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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-freetechniques 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 setsof 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 com-

monly 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.

150 G. Puxty et al. / Chemometrics and Intelligent Laboratory Systems 81 (2006) 149 – 164


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number of measured wavelengths (nλ columns). This arrange-

ment 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 c1ðt Þe1ðkÞ þ c2ðt Þe2ðkÞ þ : : : þ cncðt ÞencðkÞ¼

Xnci¼1

ciðt ÞeiðkÞ ð1Þ

The structure of Eq. (1) is such that when spectra arearranged according to the matrix Y, Beer –Lambert's law can be

elegantly expressed using matrix notation [4].

Y ¼ CA þ R ð2ÞThis 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 andA. 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)

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.

2AYk 2A

B

d½Adt ¼

−2k 2A

½A

2

;

d½Bdt ¼

k 2A

½A

2

A½ ¼ ½A01 þ 2½A0k 2At

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.

151G. Puxty et al. / Chemometrics and Intelligent Laboratory Systems 81 (2006) 149 – 164


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B½ ¼ ½B0 þ 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 dciðt Þ

dt ) 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 curvatureand 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).

ciðt þ Dt Þcciðt Þ þ dciðt Þdt

Dt ð4Þ

½At þDt c½At −2k 2A½A2t Dt ; ½Bt þDt c½Bt þ k 2A½A2t Dt 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 higher-

order 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

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 timescompared 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).

A

þ B Y

k AB

C

2CYk 2C

D ð6ÞTwo matrices are then constructed; one containing the

reactant stoichiometries Xr , and one containing the product

stoichiometries X p. 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 (k BA and k 2C). Xr

contains the stoichiometry of each species as a reactant and X pcontains the stoichiometry of each species as a product. If Xr is

subtracted from X p, the result is the matrix of stoichiometric

coefficients X (see Fig. 4).Fig. 3. Application of Euler's method to the calculation of a concentration profile.

Fig. 4. Reactant and product matrices for the reaction mechanism of Eq. (6).



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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 v j is the rate law of

the j -th elementary step in the mechanism [30].

v j ¼ k j Ynsi¼1

c xrj ;ii where j ¼ 1 to np

dci

dt ¼

Xnp j ¼1

x j ;iv j where j ¼ 1 to ns ð7Þ

The ODEs that result from the application of Eq. (7) to thereaction mechanism of Eq. (6) are given by Eq. (8). The result of

integrating the ODEs in Eq. (8) with [A]0= 1, [B]0=0.8, k BA= 2

and k 2C= 1 from 0 to 10 time units in 100 increments is shown in

Fig. 5.

v 1 ¼ k AB ½A1½B1½C0½D0

v 2 ¼ k 2C½A0½B0½C2½D0

d½Adt

¼ d½Bdt

¼ 1v 1 þ 0v 2

d½Cdt

¼ 1v 1−2v 2

d½Ddt

¼ 0v 1 þ 1v 2 ð8Þ

4. Linear and non-linear least-squares regression

4.1. Linear parameters

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. 5. Integrated concentration profiles for the mechanism of Eq. (6) using the

ODEs of Eq. (8).

Fig. 6. Flow diagram of the Newton–Gauss–Levenberg/Marquardt (NGL/M) method.



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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 pseudo-

inverse of C. It is necessary to use the pseudo-inverse as C isnot a square matrix precluding direct calculation of its

inverse.

Cþ ¼ ðCTCÞ−1CT

A ¼ CþY 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

R 2). 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 regres-

sion 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 inverseHessian 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 X

nt

i¼1X

nk

j ¼1r 2

i;

j ð10

ÞOnce 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).

R

ðp

Þ ¼ Y−CCþY

ð11

ÞIf 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 Δ p = (Δ p1, Δ p2, …, Δ pnp) (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).

R ðp0 þ DpÞcR ðp0Þ þAR ðp0Þ

A p1D p1 þAR ðp0Þ

A p2D p2

þ: : : þ AR ðp0ÞA pnp

D pnp 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.

R ðp0Þc−AR ðp0Þ

A p1D p1−

AR ðp0ÞA p2

D p2− : : : −AR ðp0ÞA pnp

D pnp

ð13ÞR (p0) is calculated as described and the partial derivative

AR ðp0ÞA pi

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 Δ pi and the

residuals are calculated to yield R (p0+Δ pi). AR ðp0Þ

A piis then

calculated by subtracting R (p0+Δ pi) from R (p0) and dividing

Δ pi (equivalent to calculating the tangent in two dimensions).

AR ðp0ÞA pi

cR ðp0 þD piÞ−R ðp0Þ

D pið14Þ

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



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matrix of vectorised AR ðp0Þ

A pi

matrices is called the Jacobian J and

results in Eq. (15).

rðp0Þ ¼ −JDp ð15ÞThis 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 ðJTJÞ−1JT

Dp ¼ −Jþrðp0Þ 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, aniterative 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.

4.4. The Marquardt modification

Generally, the Newton–Gauss method, as described so far,

converges rapidly, quadratically near the minimum. However, if

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 theHessian matrix, H = JTJ, during the calculation of the parameter

shifts, as shown in Eq. (17).

H ¼ JTJ

Dp ¼ −ðH þ mp IÞ−1JTrðp0Þ

where I ¼ the identity matrix ð17ÞWhen the value of mp is significantly larger than the elements

of H, the expression H + mp × I becomes diagonally dominant.

This means the inverse is effectively a diagonal matrix, with 1

mpin the diagonal elements. This causes Eq. (17) to collapse to Eq.

(18), which has the form of the linear descent method.

Dp ¼ 1mp

I

JTrðp0Þ ð18Þ

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 the magnitude of the

Marquardt parameter is reduced (division by ffiffiffiffiffi

10p 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).

ri ¼ rY ffiffiffiffiffiffiffi

h−1i; j

q ð19Þ

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 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ssq

nt nk−ðnk þ nc nk Þr

ð20Þ

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

Fig. 7. Vectorising and collapsing Eq. (13) into a matrix and vector product [39].



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(np + nc × nλ, that is the number non-linear and linear para-meters). 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 andthe 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

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 R tot ð21ÞThis 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 measure-ment. 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 concen-

tration profiles for a single measurement are linearly dependent.

A þ B Yk A

B ¼1

C ð22ÞFor 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 termsthat 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. 8. Application of second order global analysis using a single matrix of pure

component spectra A (global mode).

Fig. 9. Application of second order global analysis where an individual A i iscalculated for each measurement (local mode).



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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

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.



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Fig. 11. Reaction scheme used for the chlorination of benzene. The chlorination reagent is omitted.


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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 toindicate 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 discus-

sion 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 k 1,3–1,2,4 was assigned a value of

2 and for 1,3,5-trichlorobenzene to 1,2,3,5-tetrachlorobenzene

there are three possibilities so k 1,3,5–1,2,3,5=3. This modelignores 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.

Data were generated by modelling the pure spectra of the

benzenes as severely overlapping Gaussian peak s. 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. Anexample 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 concentra-

tion 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 second-order 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.



12/16

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). Thekinetic model given in Eq. (23), coupled to the protonation

equilibria given by Eq. (24) (L-cyclam), was fitted simulta-

neously 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.

Cu2þ þ LHþ Yk Cu

LH

CuL2þ þ Hþ

Cu2þ þ LH2þ2 Yk Cu

LH

2 CuL2þ þ 2Hþ ð23Þ

L þ HþW K 1 LHþ

LHþ þ HþW K 2 LH2þ2

LH2þ2 þ

HþW K 3

LH3þ3

LH3þ3 þ HþW K 4

LH4þ4 ð24ÞThe 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 multiwave-

length 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 deter-

mined 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 simulta-

neously. 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 para-

meters defined.

6.3. Acid-induced dissociation of tris(ethylenediamine)

nickel(II)

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).

NiðenÞ2þ3 YHþ

k NiðenÞ3 NiðH2OÞ2ðenÞ2þ2 þ en

Ni

ðen

Þ2þ2 Y

Hþ

k Niðen

Þ2 Ni

ðH2O

Þ4

ðen

Þ2þ

þ en

NiðenÞ2þYHþ

k NiðenÞ3 NiðH2OÞ2þ6 þ en 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

Fig. 13. Fit results for a manually mixed measurement with [Cu2+]0=3.9×

10−3 M, [LH22+]0=4.3×10

−3 M and [H+]0=1.0×10−7 M. (a) Calculated

concentration profiles and pH and (b) measured (•••) and calculated (—)

absorbance data and calculated pure component spectra (inset).



13/16

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 k Ni(en) being poorly definedand choosing a wavelength above 640 nm means k Ni(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 k Ni(en)3 =99.0±0.3 s−1, k Ni(en)2 =4.10± 0.01 s

−1 and k Ni(en)=

0.184± 0.001 s−1. Swapping of the values of k Ni(en)3 and k Ni(en)2or k Ni(en)2 and k Ni(en) results in fits of identical quality. This fast-

slow 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

wavelength data, such physically impossible spectra may not

occur or may not be detected. For example, using 640 nm and

swapping the values of k Ni(en)3 and k Ni(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 cisform [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-1,4-

benzoquinone (TBB) is considered. The experimental details

can be found elsewhere [17]. In summary, TBB and tert -butyl-

hydroperoxide (TBH) were added to a small volume (b45 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 wasthen followed by IR spectroscopy using an in situ ATR probe.

Fig. 15. Calculated pure component spectra for all species with (a) the rate

constants in the correct order and (b) with the values of k Ni(en)3 and k Ni(en)2swapped.

Fig. 14. (a) Calculated concentration profiles and (b) measured (•••) and

calculated (—) absorbance data for the dissociation of Ni(en)32+. The time axis

has been plotted with a logarithmic scale so initial fast changes are visible.



14/16

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) andthird 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 multiplespecies. 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 datahave 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

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. 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.

Fig. 16. Reaction mechanism for the epoxidation of TBB by TBH with Triton-B as a catalyst.



15/16

availability of instrumentation capable of delivering multi-

variate data, there is no reason not to take advantage of the

techniques that exist for their treatment. Furthermore, a

n umber o f simulated and real examp les h ave b een

considered to illustrate the benefits.

An additional point to note is that although the techniques

described in this tutorial relate directly to absorbance data, theycan 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 non-

isothermal 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]).

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 +Δ x (Δ x 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

+Δ x), 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.

f ð x þ D xÞ ¼ f ð xÞ þ 11!

d f ð xÞd x

ðD xÞ þ 12!

d2 f ð xÞd2 x

ðD xÞ2

þ: : : þ 1n!

dn f ð xÞdn x

ðD xÞn ð26Þ

f ð x þ D xÞ ¼ f ð xÞ þ 11!

d f ð xÞd x

ðD xÞ ð27Þ

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 +Δ x) is made.

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