Anal. Chem. 1993, 65, 2428-2436
2420
Evaluation of Electrochemical Parameters for an EC Mechanism from a Global Analysis of Current-Potential-Time Data: Application to Reductive Cleavage of Methylcobalamin Vimal T. Kumar and Ronald L. Birke' Department of Chemistry, The City College of the City University of New York, New York, New York 10031
Simultaneous evaluation of electron-transfer rate constant, P , following chemical reaction rate constant, kf,electron-transfer coefficient, a,and standard potential, E",for electron transfer coupled to a following chemical reaction (EC mechanism) is described. A mathematical model for the current response to a potential step is developed by incorporating the appropriate concentration terms into the Butler-Volmer equation. Experimental current-potential-time (;Et)surfaces are fit to this model to evaluate the parameters. Fitting individual t t or i-Ecurves did not yield unique parameter values whereas a n &Et surface constituted by several i-t o r i-E curves could be fitted to obtain unique values. A generalized kinetic zone diagram for the EC reaction is drawn by examining the limiting forms of the expression for current. Theoretical limits of measurable rate constants are estimated from the zone diagram. The three-dimensional electrochemistry described above was used to study the reductive cleavage of methylcobalamin in dimethyl sulfoxide (DMSO) solvent and 0.1 M tetrabutylammonium perchlorate supporting electrolyte. The parameters estimated are as follows: a = 0.552 f 0.004; P = 0.011 f 0.0015 cm s-l; kf= 1500 f 140 s-l; E" = -1.54 f 0.01 V. The rate constant for the following reaction, kf, in DMSO solvent is -4000fold faster than the similar process in aqueous medium. I t is suggested that this enhancement is relevant to methyl group transfer in enzymatic reactions, e.g., methionine synthase, if the enzyme mechanism involves a reductive cleavage which produces a methyl radical. INTRODUCTION Electrochemical oxidation or reduction often produces reactive intermediates that can undergo homogeneous chemical reaction. This frequently observed electrode process called an EC mechanism has been studied by various electrochemical techniques,' mainly cyclic voltammetry and chronoamperometry. While these are effective techniques to study coupled chemicalreactions, the information obtained has been typically restricted to one or two parameters. Part of the reason for this limitation is the fact that only a small part of the experimental data has been utilized when classical analog instruments are used to make the measurements. A cyclic voltammetric study, for instance, may use only peak currents and peak potentials though information is contained ~
~~
(1)Bard, A. J.; Faulkner, L. R. Electrochemical Methods, J. Wiley & Sons: New York, 1980; Chapter 11. 0003-2700/93/0365-2428$04.00/0
in the entire voltammogram. Likewise the ratio of anodic to cathodic current may be the only information used from a chronoamperometric experiment. Digital computers being widelyavailable, it is now possible to obtain more complicated experimental data sets and to extract information from much larger data sets. Such an electrochemical data set is the current-potential-time (i-E-t) surface,originally envisioned by Reinmuth? which we utilize in this paper to analyze the EC electrode process. Utilization of the i-E-t surface is the basis of some recent articles on electrochemical methodology and applications. Anderson and Bond have described the use of threedimensional normal pulse p0larography.39~Bond, Henderson, and Oldham used the three-dimensional concept in the analysis of cyclic voltammetric data.6 More recently, Lipkowski and co-workers have used three-dimensional electrochemistry to study electrode mechanisms using potential step methodscg and to study adsorption processes of organic molecules on solid electrodes.lOJ1 Papadopoulos et al.12 have described a computerized potentiostat for the routine recording of three-dimensional i-E-t curves in chronoamperometry and used the technique to study redox processes of tetrahydroxy-l,4-benzoquinone.These authors pointed out that chronoamperometry when plotted as a function of potential is a very sensitive technique since all the information obtainablefrom the electrode process is embodied in the threedimensional i-E-t surface.l2 Although, they did not treat their experimental results in a quantitative manner, Papadopoulos et al. suggested12 the desirability of such developments to enhance the usefulness of the three-dimensional viewing. It is the purpose of our present work to investigate the application of digital simulation and nonlinear regression analysis to the quantitative analysis of global data from i-E-t surfaces for the important EC mechanism. Chronoamperometry was first used to study a chemical reaction following electron transfer by Schwartz and Shain,13 who developed the theory for double-step chronoamperometry. The potential is stepped from the region where the species is not electroactive to the plateau of the i-E curve (2)Reinmuth, W. H. Anal. Chem. 1960,32, 1509. ( 3 ) Anderson, J. E.; Bond, A. M. Anal. Chem. 1981, 53, 504. (4)Anderson, J. E.;Bond, A. M. J. Electroanal. Chem. 1983,145,21. (5)Bond, A. M.; Henderson, T. L. E.; Oldham, K. B. J.Electroanal.
Chem. 1985,191, 75. (6) Seto, K.; Iannelli, A.; Love, B.; Lipkowski, J. J.Electroanal. Chem. 1987,226, 351. (7) Seto, K.; Noel, J.; Lipkowski, J.; Altounian, Z.; Reeves, R. J. Electrochem. SOC.1989, 136, 1910. (8) Sun, S. G.; Noel, J.; Lipkowski, J.; Altounian, Z. J. Electroanal. Chem. 1990,278, 205. (9)Sun,S.G.;Lipkowski, J.; Altounian, Z. J.Electrochem. SOC.1990, 137, 2443. (10)Richer, J.; Lipkowski, J. J. Electrochem. SOC.1986, 133, 121. (11)Hamelin, A.; Morin, S.; Richer, J.; Lipkowski, J. J.Electroanal. Chem. 1990,285, 249. (12)Papadopoulos, N.;Hasiotis, C.; Kokkinidis, G.; Papanastasiou, G. J. Electroanal. Chem. 1991, 308, 83. (13)Schwartz, W. M.; Shain, I. J. Phys. Chem. 1965, 69, 30. 0 1993 Amerlcan Chemical Society
ANALYTICAL CHEMISTRY, VOL. 05, NO. 18, SEPTEMBER
15, 1993 2420
and then back to the initial potential. The ratio of the anodic to cathodic current can be used to determine the rate constant of the homogeneous reaction. Later Marcoux and OBrien developed the theoretical expression for the current when potential is stepped to the rising portion of the i-E curve.14 This technique, known as potential-dependent chronoamperometry, was applied by Cheng and McCreery16 for simultaneous determination of reversible potential and the rate of the chemical step using nonlinear regression analysis for multiparameter estimation. The method described here, which could be viewed as a logical extension of potentialdependent chronoamperometry along a third dimension, evolved from our recognizing the limitations of multiparameter estimation from single current-time or current-potential curves and is the culmination of previously reported work.lBJ7 In this method, an i-E-t data set for an EC mechanism is simulated by numerical solution of an integral equation, and the electrochemical system parameters found by curve fitting the entire experimental data set by nonlinear regression analysis. The integral equation was solved by a numerical procedure given by Nicholson and Olmstead.1S Analysis of this integral equation leads to a zone diagram from which it is easy to observe the conditions necessary for the measurement of the various electrochemical parameters of the EC mechanism. Similar zone diagrams have been used in the electrochemical literature to discuss the methodology of linear sweep (cyclic) voltammetry techniques for various types of electrode processes, e.g., for electron transfer without coupled chemical reactions,1° for a reversible chemical reaction preceding electron transfer,20 and for an irreversible chemical reaction following electron transfer.21 As an experimentalexample of the use of the methodology, we investigate the electroreduction of the vitamin B12 derivative methylcobalamin in dimethyl sulfoxide solvent. Methylcobalamin is involved in enzymatic methyl grouptransfer reactions in which cleavage of the Co-C bond of the vitamin is a key step. The electrochemical reduction of methylcobalamin is a well-known example of an EC mechanism in which a one-electron transfer to give a radical anion occurs.22J3 The formation of this radical anion with an electron in a Co-C u* antibonding orbital labilizes the bond for homolytic cleavage.23~24The relevance of this process for enzymatic reactions is discussed.
EXPERIMENTAL SECTION The experimental setup consisted of an electrochemical cell, protected from light, containing methylcobalamii (Sigma Chemical Co., lot 67F-0773, >99.5% purity) in dimethyl sulfoxide (DMSO; Burdick & Jackson distilled in glass solvent) and supporting electrolyte, 0.1 M tetrabutylammonium perchlorate (Fisher Scientific, polarographic grade). The solution was maintained oxygen-free by bubbling solvent-presaturated nitrogen (prepurified grade) through a fine glass frit in the electrochemicalcell. A mercury (99.999%,Aldrich, ACS reagent grade) drop suspended from the recessed tip of a platinum wire fused inside glass served as the working electrode. A saturated calomel reference electrode (SCE) and a platinum counter electrode completed the three-electrode cell. Our instrumental (14) Marcoux, L.; O'Brien, T. J. P. J. Phys. Chem. 1972, 76, 1666. (15) Cheng, H.; McCreery, R. L. Anal. Chem. 1978,50,645. (16) Birke, R. L.; Kumar,V. T. 169th Meeting of the Electrochemical Society, Boston, MA, May 1986; Abstr. 482. (17) Kumar,V. T. Ph.D. Thesis, The City University of New York, 1992. (18) Nicholson, R. S.; Olmsted, M. L. In Computers in Chemistry; Mercel Decker: New York, 1972. (19) Matauda, H.; Ayabe, Y. 2.Elektrochem. 1955,59,494. (20) Saveant, J. M.; Vianello, E. Electrochim. Acta 1963,8,905. (21) Nadjo, L.; Saveant, J. M. J. Electroanal. Chem. 1973, 48,113. (22) Lexa, D.; Saveant, J. J. Am. Chem. SOC.1978,100,3220. (23) Kim,M. H.; Birke, R. L. J. Electroanal. Chem. 1983, 144, 331. (24) Fmke, R. G.; Martin, B. D. J. Inorg. Biochem. 1990,40, 19.
-1.20
.1.05
-1.40
VOLTS vs SCE
-1.60
-1'25
-1.'15
-1 :EO
-1.35
VOLTS vs SCE
Figure 1. (a, top) dc polarogram of methylcobalamin In DMSO: concentratlon, 1.0 X lo4 M; Supporting electrolyte, 0.1 M tetrabutylammonium perchlorate. (b, bottom) dc polarogram of methylcobalamln in water: concentratbn,0.90 X lo4 M; Supportingelectrolyte, 0.1 M tetraethylammonium perchlorate.
system included a DEC 11/73computer which digitally produces a variety of pulse wave forms for output to a potentiostat through a 16-bit DAC with a 1-ms pulse width limit. A Wenking Model 68FR0.5 fast rise time potentiostat was used to apply the potential on the cell. The 100%rise time of the pulse in our electrochemical cell was ca. 10 ps. At first the potential of the working electrode was held where no reaction takes place. A potential pulse was then applied to the rising part of the i-E curve. Around 100 points of currenttime data was collected in a Nicolet Model 206 digital storage oscilloscope over a time range of 0.4-9.0 ms. The experiment was repeated at various potential pulse heights, covering the entire i-E curve from the base line to the diffusion-limitedregion. At each potential a background i-t curve was recorded with an electrolyte solution without the electroactive species, and these data were subtracted in the Nicolet scope memory from the i-t curve taken with electroactive species present. Each currenttime datum was normalized by dividing by the diffusion-limited current. It is also possible to collect the data using sampledcurrent or normal pulse voltammetry with a variable sampling or pulse width time. Thus the i-E-t data are easily obtained in standardtwo-dimensionalelectrochemicalexperiments and after being stored digitally can be input to readily available 3D plotting and nonlinear regression programs for viewing and analysis. It was observed in the pure electrolyte solution that by 400 ps in the constant potential step experiment, the capacity (charging) current had decayed to zero. Of course, in the presence of adsorption from the electroactive species, the charging current can change,especiallyin the presenceof vitamin I312compounds.23 However, in the presence of methylcobalamin in DMSO, a potential step to a potential before the rise of the i-E curve shows the exact same capacity current decay as in the absence of the compound. Also, the dc polarogram of methylcobalamin in DMSO solvent is very well defined (Figure la) and shows no evidence of adsorption in the entire potential range, in contrast to other solvents such as water (or DMF); see Figure 1b. In aqueous medium, Figure l b illustrates the drastic effects of the adsorption of methylcobalamin on the i-E curve. Thus, charging current was not included in the simulation and appears not to be a component of the experimental total current in DMSO solvent at the times measured in these experiments. The i-E-t data collected in the memory of the Nicolet scopewere transferred either to a Macintosh I1 computer or to a V A X 11-780 for data analysis. Delta graph on a Macintosh I1 computer was used to
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ANALYTICAL CHEMISTRY, VOL. 65, NO. 18,SEPTEMBER 15, 1993
plot the three-dimensionaldata. Nonlinear least-squaresanalysis was performed with a FORTRAN 77 version of the subroutine NLLSQ on either the Macintosh I1 or the Vax 11/780 computer. The latter subroutine was written at Bell Telephone Laboratories and is an implementationof the Marquardt iterative schemez5 for nonlinear least-squares fitting and employs both gradient and Taylor series methods.
THEORETICAL ANALYSIS Current-Potential Equation for an EC Reaction. The EC reaction scheme involvinga solution-soluble redox couple 0, R can be represented as follows:
0 + ne- s R
(1)
1 '
0.8 '
"
0
g
2 c g
ne 0
0.6
e
o
E
0.4
O 0 0
0 0 0
0
e
0 0
0
O
0 0 0
:
O
0
e o
F
0
0
O
0 0
kr
R s Z
(2)
kb
The oxidized form, 0, is reduced to R at the electrode. R is then converted to electroinactive Z by a chemical reaction occurring in the solution, with equilibrium constant K = kf/ kb. The expression for the current is derived by starting with the Butler-Volmer equation for electrode kinetics, which may be written as
i = nFAkO [Co(O,t)el-anF(E-E"',/RTJ - C R ( O , t ) e ( ( l ~ ) n F ( E - E " ) / R T ) ] (3)
where k" is the standard heterogeneous rate constant, E"' the formal reduction potential, a the apparent charge-transfer coefficient, A the electrode surface area, Ci(0,t) the concentration of 0 or R at the electrode surface, and n, F, R, and 5" have their usual meaning in the electrochemical literature.' For the EC reaction the surface concentration terms, Co(0,t) and CR(0,t)are given by the followingconvolutionexpressions:
0.1
0.2
0
-0.1
-0.2
-0.3
( E-EO j/voits
Figure 2. Effect of electron-transferrate on current-potential curves: a = 0.5;P' = -1.5 V; k, = lo3 s-l;t = 8. (0)ko = 5 X lo-' cm s-1; ( 0 )ko = 1 X 10-1 cm s-1;(0) kO = 2 X 10-2 cm si;(+) ko = 4 X 10-3 cm s-1;(0) ko = 8 X lo4 cm s-'.
limited current, (Cottrell equation)
i(t), =
nFADo1/2CO* ,1/2t'/2
and defining W t ) = i(t)/i(t)d,
26
Q ( t ) represents the normalized, dimensionless current function independent of CO* and electrode surface area. The above equation was solved numerically by a step-function method as outlined in the Appendix, which has been discussed in detail by Nicholson and Olmsted.l*
RESULTS AND DISCUSSION where Ci* are bulk concentrations and Di diffusion coefficients for 0 and R. The expression for current is obtained by substituting these concentration expressions 4 and 5 into the Butler-Volmer eq 3. It is assumed that only the oxidized form is initially present and that the following reaction is irreversible (kf >> kb).
where k,, = k o e{-anF(E-E")/RT) and l=kf+k, It is more convenient to write eq 6 in terms of dimensionless current. Dividing both sides of the equation by the diffusion(25) Marquardt, D. W. J. SOC.Ind. Appl. Math. 1963, 11, 431. (26) Brown, E. R.; Large, R. F. In Techniques of Chemistry; Weissberger, A., Rossiter, B. W., Eds.; Wiley Interscience: New York, 1971.
Simulated Current-Time and Current-Potential Curves. The behavior of the numerically simulated currenttime and current-potential curves as the parameters of the system are varied over a wide range of values was well in agreement with theoretical prediction as well as documented behavior of real systems. In our case we are interested in a system with a relatively fast following reaction, so we have simulated the i-E curve for a kf = l o 3 s-l. The effect of varying the heterogeneous rate constant on the i-E curves for this homogeneous rate constant is shown in Figure 2. This curve is equivalent to a sampled current polarogram or a normal pulse polarogram with a sampling time or pulse width of 10 ms. As the electron-transfer rate increases the curves shift to the positive direction, as would be expected. The limit is reached when electron transfer becomes reversible. Figure 3 shows the effect on the i-E curve of varying the rate of the homogeneous reaction following a nearly reversible electron transfer. As the homogeneous chemical rate constant is increased, in this case for a quasi-reversible electrode reaction with k o = 0.1 cm s-l, the i-E waves shift toward positive potentials and reach a limit when kf is 109. On the other extreme, values of kf less than 1.0 have no effect on the i-E curves. The effect of kf on the shape of the i-E curve varies, depending on the value of k". For reversible electron transfer the effect is mostly a shift in the position of the curve and is thus indistinguishable from a change in E". For smaller
ANALYTICAL CHEMISTRY, VOL. 65,NO. 18, SEPTEMBER 15, 1993
2431
1 -
0.8
-
C
-
j LL
0.4
3
p--mf-uwottl
0.3
0.2
0
0.1
-0.1
-0.2
( E - EO') /volts
Figure 3. Effect of following chemical reaction on current-potential curves: a = 0.5; ko = 0.1 cm s-l; t = s. (0) kl = IO-' S-l;(0) kf = 101 S-1; (a)kf = 103 s-1; (+) kl = 105 S-1; (0)kl = 107;(0) kf = 100 s-1. 1
0.9
a
k" (cm 5-1)
kt (8-9
E"' (V)
*
a Each row in first section (a) lists the parameter values obtained by fittinga single i-Ecurve along with the true values in parentheses. The value of time used in simulating the i-E curve is shown in the firstcolumn. The second section (b)contains the results from fitting i-t curves simulated at the potential given in the first column. The last column contains the residuals.
o.6
I? 0.5 c
3e!
*
-1.45 0.237 (0.5)0.030 (0.05) 500 (500) -1.50 (-1.50) 2.4 X 10-13 -1.60 -0.51 (0.5) 0.0095 (0.01) 1 X 1OI6 (500)-1.60 (-1.60)8.1 X lo" -1.55 0.002 (0.5)0.0073 (0.005)2894 (2000) -1.63 (-1.60)2.1 X 1O-e -1.60 -0.72 (0.5) 0.022 (0.05) 1OOO (1000) -1.60 (-1.60)6.4 X 10-13
0.7
.B
-
Table I. Parameters Estimated by Fitting Simulated i E and i t Curves' a k" (cm s-1) kt (8-1) E"' (V) time ( 8 ) 5 X lW 0.463 (0.5) 0.076 (0.05) 6898 (1OOO) -1.62 (-1.60) 5.9 X lo-' 1 X lW 0.478 (0.5) 0,099(0.05) 7088 (1OOO) -1.63 (-1.60) 2.6 X lo" 2 X 1V 0.596 (0.5) 0.028 (0.05) 26.56 (1OOO) -1.58 (-1.60) 1.1 X lo" 5 X 10-3 0.650 (0.5) 0.025 (0.01) 34.38 (1000) -1.57 (-1.60) 8.5 X lW
EN)
0.8
-
SCE Flgure 5. Simulated current-voltegq-time (LE-t) surface: a = 0.5; ko = 0.05 cm s-l; kf = lo3 s-l; €' = -1.60 V.
0.4
0.3
Table 11. Parameters Obtained by Fitting an i-E Curve Simulated at Varying Times' a E"' (V) k" (cm 8-1) kf (5-1) time (8)
l e
0
5
30
55
80
Time/millisec
Figure 4. Normalized current at various values of kl: a = 0.5;ko = 0.1 cm s-1; P' = -1.50 V; E = -1.45 V. (W) kl = 10' S-'; ( 0 )kf = 101 s-1;(A)kl = io2 S-1; (a)kf = 103 S-1; (01 kf = 104 s-1;(0) kf = 105 S-1; (0)kl = 106 S-1. values of ko,the perturbation of the i-E curve is more at the lower part of the curve. The effect of kf in this case could be mimicked by a slower electron-transfer rate and a more positive Eo. These simulationexperimentsshow a high degree of correlation among the parameters, suggestingthat resolving a set of these parameters may not be straightforward. Figure 4 shows a typical set of current-time curves, in the form of the current function vs time, simulated for a wide range of kf values. This set of curves represents the situation when the potential is stepped to a value slightly positive to the Ell2 of the electrode reaction. As the krvalue is increased above 1.0 8-1, there is an increase in the normalized current as a function of time. This effect is the result of the chemical removal from the vicinity of the electrode surface of the product of the redox couple. In the limit of high rate constant, e.g., kf = 108s-1,the current function approachesthe diffusionlimited value as if the potential was stepped to a value well beyond Ellz. When these current function curves are plotted
le
103 103 10-2 10-2
0.5 (0.5) -1.5 (-1.5) 0.5 (0.5) -1.5 (-1.5) 0.5 (0.5) -1.5 (-1.5) 0.5 (0.5) -1.5 (-1.5) 0.5 (0.5) -1.5 (-1.5) 0.5 (0.5) -1.5 (-1.5)
0.04 (0.05) 1562 (1OOO) 2.4 X 10-' 0.06 (0.05) 471 (1OOO) 1.5 X 10-' 0.04 (0.05) 1084 (1OOO) 6.0 X le2 0.06 (0.05) 936 (1OOO) 2.9 X 0.04 (0.05) 1055 (1OOO) 2.4 X 1t2 0.06 (0.05) 962 (1OOO) 1.1 X 10-*
a a and E"' were held constant at their true values. k" was held constantat an erroneousvalue. kfalone was allowedtoadjust.Values in parentheses are the true values. The last column contains the residuals.
as a function of potential, a three-dimensional i-E-t surface results (Figure 5). Before analyzingsuch a surface,we consider the curve fitting analysis of the usual 2D experimental curves. Problems with Fitting Current-Time or CurrentPotential Curves. In order to test the procedure of curve fitting for an EC mechanism with quasi-reversible electron transfer and fast following reaction, data were generated by numerical solutions of eq 6 for various sets of parameters. These simulated current-time and current-potential curves were then fit to the theoretical equation without added noise using a nonlinear least squares program based of the Marquardt algorithmS25Data presented in Tables I and I1 are typical of the results obtained by fitting i-E and i-t data. In most cases, except for EO', the fitted values even without added noise are considerablydifferent from those used in the
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ANALYTICAL CHEMISTRY, VOL. 65, NO. 18, SEPTEMBER 15, 1993
Table 111. Parameters Obtained by Fitting Simulated i t , &E, and iI3-t Data' time/potential a ko kr (8-1) E" (V) Ip i-E data 150 ps 400 ps 650 ps 900 wcs 1150 ps
0.437 0.422 0.429 0.428 0.429
0.0079 0.0072 0.0062 0.0587 0.0057
15048 8257 4370 4101 4060
-1.62 -1.622 -1.616 -1.617 -1.617
2.13 X 3.79 X 6.04 X 3.91 X 3.92X
lo-' lo" lo" 1o-B 1o-B
i-1
-4.42 -0.704 0.003 1.21 1.05
0.0422 0.0607 0.3514 0.0047 0.0047
1013 2.5 X 10'5 lo00 2527 lo00
2.34 X 1P2 2.70X le' 1.60 X lo-'' 1.42 X 1P2 1.80 X
Reversible ET with Chem Step
Diffusion Limited (D)
91
( Eo',
( Eo' 1
Quasi-rev.ET with
Quasi.mversibie ET
Irrev. ET
Chem Step (kO,a) ( EO',
-1.603 -1.487 -1.60 -1.634 -1.60
I) - 1
1.5
i-t Data
-1.51 V -1.56 V -1.61 V -1.66 V -1.71 V
Reversible ET ;ChemSte;
a,ko
-1.5
.............................. lrrevenible ET
i-E-t Data ~
150-115Ops; -1.5 to -1.71 V
0.50 0.050
lo00
-1.60
true value
0.50
lo00
-1.60
0.05
4.00X 1Pl2
The first five rows are for current-voltage curves simulated at thetimeshowninthe firstcolumn. Thenextfiverowsareforcurrenttime curves simulated at the voltage given in the first column. The next row is for current-voltage-time surfacestimulatedover the same domain as the i-E and i-E curves. The true value of each parameter is shown in last row. The last column contains the residuals. a
simulation;yet the residuals are very small. It is quite evident that the parameter values that generate a given i-E or i-t curve are not unique. The curve-fitting routine compensates for the error in a parameter by adjusting other parameter(s). The adjustment of one parameter to offset the error in another while fitting three i-E curves is shown in Table 11. Parameters a and E"' are held constant at their true values, ko is held a t a value different from its true value, and the only parameter to be adjusted is kf. The optimizing program adjusted the value of kf by a different amount for each curve, in order to correct for the same error ink". This result is not entirely unexpected. The effect of kinetic parameters such ask" and kf on current depends on their magnitudes and also the time window of measurement. Nevertheless, this observation suggests a different approach for the evaluation of these parameters. If the data to be fitted were made up of the three curves taken together, the optimizing program in seeking parameters that fit all the curves is more likely to converge on the true values. Fitting a Current-Potential-Time Surface. Several current-potential curves recorded a t different sampling times, or equivalently several current-time curves recorded a t different potentials, constitute a current-potential-time (iE-t) surface. Table I11 shows the result of fitting data from an i-E-t surface. Entry 11in the table is the result of fitting the simulated i-E-t surface shown in Figure 4 while the first 10 entries were obtained by fitting i-t and i-E curves from the same time-potential domain. Entry 11, showing convergence of all four parameters on the true values, is representative of several simulation studies, demonstrating that fitting an i-E-t surface in three dimensions is a viable approach for the simultaneous evaluation of these parameters. Accessible Rate Constants a n d Kinetic Zones. Though not limited by experimental factors such as scan rate and reversal time, the competition between electron transfer, homogeneous reaction, and diffusion set the limits of measurable rate constants in the i-E-t experiment. These limits are determined by analyzing the digitized current-time equation (5A) derived in the Appendix. Now by defining dimensionless rate constant parameters A= kM(d)1/2/Do1/2 and h = dl, where d is a fixed interval of time, we rewrite eq
-3.5 -5
-4
-3
-2
-1
0
1
2
3
4
lOS(V Figure 6. Kinetic zone diagramfor EC reaction. E= f?". Parameters that control the current In each zone are listed In parentheses.
5A for the generalized EC mechanism as follows: *mX)l/' erf x ' / ~
2 ( ~ ) ' / ~m-1 C q i ( ( m- i *'I2 erf x ' / ~t = i
+ 1)'12 -
Equation 8 represents the numerical solution of integral eq 7 for the current function, 9,as a function of a discretized time, t = md, and the potential parameter, 8, where 8 = (Do/DR)ll2exp[nF(E - E"')/RTI. The second term in the denominator of eq 8 is purely a function of the following rate constant parameter A, and its magnitude controls the effect of the following chemical step on the i-E-t surface. For small A, the term goes to unity, indicating no effect of the following reaction. It is interesting to note that this term also gives the chemical rate effect for the EC mechanism in chronopotentiometry.1 The first term in the denominator of eq 8 contains both A and A, showingthat irreversibility can be imparted by a slow redox rate or a fast chemical step. The use of dimensionlessparameters A and A yields results, most general in nature, compactly displaying the characteristics of the system. Dimensioned profiles may be obtained by multiplying A by Do1/2/d1/2and X by lld. The limiting behaviors of eq 8 result when A and X assume extreme values. For a given set of these parameters, the relative weights of the denominator in eq 8 determine the regions of the kinetic zones in the zone diagram (Figure 6). We divide this diagram by considering three major dividing lines. Above the topmost line, line 1, the electron-transfer rate is reversible, while to the left of the next line, line 2, there is no effect of the following reaction because it is too slow. In this region there is general quasi-reversible electron transfer which only depends on A. Finally, to the right of line 3, the system will be irreversible by either slow electron transfer or fast following chemical kinetics. Considering these three regions, we have the following cases: (1) A m (Reversible Electron Transfer with Chemical Step). The first term in the denominator drops out and
-
ANALYTICAL CHEMISTRY, VOL. 65, NO. 18, SEPTEMBER 15, 1993
2433
eq 8 becomes
The boundary of this region was mapped in Figure 6 (line 1) by calculating values of parameters A and X for which the denominators of eqs 8 and 9 are nearly equal (within 2%); i.e.,
-
(2) X 0 (Quasi-ReversibleElectron Transfer). In this case the second term in the denominator of eq 8 becomes unity and factors in X in the numerator drop out since there is, in effect, no following chemical step so that eq 8 becomes
(Note: erf x = 2x/a1/2 for x
