Anal. Chem. 1994,66,45&463
Multicomponent Kinetic Determination of Lanthanides with Stopped-Flow, Diode Array Spectrophotometry and the Extended Kalman Filter Brett M. Quencert and S. R. Crouch' Department of Chemistry, Michigan State University, East Lansing, Michigan 48824 The application of the extended Kalman filter to multicomponent kinetic data is described. The method is based on obtaining data at multiple wavelengths over time using a linear photodiodearray detector. The extended Kalman filter is used to process the data obtained. It is shown that accurate results can be obtained even if the estimated value of the rate constant is not completely accurate or reproducible. No pH, ionic strengfh, or temperature controls were used in testing the chemical system. A system of three lanthanides reacting with 4-(2-pyridylazo)resorcinol (PAR) was used. Accurate estimates of concentrations were obtained even though the relative rate constants for the reactions of La, Pr, and Nd with PAR were kl.71.9, and a high degree of spectral overlap is present. Multicomponent kinetic determinations usually take advantage of differences in the rate constants of parallel reactions in order to obtain quantitative information about two or more analytes. Many different techniques have been developed in order to facilitate the data analysis.'-ll In the past, the most common methods used for multicomponent kinetic analyses were the methods of proportional equations and logarithmic extrapolation.14 However, the method of proportional equations is limited by the fact that only a small fraction of the data collected is used, which thereby limits the precision obtainable. Also, logarithmic extrapolation is limited by assuming that one reaction is complete while the other is still in its initial stages. More recently, newer techniques that do not suffer from these limitations have been introduced. Pefia, Rubio, and P6rez-Bendito7 have applied a new method, called the kinetic wavelength-pair method, which relies on measuring the difference in the rate of change of the absorbance with time at two preset wavelength pairs. Schechter8 has developed a new error-compensating algorithm that simultaneously determines concentrations in dual-component mixtures without 7 Present address: Research Assignments Program, The Dow Chemical Co., 2020 Dow Center, Midland, MI 48674. (1) Mark, H. B., Jr.; Rechnitz, G. A. Kinetics in Analytical Chemistry; Wiley: New York, 1968; Chapters 6-7. (2) Mottola, H. A. Kineric Aspeers OfAMlyricd Chemistry; Wiley: New York, 1988; Chapter 7. (3) PCrez-Bendito, D.; Silva, M. Kinetic Merhods in Analytical Chemisrry; Wiley: New York, 1988; Chapter 6. (4) Pardue, H. L. Anal. Chim. Acta 1989, 216, 69-107. ( 5 ) Ridder, G. M.; Margerum, D. W. AM^. Chem. 1977,49, 2090-8. ( 6 ) Willis, B. G.; Woodruff, W. H.; Frysinger, J. R.; Margerum, D. W.; Pardue, H. L. Anal. Chem. 1970,42, 1350-5. (7) Pefia, J. M.; Rubio, S.;Perez-Bendito, D. AMI. Chim. Acta 1991,244.81-8, (8) Schechter, I. Anal. Chem. 1992, 64, 729-37. (9) Cummings, R. H.; Pardue, H. L. Anal. Chim. Acra 1989, 224, 351-62. (10) Wentzell, P. D.; Karayannis, M. I.; Crouch, S.R. Anal. Chim.Acra 1989,224,
263-74.
(11) Crouch, S.R. Chemom. Intell. Lab. Syst. 1990, 8, 259-73.
450
Analytical Chemistry, Vol. 66, No. 4, February 15, 1994
prior knowledge of reaction orders and rate constants. Wentzell, Karayannis, and Crouchloapplied the linear Kalman filter to simultaneous kinetic determinations. Ultimately, however, all of these techniques suffer from drawbacks. Most previous techniques, except for the kinetic wavelength-pair method, utilize data from only a single wavelength. With the proliferation of array-type detectors, it is expected that this will change. Most previous techniques have also had difficulty in determining concentrations as the ratio of rate constants approaches unity. The current work describes a method to obtain accurate estimates of concentrations even in the presence of a high degree of spectral and kinetic overlap. This is done by utilizing data at multiple wavelengths. Three lanthanides reacting with 4-(2-pyridylazo)resorcinol (PAR) are used to test the approach. The combination of differences in spectra of the species and differences in the kinetics of their reactions allows resolution of the metals and yields accurate estimates of concentrations. This is done by applying the extended Kalman filter to the data at several wavelengths. The Kalman filter12 has recently been applied to many systems in analytical chemistry,13J4 including the area of kinetic d e t e r m i n a t i ~ n s . ~ ~However, J ~ - ~ ~ these studies were only concerned with determination of a single species,15-1* or else used the linear Kalman fi1ter,l0J9which assumes the rate constant to be invariant from run to run. By employing the nonlinear form of the filter, known as the extended Kalman filter, an invariant rate constant is no longer required. This is quite advantageous since rate constants are functions of a number of factors (i.e., pH, ionic strength, temperature). We have previously described simulations that showed the applicability of this method to two-component mixtures.20 It was found that the results obtained using the extended Kalman filter are accurate as long as there are sufficient spectral and/ or kinetic differences. The degree of difference necessary to achieve accurate results depends upon the number of wavelengths utilized for the analysis, but more overlap is tolerated with a larger number of wavelengths used. The accuracy of (12) Kalman, R. E. Trans. ASME. Ser. D: J . Basic Eng. 1960, 82, 3545. (13) Brown, S. D. Anal. Chim. Acra 1986, 181, 1-26. (14) Brown, S,D.; Barker, T. Q.; Larivcc, R. J.; Monfre, S. L.; Wilk, H. R. Anal. Chem. 1988, 60, 252R-73R. (15) Corcoran, C. A.; Rutan,S.C. AM^. Chem. 1988,60, 1146-53. (16) Corcoran, C. A.; Rutan, S.C. Anal. Chem. 1988,60, 2450-4. (17) Rutan, S.C.; Fitspatrick, C. P.; Skoug, J. W.; Weiser, W. E.; Pardue, H. L. AM^. Chim. Acta 1989, 224,243-61. (18) Wentzell, P. D.; Vandyke, S. J. AM^. Chim. Acta 1992, 257, 173-81. (19) Xiong, R.; Velasco, A.; Silva, M.; PCrez-Bendito, D. Anal. Chim. Acta 1991, 251, 313-9. (20) Quencer, B. M.; Crouch, S. R. Analyst 1993, 118, 695-701.
0003-2700/94/03660450$04.50/0
0 1994 American Chemical Soclety
determinations is very good even with a high degree of overlap and a few wavelengths, as we have shown previously.20 Systems with complete kinetic overlap were able to be resolved providing there were spectral differences at the chosen wavelengths, while systems with complete spectral overlap were successfully determined as long as sufficient kinetic overlap was present. In the case of the complete spectral overlap, rate constants needed to be different by a factor of about 2.5-3 in order to achieve acceptable results. In the cases shown where the analytes were not accurately determined, the filter provides information (e.g., by the covariance matrix) that lets the user know that the results are suspect.
MATHEMATICAL DESCRIPTION An n-component system which could be utilized for multicomponent kinetic determinations is described in the following reactions:
.
.
where C I and C2 are the species in the mixture that are to be determined, R is a general reactant, and kl and k2 are the rate constants. If we have an n-component system that follows the reactions described in eq 1 above, if R is in excess so that pseudofirst-order conditions apply, and if PI]^ = [P2l0 = ... = [Pnlo = 0, then the concentration of each species with respect to time is given by [Clll = [cllOe-k'f
[cA,=[c,J0e-'+"
If we assume that absorbances are additive and unit path length, the total absorbance at any time, t , and any wavelength, j , is given by Ajf =
simplification and collecting terms Aj, = (cjc, + cjR -
[CII,e-klf + (tjc,
+ +
[ ~ ~ ] ~ e +...'
(E&,
+ cjR - fly>
x
+ cjR - c,yn)[~n]oe-knf+
- cjR)[Cllo + (ejpz- ejR)[c210 + + (cjF'a - cjR)[Cnlo + cjRIRlo (4) We now have a model for the absorbance of the system at any time and at any wavelength. The absorbance of the system is described as a function of ( 2 n + 1) parameters, [CI],, [C2l0, ..., [Cnlo, [R],, kl, k2, ..., and k,. This model assumes that all reactions behave independently and that there are no synergistic effects. By using eq 4 as a model, information about concentrations and rate constants can be obtained from first- or pseudo-first-order reactions. Useful information can be obtained even if the spectra of the reactants and products are severely overlapped. Equation 4 is used as the model for the extended Kalman filter. As each new data point is obtained, the estimates for all parameters given above are updated using the difference between the estimated and true absorbance values to help give weights to the estimates. The updated values are used as the initial points for the next loop of the filter. Since the Kalman filter and extended Kalman filter algorithms have been presented elsewhere,13J5 only a brief description is given here. Two equations are required for the Kalman filter. The system dynamics equation describes how the measurements, (e.g., absorbance) vary with one or more independent variables. The second equation describes the measurement process, and how the measured values relate to the parameters desired. The parameters of the model are known as the state vector. Estimates of each of these parameters are adjusted after each measurement by means of a weighting function which takes into account thedifference between the actual measurements and the predicted measurements. These differences are used to adjust the values of the state vector and the estimated errors in the values, which are described in the covariance matrix. At the start of filtering, no knowledge of species concentrations is available, and therefore, the filter is not able to provide accurate estimates, but as more data points are obtained, the filter estimates become more accurate. Gaussian white noise is assumed to be present. For the system described in eq 4, the state vector would be
X=
cjC,[Cllf+ cjCz[c21f+ "' + cjC,[cnlf + cjRIRlf + cjpl[P1lf+ cjp2[P21f+
* * e
+ cjPDIPnlt(3)
If we substitute the equations for the concentration of each species with regard to time (eq 2 ) into eq 3, we obtain after Amlytical Chemistry, Vol. 66, No. 4, February 15, 1994
459
In other words, we are attempting to estimate the initial concentrations of all reactants, and the rate constants of the reactions. The reaction used to test the method described here is the complexation of three lanthanides (La, Pr, Nd) with 4-(2pyridy1azo)resorcinol. These reactions can be made pseudo first order in the metal species by maintaining PAR in excess. Since the Kalman filter is recursive in nature, there is a need to provide an initial guess for each of the parameters in the state vector. In order to avoid bias in the filter, the initial estimates for all [CJ0 values were taken to be zero. The concentration of the common reagent would be known to the experimenter under normal conditions, so this value was supplied as the initial estimate for [R],. It is also assumed that the experimenter has some knowledge of the rate constants for the reaction, so the best estimates of these were also supplied to the filter.
Table 1. CondHlons for Spectral AcqulrHlon of Data'
metal mixture
time between scans (ms)
La (II I ) Pr(II1) Nd(II1) LdPr LdNd Pr/Nd La/Pr/Nd
8.5 or 10.5 8.5 or 10.5 8.5 or 10.5 8.5 8.5 6.5 6.5
@
2.00 X
l eM
14000 6ooo
!
12000 I
delay before %tal data f i s t scan (me) collection time (ms)
13.0 or 15.0 13.0or 15.0 13.0or 15.0 13.0 13.0 11.0 11.0
863 or 1065 863 or 1065 863 or 1065 863 863 661 661
PAR used for all reactions.
*%
:f \
10000
. 8000 w
6000 EXPER IMENTAL SECT1ON 4000 Apparatus. A Tracor Northern (Model TN-6123) 5122000 element intensified linear photodiode array was used to acquire spectra in the 300-700-nm wavelength range. Software to 0 300 350 400 450 500 550 610 650 700 control the data acquisition was written in-house. A stoppedwavelength flow reagent mixing system, constructed in-house,21was used Flgure 1. Molar absorptlvkies of IanthanMes and PAR: (0)PAR, (+) to mix the reagents. La(PAR) complex, (-) Pr(PAR) complex, and (- - -) Nd(PAR) complex. Reagents. 4-(2-Pyridylazo)resorcinol (Sigma Chemical Co. P-8019) was used as the common reagent. A M stocksolution was prepared, from which a 2 X l e M working and kinetic overlap is present. The mixture of La, Pr, and Nd solution was made as needed. La(NO3)3.6H20, PrC13.6H20, fits this description well. and Nd(Cl0&6H20, were desiccated and used to prepare Certain information needs to be supplied to the filter for M stock solutions. A diluted stock solution was then it to be able to obtain accurate estimates of concentrations. prepared at lo4 M in each of the metals. From these stock Pure component spectra and an estimate of the rate constant solutions, working samples were prepared at concentrations for each reaction are required information. This is done in from 3 X 1V to 3 X M. These diluted samples were practice by having separate data files containing molar mixed 1:l with the PAR reagent previously prepared. No absorptivities for each component in a reaction. In order to buffering or ionic strength control was used. provide estimates of the rate constants to the filter, they were Procedure. Solutions of PAR were mixed with solutions first determined by fitting a nonlinear function to an individual of the appropriate mixture of metals in the stopped-flowmixing reaction of each metal. The relative rate constants for system. Progress of the reaction was followed spectropholanthanum, praseodymium, and neodymium with PAR are tometrically with the linear photodiode array (LPDA). Using 1:1.7:1.9 under the conditions explored. the LPDA, it is possible to take one complete spectrum every Single-Component Studies. In order to determine how well 3.5 ms, and up to 100 spectra may be acquired in a single run. the filter works under the conditions used in this work, it was The actual length of time between acquisition of spectra was first tested in a one-component system. varied as necessary. Varying amounts of each metal solution and PAR were Computational aspects. The extended Kalman filter mixed in the stopped-flow unit under the conditions given in programs used were written in QuickBASIC (Microsoft Corp.) Table 1. The extended Kalman filter algorithm described version 4.0 using the algorithm described p r e v i o u ~ l yand ~ ~ ~ ~ ~above was then applied to the data from each diode individually the measurement model described in eq 4 above. Calculations in order to determine which wavelength regions were best for were carried out on a 286-type computer equipped with a this study. Figure 1 shows the molar absorptivities for each math coprocessor. of the products of the complexation reaction. Figure 2 shows the estimated concentration of each of the three lanthanides vs the wavelengths used for analysis. The error bars represent RESULTS AND DISCUSSION the standard deviation in the filter estimates for triplicate Discussion will focus on the determination of mixtures of determinations. Relative standard deviations were approxlanthanides by their complexation reactions with PAR because imately 10%or less for each of the three metals in the region the rate constants of these species are so similar. The goal of the optimum wavelength, with precision improving as more of this work was to develop a method which would allow the wavelengths were used. However, the greater the number of determination of mixtures even when a large amount of spectral wavelengths used, the longer the computation time. It is apparent from Figure 2 that the area from approximately 450 ( 2 1 ) Beckwith, P. M . ; Crouch, S. R. Anal. Chem. 1972, 4 4 , 221-7. QT
480
Analytical Chemistty, Vol. 66,No. 4, February 15, 1994
1 .o 10.j
425
468.7
512.5
556.2
0
600
0.275
wavelengthlnm
0.55
0.825
1 .I
0.825
1.1
Timeis
1.7
5.0 1 0 ' ~
1.6 IO"
4.0 10.'
1.5 10.j 13.0 1 0 ' ~ h
L
&1,4 10"
a_
VI
i2.0 1 0 ' ~
'1.3 10.5
1.0 1 0 ' ~
1.2 10-j
0.0 1 oo 425
1.1 1 0 ' ~ 468.7
512.5
556.2
0
600
0.275
0.55 Timeis
wavelengthinm
1.7
(C)
I
1.6 10-j 1.5 1 0 ' ~
z21.4 1 0 ' ~
L
B
'1.3 IO" 1.2 1 0 ' ~
1.1 1 0 5 425
468.7
512.5
556.2
600
wavelengthinm
0
0.275
0.55
0.825
1.1
Timeis
M lanthanum, Flguro2. Estimatedinitial concentrationof (a) 2.88 X (b) 1.42 X M praseodymium, and (c) 1.39 X 10" M neodymium with respect to wavelength. True concentration shown by solld line: 8.5-ms scans.
Figure 3. Estimated concentration using 1 pixel at 495 nm of (a) 1.44 X M lanthanum, (b) 1.42 X M Praseodymium, and (c) 1.39 X 10" M neodymium with respect to Mme. Error bars fl standard deviation; (-) true concentration, (- -) 10% error in concentration; 10.5-ms scans.
to 550 nm is analytically useful for all metals while the rest of the spectral region studied is less useful. The spectral region below 450 nm is not useful due to the strong absorbance of PAR. Since PAR is in excess to maintain pseudo-first-order conditions, the overall absorbance in this region changes little throughout the course of the reaction. Therefore, little information is available in this region, and it is not possible toobtain accurate estimates of themetal concentration. Above 550 nm, the metal/PAR complexes exhibit little absorbance, and so again, there is little absorbance change throughout the course of the reaction, and little information for the filter to work with. Of course, in these wavelength regions, other techniques would also fail to obtain accurate results, since there is little useful analytical information.
Figure 3 shows the estimated concentrations for samples of lanthanum, praseodymium, and neodymium with respect to time. This figure represents data processed by use of only a single wavelength (495 nm). For consistency, the 495-nm wavelength was kept the same for all three determinations, even though it does not provide optimal estimates for any of the metals. However, this wavelength does provide quite accurate results for all three metals, while other wavelengths may be better for some of the metals. It is apparent that the filter settles onto an approximate value early in the reaction, and after that, the estimated concentration values vary little from point to point. Error bars are included on the plots, and these represent square roots of values taken from the diagonal of the covariance matrix, which yields the standard deviation
-
Analytical Chemistry, Vol. 66, No. 4, February 15, 1994
461
Table 2. Flnal Extended Kaiman Fllter Estimates ot Concentratlon for Mixtures of Lanthanum, Praseodymium, and Neodymium
no. of wavelengths
taken (106M )
La found (106M)
1 2 3 1 2 3 1 2 3 1 2 3 4
1.44 1.44 1.44 1.44 1.44 1.44
0.82 1.21 1.29 0.74 1.24 1.26
1.44 1.44 1.44 1.44
0.67 -0.49 0.81 1.24
error (%)
-43.0 -16.0 -10.4 -48.4 -13.7 -12.5
-53.3 -134 -43.5 -13.9
taken (106M)
Pr found (106M)
1.42 1.42 1.42
1.49 1.27 1.42
1.06 1.06 1.06 1.06 1.06 1.06 1.06
of the estimated values. As the reaction progresses and more data are available to the filter, the precision of the estimated values improves while the accuracy is essentially unchanged once the filter settles onto a stable estimate. Note that the sizeof the error bars is quite small. This provides an indication that the filter has enough information to give accurate estimates of concentrations using only a single wavelength. This is not surprising, as most multicomponent kinetic methods only use data from a single detector. However, as the number of components and the spectral and kinetic overlap increase, the errors increase substantially if only a single wavelength is utilized. Residual absorbancevalues were randomly distributed with a zero mean, which shows that the model used to fit the data in the extended Kalman filter algorithm is adequate. The only residual that is large is the very first point, which is due to the initial values provided to the filter, and not to a fitting error. The values of the rate constant provided to the filter were also varied from the calculated value. These values were varied from 50 to 150% of the true value. Although the best errors were usually observed when the calculated value of the rate constant was provided to the filter, even values that were off by as much as 50% provided excellent results. The maximum increase in error for any of the trials tested was less than 3%. This seeming insensitivity to the value of the rate constant is due to the nonlinear form of the Kalman filter. Since the rate constant value is one of the parameters that is adjusted at each iteration, as long as the rate constant provided to the filter initially is approximately correct, the filter is able to calculate accurate results. Two- and Three-Component Studies. All three possible two-component mixtures of the three lanthanides were prepared, as well as three-component mixtures. These mixtures were then reacted with the PAR reagent in the stopped-flow apparatus and the reactions followed spectrophotometrically as described above. Conditions for the data collection are described in Table 1 for each of the mixtures studied. For greater than one wavelength used, the choice of which wavelengths to utilize for the analysis was made by a simplex optimization p r ~ c e d u r e . Once ~ ~ , ~these ~ wavelengths (22) Betteridge, D.; Wade, A. P.; Howard, A. G. Talanta 1985, 32, 709-22. (23) Betteridge, D.; Wade, A. P.; Howard, A. G. Talanta 1985, 32, 723-34.
462
Analytical Chemistry, Vol. 66, No. 4, February 15, 1994
-0.39 1.06 1.17 23.6 3.05 1.99 0.87
error (%)
taken (106M)
Nd found (106M)
error
1.39 1.39 1.39 1.04 1.04 1.04 1.04 1.04 1.04 1.04
1.45 1.35 1.37 1.67 1.08 1.16 -4.69 1.41 0.98 0.97
4.3 -2.8 -1.4 60.6 4.0 11.8 -551 35.6 -6.3 -6.7
(%)
4.9 -10.6 0.0
-137 0.0 10.2 2126 188 87.7 -17.9
were determined, they were kept constant for all remaining analyses of that mixture. Table 2 summarizes the final estimated concentrations from the extended Kalman filter for each of the possible two-component mixtures when from one to three wavelengths are analyzed and the final estimated concentrations for a three-component mixture when from one to four wavelengths are analyzed. Note that, in general, the more wavelengths used for the determination, the more accurate the results from the extended Kalman filter. One notable exception to this is in the mixture of praseodymium and neodymium where the increase from two to three wavelengths actually decreases the accuracy of the determination. However, the precision of the determination always increases with more wavelengths being utilized. Another interesting result is seen in Table 2 for the analysis of lanthanum and neodymium. The estimated concentration of neodymium using only one wavelength appears to be quite accurate. However, the relative standard deviation for this number from the covariance matrix is approximately 80%. Relative standard deviations improve to approximately 5% with three or four wavelengths. Increasing the number of wavelengths used beyond three or four will yield an improvement in the precision of the determination, but will also take longer to analyze. Relative standard deviations from multiple runs of samples were generally less than 10% for twocomponent mixtures and less than 15% for three-component mixtures. Different concentrations of each of the species were tested to see the range over which the method is applicable. Table 3 summarizes the results of these studies. Larger errors at higher concentration ratios are most likely due to the limits of detection for the chemical system, and not a flaw in the method. Simulations presented in our previous paperZoshow excellent results for broader concentration ratios. Table 4 shows a comparison of the method described here and the method of proportional equations. As can be seen in the table, the extended Kalman filter shows a vast improvement over the method of proportional equations. This method was modified to be used at the two wavelengths used for the extended Kalman filter analysis shown in Table 2. The rate constants provided to the filter for two-component mixtures were varied from 50 to 150% of the true value for each of the components. The worst estimates of concentration
Table 3. Flnal Extonded Kalman Flltor Estknatos ol Concontratkn for Mlxturer of Lanthanum, PraI).odymlum, and Noodymlum
error
taken
(106MI
La found (10" M)
(%I
(106 M)
1.44 1.50 1.44
1.29 1.09 1.26
-10.4 -27.3 -12.5
taken
1.44 1.44
1.24 1.58
-13.9 9.9
Pr found (1odM)
error (%)
1.42 3.00
1.42 4.22
0 40.7
1.06 1.50 1.50 0.750 1.50 1.06 1.42
1.06 1.72 1.40 0.756 1.31 0.87 1.61
0 14.7 -6.7 0.8 -12.7 -17.9 13.7 2.5 1 0 ' ~
Table 4. Comparkon of the Mothod of Proporllonal Equatlona and the Extended Kalman Fllter
method prop eqs
extendedKF
Pr Nd taken found error taken found error (106M)(106M) (%) (106M) (106M) (%)
Nd found
taken (106M)
(106M)
error
(%I
1.39 1.04 0.75 0.30 1.50 0.15 1.04 1.39
1.37 1.08 0.72 0.29 1.35 0.25 0.97 1.51
-1.40 4.0 -3.6 -4.0 -10.0 67.0 -6.7 8.9
3 0
2.0 1 0 ' ~
0
0
a
4 . 5 10'~
I
1.06 1.06
2.25 1.06
112.3 0.0
1.04 1.04
-0.07 1.08
-106.7 4.0
I 1.0 1 0 ' ~
occurred when both rate constants were below the true values. The maximum change in the errors in concentration for the mixture of praseodymium and neodymium (described in Table 2), when two wavelengths were used, was 6% when both rate constants were underestimated significantly. The maximum error for either component under these conditions was 9.6%, compared to 4.0% when accurate rate constants for both reactions are provided to the filter. The residual absorbance values were random and centered about zero. The number of iterations that it takes the filter to settle onto accurate estimates is longer in multicomponent mixtures than it is in single-component samples. Figure 4 shows the progress of the estimated concentrations of praseodymium and neodymium with time. Again, however, once the estimated concentrations have settled, they remain quite stable.
CONCLUSIONS The extended Kalman filter has been shown to be a powerful technique for multicomponent kinetic determinations. There are several advantages to this method over most previous approaches. The basis of determination on differences in both spectral and kinetic behavior allows differentiation of more
5.0 1 0.6
0
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Time. s
Flgure 4. Estimated concentrations of praseodymium (0) and neodymlum (+) vs time using two wavelengths. True concentrations: [R] = 1.08X lo-* M, [Nd] = 1.04 X lo4 M, and [PAR] = 2.00 X
10" M.
closely3elated species. The recursive nature of the Kalman filter provides for fast computations. Finally, the use of the nonlinear, extended form of the Kalman filter avoids the most common assumption used in other methods. This assumption is that the rate constant is required to be invariant from run to run. The present method does not require this assumption be made. Previous multicomponent kinetic methods require strict control of such reaction conditions as pH, ionic strength, and temperature. These controls were not used in the determinations described in this paper. Received for review August 20, 1993. Accepted November 19, 1993.' ~~~
Abstract published in Advance ACS Abstracts, January 1 , 1994.
Ana&tIcaIChem&try, Vol. 66, No. 4, February 15, 1994
483
