2143
Anal. Chern. 1983, 55,2143-2147
Curve Fitting of Semiderivat ive Linear Scan Voltammetric Responses: Effect of Reaction Reversibility Donga ? Caster, I . Jeffrey J. Toman,' and Steven D. Brown* Department of Chemistry, Washington State University, Pullman, Washington 99164-4630
Semlderlvative methods have been used in the resolution of strongly overlapped reqponses In linear sweep vpltammetry (LSV). I t Is demonstrated that, although a simple form Is available only for a totally reversible eemlderlvatlve, the change in peak shape with decreasing charge-transfer reverslbillty is small. Thls permlts use of a slmple approxlmate equation for fitting all semiderlvatlves regardless ~1 reverslblllty, provided the charqe transfer Is uncompll6ated by other effects.
The increased use of laboratory microlcomputers and microprocessors in electrochemical experimentation is driven by two major factors. One is that computgrs can control experimental conditions precisely. This feature is especially valuable i n carrying out complex proceldures or in the application of unusual wave forms. A second factor is that the computer, when coupled to rapid analog-toLdigital converters, is capable of taking large amounts of data and storing these data in digital form. This feature makes possible the routine application of sophisticated numerical techniques for the purpose of signal-to-noise enhancement, peak resolution enhancement, or both. One such numerical technique which has become popular recently is convolution potential sweep voltammetry (1-3). Convolution potential sweep voltammetry transforms the broad, asymmetric wave shape of a linear !3canvoltammogram (LSV) into a form resembling a dc polarogram. When this charge transfer is reversible, this form, called the semiintegral, is described by a simple analytic functioln, unlike the linear scan voltammogram. Simple differentia tion transforms the reversible semiintegral response into a semiderivative response, which is also a simple analytic function. This may make possible, by means of curve fitting, both signal-to-noise enhancement and resolution enhancement. Further, because these transforms are based wholly on electrochemical theory, certain physical properties of the electrochemical system under study are easily determined from the transformed wave. Previously, we have shown that curve fitting of electrochemical data can be used to separate (closelyoverlapped, reversible linear scan voltammograms by use of the semiderivative (4). The semiderivative wave form is given for a reversible charge transfer process, as (5, 6 )
where v is the scan rate, m* is the limiting semiintegral (where E -+ -m )
m* = n F A C * , , G and m(E),the semiintegral, is
Present address: Chevron Research, Richmond, CA.
(2)
The other symbols have their usual electrochemical meaning. fiquation 1 is strictly true for a reversible charge transfer feaction with only reactant present initially, with conditions of planar diffusion. Further, it is assumed that E, is sufficiently positive that i(E,) E 0 and that no complicating kinetic steps occur either before or after the electrode reaction. It is of considerable interest to discuss the limitations inherent in using this equation routinely to fit LSV or ASV responses, regardless of tho degree of charge transfer reversibility. This paper examines in detail the shapes of semidifferentiated LSV responses for simple electrode reactions with varying charge transfer rates, and the ability of various curve fitting approaches to fit these responses and to extract physical parameters.
THEORY For the simple charge transfer reaction Ox
ko + ne- J Red
(4)
there are two limiting regions of heterogeneous kinetic behavior, depending on the size of k o , the heterogeneous charge transfer rate constant, Matsuda and Ayabe (7) describe the behavior of simple charge transfer measured by linear scan voltammetry in terms of the dimensionless rate constant A, where
and where a is the transfer coefficient for the reaction. When A > 15, the reaction described by eq 4 is considered reversible. The charge transfer is called irreversible when A < 10-2(1+a). Because of the complexity of the general solution of the boundary value problem which describes the charge transfer in the linear scan voltammogram, direct fitting of untransformed linear scan current responses is difficult. For this reason, physical parameters, such as k" and a , are not easily obtained from linear scan measurements. Saveant and co-workers (2) and Oldham and co-workers (1) have shown that convolution of the current by transforms the current output from a reversible electrode reaction into a simple form. An equivalent convolution of the current by l/(a(E,- E ) ) l l 2gives eq 3. The shape of a reversible wave is then
The derivative of this function with respect to potential generates eq 1, the reversible semiderivative function. Equations for quasi-reversible and irreversible electrode reactions have allso been reported. Goto and Oldham (8)have obtained a series solution for the semiintegral of an irreversible charge transfer, which Dalrymple-Alford et al. (9) have converted to an expression for the semiderivative. The peak potential was shown to be
E,,
E o t.0.055-RT - RT In a a n F 2anF
0003-2700/83/0355-2 143$01.50/0 -. . . 0 1983 American Chemical Society
+RT In A anF
(7)
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ANALYTICAL CHEMISTRY, VOL. 55, NO. 13, NOVEMBER 1983
with a peak width of 2.94 RT/cunF mV. Saveant and Tessier (10,1I ) have previously considered convolution methods for quasi-reversible and irreversible systems, which were then applied to studies of the reduction of tert-nitrobutane in acetonitrile and dimethylformamide. Their expressions include both semiintegral and current, making direct curve fitting using these expressions inconvenient, particularly when limiting current values cannot be obtained accurately. Curve Fitting of Simple Charge Transfer Reactions. T o demonstrate the validity of using simple functions to describe the transformed current responses from electrode reactions over a wide range of reversibility, several criteria must be met. First, the similarity in shape of the transformed data and the fitting function must be demonstrated for a wide range of values of A, the dimensionless heterogeneous rate constant. Second, the parameters returned from the fitting should agree with those expected from theoretical expressions for simple charge transfer, since the systems studied here are presumed to be free from effects such as homogeneous kinetic reactions and adsorption of reactant and/or product. Finally, the quality of fitting must be evaluated, so that the limitations of the method may be established for its application to quantitative analysis. As in the previous study of overlapped peaks (4),the fitting function
e(E) = A sech2 [B(E- C)] (8) is employed, where parameters A, B , and C are adjusted to obtain fits to the semidifferential current responses. Parameter A provides information on the height of the transformed response, B provides information on the peak width, and C provides information on the peak location. Equation 8 is strictly correct only for the case of reversible electrode kinetics. That the semiderivative of a totally irreversible reaction is sech2 in shape, however, is also implied by the work of Goto and Oldham (8) and Dalrymple-Alford et al. (9). From their work, the peak width a t half-height is expected to range between the value of 90.6111 mV for the reversible charge transfer at 25" to 75.5/an mV for the totallv irreversible chkge transfer. Equation'8 should therefore be suited to fitting of semidifferentiated data, regardless of the degree of reversibility.
EXPERIMENTAL SECTION and Mn2+ Reagents. Stock solutions of Pb2+,Cd2+, were made approximately 1.0 X M. Reagent-grade MnC12.4H20 and Cr(C10,)3.6H20 were used to prepare the stock solutions for Mn2+and Cr(Hz0)63+,respectively. The supporting el8ctrolyks used were all of analytical reagent grade. None were further purified before use. All solutions were made with water obtained from a Super-Q water purification unit; water resistivity exceeded 18 MQ. Cells were leached with reagent grade HN03 between runs. Instrumental-grade Hg was pinholed before use. Cell, Electrodes, and Equipment. The cell used in this study was obtained from Bioanalytical Systems, Inc., and had a capacity of about 15 mL. It was equipped with inlets for the three electrodes and a Teflon purge tube. The reference was an RE-6 Ag/AgCl electrode, also from Bioanalytical Systems. It had a potential -41 mV relative to the SCE. The counterelectrode was a length of Pt wire, and the working electrode was a Metrohm hanging mercury drop electrode. The mercury drop electrode capillary was siliconized prior to data collection. Nitrogen, purified by passage through a V(I1) solution, was used to deaerate samples. Temperatures were controlled at 22 f 1OC for these experiments. Equipment used for data collection included an IBM EC225 potentiostat, interfaced to the computer via an Analog Devices 1136K 16-bit digital to analog converter and a homebuilt trigger circuit. The digital-to-analogconverter was used to set the initial potentials, and the trigger circuit allowed computer control of the IBM ramp generator. The computer used for these studies was a Digital Equipment Corp. LSI-11/23 system with 128K wmds
of memory, serial and parallel 1/0 capability, a 12-bit, 35-kHz analog-to-digitalconverter, and floppy disk storage. Scan rates were calibrated by monitoring the potential ramp and fitting the ramp with a line, using least squares. Procedure. In each case, 15 mL of supporting electrolyte solution was pipetted into the cell and deaerated for 10-15 min to remove dissolved oxygen. After deaeration, a Hg drop was extruded, and voltammograms were taken of the supporting electrolyteover an appropriate potential range. The drop radius was determined by gravimetry. Various scan rates were used. The various blank voltammograms obtained were stored on disk. Scans were then taken of the analyte, generally at an analyte concentration of M in the supporting electrolyte. Again, several scan rates were , stored on disk. used, and the voltammograms obtained were Numerical Methods. Synthetic linear scan voltammetry data with varying degrees of reversibility were generated by using the explicit fiiite difference method of Feldberg (12,13),programmed in FORTRAN. Linear scan voltammetry data were collected by using a FORTRAN program coupled to a MACRO assembly language data collection subroutine. Other FORTRAN programs were used to subtract blank scans from analyte scans for background subtraction, to remove noise by means of digital filtering and to perform semiintegration and semidifferentiation by the G-1 algorithm of Grunwald (14). These routines have been previously described ( 4 , 6). Fitting of eq 8 to semidifferentiated data was accomplished in FORTRAN by using Marquardt's method for nonlinear regression (15),as given by Bevington (16). As input, the method requires an independent and dependent variable array, the number of parameters to be fit, an array containing initial guesses for the values of the parameters, and a constant factor for the Marquardt correction vector. The convergence criterion for this program was set so that the relative change of each of the fitted parameters A, B, and C in eq 8 was less than 0.1%. This method gave essentially the same results as the simplex and iterative stripping methods used earlier ( 4 ) , but it converged about twice as fast (typically 25-30 s for a single peak). Output consisted of the final values of peak potential, peak width and height, as well as the coefficient of determination.
RESULTS AND DISCUSSION Fitting Synthetic LSV Data. To evaluate the adequacy of the sech2 function (eq 8) in describing semidifferentiated LSV data with varying degrees of reversibility, a series of simulated LSV responses were generated. These are summarized in Table I. These responses were generated with constant transfer coefficient but with widely differing heterogeneous rate constants, so that the simulated LSV data encompassed charge transfer behavior ranging from strictly reversible (A > 15) to totally irreversible (A I In each simulated response, only one LSV peak was present, and all simulations used CY = 0.500, EO = -0.5000 V, n = 2, and v = 1.0 V s-l over a potential range of 0.0 V to -1.0 V. Each simulation required about 85 s to ryn on the LSI-11/23. The simulated LSV responses had a 1.95 mV spacing of sampled (simulated) current data; this was identical with that of the experimentally observed LSV data, since both involved 512 samples taken over a 1.0-V range. Table I also lists the parameters returned from fitting eq 8 to the semidifferentiated, simulated LSV responses. From the definitions of strict reversibility and total irreversibility, LSV responses in files N and 0 are seen to be reversible, while those in files A-F are totally irreversible; others fall between these two regions of limiting behavior. The quality of fitting of the semidifferentiated irreversible responses by eq 8 is very good, even though the equation is not strictly correct for these systems. Peak potentials obtained from fits agree well with those calculated from eq 7 , and peak widths are near the expected value of 75.5 mV. Slight disagreement is not doubt due to the asymmetry of the irreversible semiderivative. This asymmetry is not reflected in
ANALYTICAL CHEMISTRY, VOL. 55, NO. 13,NOVEMBER 1983
2145
Table I. Definition of Systems and Returned Parameters for Fitting Synthetic LSV Data from Simple Charge Transfers calcd peak potential, V
file
name A E(
C
n
E I?
G I-I I J
IC L M N
Aa
1.00 x 1.000 x 2.000 x 5.000 x 1.000 x i..i33 x 1.133 X 0.11 33 0.2667 0.5667 1.133 5.6678 1.133 X 1.133 X 1.133 X
10-5 10-5 10-5 10-3 lo-*
-0.8357 -0.7765 -0.7588 -0.7352 -0.7174 -0.6551 -0.5960 -0.5367 -0.5148
-0.5000 -0.5000 0 -0.5000 a Calculated from parameters used in simulation. mination. 10' lo2 IO3
peak height'
peak half width, mV
peak potential, V
1.85 76.0 1.85 76.0 1.85 76.0 1.85 76.0 1.85 76.0 1.85 76.0 1.85 76.0 1.94 73.3 2.08 68.5 2.39 59.2 2.65 53.4 3.01 47.1 3.06 46.2 3.12 45.4 3.13 45.3 Calculated from eq 7. Arbitrary
the sech2 shape used for curve fitting. In all cases the coefficients of determination approach 1, also indicative of adequate fitting. Least-squares analysis of the returned peak potentials for the LSV responses contained in files A-F as a function of In A gave an intercept of -0.483 V and a slope of 0.0257 V-l, again in excellent agreement with those values calculated (intercept = -0.490 V, slope = 0.0257 V-l) from eq 7. On the basis of' these results, it appears that the sech2 curve does adequately describe the shape of the totally irreversible semiderivative response. Results of fitting of the semidifferentiated responses for the strictly reversible LSV systems are also good, as expected from previous work (4). Peak potentials and half-widths agree with values expected from theory. Coefficients of determination also indicate an excellent fit. Fitting the sech2 function to responses outside these two limiting regions produces results which aire less easily evaluated, since no theoretical semiderivative relations exist. As might be expected for semiderivatives of these quasi-reversible charge transfer systems, the returned peak half-widths decrease with increasing reversibility, and the returned peak potentials shift toward those expected for the reversible behavior as the value of A, increases. Again, coefficients of determination are near 1 in all cases, suggesting adequate fits. The improvement in fitting is attributed to the gradual increase in semiderivative peak symmetry as A increases, making the peak closer in shape to the sech2 function used in fitting. Fitting Real LSV Data. Results of fitting experimental semiderivatives are given in Table 11. Tlhe systems studied ranged from systems that were strictly revwsible to those that were totally irreversible. In general, the trends observed in fitting the simulated systems are observed here as well. Most systems were well described by the sech2 function. For the reversible systems, Pb2+in 1.0 M HC104,for example, the peak half-widths returned were in agreement with those expected from theory. Except for data taken a t the lowest scan rate, peak potentials were constant (to f l mV) and peak heights showed a linear dependence on the (scan rate)l12 as summarized in Table 111. These results are similar to others previously observed ( 4 ) . The totally irreversible system involving reduction of Cr(H20),3+was also well described by the sech2 function. A fit of eq 8 to one of the semidifferentiated responses obtained for this system is shown in Figure 1. Again, peak heights showed a linear dependence on (scan rate)l12. Peak potentials shifted as expected with In A as given by eq 7. A least-squares fit of the relation gave a slope of 0.036 V, an intercept of -0.653 V, and r = 0.999. From these data, the formal potential (in
XI
+c-1.48
:
-1,211
r2
-0.8389 0.9980 -0.7797 0.9980 -0.7619 0.9980 -0.73 84 0.9980 -0.7206 0.9980 -0.6614 0.9980 -0.6023 0.9980 -0.5441 0.9990 -0.5289 0.9996 -0.51 43 0.9990 -0.5n78 0.9990 -0.501 1 0.9990 -0.501 1 0.9999 -0.5004 0.9999 -0.5003 0.9994 units. Coefficient of deter-
-1.88
-E,M -8.60 -8.48 POTENTIRL [ V vs Rg/RgCl)
-8.28
Flgure 1. Fitting of eq 8 to semidifferentiated irreversible linear scan voltammogram obtained for reduction of Cr(H,O):+ in 0.5 M NaC104/0.01 M HCIO, at a scan rate of 0.997VIS. Results of this fit are given in Table 11.
0.5 M NaC104/0.01 M HC1O4) for the Cr(H20)63+ reduction was calculated via eq 7 to be -0.702 V vs. SCE, in poor agreement with other studies done by Galus (19),where Eo' = -0.655 V, and by Anson et al. (29) where Eo' = -0.65 V. From the slope, the transfer coefficient was calculated to be 0.71, considerably higher than that observed by Anson (20) and Galus (19),who found values of 0.60 and 0.55, respectively. It is also considerably higher than the value observed (a = 0.61) by analyzing the same data in a different manner (23). Peak half-widths are in line with those expected; the average of these peak half-widths is 122.2 mV, a value that gives a = 0.62 using equations in ref 9. Fitting of the other systems studied, which show quasireversible charge transfer kinetics, is also good. Results of a typical fit are shown in Figure 2. Generally, peak halfwidths decrease with increasing A, as expected from the studies on simulated data reported in Table I. In all cases, linear relations for peak height w. (scan rate)l12were observed, with r values in excess of 0.99, except for the system with Mn2+ in 0.1 M KC1. These results are summarized in Table 111. The agreement of parameters obtained from the plots of peak height vs. (scan rate)'12 with values expected from theory is not easily evaluated, however. The slope obtained for the system involving a totally irreversible reduction is in excellent agreement with that expected from theory. For a totally reversible reduction, as is the case for Pb2+in 1.0 M HC104, the slope obtained is slightly less than that expected from thdory; this difference may reflect errors in published diffusion
2146
ANALYTICAL CHEMISTRY, VOL. 55, NO. 13, NOVEMBER 1983
Table 11. Semiderivative Voltammetric Fitting of Charge Transfer Systems scan rate,
v
El,,,
A
s-1
59.ga 26.5 18.9 10.9 6.00 3.47 2.67
0.01 00
0.051 0.100
0.300 0.994 2.98 5.00
298 133 94.5 38.6 29.9 17.3 13.4
0.01 00
0.051 0.100 0.600 0.994 2.98 5.00
1.6 x 4.9 x 2.9 x 2.2 x 1.9 x 1.6 x
0.100
0.997 2.99 5.01 6.98 8.98
0.100 0.501 0.994 2.01 5.00 6.98
2.87e 2.04 0.91 0.68
0.501 0.994 5.00 8.98
v
r2 f
CdZ+in 1.0 M Na,SO, -0.581 2' -0.5809 -0.5795 -0.5804 -0.5791 -0.5820 -0.5805
49.3 47.;3 47.0 46.8 47.2 48.6 49.0
1.92 4.44 6.53 10.87 19.80 33.1 41.9
0.987 0.996 0.998 0.998 0.997 Q.991 0.972
Pb2+in 1.0 M HC10, -0.374.5 -0.3721 -0.3719 -0.3712 -0.3714 -0.3 7 09 -0.3714
49.9 46.7 46.5 45.9 46.5 46.3 46.3
2.05 4.43 6.38 14.49 19.55 33.61 41.15
0.286 0.696 0.896 0.895 0.996 0.987 0.990
5.84 17.68 30.7 38.3 47.1 52.4
0.989 0.998 0.995 0.997 0.998 0.997
Pb2+in 0.9 M NaClO,/O.l M NaOH -0.6663 49.3 -0.6656 48.3 -0.6658 48.8 -0.6 6 78 50.4 -0.6699 51.6 -0.6737 55.6 -0.6 7 52 57.6
1.83 5.40 12.30 17.1 24.0 35.2 40.6
0.979 0.996 0.998 0.998 0.996 0.986 0.990
Mn2+in 0.1 M KCl -1.489 -1.495 -1.507 -1.509
19.44 25.1 61.9 71.0
0.895 0.910 0.860 0.83 0
Cr(H20);+ in 0.5 M NaC1O4/0.01M HClO, 10-3 -0.8863 121.9 10-4 -0.9301 120.5 10-4 -0.9458 122.6 10-4 -0.957 7 120.0 10-4 -0.964 2 123.3 io-, -0.9701 125.0
L9.9d 6.30 2.81 2.00 1.40 0.89 0.75
0.010
fitted Darameters peak half peak height, WAV-'" width, mV
51.0 55.0 55.2 66.7
cmz/s, DR = 16.1 X lo-' cmz/s. Calculated by a Calculated by using ref 17: k" = 0.15 cm/s, 01 = 0.30, Do = 6 X cm2/s. ' Calculated by using ref 19 and 20: cm2/s,DR = 13.6 X using ref 18: h" = 0.9 cm/s, 01 = 0.5, Do = 1 X cmz/s. Calculated by using ref 21: k o = 6.0 X cmz/s, DR = 7.9 X k" = 8.1 X cm/s, DI = 0.60, Do = 5.6 X cmz/s, DR = 13.6 X lo-' cm2/s. e Calculated by using ref 22: k" = 5.0 X lo-' cm/s, cm/s, DI = 0.51, D o = 1 X cm2/s. f Coefficient of determination. g In V vs. Ag/AgCl, 3 M NaCI. 01 = 0.35, Do = 7.2 x cm2/s,DR = 9.0 X Table 111. Dependence of Semidifferentiated LSV Peak Heights on (Scan Rate)"*
system
concn,= M
Cd2+in 1.0 M Na,SO, Pb2+in 1.0 M HClO, Cr(H20),3+in 0.5 M NaC10,/0.01 M HC10, Pb2+in 0.9 M NaC104/0.1 M NaOH Mn2+in 0.1 M KCl
1.06 X 6.86 X lo-' 4.16 x 6.86 X 10:: 1.03 X 1 0
theoretical slope, V-' 25.6,' 21.3 17.3' 21.3,' 27.2,'
9.1d 12.gd
12.9 ? 11.3 I d
fitted parameters slope, intercept, s112 V-1 &AV-I12 18.8 (0.2)e 18.6 (0.4) 17.4 (0.3) 15.3 (0.4) 24(3)
0.5 (0.3)f 0.4 (0.4) 0.3 (0.6) 1.1 (0.6) 3 (5)
r
0.9996 0.9991 0.9995 0.9985 0.9885
Calculated from least-squares fit of peak height vs. (scan rate)'". ' Calculated by a Concentration of oxidized species. use of the relation: ep = n2F2AC,*x(vD,x)"2/4RT,using diffusion constants from Table 11, and an electrode area of 2.6 X lo-' cm2. Calculated by use of the relation (9): ep = o t n Z F Z A C ~ x ( v D , x ) " 2 / 3 . 3 6 7 Rusing T , transfer coefficients and cmz. e Standard deviation of the slope is given in diffusion constants from Table 11, and an electrode area of 2.6 x parentheses. Standard deviation of the interceDt is given in parentheses.
-
constants or errors in measuring solution concentrations and electrode areas. Slopes obtained for quasi-reversible systems fall between those expected from theories assuming total reversibility and total irreversibility. Such behavior is consistent with other properties of quasi-reversible systems. The slope predicted for reduction of Pb2+in 0.9M NaC104/0.1 M NaOH assuming totally irrevefsible behavior is possibly too
large; other studies of this system give 01 = 0.37 f 0.03 and k" = 6.23 X cm/s (23). The limiting theoretical (irreversible) slope is then 9.4 pA SI/* V-l. In all cases, the intercepts are near zero, suggesting that adequate background subtraction has been achieved. The intercept obtained for the studies of Mn2+ in 0.1 M KC1 reflects the lack of background subtraction, and the error in the intercept reflects the
XI
r
Flgure 2. Fitting of eq 8 to semidifferentiated quasi-reversible llnear scan vokammcgrann obtained for reduction of Pb2+ in 0.9 M NaClOJO.1 M NaOH at a scan rate of 2.01 V/s. Results of this fit are given In Table 11.
variability in the background. Sources of Error in Fitting. The discrepancy of values for the chromiwi transfer coefficient from the same data using different methods of calculation suggests that a slight bias exists in the E,, In A relation. This bias may be a result of the effects of irreversible semiderivative asymmetry on fitting by eq 8. Increasing asymmetry as A decreases appears to produce slight errors in locating the peak potential, and thus lead to a slope somewhat lower than that expected. The incorrect slope is reflected in the transfer coefficient (here 0.71 as compared to 0.60 expected); these errors may have considerable effect on the value calculated for the formal potential. In some of the systems reported in Table I1 and discussed above, the coefficients of determination were below 0.9. This is due to the presence of a substantial baclkground component, produced by reduction of hydrogen ion. Background subtraction was only partially successful in removing this component; for the data obtained on Mn2+in 0.1M KC1, background subtraction was not attempted because the shape of the background changed between background and analyte runs. Thus, the coefficient of determination is strongly biased by the inability of eq 8 to account for the background, as can be seen in Figures 1 and 2. Generally, ithe semidifferential background component due to hydrogein ion reduction was well resolved (135 mV) from the peak being fitted. Despite this background, the parameters returned from the fitting were generally consistent with those expected from theory, although similar plots of peak height vs. concentration gave sizable intercepts. These intercepts indicate that quantitative analysis using fitting and standard additions would fail. For low scan rates, another effect not included in eq 8 became apparent. When the electrode drop radius is small or when scan rates are low, sphericity effects become significant, and a correction €or electrode sphericity (24) must be employed. Sphericity effects distort semiderivatives by
ANALYTICAL CHEMISTRY, VOL. 55, NO. 13, NOVEMBER 1983
2147
cathodically shifting peak potentials and by significantly broadening peak half-widths. Peak heights are not as strongly affected. These effects were noticeable in the Cd and Pb systems when scan rates were below 100 mV s-l. Similar effects have been noticed for semiintegrals (2),where low scan rates produce a sloping limiting current plateau.
CONCLUSIONS The sech2 function is an adequate descriptor for semiderivatives of LSV responses, regardless of the degree of charge transfer reversibility. Fitting of semiderivatives improves slightly as the degree of reversibility increases, and resolution of most overlapped semiderivative systems should be feasible even when one of the overlapped peaks is irreversible or quasi-reversible. Because the asymmetry of the nonreversible semiderivative response manifests itself in peak width errors, resolution accuracy will be enhanced by constraining peak half-widths to previously determined values. Since peak heights are of paramount importance in electroanalysis, constraining peak half-widths may not be necessary. The speed and reliability of curve fitting suggest that it should be useful in routine deconvolution of overlapped electrochemical responses, especially in anodic stripping voltammetry, where background effects are easily corrected (25, 26). Registry No. Cd, 7440-43-9; Pb, 7439-92-1; Cr(H20)63+, 14873-01-9;Mn, 7439-96-5. LITERATURE CITED Grenness, M.; Oldham, K. B. Anal. Chem. 1972, 4 4 , 121. Imbeaux, J. C.; Saveant, J. M. J. Electroanal. Chem. 1973, 4 4 , 169. Whltson, P. E.; Van den Born, H. W.; Evans, D. H. Anal. Chem. 1973, 45, 1298. Toman, J. J.; Brown, S. D. Anal. Chem. 1981, 53, 1497. Goto, M.; Ishll, D. J. Electroanal. Chem. 1975, 6 1 , 361. Toman, J. J.; Corn, R. M.; Brown, S.D. Anal. Chim. Acta 1981, 123, 182. Matsuda, H.; Ayabe, Y. 2.Elektrochem. 1955, 5 9 , 494. Goto, M.; Oldham, K. B. Anal. Chem. 1976, 48, 1671. Dalrymple, P.; Goto, M.; Oldham, K. B. J. Electroanal. Chem. 1977, 8 5 , 1. Saveant, J. M.; Tessler, D. J. Phys. Chem. 1978, 82, 1723. Saveant, J. M.; Tessier, D. J. Electroanal. Chem. 1975, 6 5 , 57. Feldberg, S. W. Electroanal. Chem. 1969, 3 , 199. Feldberg, S. W. I n “Computers In Chemistry and Instrumentation”; Mattson, J. S., Mark, H. B., MacDonald, H. C., Eds.; Marcel Dekker: New York, 1972; Vol. 2, Chapter 7. Oldham, K. B.; Spanler, J. “The Fractional Calculus”; Academic Press: New York, 1974. Marquardt, D. W. J. SOC. Ind. Appl. Math. 1983, 1 1 , 431. Bevlngton, P. R. “Data Reduction and Error Analysis for the Physical Sciences”; McGraw-HIII: New York, 1969. Bond, A. M.; O’Hailoran, R. J.; Ruzlc, I.; Smith, D. E. Anal. Chem. 1978, 5 0 , 216. DePalma, R. A.; Perone, S.P. Anal. Chem. 1979, 5 1 , 829. Ziellnska-Ignaciuk, M.; Galus, 2. J. Nectroanal. Chem. 1974, 5 0 , 41. Anson, F. C.; Rathjen, N.; Frisbee, R. D. J. Nectrochem. SOC. 1970, 117, 477. Blegler, T.; Laltlnen, H. Anal. Chem. 1965, 3 7 , 572. Gupta, J. K.; Gupta, G. M. Monatsh. Chem. 1969, 100, 2019. Brown, T. F.; Caster, D. M.; Brown, S. D., submitted for publication. Relnmuth, W. H. J. Am. Chem. SOC. 1957, 7 9 , 6358. Kryger, L.; Jagner, D.; Skov, H. J. Anal. Chim. Acta 1975, 78, 241. Brown, S. D.; Kowalskl, B. R. Anal. Chim. Acta 1979, 107, 13.
RECEIVED for review February 16, 1983. Accepted August 4, 1983.
