Anal. Chem. 1998, 70, 285-294
Voltammetry for Reduction of Hydrogen Ions from Mixtures of Mono- and Polyprotic Acids at Platinum Microelectrodes Salvatore Daniele,*,† Irma Lavagnini,‡ M. Antonietta Baldo,† and Franco Magno‡
Department of Physical Chemistry, University of Venice, Calle Larga S. Marta, 2137, 30123 Venice, Italy, and Dipartimento di Chimica Inorganica, Metallorganica e Analitica, Universita’ di Padova, Via Marzolo,1, 35131 Padova, Italy
The steady-state voltammetric behavior for reduction of several polyprotic acids and mixtures of strong and weak mono- and polyprotic acids was studied at platinum microelectrodes. The results demonstrated that over the potential range accessible to reduction of acids in water (up to ∼-1 V vs Ag/AgCl) via a preceding chemical reaction (CE mechanism), the reduction of weak polyprotic acids and mixtures of acids can produce either a single well-defined wave or two waves separated to a different extent, depending on the dissociation constant of each acidic form, on the analytical concentration of each acid, and on the mutual ratio of the acids present at equilibrium in the bulk solutions. The overall reduction mechanism for most of the mixtures examined was interpreted on the basis of a series of CE processes associated to the hydrogen evolution. This interpretation was supported by digital simulation procedures. A theoretical relationship for predicting the steady-state limiting current for any mixture of acidic species, whose dissociation steps are fast, was also derived. This equation proved valid for all those acids with equilibrium constants larger than ∼10-6. On the basis of this theoretical relationship, a simple diagnostic criterion to assess whether or not the reduction process of a mixture of acids is under a kinetic control was also established. Recently, the electrode reaction involving discharge of hydrogen ion arising from strong and weak acids at platinum microelectrodes has been the object of several investigations.1-9 Measurements have been carried out in solutions without and with †
University of Venice. University of Padova. (1) Fleischmann, M.; Lasserre, F.; Robinson, J.; Swan, D. J. Electroanal. Chem. Interfacial Electrochem.1984, 177, 97. (2) Ciszkowska, M.; Stojek, Z.; Morris, S. E.; Osteryoung, J. G. Anal. Chem. 1992, 64, 2372. (3) Troise, F. M. H; Denault, G. J. Electroanal. Chem. Interfacial Electrochem. 1993, 354, 3311. (4) Stojek, Z.; Ciszkowska, M.; Osteryoung, J. G. Anal. Chem. 1994, 66, 1507. (5) Ciszkowska, M.; Stojek, Z.; Osteryoung, J. G. J. Electroanal. Chem. Interfacial Electrochem. 1995, 398, 49. (6) Perdicakis, M.; Piatnicki, C.; Sadik, M.; Pasturaud, R.; Benzakour, B.; Bessiere, J. Anal. Chim. Acta. 1993, 273, 81. (7) Daniele, S; Lavagnini, I; Baldo, M. A.; Magno, F. J. Electroanal. Chem. Interfacial Electrochem. 1996, 404, 105. (8) Daniele, S; Baldo, M. A.; Simonetto, F. Anal. Chim. Acta 1996, 331, 117. ‡
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© 1998 American Chemical Society
varying concentrations of supporting electrolyte, and the effects of ionic strength on mass transport properties have been examined.2-6 Other studies were aimed at verifying the kinetics of the electrode process involved in the reduction of weak monoprotic acids for which a CE mechanism applies.1,7-9 It was verified that, using platinum microdisk electrodes having radius over the range 10-12.5 µm, the chemical reaction preceding the electron transfer is fast for those acids whose dissociation constant, Ka, is larger than ∼1 × 10-6.7,8 In solutions of strong acids, which are completely dissociated, it was found that the reduction wave height depends linearly on concentration over a wide range.2,3,6 On the other hand, in solutions of monoprotic weak acids the dependence of the steadystate limiting currents on the acid concentration is not strictly linear, though in restricted concentration ranges an apparent linearity can be observed.7,8 For weak monoprotic acids, whose dissociation step is fast, the steady-state limiting current, Il, can be predicted by2,7,8,10
Il ) 4nFr(DH+[H+]b + DHA[HA]b)
(1)
where DHA and [HA]b are the diffusion coefficient and the bulk equilibrium concentration of the undissociated acid, DH+ and [H+]b are the diffusion coefficient and bulk equilibrium concentration of proton, the electron number n is equal to 1, and the other symbols have their usual meanings. This relationship has been derived by resorting to the apparent diffusion coefficient of the acid, which is given by the mole fraction weighted average of H+ and HA present in the bulk solution at equilibrium, in analogy with similar situations where two or more species, involved in a homogeneous equilibrium, are characterized by different diffusion coefficient values.11-13 In fact, in water, the diffusion coefficient of proton and undissociated acid differ generally by ∼1 order of magnitude.7,14 The wave position of weak monoprotic acids was found to depend on both the equilibrium dissociation constant and the (9) Ciszkowska, M; Jaworski, A.; Osteryoung, J. G. J. Electroanal. Chem. Interfacial Electrochem. 1997, 423, 95. (10) Oldham, K. B. Anal. Chem. 1996, 68, 4173. (11) Carofiglio, T.; Magno, F.; Lavagnini, I. J. Electroanal. Chem. Interfacial Electrochem. 1994, 373, 11. (12) Rusling, J. F.; Shi, C. N.; Kumosinski T. F. Anal. Chem. 1988, 60, 1260. (13) Evans, D. H. J. Electroanal. Chem. Interfacial Electrochem. 1989, 258, 451. (14) Heyrovsky, J.; Kuta, J. Principles of Polarography, Academic Press: New York, 1966.
Analytical Chemistry, Vol. 70, No. 2, January 15, 1998 285
analytical concentration of the acid. Table 1 summarizes data obtained in previous investigations7,8 for a series of monoprotic acids. This table shows that, at a given acid concentration, the reduction wave is the more negative the lower the dissociation constant. Moreover, very weak acids show a cathodic shift, on increasing acid concentration, whereas strong acids exhibit an anodic shift. Moderate weak acids show an intermediate behavior.7,8 The dependence of the half-wave potential E1/2 on concentration for proton reduction is due to a nonunity electrode reaction mechanism,16 2H+ + 2e ) H2. The reduction of polyprotic acids at microelectrodes has found less attention. In particular, there is no report of the acid concentration effects on their voltammetric behavior. Two recent papers5,9 dealt with the effects of supporting electrolyte on the wave height. Finally, to the best of our knowledge, no report exists dealing with the reduction of acids in mixtures. Since measuring the current height of voltammetric waves arising from solutions of acids can be competitive, or perhaps, superior to potentiometry for monitoring the acid content of a solution,17-19 the voltammetric behavior both of polyprotic acids and mixtures of acids is relevant. In this paper, a comprehensive investigation concerning the voltammetric behavior of mixtures of aqueous solutions of acids (strong and weak, mono- and polyprotic) is reported. A detailed analysis of the reduction process is carried out from both an experimental and theoretical (via digital simulation) point of view. Moreover, it is shown that, under the assumption that no kinetic hindering exists in all the chemical dissociation steps of the acids, a relationship able to predict the steady-state limiting current for any mixture of acids can be used. This equation can be the basis for understanding to what extent the hydrogen evolution process from strong and weak acids can be employed for monitoring the acid content of a solution by amperometry. EXPERIMENTAL SECTION Chemicals. All the chemicals employed were of analytical reagent grade. The following reagents were used: perchloric, nitric, hydrochloric, acetic, sulfuric, malic (Merck), maleic, citric, and tartaric acids, sodium dihydrogen phosphate, disodium ethylenediaminetetraacetate (Na2H2Y), sodium acetate, sodium hydroxide, and potassium chloride (Aldrich). Milli-Q purified water was used throughout to prepare stock solutions of the reagents. Nitrogen, pure (Siad, Bergamo, Italy) 99.99%, was used for deoxygenation in the voltammetric experiments. All the acids and bases employed were previously standardized by an acidbase titration procedure; Na2H2Y was standardized by complexometric titration.20 Electrodes and Instrumentation. In order to prepare the platinum microelectrodes, wires of diameter 20 and 25 µm were (15) Martell, A. E.; Smith, R. M. Critical Stability Constants; Plenum Press: New York, 1981. (16) Shuman, M. S. Anal. Chem. 1969, 41, 142. (17) Daniele, S; Baldo, M. A.; Ugo, P. Electroanalysis 1991, 4, 93. (18) Baldo, M. A.; Daniele, S.; Mazzocchin, G. A. Anal. Chim. Acta 1993, 272, 151. (19) Baldo, M. A.; Daniele, S.; Mazzocchin, G. A.; Donati, M. Analyst 1991, 116, 933. (20) Vogel, A. I. A Text-Book of Quantitative Analysis. Including Elementary Instrumental Analysis, 3rd ed.; Longman: London, 1961
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sealed directly into glass as previously described.1 The platinum electrodes were polished mechanically with graded alumina powder (1, 0.3, to 0.05 µm) on a polishing microcloth. An electrochemical activation of the electrode surface was also made by cycling the potential 2-3 times over the range -0.1-2 V at 200 mV/s and then keeping the electrode at -0.1 V for 10-15 s.7 The effective electrode radius of the microelectrode was determined by recording the steady-state limiting current, Il, from a 1 mM ferrocene solution in acetonitrile and using the equation 21
Il ) 4nFDCr
(2)
where r is the radius of the microdisk, C the bulk concentration, and D the diffusion coefficient (2.4 × 10-5 cm2 s-1 22 ) and the other symbols have their usual meanings. The reference electrode used was an Ag/AgCl saturated with KCl. The experiments with the microelectrodes were carried out in a two-electrode cell maintained in a Faraday cage made of sheets of aluminum to avoid external noise. Linear scan voltammograms were generated using a PAR 175 function generator. A Keithley 485 picoammeter served as current-measuring device, and data were plotted by means of an X-Y recorder (HewlettPackard 7045 B). The measurements were carried out at 25 °C ((0.1) using a thermostatic bath. In all the experiments a scan rate, v, of 5 mV/s was employed. The parameter p ) (nFvr2/DRT)1/2 23 under the most unfavorable conditions, i.e., r ) 12.5 µm and D ) 5 × 10-6 cm2 s-1, was equal to 0.25, so that the diffusion limiting current deviates from the steady-state limiting value within 5%.23 All data points are mean values of at least three replicates (relative standard deviation, within 3%). All the measurements were carried out in 0.1 M KCl as supporting electrolyte. Under these conditions, the support excess, that is, the ratio of the concentration of the inert electrolyte to the concentration of hydrogen ion at equilibrium in the mixture,4,5 ensured a negligible contribution from migration (within 2%, with respect to the limiting current and 1 mV with respect to the half-wave potential)24 for any mixture of acids investigated. The essential features of the numerical approach used in the simulations have been described extensively elsewhere.11,25,26 The simulation programs were written in FORTRAN and ran on a DEC computer ALPHA AXP 3000/300. Some of the kinetic and diffusive parameters employed in the simulation were already available in the literature. Otherwise they were determined as follows. To evaluate the forward kinetic constants, it was assumed that the recombination rate constant of each step was of ∼1010 M-1 s-1, similarly to other weak acids.14 (21) Wightmann, R. M., Wipf, D. O. Bard, A. J., Eds. In Electroanalytical Chemistry; A Series of Advances Marcel Dekker: New York, 1989; Vol. 15, pp 67-353. (22) Adams, R. N. Electrochemistry at Solid Electrodes; Marcel Dekker: New York, 1969; p 221. (23) Aoki, K.; Akimoto, K., Tokuda, K.; Matsuda, H.; Osteryoung J. J. Electroanal. Chem. Interfacial Electrochem. 1984, 171, 219. (24) Myland J. C.; Oldham, K. B. J. Electroanal. Chem. Interfacial Electrochem. 1993, 347, 49. (25) Lavagnini, I. ; Pastore, P.; Magno, F; Amatore, C. A. J. Electroanal. Chem. Interfacial Electrochem. 1991, 316, 37. (26) Lavagnini, I.; Pastore, P.; Magno, F. J. Electroanal. Chem. Interfacial Electrochem. 1993, 358, 193.
Table 1. Experimental Half-Wave Potentials E1/2 for Reduction of Strong and Weak Monoprotic Acids as a Function of the Concentration -E1/2/V (vs SCE) acid
pKaa
0.5 mM
1.0 mM
5.0 mM
10 mM
HClO4b HSO4- c ClCH2COOHc CH3CH(OH)COOHc CH3COOHb H2Y2- b H2PO4- b
1.55 2.68 3.66 4.56 6.16 6.75
0.455 0.455 0.460 0.465 0.485 0.575 0.615
0.446 0.450 0.454 0.463 0.490 0.580 0.625
0.430 0.435 0.443 0.461 0.500 0.600 0.660
0.415 0.430 0.430 0.463 0.515 0.615 0.680
a
Figure 1. Experimental voltammetric waves for mixtures of 1.0 mM perchloric acid and a monoprotic weak acid at different concentrations. (a) Lactic acid concentration (mM): 0 (1); 5.0 (2); 10.0 (3). (b) Acetic acid concentration: 0 (1); 1.0 (2); 4.0 (3); 8.0 (4). Pt electrode radius, 12.5 µm. Scan rate 5 mV s-1. Supporting electrolyte 0.1 M KCl.
Ionic strength, 0.1 M.15 b Reference 7. c Reference 8.
guished provided that the differences in their half-wave potentials are sufficiently large. Since the half-wave potential of strong acids depends only on concentration, and those of weak acids also on the equilibrium constant, the prediction of whether or not the two processes can be discriminated is not straightforward. For instance, voltammetric waves recorded in mixtures of perchloric plus acetic acid, over the concentration range 0.1-1 mM, showed no separation at any weak to strong acid concentration ratio, at variance with the case illustrated in Figure 1b. This agrees with the fact that the E1/2 values of strong and weak acids become closer by decreasing the analytical (total) concentration of the acids as shown in Table 1. Measurements performed on mixtures of a strong (hydrochloric and nitric acids) and a weak acid, with a dissociation constant larger than that of acetic acid (e.g., monochloroacetic and hydrogen sulfate ions), displayed a voltammetric behavior similar to that of the mixtures perchloric plus lactic acid, over the concentration range 0.1-10 mM. (The behavior of mixtures of strong and weak acids with Ka lower than that of acetic acid will be discussed below). The reduction mechanism for mixtures of a strong and a weak acid can be presented as follows:
HA1 f H+ + A1Afterward, from the dissociation equilibrium constant, the forward rate constant was calculated. The diffusion coefficients of the weak acids were calculated from equivalent conductance at infinite dilution values and corrected by the ionic strength.8,27,28 Kinetic and diffusive parameters employed in the simulations will be indicated in the relevant tables and figures. RESULTS AND DISCUSSION Mixtures of a Strong and a Weak Uncharged Monoprotic Acid. Typical steady-state voltammograms obtained from mixtures of perchloric plus lactic acid and perchloric plus acetic acid are shown in Figure 1. Single well-defined waves can be observed in the first series of mixtures (Figure 1a), whereas two waves separated to some extent are evident in the second series of mixtures (Figure 1b), when the weak to strong acid concentration ratio is larger than 1. The different behavior observed for these mixtures indicates that the reduction of proton and undissociated weak acid, which occurs via a CE mechanism, can be distin(27) Purushottam, A. Raghava Rao, B. H. S. V. Anal. Chim. Acta 1955, 12, 589. (28) Bockris, J. O; Reddy A. K. N. Modern Electrochemistry; Plenum Press: New York, 1970; Vol. 1.
k1
(3)
HA2 {\ } H+ + A2k
(4)
2H+ + 2e- a H2
(5)
2
where HA1 is a strong acid, which is considered completely dissociated at any concentration, and HA2 is a weak monoprotic acid whose dissociation step is conditioned by the equilibrium constant Ka, which is linked to the kinetic constants k1 and k2 through the relationship Ka ) k1/k2. In order to verify from a theoretical point of view that the above mechanism produces voltammograms of the type showed in Figure 1, in the lack of analytical solution, the reaction scheme eqs 3-5 was treated by digital simulation. In the simulation, a general program, which allowed one to account for different reaction orders and kinetic hindering for the chemical reaction, was adopted. Moreover, the reduction of hydrogen ion was always considered reversible in character. Figure 2 shows simulated voltammograms obtained for some mixtures of perchloric plus lactic acid and perchloric plus acetic Analytical Chemistry, Vol. 70, No. 2, January 15, 1998
287
Figure 2. Simulated voltammograms for reduction of mixtures of a strong and a weak monoprotic acid. The potential values are reported as E-E0, where E0 is the formal potential of the electrode reaction 2H+ + 2e- a H2. (a) Mixtures of 1.0 mM HClO4 plus 8.0 mM acetic acid (1) and 1 mM HClO4 plus 10.0 mM lactic acid (2). (b, c) Effect of the concentration of the strong and the weak acids at a fixed concentration ratio on the shape of the simulated voltammograms. Because of the large differences between the currents relative to the two solutions, the simulated current values are reported in dimensionless form I/I10, where I10 ) 4FDH+Cr is the limiting current corresponding to the analytical hydrogen concentration C (C is the sum of the analytical concentration of the strong and the weak acids). (b) 0.1 mM perchloric acid and 1.0 mM lactic acid (1); 0.01 M perchloric acid and 0.1 M lactic acid (2). (c) 0.1 mM perchloric acid and 0.8 mM acetic acid (1); 0.01 M perchloric acid and 0.08 M acetic acid (2). Simulation parameters: (acetic acid, k1 ) 7.6 × 105 s-1, k2 ) 4.2 × 1010 M-1 s-1, and D ) 1 × 10-5 cm2 s-1;1 lactic acid, k1 ) 2.2 × 106 s-1, k2 ) 1.0 × 1010 M-1 s-1, and D ) 8.59 × 10-6 cm2 s-1.8 DH+ 7.91 × 10-5 and DH2 ) 4.0 × 10-5 cm2 s-1.7 Electrode radius 12.5 mm; grid 20 × 200; potential step 1 × 10-6 V per iteration.11,25 288
Analytical Chemistry, Vol. 70, No. 2, January 15, 1998
acid at different concentrations. In particular, the simulated curves 1 and 2 in Figure 2a resemble quite strictly the corresponding experimental curves, i.e., curve 3 in Figure 1a and curve 4 in Figure 1b, respectively. It must be noted, however, that the slope of the wave of perchloric acid is smaller than that predicted by simulation. This can be quantified in terms of the Tomes difference (E1/4 - E3/4).29 For a 2:1 stoichiometry (2H+:H2), the theory predicts a Tomes difference of 42 mV7 against an experimental value of 55 ( 3 mV. This discrepancy was already noticed in a previous investigation7 and can be attributed to the actual status of the electrode surface, which may inhibit to some extent the reversibility of the overall electrode process. The influence of the concentration on splitting the voltammetric curves for a mixture of acids is shown in Figure 2b,c. From these it can be observed that for a given weak to strong acid concentration ratio, a single wave is obtained at lower concentrations, whereas two waves are evident at larger concentrations. In addition, the acid with the largest Ka requires a larger concentration in order to display two waves. Obviously, the two waves can be better distinguished at larger weak to strong acid concentration ratios, owing to the great difference between the diffusion coefficients of H+ and HA2. Exemplary experimental data concerning the overall steadystate diffusion limiting currents obtained for some mixtures of a strong and a weak acid are reported in Table 2, along with the steady-state limiting currents recorded in solutions containing each acid alone. The comparison shows that the overall steady-state limiting current of a mixture of acids is always lower than the sum of the steady-state limiting currents of each acid alone at the same concentration. This is a consequence of the redistribution of species at equilibrium upon mixing the acids, which leads to a final composition made of a relatively larger concentration of the slower diffusing species HA2, and of a relatively lower concentration of the faster diffusing species H+. The second and third columns of Table 2 show the theoretical currents obtained by simulation and by eq 1, respectively. In fact, eq 1 should predict the steady-state limiting current even for a mixture of a strong and a weak acid, provided that the dissociation reaction of the weak acid is fast, since proton and undissociated acid are still the only diffusing species involved. Table 2 shows that the two sets of theoretical data are identical to three significant figures, while experimental and theoretical data agree within 5%. In general, either theoretical procedure should yield identical results regardless of both the nature and concentration of the acids present in the mixture, provided that the dissociation step of the weak acid is fast. Table 3 compares theoretical data for several mixtures of perchloric plus acetic acid and perchloric plus lactic acid, over a wider concentration range. An excellent agreement is found for all concentrations examined, meaning that no kinetics occurs in the dissociation steps of the above weak acids. It can be stated that the comparison of simulated (or experimental) and calculated by eq 1 Il/Il0 values can provide a criterion to establish whether or not the preceding chemical reaction of the weak acid is under kinetic control. In the latter instance, experimental Il/Il0 values are lower than those calculated by eq 1. (29) Bond, A. M.; Oldham, K. B.; Zoski, C. G. Anal. Chim. Acta 1989, 216, 177.
Table 2. Comparison of Steady-State Limiting Currents for Reduction of Mixtures of a Strong and a Weak Acid: Experimental, I1exp, Predicted by Simulation, I1sim, and Eq 1a limiting current (nA) mixtureb
I1exp
I1sim
I1
I1expc
I1expd
1 mM HCl + 1 mM ClCH2COOHe 1 mM HClO4 + 1 mM CH3CH(OH)OOH 1 mM HClO4 + 1 mM CH3COOH 1 mM HClO4 + 4 mM CH3COOH
63.3 49.1 42.6 56.9
62.58 48.13 43.41 59.89
62.52 48.13 43.44 59.84
37.5 37.5 39.6 39.6
31.5 20.1 9.87 32.5
a I experimental conditions: Pt microdisk, radius 12.5 µm, scan rate v ) 5 mV s-1. Supporting electrolyte 0.1 M KCl. b All concentrations given 1 are analytical concentrations. c Experimental limiting current for reduction of 1 mM strong acid alone. d Experimental limiting current for reduction of weak acid alone. e Simulation parameters: chloroacetic acid k1 ) 2.9 × 107 s-1, k2 ) 1.0 × 1010 M-1 s-1. Other simulation parameters as in Figure 2.
Table 3. Dimensionless Limiting Currents Obtained via Digital Simulation,a I1min/I10, and from eq 1, I1/I10 for Mixtures of a Strong and a Weak Acidb mixture CHClO4 (mM)
CCH3CH(OH)COOH (mM)
I1sim/I10
I1/I10
0.1 0.1 1.0 1.0 10
0.1 1.0 1.0 10 100
0.813 0.460 0.625 0.268 0.206
0.813 0.459 0.622 0.268 0.202
mixture CHClO4 (mM)
CCH3COOH (mM)
I1sim/I10
I1/I10
0.1 1.0 1.0 1.0 10 10
0.8 1.0 4.0 8.0 40 80
0.292 0.577 0.314 0.234 0.309 0.220
0.294 0.569 0.311 0.233 0.300 0.222
a Simulation parameters as in Figure 2. b I ) 4FD +Cr, where C is 10 H the sum of the strong and weak acid analytical concentration.
Mixtures of Several Monoprotic Acids and Bases. The general voltammetric behavior of mixtures of several acids did not differ from that observed for mixtures of two acids. One or more waves could be observed, depending on the number of acids in the mixture, on the dissociation constant, and on the concentration of each acid. The presence of one wave was the most common case observed when mixtures of three or more acids with equilibrium constants larger than ∼10-5 were examined. An example of a voltammetric curve obtained in a mixture of acetic, chloroacetic, and lactic acids is shown in Figure 3, curve 1. Figure 3, curve 2, shows the voltammogram relative to the reduction of a mixture of several acids to which a base such as sodium acetate was added. The shape of the voltammogram does not change, but the height and position of the wave reflect the amount of both proton and undissociated acids at equilibrium. Table 4 presents experimental and theoretical steady-state limiting currents, calculated by eq 6, obtained for a series of mixtures of acids and bases.
Il ) 4Fr(DH+[H+]b +
∑D
b HAi[HAi] )
(6)
This equation is an extension of eq 1 and accounts for the
Figure 3. Experimental voltammograms for hydrogen evolution from the mixtures 0.3 mM acetic acid, 0.4 mM lactic acid, and 0.3 mM chloroacetic acid (1); and 1 mM acetic acid, 1 mM lactic acid, 1 mM chloroacetic acid, and 5 mM sodium acetate (2). Experimental conditions as in Figure 1.
contribution to the steady-state limiting current of each undissociated acid HAi present at equilibrium in the bulk solution. A good agreement (within 3%) is evident, indicating that for all the mixtures examined the preceding chemical reactions of the weak acids are fast. Regarding this last aspect, it should be noted that very large amounts of bases with respect to their conjugated weak acids can slow down the rate of dissociation of the weak acid. For acetic acid, this effect occurred for acetate to acetic acid concentration ratio of ∼100.7 Polyprotic Acids. Typical steady-state voltammograms of sulfuric, tartaric, citric, malic, and maleic acids are shown in Figure 4, while Table 5 collects relevant pKa values, half-wave potentials, and the Tomes differences of the acids at various concentrations. From Figure 4 it is evident that maleic acid, which exhibits the largest difference between the Ka values, gives two separated waves (see curve 3). However, the extent of the separation is greater the higher the analytical concentration of the acid, as shown in Table 5. Sulfuric acid displays a well-defined wave at any concentration, but the other acids show waves that are more or less rounded in their upper regions and drawn out depending on the number of dissociable hydrogen ions, on the analytical concentration, and on the strength of the acids. The extent of the last effect can be inferred from the Tomes differences (see Table 5). A closer examination of the waves recorded in tartaric, Analytical Chemistry, Vol. 70, No. 2, January 15, 1998
289
Table 4. Comparison of Experimental Steady-State Limiting Curents, I1exp, and Calculated by Eq 6, I1, for Mixtures of Monoprotic Acids and Basesa limiting current (nA) mixtureb
I1exp
I1
1 mM CH3CH(OH)COOH + 1 mM CH3COOH 1 mM CH3CH(OH)COOH + 1 mM CH3COOH + 1 mM ClCH2COOH 0.2 mM HNO3 + 0.2 mM KHSO4 + 0.6 mM CH3CH(OH)COOH 0.3 mM ClCH2COOH + 0.3 mM CH3COOH + 0.4 mM CH3CH(OH)COOH 1 mM CH3COOH + 1.2 mM CH3CH(OH)COOH + 1 mM CH3COONa 1 mM CH3COOH + 1.2 mM CH3CH(OH)COOH + 10 mM CH3COONa 1 mM CH3COOH + 1.2 mM CH3CH(OH)COOH + 10 mM CH3COONa + 1 mM HCl 2 mM CH3COOH + 4 mM CH3CH(OH)COOH + 2 mM CH3COONa + 1 mM NaOH
24.0 44.1
22.9 44.6
22.9
23.5
18.1
18.4
12.0
12.8
9.70
9.52
14.0
13.9
29.0
27.4
a Experimental conditions as in Figure 1. b All concentrations given are analytical concentrations.
citric, and malic acid solutions suggests that rounding is due to overlapping processes that are starting to separate. These effects are more evident as the concentration of the acid increases. The effect of concentration on the shape of the waves of polyprotic acids can be regarded in a manner similar to that seen for mixtures of monoprotic acids. That is, since the second or third dissociation constants of a polyprotic acid are lower than the first one, the increase of the concentration of the acid should favor the separation of the waves. The extent of this separation is larger the greater the difference among the dissociation constants of an acid, as observed experimentally. On the basis of this view, the reduction process of polyprotic acids was simulated using the following general scheme: k1
HnA {\ } H+ + H(n-1)Ak
(7)
2
k3
H(n-1)A- {\ } H+ + H(n-2)A2k
(8)
4
k2n-1
HA(n-1)- {\ } H+ + Ank
(9)
2H+ + 2e a H2
(10)
2n
where HnA is a generic polyprotic acid with n protons. Figure 5 shows simulated voltammograms of polyprotic acids, and in this case also, simulated and experimental voltammograms look very similar. Table 6 shows experimental Il/Il0 values along with those obtained by simulation and by the general eq 11, which gives Il na
Il ) 4Fr{DH+[H+]b +
nH
∑ ∑ nD [H A ] } b
i
Figure 4. Experimental voltammograms for reduction of 5.0 mM polyprotic acids: (a) 5 mM malic (1), citric (2), and maleic acids (3). (b) 5 mM tartaric (1), and 2 mM sulfuric acids (2). Experimental conditions as in Figure 1.
n i
(11)
i)1 i)1
values for a generic polyprotic acid HnA, whose dissociation steps are fast, where na is the number of weak acids, nH is the number 290 Analytical Chemistry, Vol. 70, No. 2, January 15, 1998
of dissociable hydrogen ions undergoing the reduction process,7 and Di is the diffusion coefficient of the ith acid. For the sake of simplicity, eq 11 was derived by assuming that all the acidic forms coming from the same acid share a common diffusion coefficient value. Also, this equation was derived on the basis of the apparent diffusion coefficient definition. It is evident from Table 6 that for tartaric and malic acids the two sets of theoretical data agree within 1% at any concentration, while experimental data, compared to two significant figures with the theoretical values, agree within 5%. This suggests that no kinetics hinders the dissociation steps of these acids. On the contrary, for citric and maleic acids, a small difference between the two sets of theoretical data is found. Experimental and simulated data compare well within 6%. These facts suggest the occurrence of a slight kinetic control in the overall process, which probably arises in the third and second dissociation steps of the two above acids, respectively. These findings agree also with the results obtained for monoprotic weak acids, where a kinetic hindering was observed when the equilibrium dissociation constant was lower than that of acetic acid (i.e, pKa > 6.0).7 The dependence of steady-state limiting currents on polyprotic acid concentration was also examined to verify whether or not
Table 5. Experimental Values of Half-Wave Potential, E1/2, and Tomes Parameter, (E1/4 - E3/4), as a Function of the Concentration of Polyprotic Acidsa -E1/2 (V; Ag/AgCl) b
b
acid
pKa1
pKa2
sulfuric tartaric citric malic maleic
2.82 2.87 3.24 1.75
1.55 3.95 4.35 4.71 5.83
pKa3
b
5.69
(E1/4 - E3/4) (mV)
0.5 mM
1.0 mM
5.0 mM
0.5 mM
1.0 mM
5.0 mM
0.444 0.456 0.461 0.461 0.450c 0.610d
0.440 0.453 0.452 0.459 0.448c 0.620d
0.426 0.446 0.450 0.457 0.430c 0.640d
50 50 54 52 50
50 50 60 55 50
53 66 83 70 55
a Pt microdisk, radius 12.5 µm, scan rate v ) 5 mV s-1, and supporting electrolyte 0.1 M KCl. b Ionic strength, 0.1 M. c First wave. d Second wave.
Table 6. Comparison of Dimensionless Limiting Currents for Polyprotic Acids as a Function of the Acid Concentration: Experimental, I1exp/I10, simulated I1sim/I10, and Calculated by Eq 11, I1/I10a acid
C (mM)
I1exp/I10
I1sim/I10
I1/I10
tartaric
0.1 1.0 5.0 0.1 1.0 5.0 0.1 1.0 5.0 0.1 1.0 5.0
0.72 0.47 0.30 0.43 0.28 0.20 0.54 0.4 0.22 0.51 0.48 0.41
0.722 0.456 0.312 0.435 0.276 0.186 0.557 0.346 0.233 0.527 0.513 0.444
0.722 0.453 0.313 0.440 0.288 0.201 0.556 0.347 0.235 0.539 0.528 0.466
citric malic maleic
a I ) 4FD +C*r, where C* is equal to the product of the analytical 10 H concentration of the acid and the number of dissociable hydrogen ions. Experimental conditions as in Figure 1; simulation parameters as in Figure 2.
Figure 5. Simulated voltammograms for reduction of 5.0 mM polyprotic acids: malic (1), citric (2), tartaric (3), and maleic acids (4). Electrode radius, 12.5 µm. Simulation parameters: Malic acid, k1 ) 5.6 × 106 s-1, k2 ) 1.0 × 1010 M-1 s-1, k3 ) 6.0 × 105 s-1, k4 ) 3.1 × 1010 M-1 s-1, D ) 8.52 × 10-6 cm2 s-1. Citric acid, k1 ) 1.2 × 107 s-1, k2 ) 1.0 × 1010 M-1 s-1, k3 ) 4.5 × 105 s-1, k4 ) 1.0 × 1010 M-1 s-1, k5 ) 5.1 × 104 s-1, k6 ) 2.5 × 1010 M-1 s-1.14 D ) 5.39 × 10-6 cm2 s-1. Tartaric acid, k1 ) 1.5 × 106 s-1, k2 ) 1.0 × 1010 M-1 s-1, k3 ) 1.1 × 106 s-1, k4 ) 1.0 × 1010 M-1 s-1; D ) 8.54 × 10-6 cm2 s-1. Maleic acid, k1 ) 2.8 × 108 s-1, k2 ) 2.0 × 1010 M-1 s-1, k3 ) 3.0 × 106 s-1, k4 ) 3.0 × 1012 M-1 s-1.14 D ) 8.06 × 10-6 cm2 s-1. Other simulation conditions as in Figure 2.
linearity exists. For sulfuric acid, the diffusion limiting current increases linearly with increasing concentration (see Figure 6), and the regression analysis gives a correlation coefficient of 0.999, corresponding to a slope of 76.9 nA/mM, which is about twice that observed for strong acids at a platinum microdisk with the same radius.7 On the other hand, Figure 6 shows that the calibration plots obtained for the other polyprotic acids studied are no longer linear, similarly to the trends observed for weak monoprotic acids.7 This behavior is explainable considering that, increasing the analytical concentration, the weak acids are less dissociated and the contribution of the faster diffusing proton is replaced by the slower diffusing undissociated acidic species. Mixtures of three polyprotic acids and of a polyprotic acid and its conjugate base were also studied. A single wave was always obtained, and their broad shapes did not change much with
respect to those shown in Figure 4. Shape, position, and height of the waves reflected, as noted above, the number of acids and the level of free proton in the mixture. Table 7 shows voltammetric parameters of some mixtures considered. As for the steady-state limiting current, it can be predicted by eq 11, and as shown in Table 7, the agreement with experimental data is within 6%. Mixtures of a Strong Acid and an Ampholyte. (a) Mixture of Perchloric Acid and Sodium Dihydrogen Phosphate. The final composition (within 5%) of mixtures made by perchloric acid and dihydrogen phosphate ion is H+, H3PO4, and H2PO4-. Typical voltammograms for a series of the above mixtures at several compositions are presented in Figure 7. In these cases, two wellseparated waves can be seen at any weak to strong acid concentration ratio. On the basis of the equilibrium constants characterizing the dissociation steps of the acidic species involved, it is likely that H+ and H3PO4 (Ka1 ) 7.5 × 10-3) are reduced within the less cathodic wave, while H2PO4- (Ka2 ) 4.1 × 10-7) is reduced at the second more cathodic one. The reduction process of these mixtures was also treated by simulation, and Figure 8 shows some of the voltammograms obtained. It can be seen that simulated and experimental voltammograms exhibit the same shape. However, considering the overall steady-state limiting currents, agreement between theoretical and experimental values exists only at low concentraAnalytical Chemistry, Vol. 70, No. 2, January 15, 1998
291
Table 7. Steady-State Voltammetric Parameters for Mixtures of Polyprotic Acids mixturea
-E1/2 (V; Ag/AgCl)
(E1/4 - E3/4) (mV)
I1exp (nA)
I1 (nA)
1 mM tartaric + 1 mM citric + 1 mM malic acids 1 mM tartaric acid + 1 mM sodium hydrogen tartrate 1 mM citric acid + 1 mM sodium dihydrogen citrate
0.454 0.456 0.469
62 50 80
69.2 34.47 31.21
72.58 35.01 31.81
a
All concentrations given are analytical concentrations.
Figure 6. Dependence of experimental limiting current on analytical concentration of polyprotic acid: (0) sulfuric, ( ) maleic, (b) tartaric, * (]) citric, and (2) malic. Experimental conditions as in Figure 1.
Figure 8. Simulated voltammograms for reduction of mixtures of 1.0 mM perchloric acid and 5.0 (a) and 10.0 mM (b) sodium hydrogen phosphate. Pt electrode radius, 12.5 µm. Simulation parameters: phosphoric acid, k1 ) 1.5 × 108 s-1, k2 ) 2.0 × 1010 M-1 s-1, k3 ) 2.5 × 104 s-1, k4 ) 6.2 × 1010 M-1 s-1;7 D ) 9.60 × 10-6 cm2 s-1. Other simulation conditions as in Figure 2. Table 8. Comparison of Dimensionless Limiting Currents for Mixtures of HClO4 and NaH2PO4 as a Function of the Analytical Concentration of the Strong Acid, CHClO4, and the Weak Acid, CNaH2PO4 CHClO4 (mM)
CNaH2PO4 (mM)
I1exp/I10
I1/I10
I1sim/I10b
I1exp/I10c
0.1 0.1 1.0 1.0 1.0 1.0
0.5 1.0 1.0 2.0 5.0 10.0
0.22 0.15 0.49 0.35 0.19 0.14
0.261 0.194 0.512 0.356 0.210 0.154
0.214 0.147 0.457 0.293 0.137 0.083
0.243 0.176 0.480 0.320 0.176 0.122
a I ) 4FD +C 10 H NaH2PO4r. Pt microdisk, radius 12.5 µm, scan rate v ) 5 mV s-1, and supporting electrolyte 0.1 M KCl. b The simulations were perfomed using the value k1 ) 2.5 × 104 s-1 as forward kinetic constant. c The simulations were performed using the value k ) 4.1 × 104 s-1 1 as forward kinetic constant.
Figure 7. Experimental voltammograms for reduction of mixtures of 1.0 mM perchloric acid and 0, (1) 1.0 (2), 2.0 (3), 5.0 (4), and 10.0 mM (5) sodium dihydrogen phosphate. Pt electrode, radius 11 µm; other experimental conditions as in Figure 1.
tion of the acids, as can be seen from data collected in Table 8 (third and fifth columns). Moreover, theoretical steady-state limiting currents calculated by eq 11 do not agree with the other two sets of data. It must be noted that if an estimate of the limiting currents is made at the first wave, to which dihydrogen phosphate ion should not contribute, experimental and both theoretical predictions agree within 6%. This suggests that the discrepancies 292 Analytical Chemistry, Vol. 70, No. 2, January 15, 1998
observed arise with the species being reduced at the second wave, that is, dihydrogen phosphate ion. The reaction mechanism through which the reduction of H2PO4- occurs is somewhat debated. Both a CE process,7 involving reduction of hydrogen ion and the chemical reaction kinetically hindered, and a direct reduction of the acid5 have been considered to rationalize the experimental data. In order to shed more light on this reduction mechanism, the reduction process of the H2PO4- alone was reconsidered.
Table 9. Comparison of Dimensionless Limiting Currents for Solutions of NaH2PO4 as a Function of the Analytical Concentration, Ca C (mM)
I1sim/I10
I1exp/I10
I1/I10
I1sim/I10c
I1sim/I10d
0.1 0.5 1.0 5.0 10
0.119 0.119 0.119 0.119 0.119
0.14 0.106 0.105 0.099 0.098
0.174 0.144 0.136 0.127 0.124
0.156 0.115 0.102 0.075 0.065
0.169 0.135 0.125 0.105 0.097
a I 10 ) 4FDH+Cr. Pt microdisk, radius 11 µm, and supporting electrolyte 0.1 M KCl. b Values obtained via simulation of the direct reduction of H2PO4- through 2H2PO4- + 2e a H2 + 2HPO42-. The electrode reaction is considered reversible. c Values obtained via simulation using the value k1 ) 2.5 × 104 s-1 as forward kinetic constant. d Values obtained via simulation using the value k1 ) 4.1 × 104 s-1 as forward kinetic constant.
The overall steady-state limiting currents obtained experimentally, by simulation, by eq 11, and by assuming direct reduction of dihydrogen phosphate ion, are compared in Table 9. These data confirm that experimental (third column) and simulated values (fifth column), using k1 ) 2.5 × 104 s-1 7,14 agree satisfactorily for dilute solutions. Conversely, larger discrepancies arise for solutions more concentrated than 1 mM. In any case, the currents are lower than those predicted by eq 11 (fourth column), which indicates the occurrence of a kinetic hindering. Also, experimental data do not accord with calculations made, assuming that direct reduction occurs (second column). Assuming the occurrence of a CE mechanism, and by using a best-fitting procedure of simulated and experimental steady-state limiting current at the largest concentration of H2PO4-, a forward dissociation constant value equal to 4.1 × 105 s-11 was obtained. If the latter optimized value was employed for running simulations, experimental and theoretical data agree only at larger concentrations of the acid (see Table 9, sixth column). The same conclusion applies using the latter kinetic constant value for simulating the mixture of perchloric plus dihydrogen phosphate ion, as shown in Table 8, sixth column. To explain this anomalous behavior, it can be hypothesized that the reduction mechanism of dihydrogen phosphate ion may occur in parallel through both CE and direct reduction schemes. From the experimental results, it would seem that at low concentration of the acid, where it is more dissociated, the CE scheme prevails. To obtain evidence about this hypothesis, more experiments and new simulations are needed. A study on this matter is in progress in our laboratories. (b) Reduction of the Mixture Perchloric Acid and Disodium Ethylenediaminetetraacetate. A more complex mixture is formed upon mixing disodium ethylenediaminetetraacetate with perchloric acid. In fact, depending on the relative amount of these two acids, the main species (within 5%) present in the final compositions are H+, H2Y2-, H3Y-, and H4Y. Some typical voltammograms obtained in this case are shown in Figure 9. In this case also, two waves are evident, and on the basis of the dissociation constants of the acids involved, it is likely that H+, H4Y (Ka1 ) 1.12 × 10-2,15) and H3Y- (Ka2 ) 2.09 × 10-3,15) are reduced within the first wave, while H2Y2- is reduced at the second more cathodic one. The simulated reduction processes for these
Figure 9. Experimental voltammograms for reduction of mixtures of 1.0 mM perchloric acid and 0 (1), 1.0 (2), 2.0 (3), 5.0 (4), and 10.0 mM (5) Na2H2Y. Pt electrode, radius 11 µm; other experimental conditions as in Figure 1.
Figure 10. Simulated voltammograms for reduction of mixtures of 1.0 mM perchloric acid and 2.0 (1), 5.0 (2), and 10 mM (3) Na2H2Y. Pt electrode radius, 11 µm. Simulation parameters: Na2H2Y: k1 ) 1.1 × 108 s-1, k2 ) 1.0 × 1010 M-1 s-1, k3 ) 2.1 × 107 s-1, k4 ) 1.0 × 1010 M-1 s-1, k5 ) 1.4 × 105 s-1;7 k6 ) 1.0 × 1010 M-1 s-1, D ) 5.01 × 10-6 cm2 s-1.7 Other simulation conditions as in Figure 2.
mixtures produced the voltammograms shown in Figure 10, which look very similar to the experimental ones. Table 10 compares experimental steady-state limiting currents and those obtained by the two different theoretical procedures adopted here. It can be seen that in this case a good agreement (within 5%) exists between experimental and simulated data, while those obtained by eq 11 are larger at any concentration. This means that the process is under a kinetic control, which probably arises in the reduction of H2Y2-, as already discussed.7 The reason why in the experiments shown in Figure 10 the overall wave height decreases, while increasing the concentration of the charged weak acid added to a constant concentration of HClO4, is understandable considering that the diffusion coefficients of the undissociated acids H3Y- and H4Y, formed upon Analytical Chemistry, Vol. 70, No. 2, January 15, 1998
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Table 10. Comparison of Dimensionless Limiting Currents for Mixtures of HClO4 and Na2H2Y as a Funciton of the Analytical Concentration of the Strong Acid, CHClO4, and the Weak Acid, CNa2H2Ya CHClO4 (mM)
CH2Y2- (mM)
I1exp/I10
I1/I10
I1sim/I10b
1 1 1 1
1 2 5 10
0.37 0.21 0.093 0.060
0.398 0.234 0.112 0.078
0.381 0.216 0.097 0.064
a I ) 4FD +C 10 H Na2H2Yr. Pt microdisk, radius 11 µm, and supporting electrolyte 0.1 M KCl. b Simulation conditions as in Figure 2.
addition of H2Y2- to HClO4, are somewhat lower than those of the other undissociated acids examined (5.01 × 10-6 cm2 s-1 against an average value of ∼9 × 10-6 cm2 s-1). CONCLUSIONS This paper has shown that the reaction of hydrogen evolution arising from weak acids both mono- and polyprotic, alone and in mixture, can be described on the basis of a series of CE processes. However, in aqueous solution, at platinum microelectrodes, the number of waves that can be discerned is hardly more than two. This circumstance has been verified by considering several acids accessible to reduction before the background discharge in water. In fact it has been proved that H2PO4- is the weakest acid reducible before water discharge occurs.5,7 The overall steadystate limiting current obtainable at platinum microdisk electrodes from polyprotic and mixtures of acids can be predicted, without resorting to cumbersome digital simulation, by using relationships that have been derived from the apparent diffusion coefficient of
294 Analytical Chemistry, Vol. 70, No. 2, January 15, 1998
the acids. These equations, however, are valid provided that the dissociation steps of each weak acid are fast. On the basis of the results reported here, using platinum microdisks with radii larger than 10 µm, this requirement has been ascertained for weak acids whose dissociation constants are larger than 10-6. A simple diagnostic criterion has been proposed to establish whether or not a kinetic control hinders the reduction process of an acid. It is based on the comparison of experimental and calculated (by eq 11) Il/Il0 values. In fact, in the case of kinetic control the experimental Il/Il0 should be lower than that calculated on the basis of eq 11, which can be accomplished by using proper Ka’s and analytical concentration of the acids. Finally, from an analytical point of view, it is evident that the determination of the acid content of a solution by voltammetry can be performed straightforwardly, only for strong mono and polyprotic acids, where linear calibration plots are found. In all other cases, either in solution of a acid alone or of a mixture of acids, the determination is problematic, owing to overlapping processes of the solution species. Strategies concerning how to obtain analytical information from voltammetric measurements will be addressed in a following paper which is now in preparation. ACKNOWLEDGMENT This work was supported by National Council of Research (CNR) and the Ministry of University and Scientific Research (MURST). Received for review June 25, 1997. Accepted October 20, 1997.X AC970666K X
Abstract published in Advance ACS Abstracts, December 1, 1997.
