Anal. Chem. 1996, 68, 1842-1850
Optimization Method for Simultaneous Kinetic Analysis Liang Xu and Israel Schechter*
Department of Chemistry, TechnionsIsrael Institute of Technology, Haifa 32000, Israel
Kinetic analysis is often carried out for simultaneous determinations; thus, a theory to establish its optimal conditions is necessary. A very simple and fast model to find conditions for optimum analytical performance and to predict the quality of simultaneous kinetic analysis has been developed. It is general and applicable to any reaction order or rate constant. The model has been based on the angle between the kinetic vectors and on their norm ratio, which are readily calculated for any kinetic scheme. Evaluation of the proposed model has been carried out by detailed simulations of numerous experimental conditions and analysis by full PCR calculations (when applicable) or nonlinear least-squares fitting. An important conclusion is that analytical performance is determined to a large extent by the stability of the space spanned by the relevant component vectors (in addition to experimental noise and other factors). The quality of the analysis is governed by the angle between the kinetic vectors, while the norm ratio determines the error distribution between components. Optimum conditions for simultaneous kinetic analysis have been studied in several representative examples, regarding the timing of the kinetic monitoring and the effects of concentrations and of rate constant ratios, as well as several other factors of experimental relevance. Kinetic aspects of analytical chemistry have been well recognized, and the difference in kinetic behaviors of various components has been utilized for simultaneous determination.1-4 Many kinetic models for simultaneous analyses have been investigated; however, the first-order or pseudo-first-order reactions were studied and used most extensively. Various methods, including proportional equations and curve-fitting, have been derived for simultaneous determinations.5,6 Several sophisticated methods have been developed for handling kinetic data,7-12 yet no general
model to provide conditions for optimization of simultaneous kinetic analysis is available. Due to the incorporation of computerized data acquistion systems, numerous data can be obtained through monitoring the whole kinetic processes. Thus, questions such as how many data points should be included in data processing, how long the kinetic processes should be monitored, or which is the most informative time domain become considerably relevant. These questions, as well as conditions for kinetic analysis, have been studied separately for several kinetic models,13-19 yet no general theory to cover all possible kinetic models has been provided. The importance of such a general model is emphasized by the difficulties in optimization of individual kinetic schemes. In principle, the results of simultaneous analysis can be improved if information obtained from kinetic measurements is efficiently utilized. Therefore, systematic studies are needed on each reaction model involved in the kinetic determination. Nevertheless, even for a single kinetic scheme, it is time-consuming and probably impossible to carry out experimental studies of all relevant parameters, covering various rate constants, concentrations, and reaction orders. An apparent solution is to adopt the algorithms developed for optical spectroscopy. Indeed, kinetic data are often considered as spectra. Various criteria are available for the optimization of spectral resolution.20-31 Condition number and determinant are the generally accepted criteria and are widely used in wavelength selection.25,30 Unfortunately, these methods are based on solving linear simultaneous equations and cannot be applied to the nonlinear problems of kinetic analysis. Although kinetic data are often considered as spectra, there is a big difference between them: the shape of an optical spectrum (as a function of
(1) Otto, M. Analyst 1990, 115, 685. (2) Perez-Bendito, D. Analyst 1990, 115, 689. (3) Mottola, H. A. Kinetic Aspects of Analytical Chemistry; Wiley: New York, 1988. (4) Mark, H. B., Jr.; Rechnitz, G. A. Kinetic in Analytical Chemistry; Interscience: New York, 1968. (5) Pardue, H. L. Anal. Chim. Acta 1989, 216, 69. (6) Perez-Bendito, D. Analyst 1984, 109, 891. (7) Willis, B. G.; Woodruff, J. R.; Frysinger, J. R.; Margerum, D. W.; Pardue, H. L. Anal. Chem. 1970, 42, 1350. (8) Harner, R. S.; Pardue, H. L. Anal. Chim. Acta 1981, 127, 3. (9) Rutan, S. C.; Fitzpatrick, C. P.; Skoug, J. W.; Weiser, W. E.; Pardue, H. L. Anal. Chim. Acta 1989, 224, 243. (10) Rutan, S. C.; Brown, S. D. Anal. Chim. Acta 1985, 167, 23. (11) Gui, M.; Rutan, S. C. Anal. Chem. 1994, 66, 1513. (12) Schechter, I. Anal. Chem. 1992, 64, 729.
(13) Tahboub, Y. R.; Pardue, H. L. Anal. Chim. Acta 1985, 173, 23. (14) Tahboub, Y. R.; Pardue, H. L. Anal. Chim. Acta 1985, 173, 43. (15) Wentzell, P. D.; Karayannis, M. I.; Crouch, S. R. Anal. Chim. Acta 1989, 224, 263. (16) Johnson, E. D.; Weber, J. P.; Meites, L. Anal. Chim. Acta 1985, 178, 263. (17) Meites, L.; Hussam, A. Anal. Chim. Acta 1988, 204, 295. (18) Meites, L. Anal. Chim. Acta 1988, 211, 31. (19) Meites, L. Anal. Chim. Acta 1989, 221, 319. (20) Kaiser, H. Z. Anal. Chem. 1972, 260, 252. (21) Junker, A.; Bergmann, G. Z. Anal. Chem. 1974, 272, 267. (22) (a) Jochum, C.; Jochum, P.; Kowalski, B. R. Anal. Chem. 1981, 53, 85. (b) Kalivas, J. H.; Kowalski, B. R. Anal. Chem. 1981, 53, 2207. (23) Brown, C. W.; Lynch, P. F.; Obremski, R. J.; Lavery, D. S. Anal. Chem. 1982, 54, 1472. (24) Rossi, D. T.; Pardue, H. L. Anal. Chim. Acta 1985, 175, 153. (25) Otto, M. Anal. Chim. Acta 1986, 180, 445. (26) Frans, S. D.; Harris, J. M. Anal. Chem. 1985, 57, 2680. (27) Kalivas, J. H. Anal. Chem. 1986, 58, 989. (28) Warren, F. V., Jr.; Bidlingmeyer, B. A.; Delaney, M. F. Anal. Chem. 1987, 59, 1890. (29) Wentzell, P. D.; Wade, A. P.; Crouch, S. R. Anal. Chem. 1988, 60, 905. (30) Juhl, L. L.; Kalivas, J. H. Anal. Chim. Acta 1988, 207, 125. (31) Kalivas, J. H.; Roberts, N.; Sutter, J. M. Anal. Chem. 1989, 61, 2024.
1842 Analytical Chemistry, Vol. 68, No. 11, June 1, 1996
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© 1996 American Chemical Society
wavelength) is usually independent of concentrations, while the kinetic spectrum of a component is concentration-dependent (with exception of unity reaction order). Thus, models developed for optimization of optical spectra usually cannot be applied to kinetic optimizations. Partial solution has been provided by computer simulations: simulations offer an acceptable method to investigate the possibility of simultaneous determinations and the influence of rate constant, reaction order, concentration, noise level, and time domain. However, extensive simulations are needed for each system of interest. Thus, a simple general model to predict the major results and provide insight to the main effects is of considerable importance. In this paper, we present such a simple general model and apply it to the investigation of several kinetic schemes. We regard data obtained from kinetic measurements of a mixture as a kinetic vector, which is a linear combination of kinetic vectors of pure components. We show that the possibility to calculate the concentrations depends on the stability of the space spanned by the component vectors, which is related to the difference between the components. Norm, norm ratio, and angle between vectors are the concepts used frequently in this paper to characterize kinetic systems and are related to the expected analytical errors for the components in unknown samples. MATHEMATICAL DESCRIPTION The general form of kinetic systems consisting of several parallel reactions of any reaction order can be represented as follows:
Ai f Bi, dA/dt ) -kiAni, i ) 1, ..., N
Ait ) Ci0 e-kit
(5)
Bit ) Ci0 (1 - e-kit)
(6)
For simplicity, kinetic models for components involving only unity reaction order are termed unity models, while models for components with nonunity reaction order (general order except unity) are termed nonunity models. In the following discussion, we shall refer to spectroscopic detection, although the algorithm is general in this respect. In practice, in order to optimize the signal output, measurements are usually carried out under conditions that are sensitive either to products or to reactants. For example, in spectroscopic measurements, wavelengths are selected in a range where only products or reactants have significant absorption coefficients. Therefore, these two cases are discussed separately. Correspondingly, two mathematical models can be established, hereafter called model 1 and model 2, which represent the kinetic schemes with increasing and decreasing signals, respectively. For the simplicity of presentation, only two components will be considered, and unity sensitivity coefficients are assumed for the monitored species. Thus, in model 1, which is based on monitoring of reactants, the signal as a function of time is
St )
∑R A
i it
St ) [k1(n1 - 1)t + C10(1-n1)]1/(1-n1) + [k2(n2 - 1)t + A20(1-n2)]1/(1-n2), ni * 1 (7a)
(1) or
where N is the number of components in the mixture and ni is the reaction order. For the above reactions, the kinetic process can be monitored by following some experimental parameters directly proportional to concentrations. Thus, the detector response at time t, St, is equal to the summation of the individual contributions of all components at the moment t:
St )
∑R A
i it
+
∑β B i
it
(2)
St ) C10 e-k1t + C20 e-k2t, ni ) 1
(7b)
where Ri ) 1 and βi ) 0 are assumed. In model 2, products are monitored, and the signal at time t is
St )
∑β B i
it
St ) C10 - [k1(n1 - 1)t + C10(1-n1)]1/(1-n1) + C20 [k2(n2 - 1)t + C20(1-n2)]1/(1-n2), ni * 1 (8a)
where Ri and βi are the detector sensitivity coefficients for reactants and products and Ait and Bit are the concentrations of reactants and products at time t, respectively (constant background response is omitted). Let Ci0 be the initial concentration of Ai. The concentrations at time t are given by32 (1-ni) 1/(1-ni)
Ait ) [ki(ni - 1)t + Ci0
]
Bit ) Ci0 - [ki(ni - 1)t + Ci0(1-ni)]1/(1-ni)
(3)
(4)
or
St ) C10(1 - e-k1t) + C20(1 - e-k2t), ni ) 1
(8b)
where Ri ) 0 and βi ) 1 are assumed. Obviously, kinetic systems with one component of unity reaction order and the other of nonunity order are also included in the above equations and will not be discussed separately. Since data obtained in each kinetic run are in the form of a table, the table can be considered as a column vector and is called kinetic vector hereafter:
The above equations have a singular point at unity reaction order. In this case, the following functions should be used instead:
S ) [St1, St2, ..., StM]T
(32) Benson, S. W. The Foundations of Chemical Kinetics; McGraw-Hill: New York, 1960.
where M is the data point number, capitalized letters stand for Analytical Chemistry, Vol. 68, No. 11, June 1, 1996
(9)
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column vectors, and superscript T denotes transpose. Thus, the kinetic vector S is a linear combination of vectors representing each component. In model 1, we have
S ) A1 + A2
(10)
S ) B1 + B2
(11)
In model 2,
where Ai and Bi are vectors representing pure components of reactants and products, respectively, and are called component vectors hereafter. The possibility to resolve a component vector from a kinetic vector, which is equivalent to the possibility to calculate the concentration, relies on the stability of the space spanned by all component vectors, which is related to kinetic differences between the components in a mixture. Thus, from the linear algebra point of view, these differences can be easily characterized by parameters such as norm, norm ratio, and angle between component vectors. These parameters can be calculated from component vectors according to well-known formulas.33 For vectors x and y, Normx ) ∑xi2 and Normy ) ∑yi2. The angle between x and y is given by ∑xiyi/(Normx1/2‚Normy1/2). The norm of a component is the measure of its contribution to the total signal, while norm ratio accounts for the relative ability of two components to resist noise. Vector angle stands for the difference between components, representing the possibility to resolve one component in the presence of the other. In principle, the best analytical results are expected when maximum information on the differences between components is utilized properly in data processing. We first demonstrate the validity of the above defined parameters and then use them as criteria to evaluate the best conditions for simultaneous determination. EXPERIMENTAL SECTION This study consists of simulating numerous experimental data and recovering initial concentrations by principal component regression (PCR) or nonlinear least-squares fitting. The recovery errors are then used as reference for the simple models developed here. Simulated kinetic data have been obtained by adding Gaussian noise at a given level to the exact signal calculated from the kinetic functions. Gaussian noise has been produced by the Box-Muller method, which provides normally distributed deviates with zero mean and unit variance. A modified LevenbergMarquardt method has been employed to solve the nonlinear leastsquares problem of the general order kinetic system.34,35 In the case of unity reaction orders, the kinetic functions have been transformed to linear form, with regard to the concentrations. Therefore, the PCR algorithm could be applied for analysis of this case. Computer experiments have been performed under the various conditions of interest. The synthetic data have been produced, and then one of the above algorithms has been applied to recover the initial kinetic parameters. For each given systems, (33) Dahlguist, G.; Bjorck, A. Numerical Methods; Prentice Hall: Englewood Cliffs, NJ, 1974. (34) Levenberg, K. Q. Appl. Math. 1944, 2, 164. (35) Marquardt, D. SIAM J. Appl. Math. 1963, 11, 431.
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the quality of the analysis has been evaluated by the relative error in recovery of concentration. The PCR program has been coded according to details described elsewhere.36-38 A centering procedure has been used to pretreat the raw data during calibration. Cross-validation has been employed, and the optical number of factors has been automatically determined from prediction error sum of squares (PRESS) through F-statistic test. Outliers in the calibration set have been detected from both concentration and the spectral F-ratio, while detection of outliers in the unknown sample set relies solely on the spectral F-ratio. The above features can be considered as standard features in any modern PCR code that is automatically applied to analytical problems. To eliminate artifacts due to the randomly chosen calibration and unknown set, sufficiently large numbers of samples (30 for calibration and 100 for unknown set, in most cases) have been used. All computer programs for data generation and processing, and for evaluation of results, have been written in FORTRAN 77. Programs have been compiled and run on several computers (UNIX workstations and personal computers). Detailed procedure and conditions for simulation of general order kinetic systems, by using the above algorithm, can be found elsewhere.12 RESULTS AND DISCUSSION Algorithms for simultaneous determination of mixtures by kinetic analysis of general order reactions have been studied previously.12 Various conditions including noise level and measurement schedule have been evaluated by extensive computer simulation. In this paper, emphasis is given to the modeling of analytical performance rather than to the detailed calculation. We are interested in finding a simple model that predicts the analytical performance, so we can apply it in order to find out optimal conditions for kinetic analysis. We first establish the model and present its results. The similarity of these results to the detailed experimental simulations serves as an indicator of the validity of the model. After this stage, we discuss the details of our findings and provide explanations in terms of these concepts of this model. The validation of the above model is carried out by both PCR and nonlinear regression methods. The K-matrix method could be used equivalently, since the PCR calibration procedure is equivalent to the calculation of parameters in the K-matrix method. Since the criteria proposed in this paper are based on the angle between component vectors, they are chemical system-dependent rather than algorithm-dependent. The optical conditions for a system are determined by its chemical features (described by kinetic parameters), regardless of the algorithm used. The algorithm may be linear or nonlinear, and it may require calibration or not. To demonstrate the generality of the criteria, we use both PCR and nonlinear least-squares fitting. Prediction Model of Analytical Error. Analytical performance, as represented by the relative error in prediction of concentrations, is governed by the differences in the kinetic behavior of the various components. Thus, we are looking for a simple function of the component vectors to model the complicated relative error values. Such a function can be used to predict optimum measurement conditions (e.g., time interval), without the need to carry out numerous experiments or extensive simulations. (36) Malinowski, E. R. Factor Analysis in Chemistry; Wiley: New York, 1991. (37) Haaland, D. M.; Thomas, E. V. Anal. Chem. 1988, 60, 1193. (38) Martens, H.; Naes, T. Multivariate Calibration; Wiley: New York, 1989.
It is expected that the angle between the pure component’s vectors has a considerable effect upon the analytical performance. Parallel vectors represent linear dependent data that do not contribute any useful information, while orthogonal vectors should provide best results. Obviously, the larger the angle, the higher is the stability of the n-dimensional space, and, in turn, the less the space will be subjected to noise. Several functions of this angle have been investigated, and it has been found that the simple reciprocal of the angle provides a good model for the relative analytical errors. The reciprocal of the angle is, itself, a function of several parameters, including measurement time, reaction order, rate constants, and components’ concentrations. In the following, the effects of these factors on the performance of simultaneous kinetic analysis are modeled by the reciprocal of the angle between component vectors. It should be pointed out that this function is a function of all seven variables (for a binary system described in the above), while only a function of two variables can be easily viewed. Although various combinations have been studied, only the most interesting will be presented. Especially functions of time are presented, in order to show the best measurement time domain for each system. Rate Constant Effect. Similar species usually undergo kinetic processes of the same mechanism with slightly different reaction rate constants. Thus, the investigation of the rate constant effect, under the same reaction order, is of practical importance. Figure 1a shows the reciprocal of the components vector angle as a function of measurement time and rate constant ratio, k2/k1, for the model 1 scheme (monitoring the reactants), where reaction orders for both components are unity. The angle in given in radians. Gaussian noise of 0.01 standard deviation has been included in the simulated experimental data. The above angle has been simply calculated from the raw data: the scalar product of the component kinetic vectors divided by their norm (length) provides directly the cosine of this angle. All data points in this study have been simulated under the same conditions, as provided in the figure captions. Unless otherwise stated, the number of points is kept constant, rather than the interval between them. This easily obtained surface should be compared to relative errors obtained either from experimental measurements or from detailed calculations. Full experimental measurements are not practical, since this surface is composed of 961 different measurements of various rate constant ratios; however, detailed calculations can be carried out as described in the following. The results are shown in Figure 1, parts b (relative errors in analysis of the slow component) and c (fast component). These surfaces consist of 961 points each. Every point in these surfaces has been obtained by simulating 30 samples for calibration and another 100 for relative error calculation with PCR algorithm. For each point in the surfaces, a PCR calibration model has been derived and then used for 100 predictions. The observed spikes are a result of the Gaussian noise added to the kinetic data. On the other hand, the surface in Figure 1a has been obtained very simply and rapidly: Each point represents a different kinetic system, and the angle between component vectors A1 and A2 has been calculated from the scalar product (A1A2). The similarity between the simply calculated surface 1a and the “experimental” surfaces 1b and 1c proves that the reciprocal of the angle between component vectors is, indeed, an adequate criterion for prediction of analytical performance. The reciprocal of this angle models correctly the main features of experimental relative errors.
Figure 1. (a) Reciprocal of the angle between component vectors of model 1 kinetics as a function of time range and rate constant ratio. Time units are half-lifetimes of slow component. Relative errors for analysis of (b) the slow component and (c) the fast component. All calculations are based on a fixed time interval of 0.01 and 0.03 standard deviation Gaussian noise. Component concentrations are generated with pseudo-random number in the range of 0-1, with the mean of 0.5. The inset shows a typical kinetic signal including a noise level of 0.03.
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Figure 2. (a) Reciprocal of the angle between component vectors of model 2 as a function of time range and rate constant ratio. Relative errors for analysis of (b) the slow component and (c) the fast component. Time interval and noise level as in Figure 1.
Similar agreements between the reciprocal angle surface and the detailed calculated relative errors have been obtained also for model 2 kinetic schemes (monitoring the products). The results are shown in Figure 2. Surface 2a shows the reciprocal of the 1846
Analytical Chemistry, Vol. 68, No. 11, June 1, 1996
angle, while surfaces 2b and 2c show the relative errors in analysis of the slow and the fast components, respectively. The details of these results and the differences between the models are discussed in the following; however, please note that, for both models, the main features of the error surfaces (representing the analytical performance) are reproduced by the reciprocal of the angle, which is calculated from component vectors Ai or Bi. So far we have considered unity reaction orders; however, generalization of these conclusions requires extension to nonunity reaction orders. As described in the next section, similar results have been obtained also when nonunity reaction order are involved. In fact, results are even better if reaction orders of the two components are different, because the angles between the component vectors are even larger in this case. These effects are discussed in the following. Reaction Order Effect. The effect of reaction order in simultaneous kinetic analysis has been studied by keeping other parameters constant and changing just the reaction order of one component: Figure 3a presents the reciprocal of the vector angle of the two components, calculated under various reaction orders. In this example, a model 1 kinetic scheme has been studied, and the concentration of both components has been kept constant (value of 1). Rate constants k1 and k2 were fixed (0.2 and 5, respectively). The reaction order for one component was fixed (1.1), while that of the other was varied from 0.05 to 3. It can be seen that the surface possesses a low plateau at high reaction orders, at almost all time domains. At each time domain, the surface has a maximum, roughly at reaction order of 1.7 (a maximum in the reciprocal of vector angle indicates conditions where good analysis is not possible, and large errors are expected). Note that the maxima are not at reaction order of 1.1, which is the order of the reaction of the first component. The surface is highest at high reaction orders and short time domain. This example indicates that collinearity problem of component vectors may occur although kinetic parameters of two components are different. Such problems can be easily pointed out by this model, while this task is practically impossible using systematic experimental analysis. Again, this simple model is to be compared to the detailed calculations representing experimental results. Figure 3b has been obtained by detailed calculations, under the same conditions as in Figure 3a. Each point in this surface has been calculated by using the Levenberg-Marquardt method for nonlinear fitting. In this procedure, the experimental points (including noise) have been fitted to a multiparameter nonlinear function, where one of the parameters is the desired concentration. The model surface of Figure 3a has the same major characteristics as the experimental surfaces: a plateau at high reaction orders, a maximum in the error as a function of reaction order for each time domain, and higher errors at low reaction order and short time domains. The surfaces calculated by the nonlinear fitting procedure are more spiky, since they are more sensitive to the noise added to data. These calculations are based on fewer data sets; thus, averaging out of noise is not possible. The resemblance of the figures reconfirms that the reciprocal of component vector angle can be used as a prediction model of the analytical performance of a kinetic system. A similar study has been carried out for model 2 systems, and the results led to the same conclusion. Concentration Effect. As shown in the above, the component vector angle is a function of several parameters, including the
Figure 3. (a) Reciprocal of the angle between component vectors of model 1 kinetics as a function of time range and reaction order (of fast component). (b) Relative error for analysis of the slow component as a function of the same variables. Calculations are based on 100 data points per experiment and 0.01 standard deviation Gaussian noise.
Figure 4. (a) Reciprocal of the angle between component vectors of model 1 kinetics as a function of time range and concentration ratio. Concentration of the fast component is fixed at 5. (b) Relative error for analysis of the slow component as a function of the same variables. Number of data points and noise level as in Figure 3.
concentration. This implies that analytical performance is also affected by the concentrations of the components. Therefore, an interesting conclusion can be obtained: two components of the same reaction order and rate constant can be analyzed simultaneously if their concentrations are different. As we show in the following, that is true provided that the reaction orders of two components are nonunity. Since in nonunity order reactions the kinetic spectrum of a pure component is concentration-dependent, different concentrations will result in different shapes of kinetic spectra. As a consequence, an angle exists between the two component vectors, although the reaction orders and rate constants are identical. This angle enables simultaneous analysis of the two components. Figure 4a shows an example of a model 1 scheme, where the two desired components have identical rate constants (3.0) and reaction orders (2.0). The reciprocal of the angle between these two component vectors is shown as a function of the ration of their concentrations and time domain. As the
concentrations of the two components become similar, the surface is higher. When concentrations are identical, it is impossible to determine the two components in a mixture since the angle is zero and its reciprocal becomes infinitely large. However, they can be determined if concentrations differ significantly. These conclusions are confirmed by results of the extensive and detailed simulations, which are shown in Figure 4b. The two surfaces possess the same characteristics. The above discussion holds for reactions of nonunity orders only. The components of unity reaction orders can be analyzed only if their rate constants differ significantly, due to the reasons that will be given the following. Prediction Model of Error Distribution. Clearly, the relative intensities in a kinetic vector are related to the contribution of that component. In a binary mixture, the component that causes higher signal intensities has a better chance to be analyzed correctly. The norm (“length”) of a vector can measure the above Analytical Chemistry, Vol. 68, No. 11, June 1, 1996
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Figure 5. (a) Norm ratio of two-component vectors of model 1 kinetics as a function of time range and rate constant ratio, where norm1 and norm2 stand for the norms of the slow and fast components, respectively. (b) Calculation of the ratio of the corresponding relative errors for the two components. Time interval and noise level as in Figure 2.
signal intensities. Therefore, we tested the hypothesis that the norm ratio of the kinetic vectors of components in a mixture can be used as a criterion for modeling the distribution of errors between these components. Figure 5a shows the norm ratio (slow component/fast component) as a function of the two kinetic variables, time and rate constant ratio. The data in this plot have been calculated for a model 1 scheme of unity reaction order. The detailed calculation of the error ratio of these two components as a function of the same variables is shown in Figure 5b. The inverse ratios (fast component/slow component) are plotted, since the analytical errors should be related to the reciprocal of norm (similar to the correlation of relative error to the reciprocal angle, as previously shown). The corresponding surfaces for a model 2 kinetic scheme are significantly different and are presented in Figure 6. The simple norm ratios are given in Figure 6a, and the error ratios, obtained from the experimental simulations, are shown in Figure 6b. The (remarkable) resemblance of Figures 5a,b and 6a,b indicates that 1848 Analytical Chemistry, Vol. 68, No. 11, June 1, 1996
Figure 6. Same as in Figure 5, calculated for model 2 kinetics.
the norm ratio of component vectors provides a reasonable modeling of the distribution of analytical errors between components. Although this approach is not designed for quantitative concentration error prediction, it is interesting to observe the surprising agreement between the actual error ratio and the predicted values. Figure 7 presentes the correlation between the error ratio and the predicted errors based on the norm ratio. Results are shown for three different kinetic conditions. Actually, these results represent the correlation between parts a and b of Figure 6. Now that we have modeled both the global analytical performance and the distribution of errors between the components, we concentrate on the differences between the two presented models and provide an explanation for the details of the above surfaces. Differences between the Kinetic Models. Model 1 versus Model 2. In the above sections, only major trends in data have been considered, in order to demonstrate the linkage between analytical performance and the concepts of vector angle and norm ratio. In the following, we discuss the details of our findings, compare the two models, and provide explanations in terms of the new concepts.
Figure 7. Actual error ratio as a function of the predicted error ratio at three different kinetic conditions. These results represent the correlation between parts a and b Figure 6.
Model 1 represents kinetic models with decreasing signals that diminish asymptotically. Model 2 represents kinetics with asymptotically increasing signals. Both are derived for general order reactions, which is supposed to cover all kinetic schemes. Figures 1 and 2 show the same calculations for both kinetic models. In both cases, better results are obtained with a large rate constant ratio, which is attributed to the larger angle between the component vectors. The larger the angle, the more stable is the space spanned by the two vectors. The relative error values as a function of time are relevant to the decision on the measurement length. In most of the cases, the error decreases steeply and reaches a real minimum. (Sometimes only a plateau is observed when viewing the global surface; however, a minimum can always be located when looking into the details.) This means that one should not attempt to carry out the measurements longer than the times indicated by the minima in the figures, since no improvement can be obtained. In the cases of a distinct minimum, a longer measurement introduces considerably higher errors; thus, the time that corresponds to the minimum error is of practical importance. Observing the analytical performance as a function of kinetic measurement time domain demonstrates some of the differences between the two models. First, model 1 provides better results at short times than model 2. This fact is attributed to the reciprocal of the angle between the component vectors, as shown
in the above figures. The explanation is that, at short times, the angle between the component vectors of model 1 is larger than in model 2. Moreover, the norm of the model 1 vectors is much larger than in model 2, due to the nature of the component functions. The first starts at its maximum value and decreases with time, while the second starts at its minimum value (i.e., zero if not constant background response exists). Another reason accounting for the difference between the two models is that kinetic curves can be considered to be linear at short times. Therefore, curves of model 2, which increase from zero, are linearly dependent, i.e., component vectors form a zero angle in this case. In the same time range, model 1 vectors usually form larger angles. In both models, the relative error as a function of measurement time reaches a minimum. However, the minima in model 1 are more pronounced. This is the case for both the slow and the fast components. The explanation is based on the fact that the analytical performance depends both on the noise level and on the vector space stability. In model 1, the signal decays monotonically and more elements of the component vectors become very small as the measurement time is getting longer. These small elements make only a small contribution to the vector angle; thus, the vector angle, as well as its reciprocal, reaches a constant value (as shown in Figure 1a). That is to say that information about differences between components is mainly located at the beginning of the kinetic process, where a steep change in the reciprocal of the vector angle is observed. Accordingly, a distinct decrease in relative error occurs in the same region, as shown in Figure 1b,c. After that, no further improvement can be obtained, due to the constant vector angle. However, addition of kinetic data introduces more and more noise, due to the inevitable experimental noise. This is why the error increases with time and explain the observed minima. In both models, the noise level is the same, but in model 1, the signal-to-noise ratio at long times is much lower than in model 2. Thus, the error increases faster, and as a result, the minima are more pronounced. Actually, the reciprocal of the kinetic angle behaves differently. No minimum (at a rate constant as a function of time) is observed in model 1 (Figure 1a), while a clear minimum can be seen in the data of model 2 (Figure 2a). It is clear from the kinetic equations of the two models that, in model 1, almost zero values are added to component vectors at long times, while constant values are added in model 2. The angle in model 1 is not changed any more, and its value is the one determined by the short time domain. This is not the case in model 2. Addition of almost constant values (in the asymptotic range) increases the similarity of the vectors and reduces their angle. At very long times, most of the kinetic vectors consist of constant values, and the relative contribution of the short time domain (where the differences originate from) becomes smaller. Another observed feature is that the relative error of the fast component is higher than that of the slower one in model 1 (Figure 1b,c), while in model 2, the slow component has the larger relative error (Figure 2b,c). Since the error distribution between components is determined by the reciprocal of the corresponding norm ratio, the component of larger norm will have smaller error. In the kinetic scheme of decreasing signals, the slower component has a larger vector norm as compared to the faster one under the same conditions. On the contrary, the faster component in the increasing signal kinetic scheme has a larger vector norm. Analytical Chemistry, Vol. 68, No. 11, June 1, 1996
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Unity versus Nonunity Reaction Orders. When reaction orders of two components are unity, eqs 7b and 8b should be used for model 1 and model 2, respectively. In this case, kinetic functions can be transferred to linear form in the matrix equation as follows: For model 1:
S ) A1 + A2 ) {U1U2}‚C
(12)
For model 2:
S ) B1 + B2 ) {V1V2}‚C
(13)
where C is the vector of concentration. Ui and Vi are unity responses of the ith component in model 1 and model 2, respectively, and have the following forms:
Ui ) [e-kit1, e-kit2, e-kit3, ..., e-kitM]
(14)
Vi ) [(1 - e-kit1), (1 - e-kit2), (1 - e-kit3), ..., (1 - e-kitM)] (15) In the above expression, the kinetic vector S is represented as a linear combination of unity response vectors of each component, with the concentration as a multiplicative coefficient. That is to say, the angles between two component vectors are determined by unity response of corresponding components, which is independent of concentration. Thus, it is impossible to analyze the components that possesses identical rate constants if the reaction orders of both components are unity; this case will definitely result in zero angle between the two component vectors. It should be pointed out that algorithms like PLS and PCR, which are developed for linear systems, can be used in this case for the prediction of concentrations. CONCLUSIONS The concept of kinetic vectors has been studied in the context of performance of simultaneous determinations by kinetic analysis. The errors in kinetic analysis are found to be systematically dependent on the stability of the space spanned by the vectors of pure components (in addition to the experimental noise level). The reciprocal angle between these vectors is shown to model correctly the analytical performance (relative error), while the norm ratio models the distribution of errors between components.
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Analytical Chemistry, Vol. 68, No. 11, June 1, 1996
Best results can be expected when the maximum information about the differences between components is utilized. The models have been applied and tested for two kinetic schemes (with increasing and decreasing signals) of general order reactions. The main conclusion is that these simple models can be used to predict the analytical performance, as well as used as a guide to locate the optimal conditions for simultaneous kinetic analysis (e.g., best time domain). The results obtained in the present approach are in agreement with previous studies.12,13,15 It is clear that kinetic analysis cannot be applied to simultaneous determinations when the angle between components is too small. A rather interesting finding is that it is possible to determine simultaneously the components of a system with identical reaction orders and rate constants, provided that the reaction order is nonunity. The optimcal kinetic range can be easily found by the reciprocal of the angle between component vectors, which can be calculated using a simple spreadsheet. This procedure is of importance, since best results are obtained using data in the optimal range. In some cases (as previously shown), deletion of data outside this range practically improves the analytical results. Conclusively, the criteria proposed in this paper are simple and general (algorithmindependent) and proved to be effective in both cases of linear and nonlinear kinetic systems. The proposed criteria predict optimal conditions for a kinetic system; however, quantitative prediction of analytical errors requires the study of error propagation, which is algorithmdependent. Therefore, the current approach cannot provide the actual magnitude of errors, yet, it allows the prediction of optimum conditions. Multicomponent (n > 2) systems can also be optimized with the same principle: optimal conditions can be found where the space spanned by n-component vectors is most stable; however, new criteria should be used in this case. We have already developed a method for such optimization in multicomponent spectral systems (to be published). The extension to nonlinear systems of kinetic analysis is under investigation. Nevertheless, the criteria proposed in this paper have the advantage of being simple and effective for two-component kinetic systems. ACKNOWLEDGMENT This research was supported by the Technion V.P.R. Fund. L.X. thank the Israel Council of Higher Education for financial support. Received for review October 24, 1995. Accepted March 7, 1996.X AC951061W X
Abstract published in Advance ACS Abstracts, April 15, 1996.
