Chemometrics and Intelligent Laboratory Systems 81 (2006) 149 – 164 www.elsevier.com/locate/chemolab
Tutorial on the fitting of kinetics models to multivariate spectroscopic measurements with non-linear least-squares regression Graeme Puxty a,⁎, Marcel Maeder b , Konrad Hungerbühler a a
Institute for Chemical and Bioengineering, Safety and Environmental Technology Group, ETH Zürich, 8093 Zürich, Switzerland b Department of Chemistry, University of Newcastle, University Drive, Callaghan NSW 2308, Australia Received 22 September 2005; received in revised form 24 November 2005; accepted 1 December 2005 Available online 14 February 2006
Abstract The continuing development of modern instrumentation means an increasing amount of data is being delivered in less time. As a consequence, it is crucial that research into techniques for the analysis of large data sets continues. However, even more crucial is that once developed these techniques are disseminated to the wider chemical community. In this tutorial, all the steps involved in the fitting of a chemical model, based on reaction kinetics, to measured multiwavelength spectroscopic data are presented. From postulation of the chemical model and derivation of the appropriate differential equations, through to calculating the concentration profiles and, using non-linear regression, fitting of the rate constants of the model to measured multiwavelength data. The benefits of using multiwavelength data are both discussed and demonstrated. A number of real examples where the described techniques are applied to real measurements are also given. © 2005 Elsevier B.V. All rights reserved. Keywords: Kinetic modelling; Multivariate data; Spectroscopic data; Non-linear least-squares regression
1. Introduction In the context of this tutorial, a kinetic study is the investigation of chemical reactions and encompasses the identification of the correct reaction mechanism and the determination of the associated rate constants. We will concentrate on how to fit a chemical reaction mechanism, defined using kinetic theory, to measured multiwavelength spectroscopic data for a reaction of interest. The fitting of reaction mechanisms to multivariate data began with the work of Knutson et al. [1] in 1983. They coined the expression “global analysis” for simultaneously fitted fluorescence decay kinetics at multiple wavelengths. Further development of techniques in this field has continued in particular with the work of Maeder and coworkers [2–4], Bisjma, Smilde and coworkers [5–7] and Fürusjö and Danielsson [8,9]. With the introduction of fast scanning and diode array spectrophotometers, kinetic investigations based on multivariate data are
⁎ Corresponding author. Fax: +41 44 632 1189. E-mail address:
[email protected] (G. Puxty). 0169-7439/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.chemolab.2005.12.001
now common place. This type of data fitting has found application across a broad range of chemistry. Some recent examples include the complexation kinetics of metals [10–13], solvent-free organic reactions [14,15], thermal hazard evaluation [16], industrially relevant reactions carried out in calorimeters [17] and process modelling and optimisation [18]. In the following sections the process of fitting a kinetic model to multivariate absorption data using non-linear regression will be described. First, the structure of the multivariate data will be outlined (Beer–Lambert's law). Next, the computation of the concentration profiles for a given mechanism and rate constants is described. These computations form the core of the fitting algorithm which will be described subsequently. We then present the second-order global analysis of series of data. A collection of applications will be presented and discussed at the end of this tutorial. It should be noted that while absorption data are considered, the techniques described are applicable to any multivariate data that show a linear relationship between signal and concentration. There are several advantages to using multiwavelength data (as compared to selecting single wavelengths): (a) Appropriate analysis results in determination of the pure spectra for all
150
G. Puxty et al. / Chemometrics and Intelligent Laboratory Systems 81 (2006) 149–164
reacting species. This enables structural interpretation even for intermediates that only transiently exist during the reaction. (b) Multivariate data allow the application of a wide range of model-free analyses, from simple factor analysis [19] to indicate the number of reaction species to sophisticated complete analyses based on evolving factor analysis [20] or the alternating least-squares algorithm [21]. These model-free techniques can also be combined with the hard-modelling techniques described in this tutorial to give the benefits of both approaches [22,23]. (c) The need to determine a “good” wavelength to follow the reaction is eliminated. (d) The analysis of multiwavelength data is often significantly more robust. The disadvantages of multiwavelength data include the large number of data that are acquired in a short time and the large number of parameters that need to be fitted. Readily available personal computers with large memory solve the first problem; appropriate algorithms solve the second. The most recent development, and not yet widely accepted, is the globalisation of the analysis of several sets of kinetic data. The investigation of reasonably complex mechanisms requires the measurement of data acquired under different conditions, e.g., variation of initial concentrations. Global analysis of the complete collection of data invariably results in increased robustness, reduced model ambiguity and also significantly less operator input as only one analysis has to be performed. All of the algorithms and analyses described in this tutorial were completed using the commercially available software package Pro-KII [24] or in-house Matlab® [25] software. Other software packages offering some similar features have also been reported in the literature or are available commercially such as
OPKINE [26], KINAJDC (MW) [27,28] and SPECFIT/32™ [29]. 2. Multivariate absorption data and Beer–Lambert's law Light absorption spectroscopy is a common technique used for chemical reaction monitoring, with the ultraviolet-visible (UV–Vis, practical range of 190–700 nm) and near-infrared (near-IR, range of 700–3000 nm) wavelength regions commonly used. While measurements in the mid-IR (3000–12,500 nm) are less commonly used for kinetic studies, the use of this region is increasing with the development of faster scanning instruments. Any of the above wavelength ranges can be used to follow the time-dependent spectral changes during chemical reactions and for the subsequent determination of reaction mechanisms, its kinetic parameters and the spectra of the absorbing reacting species. These spectra can be used evaluate the structure of intermediates and products. Instruments for making absorbance measurements are widely available with bench-top spectrophotometers now being sold covering a large portion of the UV–Vis to near-IR wavelength range in a single device. Generally, experiments are made using stopped-flow techniques for rapid reaction kinetics (milliseconds–minutes), and manually mixed or batch reactor measurements for slower kinetics (minutes–hours). To follow the evolution of a reaction, spectra are acquired as a function of the reaction time. This type of measurement results in data that can be arranged into a two-dimensional matrix which will be named Y. If each measured spectrum is arranged as a row, then the resulting row dimension is the number of measured spectra (nt rows) and the column dimension is the
Fig. 1. Structure of multivariate absorbance data arranged into a matrix Y.
G. Puxty et al. / Chemometrics and Intelligent Laboratory Systems 81 (2006) 149–164
number of measured wavelengths (nλ columns). This arrangement is shown in Fig. 1. According to Beer–Lambert's law the absorbance of a solution at one particular wavelength λ at time t is the sum of the contributions from all absorbing species. Each element in Y represents this sum for a specific time t and wavelength λ. The contribution from each species to the absorbance is linearly proportional to the concentration and the optical path length (the distance travelled through the solution by the incident light beam). If the path length and proportionality constant are expressed by the single constant εi(λ) then Beer–Lambert's law and each element in Y is represented by Eq. (1) (for nc absorbing species). yðt;kÞ ¼ c1 ðtÞe1 ðkÞ þ c2 ðtÞe2 ðkÞ þ : : : þ cnc ðtÞenc ðkÞ nc X ci ðtÞei ðkÞ ¼
ð1Þ
i¼1
The structure of Eq. (1) is such that when spectra are arranged according to the matrix Y, Beer–Lambert's law can be elegantly expressed using matrix notation [4]. Y ¼ CA þ R
ð2Þ
This equation is represented graphically in Fig. 2. The columns of the matrix C contain the concentration profiles ci of the nc absorbing species at the nt measurement times. The rows of the matrix A contain the molar absorptivities εi(λ) for each species at the nλ measured wavelengths. These will be referred to as pure component spectra. Because measurements are never perfect, the matrix Y will contain some measurement noise and as such cannot be perfectly represented by the product of C and A. This difference is captured in the matrix of residuals R, which shares the same dimensions as Y. The fact that each element in Y is an application of Beer–Lambert's law of Eq. (1)
151
is illustrated by the shading of Fig. 2. Each element in Y is the vector product of the corresponding row in C and column in A plus the noise component in the matrix R. Also shown are typical plots of the type of data contained in each matrix. They were produced by simulation of simple first-order kinetics. 3. Calculation of kinetic concentration profiles The central step in the fitting of a kinetic model to multivariate absorbance data is being able to calculate the concentration profiles of all the species involved in a chemical reaction. The concentration profile of a species is its change in concentration with time. The concentration profiles of all species are arranged into column vectors and they correspond to the columns of the matrix C in Fig. 2. According to kinetic theory, the concentration profiles of the species in a reaction mechanism are defined by a system of ordinary differential equations (ODEs) [30]. With knowledge of the initial conditions, ODEs can be integrated to any time point, allowing the calculation of the concentration profiles corresponding to data acquisition times. For a number of simple first- and second-order reactions, it is possible to explicitly integrate the resulting system of ODEs. For example, consider the reaction mechanism of Eq. (3). Shown is the system of ODEs and their integrated form for this second-order mechanism. k2A
2A Y B d½A ¼ −2k2A ½A2 ; dt ½A ¼
d½B ¼ k2A ½A2 dt
½A0 1 þ 2½A0 k2A t
Fig. 2. The matrix Y of multivariate absorbance data expressed using Beer–Lambert's law. A plot of typical data is given below each matrix.
152
½B ¼ ½B0 þ
G. Puxty et al. / Chemometrics and Intelligent Laboratory Systems 81 (2006) 149–164
½A0 −½A 2
ð3Þ
For the majority of multiple step reaction mechanisms, it is not possible to explicitly integrate the resulting systems of ODEs. To overcome this, numerical integration is used. Numerical integration allows an approximation to the explicit solution to be calculated for any system of ODEs, within limits of numerical accuracy and computation time [31]. Numerical integration routines are, at least in basic principle, based on Euler's method [31]. Euler's method uses a form of the Taylor series expansion truncated to the first derivative (see the Appendix for details of the Taylor series expansion). The principle of Euler's method is represented graphically in Fig. 3. Starting with the concentration at time t, the concentration at t + Δt is estimated by moving along the tangent (the derivative dcdti ðtÞ) at t. It can be seen that the accuracy of the approximation is dependent on the magnitude of the increment size Δt and the shape of the function. The greater the curvature and the steeper the tangent, the smaller the increment size that must be used to give an accurate approximation. Eq. (4) formalises the graph and Eq. (5) shows the equation as would be applied to calculate the concentration profiles of species A and B from Eq. (3). ci ðt þ DtÞcci ðtÞ þ
dci ðtÞ Dt dt
½AtþDt c½At −2k2A ½A2t Dt; ½BtþDt c½Bt þ k2A ½A2t Dt
ð4Þ ð5Þ
In practice, Euler's method as described here is not used in modern numerical integration routines due to its lack of accuracy. Some of the modern strategies include using higherorder terms from the Taylor series expansion or calculating the first derivative at multiple points within an increment. Adaptive increment size control is also used to achieve a specified level of accuracy. The most robust of the modern numerical integration routines use the idea of extrapolating from a particular result to the value that would have been obtained if a much smaller, ideally infinitesimally small, increment size were used [31]. A system of ODEs can be considered either non-stiff or stiff depending upon the relative difference in the rate of change of the concentration profiles. If the rate of change of each concentration profile is similar, the system of ODEs is said to
Fig. 4. Reactant and product matrices for the reaction mechanism of Eq. (6).
be non-stiff. For non-stiff systems of ODEs, the fourth-order Runge–Kutta method [31] is the “workhorse” of numerical integration and it was used in this tutorial. It is called fourth order because the first derivative is calculated at four points along the increment of integration, allowing much larger increments and thus dramatically reduced computation times compared to the simple Euler method. Alternatively, if one concentration profile changes rapidly while another does not, the system of ODEs is said to be stiff. Such a situation can occur if either rate constants or concentrations vary by orders of magnitude or the mechanism is a mixture of elementary steps with different reaction orders. In this case, if a traditional method is used the increment size must be reduced to such a small value to yield accurate integration of the rapidly changing concentration profiles, that the computation time becomes excessive for the more slowly changing concentration profiles. A modified approach to carrying out the integration is then required. For stiff systems the Bulirsch–Stoer method using semi-explicit extrapolation [31] was used. To carry out numerical integration, it is first necessary to construct the system of ODEs that represent a given reaction mechanism. This can be done manually, however such an approach is both error-prone and cumbersome for mechanisms of any complexity. Rather, an automated method commonly used to construct ODEs from chemical equations can be adopted [32]. Firstly, the number of species present in the reaction mechanism and the stoichiometry of each reactant and product is determined. Consider the reaction mechanism of Eq. (6). kBA
A þ BYC k2C
2C Y D
Fig. 3. Application of Euler's method to the calculation of a concentration profile.
ð6Þ
Two matrices are then constructed; one containing the reactant stoichiometries Xr, and one containing the product stoichiometries Xp. Each matrix has ns columns, representing all the reacting species in the mechanism (A, B, C and D as distinct from the number of absorbing species nc ≤ ns), and np rows, representing reactions or rate constants (kBA and k2C). Xr contains the stoichiometry of each species as a reactant and Xp contains the stoichiometry of each species as a product. If Xr is subtracted from Xp, the result is the matrix of stoichiometric coefficients X (see Fig. 4).
G. Puxty et al. / Chemometrics and Intelligent Laboratory Systems 81 (2006) 149–164
153
integrating the ODEs in Eq. (8) with [A]0 = 1, [B]0 = 0.8, kBA = 2 and k2C = 1 from 0 to 10 time units in 100 increments is shown in Fig. 5. v1 ¼ kBA ½A1 ½B1 ½C0 ½D0
v2 ¼ k2C ½A0 ½B0 ½C2 ½D0
d½A d½B ¼ ¼ 1v1 þ 0v2 dt dt
Fig. 5. Integrated concentration profiles for the mechanism of Eq. (6) using the ODEs of Eq. (8).
Once Xr and X have been determined, the differential equations can be constructed in a completely generalised way, according to Eq. (7). The expression defining vj is the rate law of the j-th elementary step in the mechanism [30]. vj ¼ kj
ns Y
x
ci rj;i
d½D ¼ 0v1 þ 1v2 dt
ð8Þ
4. Linear and non-linear least-squares regression
where j ¼ 1 to np
i¼1
dci X ¼ xj;i vj dt j¼1
d½C ¼ 1v1 −2v2 dt
4.1. Linear parameters
np
where j ¼ 1 to ns
ð7Þ
The ODEs that result from the application of Eq. (7) to the reaction mechanism of Eq. (6) are given by Eq. (8). The result of
Once the concentration profiles have been calculated, the pure component spectra can be calculated in a single step. This is because they are linear parameters, and as such, there is no need to pass them through the non-linear optimisation routine [4,33].
Fig. 6. Flow diagram of the Newton–Gauss–Levenberg/Marquardt (NGL/M) method.
154
G. Puxty et al. / Chemometrics and Intelligent Laboratory Systems 81 (2006) 149–164
Recall that a multivariate spectroscopic measurement Y can be expressed as the product of C and A plus a matrix of residuals (Eq. (2)). This is a system of linear equations and, for such systems, there is an explicit least-squares solution [34]. Given Y and C, the best estimate for A can be calculated as shown in Eq. (9). C+ is called the pseudoinverse of C. It is necessary to use the pseudo-inverse as C is not a square matrix precluding direct calculation of its inverse. Cþ ¼ ðCT CÞ−1 CT A ¼ Cþ Y
ð9Þ
C+ can be calculated as shown in Eq. (9); this, however, is not recommended as there are numerically superior methods [31]. 4.2. Non-linear parameters The non-linear parameters to be fitted are the rate constants of the reaction mechanism that define the matrix of concentration profiles C. The rate constants are refined so as to minimize the sum of squares of the residuals matrix of Eq. (2) (ssq, the sum over the all elements in R2). The parameters are non-linear because the relationship between the parameters and the residuals is not linear. A vast range of non-linear regression algorithms exists. One of the most commonly chosen methods of non-linear regression is the Newton–Gauss–Levenberg/Marquardt (NGL/M) method [4,35,36], and it will be described in detail here. This method is a gradient method, which means it relies on calculation of the derivative of the function being optimised (the residuals). Methods also exist that do not rely on calculation of the derivative, such as the Simplex method [37] or genetic algorithms [38]. In general, the convergence of gradient methods is superior to other methods if the initial parameter estimates lie in the region of the global optimum. Fig. 6 is a flow diagram showing the NGL/M method. The NGL/M method is able to vary smoothly between an inverse Hessian method and a linear decent method. It evolved in essentially two stages. The inverse Hessian Newton–Gauss method was developed first; this will be the starting point for a description of the NGL/M method. 4.2.1. Newton–Gauss method The first step in any gradient method is to define initial estimates for the non-linear parameters, the rate constants, to be refined. The next step is to evaluate the target function to be minimised, the ssq over the residuals. To do this, the concentration profiles must be calculated according to Section 3 using the initial rate constants. The linear parameters, the pure component spectra, can then be calculated by Eq. (9). This gives the matrices C and A.
The residuals matrix and its ssq can now be calculated by rearranging Eq. (2) to yield Eq. (10). R ¼ Y−CA ssq ¼
nt nk X X i¼1
ð10Þ
2 ri;j
j¼1
Once the ssq has been determined, the next step is to calculate a shift in the non-linear parameters in such a way that the ssq moves towards its minimum value. To do this, it needs to be emphasised that R, and subsequently the ssq, are functions of the non-linear parameters only. By substituting Eq. (9) into Eq. (10), R can be written as a function of C only, which is itself a function of the rate constants. If the non-linear parameters to be fitted, p1 to pnp, are arranged into a vector p = (p1, p2, …, pnp), this relationship is given by Eq. (11). RðpÞ ¼ Y−CCþ Y
ð11Þ
If the initial parameter estimates are given by the vector p0, the Taylor series expansion can be used to estimate R following a small shift in the parameters Δp = (Δp1, Δp2, …, Δpnp) (see the Appendix for details of the Taylor series expansion). If only the first derivative of R is used a linear expression results and is given by Eq. (12). Rðp0 þ DpÞcRðp0 Þ þ þ: : : þ
ARðp0 Þ Dpnp Apnp
ARðp0 Þ ARðp0 Þ Dp1 þ Dp2 Ap1 Ap2 ð12Þ
While this is a crude approximation, the fact that it is a linear expression makes it easy to deal with. The goal is to determine the vector of parameter shifts that moves R(p0 + Δp) towards zero. So, if R(p0 + Δp) is replaced by zero and Eq. (12) is rearranged, Eq. (13) results. Rðp0 Þc−
ARðp0 Þ ARðp0 Þ ARðp0 Þ Dp1 − Dp2 − : : : − Dpnp Ap1 Ap2 Apnp ð13Þ
R(p0) is calculated as described and the partial derivative can be calculated by the method of finite differencing according to Eq. (14). To calculate the partial derivative for parameter pi, pi is shifted by a small amount Δpi and the 0Þ residuals are calculated to yield R(p0 + Δpi). ARðp is then Api calculated by subtracting R(p0 + Δpi) from R(p0) and dividing Δpi (equivalent to calculating the tangent in two dimensions).
ARðp0 Þ Api
ARðp0 Þ Rðp0 þ Dpi Þ−Rðp0 Þ c Api Dpi
ð14Þ
In its current form, it is not clear how Δp can be calculated using Eq. (13) as it is not a single matrix–vector product. One solution to this problem is to vectorise (unfold into long column vectors) the residuals and partial derivative matrices. This expression can then be easily collapsed into a matrix–vector product. This procedure is illustrated graphically in Fig. 7. The
G. Puxty et al. / Chemometrics and Intelligent Laboratory Systems 81 (2006) 149–164
155
the initial estimates are poor, the functional approximation by the Taylor series expansion and the linearization of the problem becomes invalid. This can lead to divergence of the ssq and failure of the algorithm. The modification suggested by Marquardt [36], based on the ideas of Levenberg [35], was to add a certain number, the Marquardt parameter mp, to the diagonal elements of the Hessian matrix, H = JTJ, during the calculation of the parameter shifts, as shown in Eq. (17). H ¼ JT J Dp ¼ −ðH þ mp IÞ−1 JT rðp0 Þ where I ¼ the identity matrix
ð17Þ
When the value of mp is significantly larger than the elements of H, the expression H + mp × I becomes diagonally dominant. 1 This means the inverse is effectively a diagonal matrix, with mp in the diagonal elements. This causes Eq. (17) to collapse to Eq. (18), which has the form of the linear descent method. 1 I JT rðp0 Þ Dp ¼ ð18Þ mp
Fig. 7. Vectorising and collapsing Eq. (13) into a matrix and vector product [39]. 0Þ matrix of vectorised ARðp matrices is called the Jacobian J and Api results in Eq. (15).
rðp0 Þ ¼ −JDp
ð15Þ
This equation is now in a form that has a structure that can be solved by the linear least-squares method for Δp. This solution is given in Eq. (16). Jþ ¼ ðJT JÞ−1 JT Dp ¼ −Jþ rðp0 Þ
ð16Þ
Since both the truncated Taylor series expansion and the partial derivatives used are only approximations, the calculated parameter shifts of Eq. (13) will not be perfect. Thus, an iterative procedure is adopted, where the approximations are successively improved. The calculated shifts are applied to the parameters (p0,j+1 = p0,j + Δp going from the j-th to the j + 1-th iteration) and the process of calculating C through to Δp is repeated. This process is iterated until the relative change in the ssq from one iteration to the next falls below some threshold value (ssqold ≈ ssq). A relative change of less then 10− 4 is generally appropriate.
When the value of mp is small, Eq. (17) reverts to that of the inverse Hessian method. The Marquardt parameter is initially set to zero. There are many strategies to manage the Marquardt parameter, ours is the following. If divergence of the ssq occurs, then the Marquardt parameter is introduced (given a value of 1) and increased (multiplication by 10 per iteration) until the ssq begins to converge. Once the ssq converges thepmagnitude of the ffiffiffiffiffi Marquardt parameter is reduced (division by 10 per iteration) and eventually set to zero when the break criterion is reached. 4.5. Error estimates and correlation coefficients A spin-off from the NGL/M algorithm is that it allows direct estimation of the errors in the non-linear parameters. The inverted Hessian matrix H− 1, without the Marquardt parameter added, is the variance–covariance matrix of the parameters. The diagonal elements contain information on the parameter variances and the off-diagonal elements the covariances. The formula for the standard error σi in parameter pi is given by Eq. (19). qffiffiffiffiffiffiffi ð19Þ ri ¼ rY h−1 i;j h− 1i,i is the i-th diagonal element of the inverted Hessian matrix H− 1 and σY is the standard deviation of the residuals R (see Eq. (20)). rY ¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ssq nt nk−ðnk þ nc nkÞ
ð20Þ
4.4. The Marquardt modification Generally, the Newton–Gauss method, as described so far, converges rapidly, quadratically near the minimum. However, if
The denominator of σY is the number of degrees of freedom. This equals the number of experimental values, the number of elements in R (nt × nλ), minus the number of fitted parameters
156
G. Puxty et al. / Chemometrics and Intelligent Laboratory Systems 81 (2006) 149–164
matrices, to form one large matrix Ctot. Although multiple multivariate measurements have been concatenated, because each measurement obeys the same reaction mechanism, Eq. (2) can still be applied using Ytot and Ctot (see Eq. (21)). Ytot ¼ Ctot A þ Rtot
Fig. 8. Application of second order global analysis using a single matrix of pure component spectra A (global mode).
(np + nc × nλ, that is the number non-linear and linear parameters). If H is normalised to one in the diagonal elements, this yields the correlation coefficients between parameters [40]. The element hi,j of the normalised H is the correlation coefficient between parameter i and j. This is the cosine of the angle between the columns in J for these parameters. The closer the value of the correlation coefficient to one (the cosine of 0°) the more correlated the parameters. High correlation between parameters means that the two parameters cannot be distinguished from one another and one needs to be removed from the fit. 5. Second-order global analysis Up to this point, the analyses discussed have involved a single multivariate measurement. This approach has been dubbed first-order global analysis [1,4]. A powerful extension of this approach is second-order global analysis [2,41]. In second-order global analysis, the procedures already described are applied simultaneously to multiple measurements of the same system made under different conditions. Typically, measurements where the initial concentrations have been varied. Thus, a global model is simultaneously fitted to multiple measurements. This has a number of advantages, including more robust determination of the parameters, determination of parameters that are not defined in any single measurement and the breaking of linear dependencies [2,42]. Also, by fitting a global model to multiple measurements made under different conditions, more confidence can be had in the model. A convenient way of organising a second-order global analysis process is to organise the data sets in such a way that existing matrix based software can be adapted. If nm multivariate measurements have been made, the first step for carrying out second-order global analysis is to concatenate (stack one atop the other) the data matrices to form one large matrix Ytot. It is assumed that each measurement covers the same wavelength range, and thus, each data matrix has the same number of columns. The individual concentration profile matrices are then concatenated in a similar manner to the data
ð21Þ
This equation is represented in Fig. 8. This means calculation of the linear parameters, and fitting of the non-linear parameters, can be carried out in the same manner as previously described by replacing Y with Ytot and C with Ctot. Second-order global analysis is at its most powerful when a single matrix of pure component spectra, A, can be calculated for Ytot and Ctot, as shown in Fig. 8. This is known as the global spectra mode of analysis. It is not always possible however. If there is baseline drift, or some other inconsistency between the measurements such as temperature-dependent spectra, a single A matrix cannot be used. This is because all measurements will no longer share the same pure component spectra. In this situation, a separate A matrix is calculated for each measurement. This arrangement is shown in Fig. 9, and is known as the local spectra mode of analysis. To illustrate the power of second-order global analysis, consider the second-order reaction Eq. (22) where the concentration profiles for a single measurement are linearly dependent. kBA ¼1
AþB Y C
ð22Þ
For a single multivariate measurement of this mechanism, starting with the initial concentrations [A]0 and [B]0, the concentration profiles of the species A, B and C are linearly dependent. The smallest number of linearly independent terms that can be used to represent the data is called the rank of the data. The rank of the data can be estimated by carrying out factor analysis [19] and in this case the rank of C and subsequently Y is two and the calculation of C+ and thus A, Eq. (9), is not possible. However, if a second measurement of the same reaction, having different initial concentrations for A and B, was also made the linear dependence can be broken. By fitting the mechanism
Fig. 9. Application of second order global analysis where an individual Ai is calculated for each measurement (local mode).
G. Puxty et al. / Chemometrics and Intelligent Laboratory Systems 81 (2006) 149–164
using second-order global analysis to both measurements, using a single A matrix, the pure component spectrum of all the species can be resolved. The rank of the resulting combined data matrix Ytot is three as it can no longer be represented by two linearly independent terms. This is illustrated in Fig. 10, which shows simulated concentration profiles and data for the mechanism of Eq. (22), with one set of initial concentrations in part (a) and with two different sets and second-order global analysis in part (b). Both concentration profiles have [A]0 = 0.5 and for the first [B]0 = 0.2 and the second [B]0 = 0.4. A single Gaussian peak was used for the pure component spectrum of each species and the absorbance data was calculated as Y = CA. If one single data file and matrix of concentration profiles are used to calculate the pure component spectra according to Eq. (9) ((a) of Fig. 10), the
157
pure component spectra cannot be resolved. However, if both data files and concentration profiles are used according to second-order global analysis ((b) of Fig. 10), the spectra of all three species can be calculated and are well defined. 6. Examples In this section a number of examples are given where the techniques described have been applied, first to a complex simulated example, and then to three real examples. Each example will be used to highlight particular benefits of the global analysis of multivariate data. This tutorial does not deal directly with how to discriminate between different models. However, in summary, the basic approach adopted is that the residuals are used as an indication of the validity of the
Fig. 10. Calculation of pure component spectra using (a) one single measurement and (b) two measurements and second order global analysis using global spectra.
158 G. Puxty et al. / Chemometrics and Intelligent Laboratory Systems 81 (2006) 149–164
Fig. 11. Reaction scheme used for the chlorination of benzene. The chlorination reagent is omitted.
G. Puxty et al. / Chemometrics and Intelligent Laboratory Systems 81 (2006) 149–164
underlying model. In all cases, the simplest model (in terms of number of parameters) that yields residuals with a noise level and error structure expected from the instrumentation is chosen. The identifiabilty of model parameters is estimated based on the error values and correlation coefficients calculated during the non-linear regression. If the estimated error in a parameter is large or two parameters are highly correlated then this is used to indicate that the model is incorrect or overly complex. Convergence of the model parameters to the same values from different starting guesses is also used as an indication of model correctness and parameter identifiability. For a discussion of this topic, see Vajda and Rabitz [43]. 6.1. Simulated chlorination of benzene As a first example of the power of fitting kinetic models to multivariate data a simulation will be considered. The chlorination of benzene to hexachlorobenzene involves a complex series of reactions. The scheme is represented in Fig. 11. To highlight the complexity of the mechanism, consider the third chlorination step. Three dichloro isomers react to produce three trichloro isomers. 1,2-dichlorobenzene reacts only to the 1,2,3- and 1,2,4-trichlorobenzenes; 1,3-dichlorobenzene reacts to form all three isomers and 1,4-dichlorobenzene reacts only to form 1,2,4-trichlorobenzene. Data were generated for the mechanism by assigning hypothetical rate constants defined by the probability of the reaction given by the number of substitution possibilities that leads to a particular product. For example, for 1,3-dichlorobenzene to 1,2,4-trichlorobenzene there are two possibilities so k1,3–1,2,4 was assigned a value of 2 and for 1,3,5-trichlorobenzene to 1,2,3,5-tetrachlorobenzene there are three possibilities so k1,3,5–1,2,3,5 = 3. This model ignores electronic effects of the substituents which in reality slow down additional substitutions. This is not relevant for the present purpose. There are a total of 13 species and 20 rate constants in the mechanism.
159
Data were generated by modelling the pure spectra of the benzenes as severely overlapping Gaussian peaks. The chlorination reagent was assumed to be non-absorbing. Absorbance data was generated by multiplying simulated concentration profiles by the simulated pure spectra. White noise with a standard deviation of 0.001 (a realistic value for UV–Vis spectrophotometers) was added to the data. An example of the generated data is shown in Fig. 12. Due to the complexity of this system it cannot be resolved from a single measurement, even with perfect noise-free data as serious rank deficiencies result and simplified models can be fit. 10 measurements were generated, each one starting with the unsubstituted benzene and with one of the isomers in the steps involving three isomers (di-, tri- and tetra-chloro isomers). This was necessary to break the rank deficiencies in the concentration profiles. A concentration of 1 with a ten-fold excess of chlorination reagent was used, with 100 spectra generated at equal intervals between 0 and 0.3 time units. All 10 measurements were analysed globally using secondorder global analysis. The determined rate constants were essentially correct (exactly correct to two significant figures). Such an analysis would be practically impossible without the use of multivariate data and second-order global analysis. Due to the severely overlapped spectra, there is no single wavelength that could be chosen to follow a single species. Furthermore, due to the complexity of the mechanism, there is no single measurement or subset of the measurements that could be made to elucidate all the rate constants and spectra due to the severe rank deficiencies of the concentration profiles. 6.2. Complexation of Cu(II) by cyclam using stopped-flow and standard spectrometry In this example the reaction between Cu(II) and the macrocyclic ligand cyclam (1,4,8,11-tetraazacyclotetradecane) in aqueous solution was investigated. The details of the
Fig. 12. Generated data for the benzene example with a 1 : 10 ratio between benzene and the chlorination reagent: (a) calculated concentration profiles and pure spectra (inset) and (b) calculated absorbance data.
160
G. Puxty et al. / Chemometrics and Intelligent Laboratory Systems 81 (2006) 149–164
K1
L þ Hþ W LHþ K2
LHþ þ Hþ W LH2þ 2 K3
þ 3þ LH2þ 2 þ H W LH3 K4
þ 4þ LH3þ 3 þ H W LH4
ð24Þ
The calculated concentration profiles, calculated pH as a function of reaction time (calculated as − log10([H+])), fits of the absorbance data at two wavelengths and the calculated pure component spectra of the absorbing species are shown in Fig. 13. In this case, it was necessary to calculate the pure component spectra in the local mode due baseline shifts between measurements. However, because of the multiwavelength data and the fact that only Cu2+ and CuL2+ absorbed the calculated pure component spectra were well defined for all the measurements. This allowed measurement to measurement comparison as well as comparison with independently determined spectra for verification purposes. However, the real benefit for the study of this system comes from the fitting of a global reaction model to a number of measurements simultaneously. All the parameters associated with this mechanism cannot be defined by a single measurement due to the effect of pH on the reaction velocity. Only through fitting multiple multivariate measurements simultaneously are all the parameters defined. 6.3. Acid-induced dissociation of tris(ethylenediamine) nickel(II) Fig. 13. Fit results for a manually mixed measurement with [Cu2+]0 = 3.9 × −3 M and [H+]0 = 1.0 × 10− 7 M. (a) Calculated 10− 3 M, [LH2+ 2 ]0 = 4.3 × 10 concentration profiles and pH and (b) measured (•••) and calculated (—) absorbance data and calculated pure component spectra (inset).
experiments and results can be found elsewhere [13]. Cyclam is a tetradentate ligand and forms a 1 : 1 complex with Cu(II) in aqueous solution. The reaction rate depends upon the level of protonation of cyclam, and as such is strongly pH-dependent. Multiwavelength kinetic data of the reaction was collected using both stopped-flow and manually mixed measurements with [Cu2+]0 = 3.9 × 10− 3 M and [cyclam]0 = 4.3 × 10− 3 M and different starting acid concentrations (1 × 10− 7 − 0.057 M). The kinetic model given in Eq. (23), coupled to the protonation equilibria given by Eq. (24) (L-cyclam), was fitted simultaneously to all the measurements. A description of the method used to couple the kinetic model and protonation equilibria is beyond the scope of this tutorial. The reader is referred to Maeder et al. [13] for details. However, beyond some modification to the method used to calculate the concentration profiles, the fitting was done as described in this tutorial. Cu kLH
Cu2þ þ LHþ Y CuL2þ þ Hþ Cu
2þ
þ
LH2þ 2
Cu kLH
Y CuL2þ þ 2Hþ 2
ð23Þ
In this example, the acid-induced dissociation of tris (ethylenediamine) nickel(II) (Ni(en)32+) in aqueous solution is considered. This reaction has been studied in detail [44–47], and in the presence of an excess of acid the Ni(en)32+ complex undergoes irreversible dissociation in the three first-order steps given by Eq. (25). Hþ
2þ NiðenÞ2þ kNiðenÞ NiðH2 OÞ2 ðenÞ2 þ en 3 Y 3
Hþ
2þ NiðenÞ2þ þ en kNiðenÞ NiðH2 OÞ4 ðenÞ 2 Y 2
Hþ
2þ NiðenÞ2þ Y kNiðenÞ NiðH2 OÞ6 þ en 3
ð25Þ
The reaction proceeds as the bidentate en ligand becomes protonated and dissociates from the metal centre. Multiwavelength kinetic data of this reaction was measured using a stopped-flow spectrophotometer. Initially a 0.040 M solution of Ni(en)32+ was prepared by dissolving Ni2+ in the presence of an excess of en and 1 M sodium perchlorate. This solution was then mixed with a 1 M solution of perchloric acid in the stopped-flow and the reaction was followed between 430 and 640 nm for 10 s. Experimental details can be found elsewhere [48]. The reaction mechanism of Eq. (25) was fitted
G. Puxty et al. / Chemometrics and Intelligent Laboratory Systems 81 (2006) 149–164
161
wavelength data, such physically impossible spectra may not occur or may not be detected. For example, using 640 nm and swapping the values of kNi(en)3 and kNi(en)2 still results in all positive molar absorptivities and reasonable values. The last advantage is that the calculated pure component spectra allow structural determination of the intermediate species. In this case, the determined spectrum of Ni(en)22+ indicates that it is in the cis form [50]. 6.4. Epoxidation of 2,5-di-tert-butyl-1,4-benzoquinone As a final example the epoxidation of 2,5-di-tert-butyl-1,4benzoquinone (TBB) is considered. The experimental details can be found elsewhere [17]. In summary, TBB and tert-butylhydroperoxide (TBH) were added to a small volume (b 45 mL) stirred and thermostated reactor. The solvent used was a mixture of 1,4-dioxane, ethanol and water. The reaction was initiated by addition of a catalyst, Triton-B (benzyltrimethylammonium hydroxide), in methanol. The two-step epoxidation reaction was then followed by IR spectroscopy using an in situ ATR probe.
Fig. 14. (a) Calculated concentration profiles and (b) measured (•••) and calculated (—) absorbance data for the dissociation of Ni(en)2+ 3 . The time axis has been plotted with a logarithmic scale so initial fast changes are visible.
to the data and Fig. 14 shows the resulting calculated concentration profiles and fits at selected wavelengths. For this example the benefits of using multivariate data are significant and three-fold. Firstly, without knowledge of the spectrum of the intermediate species selection of a single wavelength to follow the reaction is difficult. Choosing a single wavelength below 500 nm results in kNi(en) being poorly defined and choosing a wavelength above 640 nm means kNi(en)3 is poorly defined. The second advantage is related to the fact that the mechanism consists of three first-order consecutive reactions. The parameter values determined for the mechanism are kNi(en)3 = 99.0 ± 0.3 s− 1, kNi(en)2 = 4.10 ± 0.01 s− 1 and kNi(en) = 0.184 ± 0.001 s− 1. Swapping of the values of kNi(en)3 and kNi(en)2 or kNi(en)2 and kNi(en) results in fits of identical quality. This fastslow ambiguity exists for all consecutive first-order reactions and it has been well documented in literature [49]. However, the correct ordering of the rate constants is immediately apparent upon examination of the pure component spectra, as can be seen in Fig. 15. When the wrong ordering is used, severely distorted and often negative spectra will result. When using single
Fig. 15. Calculated pure component spectra for all species with (a) the rate constants in the correct order and (b) with the values of kNi(en)3 and kNi(en)2 swapped.
162
G. Puxty et al. / Chemometrics and Intelligent Laboratory Systems 81 (2006) 149–164
Calorimetry data were measured simultaneously but were not considered here. To highlight the ability of second-order global analysis to break rank deficiencies two measurements made at 30 °C, with different initial concentrations of reagents have been chosen. The reaction mechanism used to fit the data is shown in Fig. 16. The reaction was fitted as irreversible (TBH is in excess) and third order in each step (with the Triton-B catalyst as a reactant and product) to allow inclusion of the catalyst. If the reaction mechanism of Fig. 16 is fitted to a single measurement with TBB, TBH, monoEp and diEp set as absorbing species ([TBB]0 = 0.22 M, [TBH]0 = 2.2 M and [Triton-B]0 = 0.064 M), the calculated pure component spectra of Fig. 17(a) result. The poor outcome is due to linear dependencies amongst the concentration profiles. The rank of the concentration profiles matrix is three, but four species have been set as absorbing. To address this, one of the species could be set as non-absorbing. The resulting calculated pure component spectrum would then represent a mixture of multiple species. A better alternative is to include a second measurement into the analysis, with different initial concentrations ([TBB]0 = 0.24 M, [TBH]0 = 1.5 M and [Triton-B]0 = 0.072 M), and calculate a global pure component spectra matrix according to the method of second-order global analysis. The result of doing this is shown in Fig. 17(b). The linear dependence of the concentration profiles is now broken and the pure component spectrum of all the species can be resolved. 7. Conclusion The methods required to carry out the fitting of a kinetic chemical model to measured multivariate spectroscopic data have been outlined in detail. From the postulation of the model and the derivation of the differential equations, through the numerical integration of the model to yield the concentration profiles and finally the calculation of the pure component spectra and fitting of the model's rate constants to measured data. The benefits of fitting kinetic models to multivariate data have been explained and demonstrated by simulated and real examples. These benefits include: more robust model and parameter determination; calculation of pure component spectra; the breaking of linear dependencies (second-order
Fig. 17. Calculated pure component spectra with TBB, TBH, monoEp and diEp as coloured for (a) calculation with a single measurement and (b) calculation using two measurements with different initial concentrations and global spectra (shaded area shown in inset). NB: The absorbance data has not been divided by the optical path length for visual clarity so the absorptivity is in M− 1 rather than M− 1 cm− 1.
global analysis); and elimination of the need for single wavelength selection and a reduction in the number of measurements required for analysis. Probably the most important point of all is that, particularly with the
Fig. 16. Reaction mechanism for the epoxidation of TBB by TBH with Triton-B as a catalyst.
G. Puxty et al. / Chemometrics and Intelligent Laboratory Systems 81 (2006) 149–164
availability of instrumentation capable of delivering multivariate data, there is no reason not to take advantage of the techniques that exist for their treatment. Furthermore, a number of simulated and real examples have been considered to illustrate the benefits. An additional point to note is that although the techniques described in this tutorial relate directly to absorbance data, they can be applied to any multivariate data that shows a linear response to the concentration of species in the reaction mechanism. For example the techniques can just as easily be applied to fluorescence or time resolved NMR data. It is also straightforward to extend the kinetic models used for nonisothermal conditions using either Arrhenius of Erying based rate constant temperature dependencies [3,15] (although careful attention must be paid to the temperature dependence of spectra). It is also possible to mix measurements of different types (for example, absorbance data covering different wavelength ranges or absorbance and fluorescence data) or use different types of modelling to calculate the concentration profiles (for example, equilibria models [42]).
[4] [5]
[6]
[7]
[8]
[9]
[10]
[11]
Appendix A. The Taylor series expansion The Taylor series expansion [51] is a mathematical means of approximating the value of a function that cannot be explicitly calculated. It is used both for the numerical integration of a kinetic model and during the calculation of parameter shifts during non-linear regression. It relies on knowing the value taken by a function at some point x. The derivative(s) of the function at x is (are) then used to extrapolate the value taken by the function at some other nearby point, x + Δx (Δx is some small increment in x). As an example, consider some function f, for which the value of f(x) is known. The full form of the Taylor series expansion as would be used to calculate its value at f(x + Δx), and its truncated form as used in Euler's method and the Newton–Gauss–Levenberg/Marquardt method, are shown in Eqs. (26) and (27), respectively. 1 df ðxÞ 1 d2 f ðxÞ ðDxÞ þ ðDxÞ2 1! dx 2! d2 x 1 dn f ðxÞ ðDxÞn þ: : : þ n! dn x
[3]
f ðx þ DxÞ ¼ f ðxÞ þ
[12]
[13]
[14]
[15]
[16]
[17]
ð26Þ [18]
f ðx þ DxÞ ¼ f ðxÞ þ
1 df ðxÞ ðDxÞ 1! dx
ð27Þ
[19] [20]
For practical reasons, generally only the first or second derivative of the function is used. However, the higher the order of derivative that is used, the greater the accuracy with which the prediction of f(x + Δx) is made. References [1] J.R. Knutson, J.M. Beechem, L. Brand, Simultaneous analysis of multiple fluorescence decay curves: a global approach, Chem. Phys. Lett. 102 (1983) 501–507. [2] P. Bugnon, J.C. Chottard, J.L. Jestin, B. Jung, G. Laurenczy, M. Maeder, A.E. Merbach, A.D. Zuberbühler, Second-order globalisation for the
[21] [22]
[23]
[24]
163
determination of activation parameters in kinetics, Anal. Chim. Acta 298 (1994) 193–201. M. Maeder, K.J. Molloy, M.M. Schumacher, Analysis of non-isothermal kinetic measurements, Anal. Chim. Acta 337 (1997) 73–81. M. Maeder, A.D. Zuberbühler, Nonlinear least-squares fitting of multivariate absorption data, Anal. Chem. 62 (1990) 2220–2224. S. Bijlsma, H.F.M. Boelens, H.C.J. Hoefsloot, A.K. Smilde, Constrained least squares methods for estimating reaction rate constants from spectroscopic data, J. Chemom. 16 (2002) 28–40. S. Bijlsma, H.F.M. Boelens, A.K. Smilde, Determination of rate constants in second-order kinetics using UV–Visible spectroscopy, Appl. Spectrosc. 55 (2001) 77–83. S. Bijlsma, A.K. Smilde, Estimating reaction rate constants: comparison between traditional curve fitting and curve resolution, Anal. Chim. Acta 419 (2000) 197–207. E. Furusjö, L.-G. Danielsson, A method for the determination of reaction mechanisms and rate constants from two-way spectroscopic data, Anal. Chim. Acta 373 (1998) 83–94. E. Furusjö, O. Svenssen, L.-G. Danielsson, Estimation of kinetic parameters from non-isothermal batch experiments monitored by in situ vibrational spectroscopy, Chemom. Intell. Lab. Syst. 66 (2003) 1–14. P.G. Lye, G.A. Lawrance, M. Maeder, Formation kinetics of pendant arm polyamine macrocycles with copper(II), J. Chem. Soc., Dalton Trans. 16 (2001) 2376–2382. D. Chatterjee, M.S.A. Hamza, M.M. Shoukry, A. Mitra, S. Deshmukh, R.v. Eldik, Interaction of [RuIII(edta)(H2O)]- with amino acids in aqueous solution. Equilibrium, kinetic and protease inhibition studies, J. Chem. Soc., Dalton Trans. (2003) 203–209. S. Yamada, T. Kido, M. Tanaka, Enhanced reactivity of nickel(II) complexes involving multidentate ligands in their complexation, Inorg. Chem. 23 (1984) 2990–2993. M. Maeder, Y.-M. Neuhold, G. Puxty, P. King, Analysis of reactions in aqueous solution at non-constant pH: no more buffers? Phys. Chem. Chem. Phys. 5 (2003) 2836–2841. L. Balland, N. Mouhab, J.-M. Cosmao, L. Estel, Kinetic parameter estimation of solvent-free reactions: application to esterification of acetic anhydride by methanol, Chem. Eng. Process. 41 (2002) 395–402. P. Gemperline, G. Puxty, M. Maeder, D. Walker, F. Tarczynski, M. Bosserman, Calibration-free estimates of batch process yields and detection of process upsets using in situ spectroscopic measurements and nonisothermal kinetic models: 4-(dimethylamino)pyridine-catalyzed esterification of butanol, Anal. Chem. 76 (2004) 2575–2582. Y.-S. Duh, C.-C. Hsu, C.-S. Kao, S.W. Yu, Applications of reaction calorimetry in reaction kinetics and thermal hazard evaluation, Thermochim. Acta 285 (1996) 67–79. A. Zogg, U. Fischer, K. Hungerbühler, Identification of kinetic and thermodynamic reaction parameters from online calorimetric and IR-ATR data using a new combined evaluation algorithm, Chem. Eng. Sci. 59 (2004) 5795–5806. G. Donnati, R. Paludeto, Batch and semibatch catalytic reactors (from theory to practice), Catal. Today 52 (1999) 183–195. E.R. Malinowski, Factor Analysis in Chemistry, John Wiley & Sons, Inc., New York, 1991. M. Maeder, A.D. Zuberbühler, The Resolution of overlapping peaks by evolving factor analysis, Anal. Chim. Acta 181 (1986) 287–291. R. Tauler, Multivariate curve resolution applied to second order data, Chemom. Intell. Lab. Syst. 30 (1995) 133–146. A. de Juan, M. Maeder, M. Martinez, R. Tauler, Combining hard- and softmodelling to solve kinetic problems, Chemom. Intell. Lab. Syst. 54 (2000) 123–141. A. de Juan, M. Maeder, M. Martinez, R. Tauler, Application of a novel resolution approach combining soft- and hard-modelling features to investigate temperature-dependent kinetic processes, Anal. Chim. Acta 442 (2001) 337–350. G. Puxty, M. Maeder, Y.-M. Neuhold, P. King, Pro-Kineticist II, Applied Photophysics Ltd, Leatherhead, Surrey, 2001 http://www.photophysics. com.
164
G. Puxty et al. / Chemometrics and Intelligent Laboratory Systems 81 (2006) 149–164
[25] MATLAB 7.1. The Mathworks Inc., Natick, MA, 2005, http://www. mathworks.com. [26] J.J.B. Baeza, G.R. Ramos, F.P. Pla, R.V. Molina, Multicomponent analysis using OPKINE, a program for the nonlinear treatment of kinetic problems, Analyst 115 (1990) 721–724. [27] M.M. Canedo, J.L. Gonzalez-Hernandez, A new computational application of the AGDC algorithm for kinetic resolution of multicomponent mixtures (static and dynamic), Chemom. Intell. Lab. Syst. 66 (2003) 63–78. [28] J.L. Gonzalez, M.M. Canedo, C. Grande, Optimization of kinetic parameters. Multipurpose KINAGDC(MW) non-linear regression program, Chemom. Intell. Lab. Syst. 39 (1997) 77–84. [29] SPECFIT/32. Spectrum Software Associates, Marlborough, MA, 2004, http://www.bio-logic.info/rapid-kinetics/specfit.html. [30] J.H. Espensen, Chemical Kinetics and Reaction Mechanisms, McGrawHill, New York, 1981. [31] W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing, Cambridge University Press, Cambridge, 1992. [32] R. Dyson, M. Maeder, G. Puxty, Y.-M. Neuhold, Simulation of complex kinetics, Inorg. React. Mech. 5 (2003) 39–46. [33] A.D. Zuberbühler, T.A. Kaden, Handling of electronic absorption spectra with a desk-top computer: II. Calculation of stability constants from spectrophotometric titrations, Talanta 26 (1979) 1111–1118. [34] D.C. Lay, Linear Algebra and Its Applications, Addison-Wesley Publishing Company, New York, USA, 2000. [35] K.Q. Levenberg, A method for the solution of certain problems in least squares, Appl. Math. 2 (1944) 164–168. [36] D.W. Marquardt, An algorithm for least-squares estimation of nonlinear parameters, J. Soc. Ind. Appl. Math. 11 (1963) 431–441. [37] J.C. Lagarias, J.A. Reeds, M.H. Wright, P.E. Wright, Convergence properties of the Nelder–Mead simplex method in low dimensions, SIAM J. Optim. 9 (1998) 112–147. [38] R. Leardi, Genetic algorithms in chemometrics and chemistry: a review, J. Chemom. 15 (2001) 559–569. [39] G. Puxty, M. Maeder, P. Gemperline, R.R. Rhinehart, A. Samir, S. Moore, Modeling of batch reactions with in-situ spectroscopic measurements and calorimetry. J. Chemom. Accepted Copyright© 2005 John Wiley & Sons Ltd.
[40] J. Lumsden, Elementary Statistical Method, University of Western Australia Press, Nedlands, 1971. [41] R. Dyson, M. Maeder, Y.-M. Neuhold, G. Puxty, Analyses of three-way data from equilibrium and kinetic investigations, Anal. Chim. Acta 490 (2003) 99–108. [42] R.M. Dyson, S. Kaderli, G.A. Lawrance, M. Maeder, A.D. Zuberbühler, Second order global analysis – the evaluation of series of spectrophotometric titrations for improved determination of equilibrium constants, Anal. Chim. Acta 353 (1997) 381–393. [43] S. Vajda, H. Rabitz, Identifiability and distinguishability of general reaction systems, J. Phys. Chem. 98 (1994) 5265–5271. [44] A.K.S. Ahmed, R.G. Wilkins, Factors influencing the rates of dissociation of metal complexes: Part I. The mechanism of dissociation of mono (ethylenediamine)–nickel(II) ion, J. Chem. Soc. (1959) 3700–3708. [45] A.K.S. Ahmed, R.G. Wilkins, Factors influencing the rates of dissociation of metal complexes: Part III. The dissociation of bis(ethylenediamine)– and tris(ethylenediamine)–nickel complexes and of the racemic 2,3diaminobutane analogues, J. Chem. Soc. (1960) 2901–2906. [46] A.K.S. Ahmed, R.G. Wilkins, Factors influencing the rates of dissociation of metal complexes: Part II. The dissociation of complexes of nickel with various diamines, J. Chem. Soc. (1960) 2895–2900. [47] K.J. Wannowius, K. Krimm, H. Elias, Visible spectra of the complex 2+ 2+ species Ni(en)2+ 3 , Ni(en)2(H2O)2 and Ni(en)(H2O)4 from kinetic studies, Inorg. Chim. Acta 127 (1987) L43–L44. [48] G. Puxty, Novel Chemometric Methods for the Hard Modelling of Multivariate Spectroscopic Data, The University of Newcastle, Newcastle, Australia, 2004. [49] W.G. Jackson, J.M. Harrowfield, P.D. Vowles, Consecutive, irreversible first-order reactions. Ambiguities and practical aspects of kinetic analyses, Int. J. Chem. Kinet. 9 (1977) 535–548. [50] T.J. Kemp, P. Moore, G.R. Quick, Kinetic and spectroscopic studies with a rapid-scanning Spectrometer: Part 1. Transient intermediates in the dissociation of nickel(II) polyamine complexes, J. Chem. Soc. 92 (1979) 1377–1383. [51] C.H. Edwards, D.E. Penney, Calculus with Analytical Geometry, Prentice Hall International Inc., New Jersey, 1994.
