Anal. Chem. 1997, 69, 2063-2069
Determination of the Diffusion Coefficient of Hydrogen in Aqueous Solution Using Single and Double Potential Step Chronoamperometry at a Disk Ultramicroelectrode Julie V. Macpherson and Patrick R. Unwin*
Department of Chemistry, University of Warwick, Coventry CV4 7AL, U.K.
An assessment is made of single and double potential step chronoamperometry (SPSC and DPSC, respectively) at Pt disk ultramicroelectrodes (UMEs) as methods for determining the value of the diffusion coefficient of hydrogen in aqueous solutions. In SPSC, measured currents for the oxidation of dissolved hydrogen (at concentrations close to saturated solution values) comprise a significant contribution, at short to moderate times, from the oxidative desorption of adsorbed hydrogen as well as the diffusion-controlled oxidation of the solution species. Provided that the electrode is preconditioned using a welldefined potential cycling procedure, the behavior for the oxidative desorption step alone can be established in an Ar-saturated solution. The chronoamperometric characteristics for the solution diffusion-controlled process may then be determined, from which the diffusion coefficient of hydrogen can be measured. In DPSC, a locally supersaturated solution of hydrogen is created transiently through the diffusion-controlled reduction of a known concentration of protons in an initial potential step. Hydrogen is subsequently collected back through oxidation to protons; the current flowing depends on the diffusion coefficients of the two species and the duration of the forward step. Under these conditions, the contribution from surface electrochemical processes to the forward and reverse chronoamperommograms is shown to be negligible. By solving the mass transport problem for DPSC with arbitrary diffusion coefficients of the redox species, the diffusion coefficient of hydrogen is readily determined. Both methods yield a consistent value for the diffusion coefficient of hydrogen, DH2, in 0.1 mol dm-3 KNO3 of 5.0 × 10-5 cm2 s-1. Despite a large number of studies,1-4 “reliable measurements of diffusion coefficients of H2 molecules in aqueous solutions are still missing.” 1a A comprehensive summary of the available data up to 1964 was compiled by Himmelblau.1b Many of the published results are old1b (dating from 1930 and earlier) and were typically obtained by the measurement of hydrogen absorption in steadystate laminar flow or quiescent systems. Even the few later measured values, obtained using rotating disk electrode (1) For reviews, see, for example: (a) Appleby, A. J.; Chemla, M.; Kita, H.; Bronoe¨l, G. In Encyclopedia of Electrochemistry of the Elements; Bard, A. J., Ed.; Marcel Dekker: New York, 1973; Part IXa, p 383. (b) Himmelblau, D. M. Chem. Rev. 1964, 64, 527. (c) Adams, R. N. Electrochemistry at Solid Electrodes; Marcel Dekker: New York, 1969; p 221. S0003-2700(96)01211-5 CCC: $14.00
© 1997 American Chemical Society
voltammetry,1c,2-4 show discrepancies of almost 1 order of magnitude. Although the differences may, in part, be due to variations in the composition and concentration of the supporting electrolytes employed, incorrect assumptions concerning the concentration of highly volatile hydrogen in the solutions studied may also be important. Through the use of single potential step chronoamperometry (SPSC) at an ultramicroelectrode (UME), it is possible to determine the value of the diffusion coefficient, D, of an electroactive species which undergoes a simple electron transfer process, independent of a knowledge of the concentration.5-9 This technique was pioneered by Winlove et al.7a to measure the concentration and diffusion coefficient of oxygen in phosphatebuffered sucrose and albumin solutions. The current (i)-time (t) behavior for the electrolysis of the electroactive species following a potential step at a disk UME, of known radius a, from a value where no electrode reactions occur to one where electrolysis occurs at a diffusion-controlled rate, can be described analytically with an expression in which D is the only remaining unknown:10
( )
i/i(∞) ) 0.7854 + 0.4431
tD a2
-1/2
+
(
0.2146 exp -0.3911
( ) ) tD a2
-1/2
(1)
In eq 1, i(∞) is the limiting, steady-state current which flows at the disk UME. A fit of the experimental data to the theoretical i-t-1/2 expression generated using eq 1 leads to a value for the diffusion coefficient of the electroactive species. The SPSC procedure is only effective in providing a value for D when the diffusional current signal, resulting from the electrolysis of the electroactive species at the electrode, is sufficiently high to suppress any contribution to the i-t response from interfacial processes, such as electrode surface reactions and (2) Lewis, G. P.; Ruetschi, P. J. Phys. Chem. 1963, 67, 65. (3) Breiter, M.; Hofmann, K. Z. Elektrochem. 1960, 64, 462. (4) Aykazyan, E. A.; Fedorova, A. I. Dokl. Akad. Nauk SSSR 1952, 86, 1137. (5) Denualt, G.; Mirkin, M. V.; Bard, A. J. J. Electroanal. Chem. 1991, 86, 27. (6) Bard, A. J.; Denuault, G.; Friesner, R.; Dornblaser, B. C.; Tuckerman, L. S. Anal. Chem. 1991, 63, 1282. (7) (a) Winlove, C. P.; Parker, K. H.; Oxenham, R. K. C. J. Electroanal. Chem. 1984, 170, 293. (b) O’Hare, D.; Winlove, C. P.; Parker, K. H. J. Biomed. Eng. 1991, 13, 304. (8) Krasinski, P.; Galus, Z. J. Electroanal. Chem. 1993, 346, 135. (9) Moressi, M. B.; Fernandez, H. J. Electroanal. Chem. 1994, 369, 153. (10) Shoup, D.; Szabo, A. J. Electroanal. Chem. 1982, 140, 237.
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double-layer charging.1c,11 For example, Denuault et al.5 measured the diffusion coefficient of ferrocyanide ions in aqueous solution at a Pt disk UME, but a concentration of 50 × 10-3 mol dm-3 ferrocyanide ions was employed to ensure a negligible contribution of surface processes to the chronoamperometric response. When the diffusional flux is small, for example when the solubility of the species of interest is low, the direct application of SPSC may be more difficult. This is potentially the situation for hydrogen in aqueous solutions, for which the solubility is typically ∼0.8 × 10-3 mol dm-3 at 298 K and 1 atm of H2.1a Furthermore, hydrogen adsorption on Pt electrodes in aqueous media1c,2,12,13 and absorption into bulk Pt after exposure to aqueous hydrogen solutions1c,2,12,13 are significant processes. The SPSC response for hydrogen oxidation at a Pt UME is thus expected to be subject to sizeable residual currents, particularly at short times in a chronoamperometric experiment. Double potential step chronoamperometry (DPSC)14 involves generating a species of interest in an initial potential step (usually, but not exclusively, through the electrolysis of a precursor species in solution) and recollecting it by reversing the potential. The methodology has been applied in conjunction with UMEs15 to characterize the lifetime of electrogenerated materials under very short time conditions so that planar diffusion models14 can be employed. To the best of our knowledge, there has been no reported work on the DPSC response under conditions where radial diffusion is important and the diffusion coefficients of the two electroactive species are unequal. In this article, we consider the use of both SPSC and DPSC in the measurement of the diffusion coefficient of hydrogen and extend the theoretical models for DPSC to include radial diffusion under conditions where the diffusion coefficients of the two electroactive species are different. For SPSC, it is shown that the surface processes which occur during the diffusion-controlled oxidation of hydrogen can be quantified. The contribution to the i-t response from diffusional processes alone can then be established, enabling the measurement of the diffusion coefficient of hydrogen. With DPSC, the aim is to make the nondiffusional current responses relatively negligible by increasing the flux of hydrogen to the UME during the diffusion-controlled oxidation of hydrogen. This is achieved by amperometrically generating hydrogen at supersaturated concentrations, in the vicinity of a UME, from the diffusion-controlled reduction of a known concentration of protons. Upon reversing the potential, hydrogen is reoxidized, and the i-t behavior provides quantitative information on the diffusion coefficient of the species collected with high precision. EXPERIMENTAL SECTION Apparatus and Instrumentation. All measurements were made at 298 K using a two-electrode arrangement with a 25 µm (11) Heinze, J. Angew. Chem., Int. Ed. Engl. 1993, 32, 1268. (12) Franklin, T. C.; Cooke, S. L. J. Electrochem. Soc. 1960, 107, 556. (13) (a) Eucken, A.; Weblus, B. Z. Elektrochem. 1951, 55, 114. (b) Wicke, E.; Weblus, B. Z. Elektrochem. 1952, 56, 169. (c) Breiter, M.; Knorr, C. Z. Elektrochem. 1955, 59, 153. (d) Breiter, M.; Knorr, C. Z. Elektrochem. 1955, 59, 681. (14) Bard, A. J.; Faulkner, L. R. Electrochemical Methods; John Wiley: New York, 1980; p 176. (15) (a) Andrieux, C. P.; Hapiot, P.; Save´ant, J. M. J. Phys. Chem. 1988, 92, 5992. (b) Andrieux, C. P.; Save´ant, J. M. In Investigations of Rates and Mechansims of Reactions; Bernasconi, C., Ed.; Wiley: New York, 1986; Vol. 6, 4/E, Part 2, p 305.
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diameter Pt disk UME serving as the working electrode and an Ag wire acting as a quasi reference electrode (AgQRE). All potentials are quoted with respect to this reference electrode. The electrochemical cell utilized has been described previously.16 The lid of the cell contained several holes which accommodated perspex tubing through which solutions could regularly be purged with Ar or H2. Various electrochemical and nonelectrochemical cleaning pretreatments were used to activate the electrode and obtain reproducible H2/H+ electrochemistry, mainly in accordance with recent studies.17-19 The most effective methods for SPSC and DPSC were found to be as follows. In SPSC studies, where soluble hydrogen was oxidized, the electrode potential was cycled between -0.95 and +0.95 V, 5-10 times, in H2-purged solution prior to each chronoamperometric measurement. For the DPSC studies, where H+ was reduced to H2, which was subsequently reoxidized, prior to each transient measurement the UME was polished for 30 s with 0.05 µm alumina on a polishing pad, made slightly damp with water, and then thoroughly rinsed with Milli-Q reagent water (Millipore Corp.). The UME potential was controlled with a purpose-built triangular wave/pulse generator (Colburn Electronics, Coventry, UK) and the current measured with a home-built current follower (gains of 10-5-10-9 A/V). Current-potential characteristics were recorded directly on either an x-y recorder (PL3, Lloyd Instruments, Southampton, UK) or a 486DX personal computer equipped with a Lab-PC (National Instruments, Austin, TX) data acquisition card. Current-time measurements were acquired using an NIC310 (Nicolet) digital storage oscilloscope. Materials. All solutions were prepared using Milli-Q reagent water. For the SPSC experiments, solutions contained 0.1 mol dm-3 KNO3 (Aldrich, 99.99%). Prior to use, these solutions were rigorously deaerated with Ar (BOC, 99.98% volume) for 30 min and then purged with H2 (BOC, 99.99% volume) for the same period of time. For the DPSC experiments, solutions contained 0.002 or 0.003 mol dm-3 HNO3, made by dilution of a volumetric standard (Aldrich), and 0.1 mol dm-3 KNO3. Prior to use, the solutions were deaerated with Ar for 30 min. During all electrochemical measurements, the solutions were blanketed with a stream of the appropriate gas (Ar or H2). RESULTS AND DISCUSSION Determination of DH2 via the Direct Oxidation of H2. Figure 1 shows a cyclic voltammogram (CV) recorded after five potential cycles at a 25 µm diameter Pt disk UME in an aqueous solution containing 0.1 mol dm-3 KNO3 and deaerated with Ar. The starting potential was +0.95 V, and, in the first instance, the scan was in the negative direction at a sweep rate of 0.1 V s-1. Over the potential window available at the Pt UME, there are clearly a variety of surface processes which produce a current flow. (16) (a) Macpherson, J. V.; Unwin, P. R. J. Phys. Chem. 1995, 99, 3338. (b) Macpherson, J. V.; Beeston, M. A.; Unwin, P. R. J. Chem. Soc., Faraday Trans. 1995, 91, 899. (c) Macpherson, J. V.; Unwin, P. R. J. Phys. Chem. 1995, 99, 14824. (d) Macpherson, J. V.; Unwin, P. R. J. Phys. Chem. 1994, 98, 3109. (e) Macpherson, J. V.; Unwin, P. R. J. Phys. Chem. 1994, 98, 1704. (17) (a) Ciszkowska, M.; Stojek, Z.; Morris, S. E.; Osteryoung, J. G. Anal. Chem. 1992, 64, 2373. (b) Troise Frank, M. H.; Denuault, G. J. Electroanal. Chem. 1993, 354, 331. (18) Daniele, S.; Lavagnini, I.; Baldo, M. A.; Magno, F. J. Electroanal. Chem. 1996, 404, 105. (19) Unwin, P. R.; Bard, A. J. J. Phys. Chem. 1992, 96, 5035.
Figure 1. Cyclic voltammogram, recorded at a sweep rate of 0.1 V s-1, for a 25 µm diameter Pt UME in an Ar-purged unbuffered 0.1 mol dm-3 KNO3 solution.
The CV depicted in Figure 1 contains all the salient anodic and cathodic features observed in a previous study of Pt UMEs in unbuffered aqueous solution, where the electrolyte was 1 mol dm-3 Na2SO4.20 In Figure 1, the scan toward negative potentials shows a well-defined peak at -0.4 V corresponding to the reduction of platinum oxide. Further negative, two broad H adsorption peaks are apparent at -0.74 and -0.92 V, corresponding respectively to strongly and weakly bound hydrogen states21 resulting from the underpotential deposition of hydrogen.22 Upon sweeping positive, a peak at -0.51 V and a shoulder on the peak at -0.4 V are due to the oxidative desorption of hydrogen, further positive, at +0.6 V, the oxidation of Pt surface is responsible for the broad peak observed. A typical linear sweep voltammogram (LSV), recorded at a sweep rate of 0.02 V s-1 for the oxidation of hydrogen in an aqueous 0.1 mol dm-3 KNO3 solution, purged with hydrogen, is given in Figure 2. The steady-state maximum current corresponds to the diffusion-controlled oxidation of hydrogen. It is important to reemphasise that it is not possible to determine accurately DH2 from these data alone, since the concentration of H2 in the solution is undefined. It follows from Figure 2 that, in order to effect the diffusioncontrolled oxidation of hydrogen in SPSC, the potential should be stepped from -0.9 to -0.3 V. Figure 1 demonstrates that, at -0.9 V, the Pt UME surface will initially be saturated with electroadsorbed hydrogen (Hads). In an aqueous solution containing hydrogen, coverage of the Pt surface with Hads can occur via two routes: the reduction of water (Figure 1) and the dissociative chemisorption of H2.22,23 Charging current and CV measurements23,24 have shown that up to a monolayer of Hads is deposited due to these processes. (20) Pletcher, D.; Sotiropoulos, S. J. Chem. Soc., Faraday Trans. 1994, 90, 3663. (21) Clavilier, J.; Orts, J. M.; Gomez, R.; Feliu, F.; Aldaz, A. J. Electroanal. Chem. 1996, 404, 281. (22) Jerkiewicz, G.; Zolfaghari, A. J. Electrochem. Soc. 1996, 143, 1240. (23) Christman, K. Surf. Sci. Rep. 1988, 9, 1. (24) (a) Will, F. G.; Knorr, C. A. Z. Elektrochem. 1960, 64, 258. (b) Elam, M.; Conway, B. E. J. Electrochem. Soc. 1988, 135, 1678. (c) Bo¨ld, W.; Breiter, M. W. Z. Elektrochem. 1960, 64, 897. (d) Breiter, M. W.; Kennel, B. Z. Elektrochem. 1960, 64, 1180. (e) Conway, B. E. Theory and Principles of Electrode Processes; Ronald Press, London, 1965.
Figure 2. Linear sweep voltammogram, recorded at a sweep rate of 0.02 V s-1, for the oxidation of H2 in a 0.1 mol dm-3 KNO3 solution purged with H2.
Upon stepping the potential at the UME to -0.3 V, Hads is oxidatively desorbed:20
Pt-Hads - e- f Pt + H+
(2)
During the diffusion-controlled oxidation of soluble hydrogen via SPSC, eq 2 will thus also contribute to the current flow. Assuming that the current signals for the diffusion-limited process and the oxidative desorption process are additive, subtraction of the i-t response measured with H2 present from that measured in the absence of dissolved H2 should yield a chronoamperometric response due to the flux of H2 at the UME only. This procedure will be valid provided that absorption of H2 in the Pt UME is negligible over the period of the experiments. The presence of H2 in the aqueous solution does not affect the solution pH, so the current-potential characteristics for the surface processes are expected to be similar with and without H2 present. The typical i-t-1/2 response following a step in the potential from -0.9 to -0.3 V, in a solution saturated with respect to hydrogen and containing 0.1 mol dm-3 KNO3, is shown in Figure 3. The current has been normalized with respect to the steadystate diffusion-limited current, i(∞), for the electrolysis of the solution species, H2. Also shown is the i-t-1/2 response under the same potential conditions for electrolysis in 0.1 mol dm-3 KNO3 purged with Ar. The sizeable current ratio recorded in the latter experiment, especially at short times, indicates that the residual anodic surface process contributes significantly to the total current. It should, perhaps, be noted that the contribution to the total and background current responses from double-layer charging is expected to be negligible for a UME of the dimensions used on the time scale of these measurements.11 The charge passed during the oxidative desorption of Hads is ∼0.72 × 10-9 C from the potential step measurements (Figure 3). The independent CV measurement (Figure 1) is in reasonable agreement, yielding a value of 0.94 × 10-9 C, although there is a greater uncertainty in integrating the charge under the currentpotential peak compared to evaluating the charge in the chronoamperometric measurement. The mean of the two charge Analytical Chemistry, Vol. 69, No. 11, June 1, 1997
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Figure 3. Chronoamperometric data, displayed as i/i(∞) vs t-1/2, following a step in the UME potential from -0.9 to -0.3 V vs AgQRE for a KNO3 solution purged with H2 (4) and Ar (0). The difference chronoamperometric characteristics (O) are in good agreement with eq 1 for DH2 ) 5.0 × 10-5 cm2 s-1 (s).
Figure 4. Linear sweep voltammogram, recorded at a sweep rate of 0.02 V s-1, for the reduction of 3.0 × 10-3 mol dm-3 H+ in a 0.1 mol dm-3 KNO3.
where n is the number of electrons involved in the electrode process and c* is the concentration of electroactive species in the solution. Armed with a knowledge of the diffusion coefficient of hydrogen, determined above, the concentration of H2 present under the conditions of the experiment above is deduced readily.
From Figure 2, given that the limiting current is 29 nA, n ) 2, and a ) 12.5 × 10-4 cm, c* is determined as 6.0 × 10-4 mol dm-3. The limiting currents recorded in a series of experiments gave c* values in the range (5.0-7.1) × 10-4 mol dm-3. These values are slightly lower than the previously reported value of 8.1 × 10-4 mol dm-3 for a saturated solution of hydrogen at 293 K in a 0.1 mol dm-3 solution of KNO3.28 The lower values measured may partly be accounted for by the higher temperature used in the experiments described herein. However, the variation in the values reflects the fact that it is difficult to achieve a saturated solution of hydrogen, over prolonged periods of time, under the experimental conditions in which measurements could only be made by first purging the solution with H2 and then passing the gas over the top of the solution during voltammetry and chronoamperometry. These observations reemphasize that, in this application, a technique which can determine the diffusion coefficient of an electroactive species, independently of its solution concentration, is particularly valuable. Determination of DH2 via DPSC. With the DPSC method, for the experiment of interest, a locally supersaturated solution of hydrogen is generated close to the UME through the diffusioncontrolled reduction of a known concentration of H+. Hydrogen is subsequently collected back through diffusion-controlled oxidation, with the corresponding i-t behavior providing quantitative information on DH2. The potential limits for the DPSC measurements were determined from LSVs for H+ reduction. DPSC experiments were performed with both 2 × 10-3 and 3 × 10-3 mol dm-3 H+ (HNO3). A typical voltammogram for the reduction of 3.0 × 10-3 mol dm-3 H+ (HNO3) is shown in Figure 4. It is clear that a potential step from -0.4 to -0.95 and back to -0.4 V should allow DPSC measurements to be made under diffusioncontrolled conditions. Using eq 3, the diffusion coefficient for H+, DH+, was determined as 7.9 × 10-5 cm2 s-1 from the voltammogram in Figure 4, in excellent agreement with values quoted under similar experimental conditions.17a,18,19,29 A typical forward and reverse transient, for the diffusioncontrolled reduction of H+ and reoxidation of H2, is shown in Figure 5 for a solution which comprised 2 × 10-3 mol dm-3 HNO3. The duration of the forward step was 325 ms. All of the
(25) Adamson, A. W. Physical Chemistry of Surfaces; Wiley: New York, 1990. (26) Saito, Y. Rev. Polarogr. 1968, 15, 177. (27) Newman, J. J. Electrochem. Soc. 1966, 113, 501.
(28) Linke, W. F. Solubilities of Inorganic and Metal Organic Compounds; American Chemical Society: Washington, DC, 1958; Vol. I, p 1081. (29) Yang, Y.-F.; Denuault, G. J. Chem. Soc., Faraday Trans. 1996, 92, 3791.
measurements equates to 170 µC cm-2 or 1.8 × 10-9 mol cm-2 based on the geometric area of the UME. This value is typical of that expected for monolayer surface coverage, taking account of the likely roughness of the electrode,25 providing supporting evidence that the background surface process is the oxidative desorption of Hads. The surface coverage is slightly lower than that measured previously on a Pt electrode extensively cycled between +0.05 and +1.35 V (vs a standard hydrogen electrode) in 0.5 mol dm-3 H2SO4 solution.21 However, the latter pretreatment is likely to cause an increase in the specific surface area of the electrode. The response obtained when the background current, measured in the absence of dissolved hydrogen, is subtracted from the chronoamperometric response measured with H2 present is also given in Figure 3. These data have been analyzed in terms of the theoretical response predicted for a simple diffusioncontrolled electrolysis, given by eq 1, which is seen to provide a good description with a value of DH2 ) 5.0 × 10-5 cm2 s-1. Together these data demonstrate that when surface processes are accounted for, the oxidation of soluble hydrogen is effectively diffusion-controlled on a time scale of ∼200 µs and longer. It should be noted that the long-time response (t-1/2 < 5 s-1/2), without background subtraction, is coincident with the theory for a diffusion-controlled process, and hence the limiting current response can be used in steady-state analysis. The following equation describes the diffusion-limited current at a disk UME: 11,26,27
i(∞) ) 4naFDc*
2066 Analytical Chemistry, Vol. 69, No. 11, June 1, 1997
(3)
Figure 6. Cyclic voltammogram, recorded at a sweep rate of 0.1 V s-1, for a 25 µm diameter Pt UME in an Ar-purged 0.1 mol dm-3 KNO3 solution containing HNO3 at a concentration of 2.0 × 10-3 mol dm-3.
Figure 5. DPSC characteristics for the reduction of H+ (2.0 × 10-3 mol dm-3) and the reoxidation of the generated species H2. The initial and final potentials were -0.4 V, and the step potential was -0.95 V. The characteristics for (a) the forward step (O) are in good agreement with eq 1 with DH+ ) 7.8 × 10-5 cm2 s-1 (s); those for (b) the reverse step (0) are in good agreement with the theory for DH2 ) 5.0 × 10-5 cm2 s-1 (s). Also shown is the i-t-1/2 response for (a) the forward (3) and (b) the reverse (]) steps over the same potential region in Ar-purged 0.1 mol dm-3 KNO3 solution.
chronoamperometric data shown have been normalized with respect to the steady-state diffusion-limited current, i(∞), measured for the initial solution species, H+. The time scale refers to times following the initial (forward) step for the H+ reduction data (Figure 5a) and for times following the reverse step for the H2 oxidation data (Figure 5b). The chronoamperometric reduction of H+ can be described by eq 1, with a best fit of experiment and theory achieved with DH+ ) 7.8 × 10-5 cm2 s-1, in good agreement with the value derived above from steady-state voltammetry. The best fit to theory of the reverse experimental transient, simulated using the method outlined in brief in the Appendix and in full elsewhere,30 was obtained with a value of DH2 ) 5.0 × 10-5 cm2 s-1. In Figure 5b, there is seen to be good agreement of the experimental data with the theoretical response for a diffusion-controlled process for a time scale of 200 µs and longer. Very similar values for DH+ and DH2 were obtained from measurements on the 3 × 10-3 mol dm-3 HNO3 solution. (30) Slevin, C. J.; Unwin, P. R., manuscript in preparation.
In contrast to those obtained with the SPSC method, the diffusion coefficients for H2 and H+ have been determined from DPSC without any correction of the data to account for surface processes. This is seen to be valid from Figure 5, which also shows the current responses when the potential is jumped from -0.4 to -0.95 and back to -0.4 V in a double step fashion, for an Ar-purged solution containing only supporting electrolyte. Under these conditions, the residual current flowing during the forward and reverse potential steps is negligible compared to the total current flow with H+ and H2 present. There are two reasons for the minimal relative contribution of electrode surface processes to the current response under the conditions of the DPSC experiments. First, the local concentration of H2 generated in the forward step is higher than the saturated solution value, producing an enhanced signal for H2 oxidation during the reverse step. Based on the electrode reaction stoichiometry, however, the maximum local H2 concentration (close to the electrode) is only a factor of 2-3 times higher than the bulk value in the SPSC experiments. The second factor is that the hydrogen adsorption/desorption process is not as significant as that for the SPSC experiment, given the potential values in the step experiment compared to the potential range of the adsorption/desorption process. This point is well-illustrated by reference to Figure 6, which shows a CV recorded after five cycles in an aqueous 0.1 mol dm-3 KNO3 solution containing HNO3 at a concentration of 2.0 × 10-3 mol dm-3. The starting potential was +0.95 V, and in the first instance the scan was in the negative direction at a potential scan rate of 0.1 V s-1. The main features described for Figure 1 are present, but the hydrogen adsorption and desorption processes have become reversible and shifted to more positive potentials.20 A very similar CV response was observed with 3.0 × 10-3 mol dm-3 HNO3. Careful examination of the CV in Figure 6 demonstrates that, at the starting potential of -0.4 V, the surface will already be partly covered with Hads, so that when the potential is jumped to drive the reduction of H+, the charge required to saturate the electrode surface with Hads is less than that for monolayer coverage. Conversely at the final potential of -0.4 V, the reoxidation of solution hydrogen is driven at a diffusion-controlled rate, but this potential is not sufficiently positive to promote the rapid oxidative desorption of Hads. Analytical Chemistry, Vol. 69, No. 11, June 1, 1997
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CONCLUSIONS Both SPSC and DPSC at UMEs have been shown to be useful methods for the measurement of DH2 in aqueous solutions. In SPSC, where hydrogen in solution is oxidized to protons, the chronoamperometric characteristics may be analyzed to provide a value for DH2, independent of a knowledge of the concentration, provided that the current response is corrected for the oxidative desorption of adsorbed hydrogen which occurs over the same potential range. In contrast, DPSC provides a route for transiently generating hydrogen at supersaturated concentrations, in an initial potential step, which is subsequently collected when the potential is reversed. The enhanced concentrations of H2 and diminution in the chronoamperometric response for the hydrogen oxidative desorption process under these conditions allows DH2 to be determined directly from the reverse current response. More generally, the DPSC procedure described should prove useful for determining the diffusion coefficients of electrogenerated species. APPENDIX We outline, in brief, the model employed to simulate the DPSC response at a disk UME, the results of which were used to analyze the experimental data in Figure 5. The solution initially contains only a precursor species, A. At time t ) 0, the potential of the UME probe is jumped to a value to cause the following diffusioncontrolled electrolysis:
(A1)
A ( ne- f xB
resulting in the generation of species B. For the studies of interest, A ) H+, B ) H2, x ) 1/2, and n ) 1. The transport of A and B in the axisymmetric cylindrical geometry of a disk UME is governed by the following general dimensionless diffusion equation (where the subscript i denotes A or B):
[
]
∂2Ci 1 ∂Ci ∂2Ci ∂Ci ) λi + 2 + ∂τ ∂R2 R ∂R ∂Z
(A2)
The dimensionless terms in eq A2 are defined as follows. R is the radial coordinate (starting at the center of the UME) normalized with respect to the electrode radius, a. Z is the coordinate normal to the UME surface normalized with respect to a. Dimensionless time is defined by
τ ) tDA/a2
(A3)
0 e τ < τswitch, Z ) 0, 1 < R < ∞: ∂CA/∂Z ) 0, ∂CB/∂Z ) 0
(A8)
0 e τ < τswitch, R ) 0, all Z: ∂CA/∂R ) 0, ∂CB/∂R ) 0
(A9)
0 e τ < τswitch, R f ∞, Z f ∞: CA ) 1, CB ) 0
(A10)
The boundary conditions have the following meanings. Equations A7 and A8 denote that the electrolysis of A to B at the UME occurs at a diffusion-controlled rate but that both species are inert on the insulating sheath surrounding the electrode. Equation A9 is a consequence of the axisymmetric cylindrical geometry, while the semiinfinite condition eq A10 dictates that both species recover their bulk solution concentrations far from the UME surface. For the period of the reverse step, in which B is electrolyzed to A at the UME at a diffusion-controlled rate, it is only necessary to evaluate the concentration profile for B, by solving eq A2 with i ) B subject to the following boundary conditions:
τ g τswitch, Z ) 0, 0 e R e 1:
CB ) 0
(A11)
τ g τswitch, Z ) 0, 1 < R < ∞: ∂CB/∂Z ) 0
(A12)
τ g τswitch, R ) 0, all Z:
∂CB/∂R ) 0
(A13)
τ g τswitch, R f ∞, Z f ∞:
CB ) 0
(A14)
The aim of the model is to provide a solution for the currenttime behavior during the forward and reverse potential steps. For the forward step, the current, i, normalized with respect to the steady-state current for the diffusion-controlled electrolysis of A, i(∞) (eq 3), is given by 1
Z)0R
A
0
dR
(A15)
(A4)
CA ) [A]/[A]∞
(A5)
CB ) [B]/x[A]∞
(A6)
Analytical Chemistry, Vol. 69, No. 11, June 1, 1997
CA ) 0, -λB(∂CB/∂Z) ) ∂CA/∂Z (A7)
∫ (∂C /∂Z)
reflects the fact that A and B have different diffusion coefficients. The concentrations of A and B have been normalized with respect to the bulk concentration of A, [A]∞, such that
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0 e τ < τswitch, Z ) 0, 0 e R e 1:
i/i(∞) ) (π/2)
where DA is the diffusion coefficient of A. The parameter
λi ) Di/DA
The following boundary conditions are true following the initial potential step at τ ) 0 and prior to the reverse step at τ ) τswitch:
For the period of the reverse step, the normalized current ratio is given by
i/i(∞) ) -λB(π/2)
∫ (∂C /∂Z) 1
0
B
Z)0R
dR
(A16)
Chronoamperometric characteristics were calculated numerically using a FORTRAN program based on the alternating
direction implicit finite difference method algorithm,31-33 which has been widely used to solve previous UME diffusion problems.34,35 To optimize the efficiency of the calculations, exponentially expanding grids were applied as described elsewhere.30,35,36 The forward step characteristics were checked against and found to be in agreement with eq 1. In the analysis of experimental
data, the step period and value of DA were defined by the experiment, so that the only unknown was λB. The best fit to the experimental data was achieved by comparison with the simulated current response with a range of λB.
(31) Peaceman, D. W.; Rachford, H. H. J. Soc. Ind. Appl. Math. 1955, 3, 28. (32) Lapdius, L.; Pinder, G. F. Numerical Solution of Partial Differential Equations in Science and Engineering; Wiley: New York, 1982. (33) Ames, W. F. Numerical Methods of Partial Differential Equations; Wiley: New York, 1977. (34) (a) Heinze, J. J. Electroanal. Chem. 1981, 124, 73. (b) Heinze, J.; Sto ¨rzbach, M. Ber. Bunsenges. Phys. Chem. 1986, 90, 1043. (c) Heinze, J. Ber. Bunsenges. Phys. Chem. 1981, 85, 1096. (35) Unwin, P. R.; Bard, A. J. J. Phys. Chem. 1991, 95, 7814. (36) Britz, D. Digital Simulation in Electrochemistry; Springer-Verlag: New York, 1988.
Received for review December 2, 1996. Accepted March 13, 1997.X AC961211I
X
Abstract published in Advance ACS Abstracts, May 1, 1997.
Analytical Chemistry, Vol. 69, No. 11, June 1, 1997
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