem. 1988, 60, 1135. (33) Bockris, J. O’M.; Reddy, A. K. N. Modern Electrochemistry; Plenum Press: New York, 1977; p 384. (34) (a) Unwin, P. R.; Bard, A. J. J. Phys. Chem. 1992, 96, 5035. (b) Macpherson, J. V.; Unwin, P. R. Anal. Chem. 1997, 69, 2063.

Figure 1. Voltammetric waves of four strong acids [p-toluenesulfonic (a), nitric (b), trichloroacetic (c), and hydrochloric (d)] obtained for their 1 mM solutions at a Pt disk microelectrode (r0 ) 11 µm, v ) 20 mV s-1).

relation is followed.35 A more thorough investigation of this phenomenon will be the subject of a separate paper. In this work, we only use this information as a basis for the assumption that under the conditions of reduction of hydrogen cation and in the absence of supporting electrolyte the anions do not affect the transport of hydrogen cations. The concentration profiles of the anion are, on the other hand, strictly determined by those of hydrogen ion. Theory. In the model, we assume that the reaction at the electrode surface is

2H+ + 2e- ) H2

(1)

The kinetics of the hydrogen discharge reaction are quite complex; the reaction mechanism can be considered either as a two-step or as a three-step reaction.36 To avoid this complicated situation, we sought and found conditions under which reaction 1 is reversible.37 Therefore, the electrode potential is described by the Nernst equation: 0

EH+/H2 ) E0,cH+/H2 +

RT (a H+) ln 0 2F a

2

(2)

H2

where a0H+ and a0H2 are the activities of H+ and H2 in solution, respectively. The reference potential in our case is the potential of the H+/ H2 system for unit activity of both oxidized and reduced forms in the solution. This potential differs from that of the normal hydrogen electrode (NHE), which is a mixed activity-pressure potential. (35) Newman, J. Electrochemical Systems, 2nd ed.; Prentice Hall: Englewood Cliffs, NJ, 1991; p 249. (36) Appleby, A. J.; Kita, H.; Chemla. M.; Bronoe¨l, G. Hydrogen. In Encyclopedia of Electrochemistry of the Elements; Bard, A. J., Ed.; Marcel Dekker: New York, 1982; Vol. IXA. (37) Jaworski, A.; Donten, M.; Stojek, Z.; Osteryoung, J. Anal. Chem., in press.

Analytical Chemistry, Vol. 71, No. 1, January 1, 1999

169

The following equations, in spherical coordinates, describe transport (eq 3) and the potential gradient (eq 4)

∂Ck ) ∂t Dk

(

∂2Ck ∂r

2

+

∂φ ) ∂r

(

))

∂2φ 2 ∂Ck zkF 2Ck ∂φ ∂Ck ∂φ + + + Ck 2 r ∂r RT r ∂r ∂r ∂r ∂r

(∑

-1 RT F z2kCkDk F2

∑

z kD k

)

∂Ck i + ∂r 2πr2

(3)

(4)

The quantities Ck, Dk, and zk are the local concentration, diffusion coefficient, and charge, respectively, of the kth species, and the other symbols have their usual meaning. Dissociation Rate of the Acids and Other Key Assumptions. The analytical concentration of the weak acid is Ca ) C*HA + C*A- ) C*HA + C*H+. The acid dissociates according to HA ) H+ + A-. We assume that the acid dissociation rate is fast enough to maintain equilibrium everywhere in the solution. The appropriate equilibrium constant expression is Ka ) CH+CA-/CHA. The assumption about sufficiently fast acid dissociation is based on the experimental evidence for steady-state voltammetry at microdisk electrodes of various radii, r0.17,20 Plots of il (limiting current) are proportional to r0. These linear relations show that the dissociation is sufficiently fast to maintain equilibrium at the largest current densities observed. Deviation from linearity at smaller values of r0 for H2PO4- limits the range over which equilibrium holds to values of p(Ka/Ca) less than 3, a range that covers a large group of practically important weak acids. A simple diffusion layer treatment also supports this conclusion.28 Our findings regarding kinetics of acid dissociation also agree with those of Daniele et al.21,22 We stress here that, in buffered solutions, in which the anion of the supporting electrolyte is identical to that of the acid, the overall recombination rate (kbCH+CA-) increases, and therefore, the rate of production of HA also increases; this finally may allow one to determine the dissociation rate constant.21,29 On the other hand, without supporting electrolyte, the concentration of anion in the depletion layer decreases (due to the electroneutrality principle), and the overall recombination rate is much slower, which helps to reduce acetic acid completely at the electrode surface through the mechanism of prior dissociation. We assume also that for electrodes of size in the range of micrometers and the concentration of ions present in solutions of weak acids the electroneutrality principle is obeyed everywhere, that is

∑z C

k k

)0

(5)

k

The above assumption is justified by the following estimate.38 The depletion layer thickness should approximately equal 10r0,39 and the thickness of the double layer can be approximated as 1.5κ-1, where κ-1 is the Debye-Hu¨ckel length. For a 1:1 (38) Palys, M.; Stojek, Z.; Bos, M.; van der Linden, W. E. J. Electroanal. Chem. 1995, 383, 105. (39) Norton, J. D.; White, H. S.; Feldberg, S. W. J. Phys. Chem. 1990, 94, 6772.

170

Analytical Chemistry, Vol. 71, No. 1, January 1, 1999

electrolyte in water the approximate κ-1 values for concentrations 10-3, 10-5, and 10-7 M are 10 nm, 100 nm, and 1 µm, respectively. These thicknesses are much lower if the electrode potential is far from the potential of zero charge. If this criterion is applied to the most widely used sizes of electrodes (r0 larger than 1 µm), the double-layer thickness can be neglected if the concentration of the electrolyte is not lower than 10-6 M. The limit of 10-6 M is very often that of solutions without deliberately added supporting electrolyte. The potential gradient outside the double-layer region, as calculated using the Poisson equation, corresponds to charge separations not exceeding the micromolar level. The initial conditions apply identically for any method described below, both with excess and without supporting electrolyte, and for all spatial points away from the electrode surface:

CiH2 ) 0; CiHA ) (1 - R*)Ca; CiH+ ) CiA- ) R*Ca

(6)

where R* is the fraction of acid dissociated in the bulk solution. The initial conditions at the electrode surface are calculated using the initial electrode potential. The boundary conditions at the boundary of the depletion layer remain constant throughout and are identical with initial conditions. Two sets of equations follow, those for the case of pure weak acid without added electrolyte and those for the case in which an electrolyte is added. No Supporting Electrolyte. The equations for the fluxes at the electrode surface when no electrolyte is added have the form 0 + JH /HA ) -DH+

JA0 -/HA ) -DA-

( )

- DHA

( )

- DHA

∂CH+ ∂r

∂CA∂r

0

0

( )

-

( )

+

∂CHA ∂r

∂CHA ∂r

0

0

()

0 +F DH+CH ∂φ ) RT ∂r 0 -i (7) 2πr02F

()

DA-CA0 -F ∂φ RT ∂r

0

)0 (8)

in which the Nernst-Einstein relation is assumed. Equation 9 describes the flux of dihydrogen.

( )

0 ) -2DH2 JH 2

∂CH2 ∂r

0

) i/2πr02F

(9)

Summing both sides of eqs 7-9, discretizing them, and multiplying by ∆r, one obtains 0 0 0 + + C -) ) 2DH2CH + 2DHAC0HA + D(CH A 2 1 1 1 + + C -) (10) 2DH2CH + 2DHAC1HA + D (CH A 2

where C0 indicates the concentration at the electrode surface and C1 the concentration at distance ∆r from the surface. The symbol D is the diffusion coefficient of H+. It is considered to be also the diffusion coefficient of A-, which is a consequence of the conclusion drawn from the preliminary experiments with strong acids. The boundary conditions are calculated using eq 10, the

Nernst equation, Ka, and the electroneutrality principle. The concentrations in other space points can be calculated using the H2 concentration profile and the generalized mass balance equation: i i i + + C -) ) 2DH2CH + 2DHACiHA + DH+(CH A 2 i-1 i-1 i-1 2DH2CH + 2DHACi-1 HA + DH+(CH+ + CA- ) (11) 2

or the appropriately modified transport equation (eq 3). Finally the concentrations are corrected at each iteration to fit Ka and the electroneutrality principle. The ohmic drop and the true electrode potential were calculated using the method of Jaworski et al.40,41 Excess Supporting Electrolyte. The equations for the fluxes with excess supporting electrolyte are formed by removing terms in ∂φ/∂r (the migration terms) from eqs 7 and 8. Equation 9 for the flux of product, H2, is unchanged. The boundary conditions are calculated from Ka, the Nernst equation, and eqs 12 and 13 which follow. Equation 12 is obtained by summing the differential forms of the flux equations for H+ (modified eq 7) and H2 (eq 9).

Figure 2. Comparison of theoretical and experimental cyclic voltammetric curves at 0.8-mm Pt disk electrode for excess supporting electrolyte (0.1 M LiClO4); 5 mM CH3COOH; experimental points (O), theory (s).

A

0 0 0 + + D 2DH2CH + DH+CH HACHA ) 2 1 1 1 + + D 2DH2CH + DH+CH HACHA (12) 2

Equation 13 is obtained by discretizing modified eq 9 for the flux of A-.

DA-CA0 - + DHAC0HA ) DA-CA1 - + DHAC1HA

(13)

The relations 12 and 13 are also valid at any distance from the electrode surface. The concentrations in consecutive space points are calculated using the concentration profile for H2, eqs 12 and 13, and Ka. The concept of equal mobilities of H+ and A- is not used in the case of excess supporting electrolyte, as H+ and A- diffuse independently and other ions take charge of preserving electroneutrality. Under conditions of excess supporting electrolyte the electrode potential is equal to the applied potential from the potentiostat; iR drop in the solution is negligibly small. Calculations and Experimental Results. The first calculations were done for a large electrode radius, so that the diffusion was predominantly linear. A comparison between an experimental curve, obtained for CH3COOH with a regular Pt disk electrode with excess supporting electrolyte, and the theoretical curve, calculated using the theory developed here, is presented in Figure 2. The good agreement supports the assumptions that the charge transfer and the homogeneous reaction are both reversible under the experimental conditions used. The currents calculated for microelectrodes were rendered dimensionless by dividing by the steady-state, diffusion-controlled current for excess electrolyte, which can also be expressed by the equation is ) 2πr0FDaCa, for a hemispherical microelectrode. (40) Jaworski, A.; Donten, M.; Stojek, Z. J. Electroanal. Chem. 1996, 407, 75. (41) Jaworski, A.; Donten, M.; Stojek, Z. J. Electroanal. Chem. 1997, 420, 307.

B

Figure 3. (A) Theoretical voltammograms at microelectrodes for various Ka. Excess supporting electrolyte, r0 ) 7 µm, Ca ) 1 × 10-3 M, and v ) 20 mV s-1. pKa: -1.2 (a), 1.8 (b), 2.8 (c), 3.8 (d), 4.8 (e), and 5.8 (f). (B) Theoretical steady-state half-wave potential, E1/2, plotted vs pKa, v ) 2 mV s-1. Other conditions as in (A).

The predicted voltammetric waves for acids of various strengths with excess electrolyte (0.1 M LiClO4) are presented in Figure 3A. Activity effects due to the high salt concentration are included in the calculation. If the acid has an equilibrium constant 1000 times greater than that of acetic acid, its voltammetric wave is indistinguishable from that of a strong acid. This phenomenon depends of course on extent of dissociation, and hence on concentration, as well. For Ka/Ca greater than about 15, acids are Analytical Chemistry, Vol. 71, No. 1, January 1, 1999

171

Figure 4. Comparison of experimental (b) and theoretical (s) voltammetric waves. Excess supporting electrolyte. 5 mM acetic acid. Other conditions as in Figure 3A.

Figure 6. (A) Theoretical voltammograms at microelectrodes for various concentrations of acetic acid; r0 ) 7 µm, v ) 20 mV s-1. No supporting electrolyte. Acid concentration: 0.1 (a), 0.5 (b), 1 (c), 5 (d), 10 (e), and 50 mM (f). (B) Theoretical voltammograms at microelectrodes for various concentrations of acetic acid plotted vs true electrode potential; r0 ) 7 µm, v ) 20 mV s-1. No supporting electrolyte. Acid concentration from right to left: 0.1, 0.5, 1, 5, 10, and 50 mM.

Figure 5. (A) Theoretical voltammetric waves at microelectrodes for various Ka; r0 ) 7 µm, Ca ) 1 mM, and v ) 20 mV s-1. No supporting electrolyte. pKa: -1.2 (a), 1.8 (b), 2.8 (c), 3.8 (d), 4.8 (e), and 5.8 (f). (B) Theoretical voltammograms at microelectrodes for various Ka plotted vs true electrode potential. No supporting electrolyte. pKa and other parameters as in Figure 5A.

sufficiently dissociated to behave as strong acids. The weaker the acid the more negative is the half-wave potential. These results agree with those of Daniele et al.21 The shift of the half-wave potential is illustrated in Figure 3B. The results in Figure 3B were computed for a very small scan rate (1 mV s-1), so that the halfwave potentials did not differ from the true steady-state values by 172 Analytical Chemistry, Vol. 71, No. 1, January 1, 1999

more than 1 mV. The slope of the line in Figure 3B reaches 56 mV per pKa unit for the highest pKa values. At larger values of pKa, the model does not apply, because the dissociation reaction is not in equilibrium. A simulated voltammogram is compared with experimental results for 5 mM acetic acid in excess supporting electrolyte in Figure 4. Theoretical and experimental curves are similar in shape and position. A small difference appears in the slope of the voltammogram at the foot of the wave. Generally, good agreement between theoretical and experimental voltammograms was obtained for a wider scan rate range than that for regular size electrodes, that is, from 1 to 100 mV s-1. This wider range is apparently connected with the particular property of microelectrodes, which is the faster transport, compared with that of regular electrodes, of the reaction products from the electrode surface. The influence of the value of the acid dissociation constant is more complex in voltammetry without supporting electrolyte. The half-wave potentials and the wave slope depend on both dissociation constant and concentration of acids. The contributions of dissociation constant and acid concentration to the obtained theoretical dependencies cannot be easily separated, since the stoichiometry of the electrode process is 2:1. Calculated voltam-

mograms for various values of Ka and Ca are shown in Figures 5 and 6, respectively. Figures 5A and 6A show the dependence of wave shape and position on potential. The somewhat confused result for concentration dependence (Figure 6A) is readily rationalized by noting the importance of ohmic drop without electrolyte. Figures 5B and 6B show the waves from Figures 5A and 6A, respectively, with the true electrode potential (i.e., applied potential less ohmic potential drop) as abscissa. The slope and position of the waves in Figure 5B resemble those obtained with excess electrolyte (Figure 3A). Figure 6B is particularly interesting because the sequence of voltammograms is reversed on the potential axis, in comparison with that in Figure 6A. The voltammogram for the highest acid concentration occurs at the most positive potentials, because the stoichiometry of the processs is 2:1, and hence, the half-wave potential depends on the hydrogen ion concentration, as predicted by eq 2. This reversal is not seen in Figure 5A and B, since the acid concentration is identical for all waves. The change in E1/2 in Figure 6B is only 25 mV, for a change in log Ca of 2.7. The change in log R*Ca is much less, only 1.4. However, even at lower concentrations, where E1/2 depends linearly on log R*Ca, the slope of the dependence is only 25 mV per log unit. For reduction of a strong acid in excess supporting electrolyte, the slope would be 60 mV per log unit. The agreement between theoretical and experimental voltammograms for acetic acid calculated for and measured in the absence of supporting electrolyte is good over the entire range of potential. The concentration of 0.5 mM was chosen for the comparison shown in Figure 7. The small differences between theoretical and experimental waves are usually located, as is seen in Figure 7, in the upper part of the voltammograms. This model predicts accurately the time-dependent and potentialdependent current, including the transport-controlled limiting value, for strong or moderately weak acids. The key assumption that makes the theory tractable is that the anion of a weak acid does not influence the transport of the hydrogen ion in the absence of supporting electrolyte. In other words, only the diffusion coefficients of hydrogen ion and the undissociated molecule determine the entire voltammograms of reduction of weak acids. Our previous concept of apparent diffusion coefficient19 leads to correct wave heights; however, it cannot be used to predict the potential dependence of the current. The results of calculations are in accord with experimental results and with extensive prior experimental17-19 and theoretical19,28 work on transport-limited currents. We note also that the

Figure 7. Comparison of experimental (∇) and theoretical (s) voltammograms at microelectrodes; 0.5 mM acetic acid, rd ) 11 µm, and v ) 20 mV s-1. No supporting electrolyte.

equations used to specify this problem are equivalent to those of Xie et al.28 and are not equivalent to those of Oldham,26 Kharkats,23-25 and Hicks and Fedkiw.27 The present treatment is not convenient for routine use; for that purpose, simplified treatments of steady-state limiting current are usually sufficient. However, it has the merit of providing a more complete description of the expected response and thus permits a more detailed experimental test of the prediction. Systems of complex ions present a formally equivalent problem, which, because of near-equivalence of diffusion coefficients, do not exhibit the striking behavior of weak acid systems. However, in the special case of polyelectrolytes or other macroions, diffusion coefficients of cations and anions may display large differences. Many classes of polyanions are basic and form complexes with metal ions (including hydrogen ion). A fundamental understanding of the weak acid system described here provides the basis for understanding complex ion equilibria and electrostatic interactions in such macroionic systems. ACKNOWLEDGMENT This work was supported in part by the National Science Foundation under Grants CHE9208987 and DMR9711205 and by Grant BST-562/5 from the University of Warsaw. Received for review April 22, 1998. Accepted October 21, 1998. AC980425S

Analytical Chemistry, Vol. 71, No. 1, January 1, 1999

173