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Resolution of strongly overlapped responses in square-wave voltammetry by using the Kalman filter
Analytica Chimica Acta, 166 (1984) 253-260 Elsevier Science Publishers B.V., Amsterdam -Printed
in The Netherlands
RESOLUTION OF STRONGLY OVERLAPPED RESPONSES IN SQUARE-WAVE VOLTAMMETRY BY USING THE KALMAN FILTER
CAROLINE
A. SCOLARI and STEVEN D. BROWN*
Department (U.S.A.)
of Chemistry-4630,
Washington
State
University,
Pullman,
WA 99164
(Received 9th May 1984)
SUMMARY Various approaches have been proposed for resolving overlapped voltammograms. Such methods have become increasingly important since the advent of small computers and commercial electrochemical instrumentation capable of rapid-scan techniques. The combination of high-resolution, rapid-scan square-wave voltammetry and a linear, recursive estimator known as the Kalman filter is described. Results of the application of this combination to mixtures of thallium(I) and lead(I1) in 0.9 M nitric acid are shown. Practical considerations for filter usage such as the choice of models, variances, and initial guesses are discussed, as are limits to the filter. With the Kalman filter, it was possible to quantify thallium in solutions containing as much as a 30-fold excess amount of lead, and it was possible to quantify lead in solutions containing a 60-fold excess amount of thallium. The filter is thus a useful tool for use with empirical models in multicomponent quantitation.
Microprocessor-based electrochemical instrumentation offers a number of significant advantages over more conventional equipment. One of these is the opportunity to use the peak resolution methods developed over the last fifteen years. Most approaches to peak resolution rely on curve-fitting of some model to the overlapped electrochemical response. The model can be an empirical description of the responses of individual components suspected in the mixture, in which case linear regression is used [l-5], or it can be a more complex expression, often based directly on electrochemical theory [6-g]. For the latter case, nonlinear least-squares fitting is usually necessary. Although many of the methods proposed for peak resolution work well on closely overlapped systems when component ratios are near unity, surprisingly little has appeared on resolution of such systems when component ratios differ substantially from unity. Binkley and Dessy [3] estimated that Tl(I)/Pb(II) mixtures with concentration ratios of up to 1:15 should be resolvable, while Bondreau and Perone [7] observed that a 17.6:1 ratio of Tl(I)/Cd(II) was resolvable. In the In(III)/Cd(II) system, successful resolution of 5: 1 ratios was possible [ 1,2] . In view of the frequency with which unequal mixtures of closely overlapped components arise in practise [ 41, evaluation of an approach with unequal concentrations is a critical and realistic test.
0003-2670/84/$03.00
0 1984 Elsevier Science Publishers B.V.
254
The Kalman filter has been used previously to resolve overlapped electrochemical responses [5,10,11]. The filter is ideal for use on a microprocessor because it has minimal memory requirements and it is rapid. Studies of synthetic data showed that successful resolution of arbitrarily closely overlapped responses could be achieved by using the filter, given suitably high point densities [5]. This paper reports the use of the filter in resolving a closely overlapped system in which peak height ratios are adjusted from values close to unity to values significantly different from unity, GENERAL
CONSIDERATIONS
The details of the use of the Kalman filter in multicomponent quantitation have been reported elsewhere [5]. To clarify the approach, however, some explanation is necessary. The Kalman filter is a recursive linear estimator which requires models for the system dynamics and the measurement process. Because the concentrations of the components do not change during the electrochemical scan, a simple system model can be used. The measurement process model Z(k) = ST ’ X(k)
+ v(k)
(1)
relates the measurement of current at potential k, Z(k), to the component concentrations by a proportionality vector ST. This vector is simply the slope of the current/concentration relation obtained for each of the N pure components included in the model. Here, X(k) is the vector of the N component concentrations (1 X N) at potential 12, and v(k) is the measurement noise present at potential k. In essence, this method provides a point-by-point linear least-squares fit of the model component responses to the experimental data. The output is a statistically optimal estimate of the N component concentrations, provided that certain noise assumptions [5, lo] are met, although substantial deviations from the noise requirements appear to have little effect on the quality of the results [12]. The formulation of the Kalman filter described above allows rapid and efficient peak resolution because the usual matrix inversion needed for leastsquares fits [13] is reduced to a mathematical division. Computer memory requirements are reduced because data are processed in a recursive, rather than a batch, mode. Only the pure component responses which describe the model need be kept in memory. These empirical models are limited (either the number of components or the point density used) only by the size of the memory; the model components can even be stored on disk for recall. Thus, a library of electrochemical responses can be generated for pure components under well-defined conditions, and this library can be used to provide empirical models containing all possible components suspected in an overlapped, multicomponent response. Fitting of such models to the overlapped response will return concentrations of zero for those components which are not present (subject to the limitations discussed below), while non-zero
255
concentrations will be assigned to components detected in the overlapped response. Omission of components in the model, or erroneous model components are easily detected [5, 10-121, and methods for compensating for these errors have been proposed [ 10,111. Two other aspects of the filter are appropriate for discussion here. If strongly overlapped components are to be separated, it must be established that the components are observable [ 51. The criterion of observability insures that sufficient information exists in the measurement to allow estimation of the state vector, X. Here, the observability criterion demands that component (model) voltammograms all must differ by at least one point from each other. Thus, the point density of the digitized model and multicomponent data determine the theoretical limits in resolving overlapped systems with the filter. A second factor is the accuracy of the initial guess made for the covariance matrix. This variance estimate determines whether it is possible to distinguish small perturbations on large, noisy signals, as might happen when a small amount of one component coexists with a large amount of a second component. Careful estimates of the within-run measurement variance, o:, are necessary to distinguish signal from noise in this case. Various methods have been used to estimate the diagonal elements of the covariance matrix [lo]. Previously, the experimental reproducibility has been used; however, such a practice may lead to biased estimates if the initial guesses for the covariance matrix are low. Here, the initial estimate for the covariance matrix is based on the level of confidence in the initial guess of concentrations, a more reliable approach [lo] . EXPERIMENTAL
Reagents, cells, electrodes and equipment A stock solution of 1.877 X lo-’ M lead(I1) was prepared by dissolving reagent-grade lead nitrate in reagent-grade nitric acid and then diluting with Super-Q water. A stock solution of 2.24 X 10s2 M thallium(I) was prepared by dissolving reagent-grade thallium(I) sulfate (99.5%) in dilute nitric acid. These solutions were standardized with EDTA using methods described by Schwarzenbach [ 141. Lead(I1) and thallium(I) stock solutions were added to a supporting electrolyte of 0.9 M nitric acid, prepared by diluting reagentgrade acid with Super-Q water. A Bioanalytical Systems cell was used for all runs. This cell was thermostatted at 234°C. The counter electrode was a length of platinum wire, and the reference electrode was a Bioanalytical Systems Ag/AgCl (3 M NaCl) electrode. The working electrode was a Princeton Applied Research Model 9223 HMDE, which had a capillary siliconized just prior to the experiments reported here. All data were collected in triplicate. A new drop was used for each run. The reproducibility observed for repetitive measurements with new drops was 1.4%. Potentials were applied to the cell from an IBM Model 225 potentiostat which was interfaced to a Digital Equipment Corp. LSI-11/23
256
computer via an isolated 16-bit digital-to-analog converter and a 12-bit analog-to-digital converter. Details of the equipment used have been previously reported [ 91. Computer programs Data were obtained using SQWV, a FORTRAN and MACRO-11 assembly language program which runs on the LSI-11/23 under the RT-11 monitor. FORTRAN-based subroutines comprising SQWV allow input of square-wave parameters from either disk file or the keyboard, run checks on the parameters, generate the voltage sequence to be applied, and allow both plotting and storage of the experimental data. The MACRO-11 subroutines allow an external device to trigger the square-wave scan, to control the potential scan, and to collect the current responses. The point density for a square-wave scan is calculated by the program from the various parameters input to it. Typical square-wave scans include about 500 current difference measurements made in 2 s, within a potential scan of about 1 V. For faster scans, or for potential scans wider than 1 V, fewer current differences are taken. For the studies reported here, a voltage range of 0.600 V was used, with a scan time of 10.0 s, giving 491 current differences across the potential range. The point density was thus 1.22 mV per point in this study. The square-wave height was 50.0 mV. In addition to starting potential, potential range, and scan time, parameters input to the program include square-wave height, forward and backward sampling delays, and square-wave symmetry parameter (the ratio of the forward pulse time to the sum of the times taken by both the forward and background pulses). A variable pre-scan delay, and point and scan averaging are also available. The voltammograms were Fourier-filtered and backgroundsubtracted prior to application of the Kalman filter. Details of these methods have been reported previously [ 5,8] . The Kalman filter was a modified version of the FORTRAN program previously reported [5]. This interactive version was not optimized for either speed or size. Processing of experimental voltammograms for a twocomponent model required about 3 s. The program size was about 14 kwords, including storage for a sixcomponent model measured at 512 points, and allowing for space for the FORTRAN run-time monitor. RESULTS
AND DISCUSSION
The thallium(I)/lead(II) system has been studied extensively by a variety of methods [ 3,4,6] . In 0.9 M nitric acid supporting electrolyte, the observed peak-to-peak separation for the thallium(I)/lead(II) system is 68 mV, and the peak widths at half-height for thallium(I) and lead(D) in 0.9 M nitric acid are 94 mV and 58 mV, respectively; all measurements refer to data collected at 0.060 V s-l scan rates and 50 mV square-wave height.
257
Model Empirical models were used here, These models were obtained by measurement of the square-wave voltammetric response of single species in 0.9 M nitric acid, under the same conditions as those used in the measurement of the overlapped voltammetric responses. Measurements of lead(II) standards over the concentration range 6.25 X 10”-6.06 X lo4 M gave peak current responses with a slope of (1.108 * 0.006) X lo5 PA M-’ and an intercept of (-2 * 2) X lo-’ PA. The correlation coefficient (n = 15) was 0.9996. For thallium(I), measurement of standards over the concentration range 7.46 X 10-6-l.66 x 10v3 M gave peak current responses with a slope of (5.455 * 0.009) x lo3 PA M-l and an intercept of (-2.2 ? 0.8) X 10-l PA. The correlation coefficient (n = 19) was 0.9999. Models were chosen from these standards by selecting the responses closest to the least-squares line, the slope of which is used in the S vector. The models used are reported in Table 1. In all of the mixtures, the initial value for the covariance matrix, one part of the initial guess required to start the filter, was taken as 1 X lo+ pA2. This value was obtained by fitting thallium(I) standards to a sech2 waveform by means of a non-linear least-squares routine [8]. The sech2 model is an adequate descriptor of the square-wave peak shape for thallium(I) because the response is close to that expected for a reversible system, as evidenced by peak half-width. A reversible one-electron process should have a peak halfwidth of 91 mV [ 151; the observed thallium peak half-width was 94 mV. The other part of the initial guess required to start the filter is an estimate of the component concentrations for all species included in the model. Various initial guesses were used. Initial guesses of zero for all components gave results identical to those obtained with initial guesses close to the true value. Initial guesses which were larger than the true values were not used, because these guesses are known to provide non-optimal results [lo], and it is easy to avoid high guesses by guessing zero. Fitting of overlapped responses Two sets of data were processed with the Kalman filter, one with thallium(I) as the major component and the other with lead(I1) as the major component. TABLE
1
Models used in peak resolution Mixture
Model*
l-4 5-7 8-13 14-17
5.96 1.48 1.53 1.53
x x x x
lo+ lo4 10-j lo-’
M M M M
Tl, Tl, Tl, Tl,
1.861 1.861 4.986 1.861
x x x x
lo-’ M Pb lo+ M Pb 1O-5 M Pb lo-’ M Pb
Woncentrations of pure species in 0.9 M nitric acid for which model responses were collected. Model consists of voltammetric responses for both components.
258
All concentrations were corrected for the small volume changes. The experimental conditions above were chosen for convenience; other mixtures are equally amenable to processing by the filter. Results obtained in fitting these data are shown in Tables 2 and 3. In all cases, the quality of fits is good, as is indicated by the coefficients of determination given in the Tables. A typical fit obtained by use of the filter is shown in Fig. 1, for which the thallium and lead are present at a ratio of 6: 1; the difference between the observed (curve A) and the fitted (curve E) voltammograms is very small. The error in the estimated concentrations is TABLE 2 Results for Pb/Tl mixtures with lead as major component Mixture
1 2 3 4 5 6 7
Molarity ratio (Pb:Tl)
5O:l 33:l 25:l 13:l 3:l 2:l 1:l
Peak height ratio (Pb:Tl)
113:l 74:l 55:l 28:l 6:l 4:l 3:l
Concentration Prepared (1O-5 M)
Computeda ( 1O-5 M)
Pb
Tl
Pb
Tl
36.8 24.6 36.8 36.8 36.6 36.5 36.4
0.732 0.736 1.46 2.92 13.1 20.3 27.5
36.6 24.7 37.7 36.0 36.0 37.3 37.9
0.656 0.758 1.40 2.75 13.3 20.6 28.4
Error (%) Pb
-0.5
to.4 t2.4 -2.2 -1.6 +2.2 +4.1
aIn ail mixtures, the uncertainty in the computed concentrations
Tl
-10.0 t 3.0 -4.1 -5.8 t 1.5 + 1.5 +3.3
Coefficient of determination
0.9999 0.9999 0.9999 0.9999 0.9998 0.9998 0.9997
was 1 X lo-’
M.
Error (%)
Coefficient of determination
TABLE 3 Results for Pb/Tl mixtures with thallium as major component Mixture
8 9 10 11 12 13 14 15 16 17
Molarity ratio (Pb:Tl)
1:60 1:54 1:48 1:40 1:30 1:20 1:15 1:12 1:s 1:6
Peak height ratio (Pb:Tl)
1:29 1:27 1:24 1:20 1:14 1:9 1:7 1:6 1:4 1:3
Concentration Prepared (1O-5 M)
Computed* (lo-$ M)
Pb
Tl
Pb
Tl
2.34 2.36 2.37 1.82 4.68 7.01 9.34 11.6 17.4 23.2
140 127 113 72.2 140 139 139 139 139 138
2.20 2.19 2.42 1.72 4.75 7.11 9.27 11.5 17.4 23.1
140 127 113 70.8 140 140 141 139 138 138
aIn all mixtures, the uncertainty in the computed concentrations
Pb
-6.0
-7.2 t 2.1 -5.5 +1.5 t1.4 -0.7 -0.9 0.0 -0.4
Tl
0.0 0.0 0.0 -1.9 0.0 to.7 +1.4 0.0 -0.7 0.0
was 1 X lo-’
1.000 1.000 1.000 0.9999 1.000 1.000 1.000 1.000 0.9999 1.000 M.
259
zi z
‘-0.64 -0.56 -0.48 -i,4E -0.32 -0.24 -a.!6 POWiR!M
1
-8.08 -1.80
Fig. 1. Deconvolution of overlapped lead and thallium responses: (A) experimental data obtained for mixture 17 (see Table 3); (B) fitted thallium response; (C) fitted lead response; (D) residuals of the fitting; (E) sum of the fitted components.
generally below 5%, with the error in the estimated concentration for the minor component increasing as the concentration of the minor component decreases. The error in the concentration of the major component remains fairly constant. These errors reflect the degree to which the models chosen describe the individual components of the overlapped response. Models for thallium perform quite well. When thallium is the major component, fitting errors of about 1% are typical; when thallium is the minor component, the accuracy of peak resolution remains good up to a Pb:Tl peak height ratio of 113 : 1, where the lead is present in a fifty-fold excess amount. Models for lead are not as suitable. When lead is the major component, fitting errors of about 2% are observed, while errors of 2-5s are observed when lead is the minor component. However, even with a Tl:Pb molarity ratio of 60:1, the error in fitting increases only to about 7%, a remarkable result given the relative widths of the two peaks and the peak separation. In all cases, initial guesses of zero concentration input to the filter did not affect the accurate determination of either of the components. The filter remains useful for quantitative work down to concentration levels near the detection limit of the minor component. The insensitivity of fits using the Kalman filter to components of the model which are not relevant to describing the experimental measurement was demonstrated by the inclusion of a cadmium(I1) response in the model. In no case did the cadmium concentration found by using the filter differ significantly from zero; typical values found ranged near -1 X 10” M, with filter estimation uncertainty [5] of about lo-’ M. The observed results for lead and thallium were unchanged from those obtained without cadmium in the model.
260
The speed, small memory requirements, and accuracy of the Kalman filter make it attractive for use with empirical models in multicomponent quantitation. The lack of significant sensitivity of the method to additional components or to the initial guess makes it useful for qualitative as well as quantitative applications. This work was supported, in part, by the Graduate School, Washington State University, through a Grant-in-Aid. Additional funding was provided by the Department of Energy (DE-FG06-84ER13202). REFERENCES 1 W. F. Gutknecht and S. P. Perone, Anal. Chem., 42 (1970) 906. 2 L. B. Sybrandt and S. P. Perone, Anal. Chem., 43 (1971) 382. 3 D. P. Binkley and R. E. Dessy, Anal. Chem., 52 (1980) 1335. 4 A. M. Bond and B. S. Grabaric, Anal. Chem., 48 (1976) 1624. 5 T. F. Brown and S. D. Brown, Anal. Chem., 53 (1981) 1410. 6 L. Meites and L. Lampugnani, Anal. Chem., 45 (1973) 1317. 7 P. A. Boudreau and S. P. Perone, Anal. Chem., 51(1979) 811. 8 J. J. Toman and S. D. Brown, Anal. Chem., 53 (1981) 1497. 9 D. M. Caster, J. J. Toman and S. D. Brown, Anal. Chem., 55 (1983) 2143. 10 S. C. Rutan and S. D. Brown, Anal. Chim. Acta, 160 (1984) 99. 11 T. P. Brown, 0. M. Caster and S. D. Brown, Natl. Bur. Stand. (U.S.), Spec. Publ., No. 618 (1981) 163. 12 T. F. Brown, Ph.D. Dissertation, University of Washington, 1982. 13 J. V. Beck and K. J. Arnold, Parameter Estimation in Engineering and Science, Wiley, New York, 1977. 14 G. Schwarzenbach, Complexometric Titrations, Methuen, London, 1969. 15 L. Ramaley and M. S. Krause, Anal. Chem., 41(1969) 1362.
in The Netherlands
RESOLUTION OF STRONGLY OVERLAPPED RESPONSES IN SQUARE-WAVE VOLTAMMETRY BY USING THE KALMAN FILTER
CAROLINE
A. SCOLARI and STEVEN D. BROWN*
Department (U.S.A.)
of Chemistry-4630,
Washington
State
University,
Pullman,
WA 99164
(Received 9th May 1984)
SUMMARY Various approaches have been proposed for resolving overlapped voltammograms. Such methods have become increasingly important since the advent of small computers and commercial electrochemical instrumentation capable of rapid-scan techniques. The combination of high-resolution, rapid-scan square-wave voltammetry and a linear, recursive estimator known as the Kalman filter is described. Results of the application of this combination to mixtures of thallium(I) and lead(I1) in 0.9 M nitric acid are shown. Practical considerations for filter usage such as the choice of models, variances, and initial guesses are discussed, as are limits to the filter. With the Kalman filter, it was possible to quantify thallium in solutions containing as much as a 30-fold excess amount of lead, and it was possible to quantify lead in solutions containing a 60-fold excess amount of thallium. The filter is thus a useful tool for use with empirical models in multicomponent quantitation.
Microprocessor-based electrochemical instrumentation offers a number of significant advantages over more conventional equipment. One of these is the opportunity to use the peak resolution methods developed over the last fifteen years. Most approaches to peak resolution rely on curve-fitting of some model to the overlapped electrochemical response. The model can be an empirical description of the responses of individual components suspected in the mixture, in which case linear regression is used [l-5], or it can be a more complex expression, often based directly on electrochemical theory [6-g]. For the latter case, nonlinear least-squares fitting is usually necessary. Although many of the methods proposed for peak resolution work well on closely overlapped systems when component ratios are near unity, surprisingly little has appeared on resolution of such systems when component ratios differ substantially from unity. Binkley and Dessy [3] estimated that Tl(I)/Pb(II) mixtures with concentration ratios of up to 1:15 should be resolvable, while Bondreau and Perone [7] observed that a 17.6:1 ratio of Tl(I)/Cd(II) was resolvable. In the In(III)/Cd(II) system, successful resolution of 5: 1 ratios was possible [ 1,2] . In view of the frequency with which unequal mixtures of closely overlapped components arise in practise [ 41, evaluation of an approach with unequal concentrations is a critical and realistic test.
0003-2670/84/$03.00
0 1984 Elsevier Science Publishers B.V.
254
The Kalman filter has been used previously to resolve overlapped electrochemical responses [5,10,11]. The filter is ideal for use on a microprocessor because it has minimal memory requirements and it is rapid. Studies of synthetic data showed that successful resolution of arbitrarily closely overlapped responses could be achieved by using the filter, given suitably high point densities [5]. This paper reports the use of the filter in resolving a closely overlapped system in which peak height ratios are adjusted from values close to unity to values significantly different from unity, GENERAL
CONSIDERATIONS
The details of the use of the Kalman filter in multicomponent quantitation have been reported elsewhere [5]. To clarify the approach, however, some explanation is necessary. The Kalman filter is a recursive linear estimator which requires models for the system dynamics and the measurement process. Because the concentrations of the components do not change during the electrochemical scan, a simple system model can be used. The measurement process model Z(k) = ST ’ X(k)
+ v(k)
(1)
relates the measurement of current at potential k, Z(k), to the component concentrations by a proportionality vector ST. This vector is simply the slope of the current/concentration relation obtained for each of the N pure components included in the model. Here, X(k) is the vector of the N component concentrations (1 X N) at potential 12, and v(k) is the measurement noise present at potential k. In essence, this method provides a point-by-point linear least-squares fit of the model component responses to the experimental data. The output is a statistically optimal estimate of the N component concentrations, provided that certain noise assumptions [5, lo] are met, although substantial deviations from the noise requirements appear to have little effect on the quality of the results [12]. The formulation of the Kalman filter described above allows rapid and efficient peak resolution because the usual matrix inversion needed for leastsquares fits [13] is reduced to a mathematical division. Computer memory requirements are reduced because data are processed in a recursive, rather than a batch, mode. Only the pure component responses which describe the model need be kept in memory. These empirical models are limited (either the number of components or the point density used) only by the size of the memory; the model components can even be stored on disk for recall. Thus, a library of electrochemical responses can be generated for pure components under well-defined conditions, and this library can be used to provide empirical models containing all possible components suspected in an overlapped, multicomponent response. Fitting of such models to the overlapped response will return concentrations of zero for those components which are not present (subject to the limitations discussed below), while non-zero
255
concentrations will be assigned to components detected in the overlapped response. Omission of components in the model, or erroneous model components are easily detected [5, 10-121, and methods for compensating for these errors have been proposed [ 10,111. Two other aspects of the filter are appropriate for discussion here. If strongly overlapped components are to be separated, it must be established that the components are observable [ 51. The criterion of observability insures that sufficient information exists in the measurement to allow estimation of the state vector, X. Here, the observability criterion demands that component (model) voltammograms all must differ by at least one point from each other. Thus, the point density of the digitized model and multicomponent data determine the theoretical limits in resolving overlapped systems with the filter. A second factor is the accuracy of the initial guess made for the covariance matrix. This variance estimate determines whether it is possible to distinguish small perturbations on large, noisy signals, as might happen when a small amount of one component coexists with a large amount of a second component. Careful estimates of the within-run measurement variance, o:, are necessary to distinguish signal from noise in this case. Various methods have been used to estimate the diagonal elements of the covariance matrix [lo]. Previously, the experimental reproducibility has been used; however, such a practice may lead to biased estimates if the initial guesses for the covariance matrix are low. Here, the initial estimate for the covariance matrix is based on the level of confidence in the initial guess of concentrations, a more reliable approach [lo] . EXPERIMENTAL
Reagents, cells, electrodes and equipment A stock solution of 1.877 X lo-’ M lead(I1) was prepared by dissolving reagent-grade lead nitrate in reagent-grade nitric acid and then diluting with Super-Q water. A stock solution of 2.24 X 10s2 M thallium(I) was prepared by dissolving reagent-grade thallium(I) sulfate (99.5%) in dilute nitric acid. These solutions were standardized with EDTA using methods described by Schwarzenbach [ 141. Lead(I1) and thallium(I) stock solutions were added to a supporting electrolyte of 0.9 M nitric acid, prepared by diluting reagentgrade acid with Super-Q water. A Bioanalytical Systems cell was used for all runs. This cell was thermostatted at 234°C. The counter electrode was a length of platinum wire, and the reference electrode was a Bioanalytical Systems Ag/AgCl (3 M NaCl) electrode. The working electrode was a Princeton Applied Research Model 9223 HMDE, which had a capillary siliconized just prior to the experiments reported here. All data were collected in triplicate. A new drop was used for each run. The reproducibility observed for repetitive measurements with new drops was 1.4%. Potentials were applied to the cell from an IBM Model 225 potentiostat which was interfaced to a Digital Equipment Corp. LSI-11/23
256
computer via an isolated 16-bit digital-to-analog converter and a 12-bit analog-to-digital converter. Details of the equipment used have been previously reported [ 91. Computer programs Data were obtained using SQWV, a FORTRAN and MACRO-11 assembly language program which runs on the LSI-11/23 under the RT-11 monitor. FORTRAN-based subroutines comprising SQWV allow input of square-wave parameters from either disk file or the keyboard, run checks on the parameters, generate the voltage sequence to be applied, and allow both plotting and storage of the experimental data. The MACRO-11 subroutines allow an external device to trigger the square-wave scan, to control the potential scan, and to collect the current responses. The point density for a square-wave scan is calculated by the program from the various parameters input to it. Typical square-wave scans include about 500 current difference measurements made in 2 s, within a potential scan of about 1 V. For faster scans, or for potential scans wider than 1 V, fewer current differences are taken. For the studies reported here, a voltage range of 0.600 V was used, with a scan time of 10.0 s, giving 491 current differences across the potential range. The point density was thus 1.22 mV per point in this study. The square-wave height was 50.0 mV. In addition to starting potential, potential range, and scan time, parameters input to the program include square-wave height, forward and backward sampling delays, and square-wave symmetry parameter (the ratio of the forward pulse time to the sum of the times taken by both the forward and background pulses). A variable pre-scan delay, and point and scan averaging are also available. The voltammograms were Fourier-filtered and backgroundsubtracted prior to application of the Kalman filter. Details of these methods have been reported previously [ 5,8] . The Kalman filter was a modified version of the FORTRAN program previously reported [5]. This interactive version was not optimized for either speed or size. Processing of experimental voltammograms for a twocomponent model required about 3 s. The program size was about 14 kwords, including storage for a sixcomponent model measured at 512 points, and allowing for space for the FORTRAN run-time monitor. RESULTS
AND DISCUSSION
The thallium(I)/lead(II) system has been studied extensively by a variety of methods [ 3,4,6] . In 0.9 M nitric acid supporting electrolyte, the observed peak-to-peak separation for the thallium(I)/lead(II) system is 68 mV, and the peak widths at half-height for thallium(I) and lead(D) in 0.9 M nitric acid are 94 mV and 58 mV, respectively; all measurements refer to data collected at 0.060 V s-l scan rates and 50 mV square-wave height.
257
Model Empirical models were used here, These models were obtained by measurement of the square-wave voltammetric response of single species in 0.9 M nitric acid, under the same conditions as those used in the measurement of the overlapped voltammetric responses. Measurements of lead(II) standards over the concentration range 6.25 X 10”-6.06 X lo4 M gave peak current responses with a slope of (1.108 * 0.006) X lo5 PA M-’ and an intercept of (-2 * 2) X lo-’ PA. The correlation coefficient (n = 15) was 0.9996. For thallium(I), measurement of standards over the concentration range 7.46 X 10-6-l.66 x 10v3 M gave peak current responses with a slope of (5.455 * 0.009) x lo3 PA M-l and an intercept of (-2.2 ? 0.8) X 10-l PA. The correlation coefficient (n = 19) was 0.9999. Models were chosen from these standards by selecting the responses closest to the least-squares line, the slope of which is used in the S vector. The models used are reported in Table 1. In all of the mixtures, the initial value for the covariance matrix, one part of the initial guess required to start the filter, was taken as 1 X lo+ pA2. This value was obtained by fitting thallium(I) standards to a sech2 waveform by means of a non-linear least-squares routine [8]. The sech2 model is an adequate descriptor of the square-wave peak shape for thallium(I) because the response is close to that expected for a reversible system, as evidenced by peak half-width. A reversible one-electron process should have a peak halfwidth of 91 mV [ 151; the observed thallium peak half-width was 94 mV. The other part of the initial guess required to start the filter is an estimate of the component concentrations for all species included in the model. Various initial guesses were used. Initial guesses of zero for all components gave results identical to those obtained with initial guesses close to the true value. Initial guesses which were larger than the true values were not used, because these guesses are known to provide non-optimal results [lo], and it is easy to avoid high guesses by guessing zero. Fitting of overlapped responses Two sets of data were processed with the Kalman filter, one with thallium(I) as the major component and the other with lead(I1) as the major component. TABLE
1
Models used in peak resolution Mixture
Model*
l-4 5-7 8-13 14-17
5.96 1.48 1.53 1.53
x x x x
lo+ lo4 10-j lo-’
M M M M
Tl, Tl, Tl, Tl,
1.861 1.861 4.986 1.861
x x x x
lo-’ M Pb lo+ M Pb 1O-5 M Pb lo-’ M Pb
Woncentrations of pure species in 0.9 M nitric acid for which model responses were collected. Model consists of voltammetric responses for both components.
258
All concentrations were corrected for the small volume changes. The experimental conditions above were chosen for convenience; other mixtures are equally amenable to processing by the filter. Results obtained in fitting these data are shown in Tables 2 and 3. In all cases, the quality of fits is good, as is indicated by the coefficients of determination given in the Tables. A typical fit obtained by use of the filter is shown in Fig. 1, for which the thallium and lead are present at a ratio of 6: 1; the difference between the observed (curve A) and the fitted (curve E) voltammograms is very small. The error in the estimated concentrations is TABLE 2 Results for Pb/Tl mixtures with lead as major component Mixture
1 2 3 4 5 6 7
Molarity ratio (Pb:Tl)
5O:l 33:l 25:l 13:l 3:l 2:l 1:l
Peak height ratio (Pb:Tl)
113:l 74:l 55:l 28:l 6:l 4:l 3:l
Concentration Prepared (1O-5 M)
Computeda ( 1O-5 M)
Pb
Tl
Pb
Tl
36.8 24.6 36.8 36.8 36.6 36.5 36.4
0.732 0.736 1.46 2.92 13.1 20.3 27.5
36.6 24.7 37.7 36.0 36.0 37.3 37.9
0.656 0.758 1.40 2.75 13.3 20.6 28.4
Error (%) Pb
-0.5
to.4 t2.4 -2.2 -1.6 +2.2 +4.1
aIn ail mixtures, the uncertainty in the computed concentrations
Tl
-10.0 t 3.0 -4.1 -5.8 t 1.5 + 1.5 +3.3
Coefficient of determination
0.9999 0.9999 0.9999 0.9999 0.9998 0.9998 0.9997
was 1 X lo-’
M.
Error (%)
Coefficient of determination
TABLE 3 Results for Pb/Tl mixtures with thallium as major component Mixture
8 9 10 11 12 13 14 15 16 17
Molarity ratio (Pb:Tl)
1:60 1:54 1:48 1:40 1:30 1:20 1:15 1:12 1:s 1:6
Peak height ratio (Pb:Tl)
1:29 1:27 1:24 1:20 1:14 1:9 1:7 1:6 1:4 1:3
Concentration Prepared (1O-5 M)
Computed* (lo-$ M)
Pb
Tl
Pb
Tl
2.34 2.36 2.37 1.82 4.68 7.01 9.34 11.6 17.4 23.2
140 127 113 72.2 140 139 139 139 139 138
2.20 2.19 2.42 1.72 4.75 7.11 9.27 11.5 17.4 23.1
140 127 113 70.8 140 140 141 139 138 138
aIn all mixtures, the uncertainty in the computed concentrations
Pb
-6.0
-7.2 t 2.1 -5.5 +1.5 t1.4 -0.7 -0.9 0.0 -0.4
Tl
0.0 0.0 0.0 -1.9 0.0 to.7 +1.4 0.0 -0.7 0.0
was 1 X lo-’
1.000 1.000 1.000 0.9999 1.000 1.000 1.000 1.000 0.9999 1.000 M.
259
zi z
‘-0.64 -0.56 -0.48 -i,4E -0.32 -0.24 -a.!6 POWiR!M
1
-8.08 -1.80
Fig. 1. Deconvolution of overlapped lead and thallium responses: (A) experimental data obtained for mixture 17 (see Table 3); (B) fitted thallium response; (C) fitted lead response; (D) residuals of the fitting; (E) sum of the fitted components.
generally below 5%, with the error in the estimated concentration for the minor component increasing as the concentration of the minor component decreases. The error in the concentration of the major component remains fairly constant. These errors reflect the degree to which the models chosen describe the individual components of the overlapped response. Models for thallium perform quite well. When thallium is the major component, fitting errors of about 1% are typical; when thallium is the minor component, the accuracy of peak resolution remains good up to a Pb:Tl peak height ratio of 113 : 1, where the lead is present in a fifty-fold excess amount. Models for lead are not as suitable. When lead is the major component, fitting errors of about 2% are observed, while errors of 2-5s are observed when lead is the minor component. However, even with a Tl:Pb molarity ratio of 60:1, the error in fitting increases only to about 7%, a remarkable result given the relative widths of the two peaks and the peak separation. In all cases, initial guesses of zero concentration input to the filter did not affect the accurate determination of either of the components. The filter remains useful for quantitative work down to concentration levels near the detection limit of the minor component. The insensitivity of fits using the Kalman filter to components of the model which are not relevant to describing the experimental measurement was demonstrated by the inclusion of a cadmium(I1) response in the model. In no case did the cadmium concentration found by using the filter differ significantly from zero; typical values found ranged near -1 X 10” M, with filter estimation uncertainty [5] of about lo-’ M. The observed results for lead and thallium were unchanged from those obtained without cadmium in the model.
260
The speed, small memory requirements, and accuracy of the Kalman filter make it attractive for use with empirical models in multicomponent quantitation. The lack of significant sensitivity of the method to additional components or to the initial guess makes it useful for qualitative as well as quantitative applications. This work was supported, in part, by the Graduate School, Washington State University, through a Grant-in-Aid. Additional funding was provided by the Department of Energy (DE-FG06-84ER13202). REFERENCES 1 W. F. Gutknecht and S. P. Perone, Anal. Chem., 42 (1970) 906. 2 L. B. Sybrandt and S. P. Perone, Anal. Chem., 43 (1971) 382. 3 D. P. Binkley and R. E. Dessy, Anal. Chem., 52 (1980) 1335. 4 A. M. Bond and B. S. Grabaric, Anal. Chem., 48 (1976) 1624. 5 T. F. Brown and S. D. Brown, Anal. Chem., 53 (1981) 1410. 6 L. Meites and L. Lampugnani, Anal. Chem., 45 (1973) 1317. 7 P. A. Boudreau and S. P. Perone, Anal. Chem., 51(1979) 811. 8 J. J. Toman and S. D. Brown, Anal. Chem., 53 (1981) 1497. 9 D. M. Caster, J. J. Toman and S. D. Brown, Anal. Chem., 55 (1983) 2143. 10 S. C. Rutan and S. D. Brown, Anal. Chim. Acta, 160 (1984) 99. 11 T. P. Brown, 0. M. Caster and S. D. Brown, Natl. Bur. Stand. (U.S.), Spec. Publ., No. 618 (1981) 163. 12 T. F. Brown, Ph.D. Dissertation, University of Washington, 1982. 13 J. V. Beck and K. J. Arnold, Parameter Estimation in Engineering and Science, Wiley, New York, 1977. 14 G. Schwarzenbach, Complexometric Titrations, Methuen, London, 1969. 15 L. Ramaley and M. S. Krause, Anal. Chem., 41(1969) 1362.