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This is to certify that the

dissertation entitled

CHEMOMETRIC DATA PROCESSING TECHNIQUES FOR
KINETIC—SPECTROPHOTOMETRIC DETERMINATIONS

presented by

Thomas Francis Cullen

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Chemistry

 

 

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Date “(z/l /E’:’

MSU i: an Affirmative Action/Equal Opportunity Institution 0-12771

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LIBRARY
Michigan State
University

 

 

 

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W81 @2200?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

CHEMOMETRIC DATA PROCESSING TECHNIQUES FOR
KINETIC-SPECTROPHOTOMETRIC DETERMINATIONS

By

Thomas Francis Cullen

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Chemistry

1999

ABSTRACT

CHEMOMETRIC DATA PROCESSING TECHNIQUES FOR KINETIC-
SPECTROPHOTOMETRIC DETERMINATIONS

By

Thomas Francis Cullen

A variety of chemometric data processing techniques were applied to
kinetic-spectrophotometric data. In order to a acquire these data, a new data
acquisition system was designed and built. This redesign of the existing system
involved the fabrication of a new optical path for a stopped-flow apparatus as well
as the creation of a new computerized interface for a diode array detection system.

A series of simulated experiments were performed. In these simulation
studies the effect of an array of experimental variables on the accuracy of a
kinetic-spectrophotometric determination of a two component mixture were
explored. Methods for quantifying the amount of kinetic and spectral information
present in kinetic-spectrophotometric data are discussed. The kinetic and spectral
angles were introduced and shown to be good measures of the quantity of
information available in each dimension. Kinetic and spectral net analyte signals
were also developed and were shown to be good predictors of the relative
accuracy with which analytes can be determined. The kinetic angle was shown to

have several contributing factors. The ratio of the analyte rate constants is the

largest contributor, but the fraction of the slower reaction for which data is
acquired and the number of spectra acquired also impact the kinetic angle.

Ga(III) and Ni(H) were the subjects of a kinetic-spectrophotometric
determination. In the studies discussed, the effect of the kinetic angle was explored
experimentally; the determination was carried out at two different values of
solution pH where the ratios of the reaction rate constants are different. A
comparison was drawn between the various chemometric algorithms; continuum
regression and multiway partial least squares regression proved most promising.

The effect of kinetic non-linearity on kinetic-spectrophotometric
determinations was examined in both simulations and in experimental
determinations. In general, it was found that most of the techniques used were
fairly tolerant of nonlinear kinetics, though they began to fail in highly nonlinear
systems. Again, the various chemometric algorithms were compared, and similar
results to those found in the initial studies were obtained. Continuum regression
and nPLS proved best suited to handling the nonlinear kinetic data. The degree of
kinetic non-linearity was measured as an angle from linearity. This angle was
shown to be a good predictor of the accuracy with which a determination could be
performed.

Cu(II) and Zn(H) were determined in a real sample with environmental
relevance. The accuracy of the determination was within acceptable limits for both
analytes. The kinetic and spectral angles and net analyte signals were used to

explain the relative accuracy with which the analytes were determined.

To Melisa and Ceara, and to Mom and Dad

With Love

iv

ACKNOWLEDGEMENTS

I use not only all the brains I have, but
all I can borrow.

--Woodrow Wilson

There are many people who deserve my thanks and gratitude. Without their
help this dissertation would never have been written. First, I should express my
heartfelt thanks to Dr. Stan Crouch, under whose guidance the work described in this
document was performed. Stan is an exceptional teacher and I’ve learned more in idle
conversation with him than I could ever glean from a thousand textbooks. His
scientific guidance and personal support were invaluable.

Others at Michigan State have played large parts in whatever success I’ve
enjoyed. I’m not sure I would have survived my first year if not for the friendship and
camaraderie of Dale Lecaptain and Wendy Flory. Together we weathered the storms
that are common to the first years of graduate school, and I thank them for their
support and friendship. Others whose friendship during those early years is much
appreciated include Eric Meyer, Lykourgos Iordanidis, Rhonda Patschke and Matt
Prater (who’s also appreciated for dragging me back onto the ice after several years of
hockey-less existence).

Dana Spence, too, played a large part in my success during those early years.
He took a first-year student under his wing and showed him how things worked in the

Crouch group, and in the department. The time and sanity saving tips and tricks he

passed on were invaluable, and his companionship during those late nights of data
acquisition were much appreciated. We learned much of each other’s projects; almost
all I know of flow injection I learned from him.

In the latter years of my career I’ve come to be good friends with several
members of the McGuffin group. John Goodpaster and Pete Krouskop are especially
appreciated for being supportive during some particularly rough stretches. Sam
Howerton is a newcomer who is appreciated for his friendship and for many
stimulating intellectual discussions.

Much of the work described in this dissertation was completed with the skilled
help of the department’s support staff. Russ Geyer, Sam Jackson, and Glenn Wesley
of the machine shop are thanked for their assistance with the repair and refurbishing
of the stopped-flow. The efforts of Ron Haas, Scott Sanderson and especially John
Rugis were exceedingly valuable during the design and construction of the diode
array interface, and they have earned my gratitude. Scott Bankroff and Manfred
Langer lent their glassblowing expertise to the project, and so are thanked for their
help. The assistance of Tom Atkinson and Paul Reed with a host of computer-related
difficulties is likewise appreciated.

In the course of my work I have on several occasions had the opportunity to
interact with scientists whose understanding of chemometrics far outstrips my own.
Dr. Pete Wentzell’s friendly advice on the subjects of both chemometrics and

graduate school was greatly appreciated. Barry Wise and Neal Gallagher provided

vi

tireless assistance to a novice chemometrician, and are owed my humble thanks.
Likewise, the patient efforts of Rasmus Bro and Claus Andersson were appreciated.

The assistance of Anthony Cochran with the acquisition of the equilibrium
spectrophotometric data described in chapter four is gratefully acknowledged.

Many people’s contribution to my success came well before I ever set foot
onto MSU’s beautiful campus. The love and support showed by my parents, Frank
and Eileen Cullen, has sustained me through many trials and tribulations. They
instilled in me a love of learning and provided me with the opportunity to pursue my
goals. They did this, in part, by allowing me to attend Fenwick High School where
Marcus McKinley and Ramzi Farran introduced me to chemistry and so set me on a
path that would impact the rest of my life. It was at Fenwick, too, that John Quinn and
Fr. John Gambro, O.P. were instrumental as I learned how to learn.

At Illinois Benedictine College my love of chemistry was confirmed. The
efforts of Mike Winkler, Dave Sonnenberger and Wayne Wesolowski formed me as a
chemist and provided me with the skills necessary for success in graduate school. The
Scholars program formed me as a person and endowed me with a global perspective.

Finally, my most heartfelt and sincere thanks must go to my wife, Melisa, and
to our daughter Ceara. Without their constant love and support this dissertation could
not have been written. Their understanding and acceptance of the long hours

necessary are greatly appreciated, and I offer them my love and gratitude.

vii

TABLE OF CONTENTS

LIST OF TABLES ......................................................................................... x
LIST OF FIGURES ....................................................................................... xi
LIST OF ABBREVIATIONS ............................................................................ xv
1 INTRODUCTION ....................................................................................... 1

1 .1 . CHEMOMETRICS ................................................................................ 2
1.1.1. Nomenclature and Conventions .............................................................. 2
1.1.2. Multivariate calibration ........................................................................ 2
1.1.3. Factor Analysis ................................................................................. 13
1.1.4. Artificial Neural Networks .................................................................... 14
1.1.5. Multiway Methods ............................................................................. 15

1.2. Kinetic Determinations ............................................................................. 16
1.2.1. Kinetic Determinations using Hard Modeling Techniques ............................... 17
1.2.2. Kinetic Determinations using Soft Modeling Techniques. . . . . . . 19

1.3. REFERENCES ....................................................................................... 23

2 INSTRUMENT DESIGN AND CHARACTERIZATION ......................................... 34

2.1. DESIGN OF DIODE ARRAY INTERFACE ................................................... 34

2.2. REDESIGN OF STOPPED-FLOW OBSERVATION CELL AND OPTICS. . . . . .. 39

2.3. CHARACTERIZATION OF DATA ACQUISITION SYSTEM ............................ 40
2.3.1. Determination of Delay Time and Maximum Acquisition Rate........................ .. 40
2.3.2. Determination of Signal/Noise Ratio ........................................................ 42
2.3.3. Determination of Linear Range ............................................................... 46

2.4. REFERENCES ...................................................................................... 48

3 INITIAL SHVIULATION STUDIES: STUDY OF THE EFFECT OF 49

EXPERIMENTAL VARIABLES .....................................................................

3.1. EXPERIMENTAL ................................................................................. 50
3.1.1. Generation of Simulated Data ................................................................. 50
3.1.2. Data Processing ................................................................................. 50

3.2. METHODS FOR QUANTIFYING KINETIC AND SPECTRAL

DIFFERENCES FOR TWO COMPONENTS .................................................... 51
3.2.1. Methods for Quantifying Kinetic Differences ............................................. 51
3.2.2. Methods for Quantifying Spectral Differences ............................................. 56

3.3. EFFECT OF EXPERIMENTAL VARIABLES ............................................... 58
3.3.1. Effect of Kinetic and Spectral Angles ....................................................... 60
3.3.2. Effect of Number of Data Points Acquired (Time Points) ................................. 68
3.3.3. Effect of Instrumental Noise .................................................................. 68
3.3.4. Effect of Rate Constant Fluctuations ......................................................... 73

3.4. EXPLORATION OF EFFECT OF EXPERIMENTAL VARIABLES ON
SIMULATIONS OF THE Ga(III)-Ni(II) SPECTRAL SYSTEM . 76
3.4.1. Effect of Kinetic Angle ........................................................................ 77
3.5. References ........................................................................................... 8O

viii

4 DETERMINATION OF GALLIUM(III) AND NICKEL (II) ..................................... 81

4.1 EXPERIMENTAL .................................................................................. 82
4.1.]. Solution Preparation ............................................................................ 82
4.1.2. Spectrophotometric (Equilibrium) Data Collection ....................................... 86
4.1.3. Kinetic-Spectrophotometric Data Collection ............................................... 87
4.1.4. Data Processing ................................................................................. 87

4.2. DETERMINATION OF RATE CONSTANTS AND ABSORPTION

SPECTRA ............................................................................................ 88

4.3. RESULTS OF THE DETERMINATION OF GALLIUM(III) AND

NICKEL (H) .......................................................................................... 89
4.3.1. Spectrophotometric (Equilibrium) Determination of Gallium (III)
and Nickel (11) ................................................................................... 89

4.3.2. Kinetic Determination of Gallium (III) and Nickel (II) 93
4.3.3. Kinetic-Spectrophotometric Detemiination of Gallium (III)

and Nickel (II) .................................................................................. 94
4.4. REFERENCES ...................................................................................... 97
5 KINETIC-SPECTROPHOTOMETRIC DETERMINATIONS IN SYSTEMS WITH
NONLINEAR KINETICS .............................................................................. 99
5.1. EXPERIMENTAL .................................................................................. 102
5.1.1. Simulations ...................................................................................... 102
5.1.2. Solution Preparation ............................................................................ 105
5.1.3. Kinetic-Spectrophotometric Data Collection .............................................. 106
5.1.4. Data Processing ................................................................................ 107
5.2. SIMULATION STUDIES INVOLVING SYSTEMS WITH NONLINEAR
KINETICS ........................................................................................... 107
5.2.1. Systems Where The Reagent Does Not Absorb ............................................ 107
5.2.2. Systems Where The Reagent Does Absorb ................................................. 112
5.3. DETERMINATION OF GALLIUM(III) AND NICKEL(II) IN A SYSTEM
WITH NONLINEAR KINETICS ................................................................ 114
5.4. REFERENCES ...................................................................................... l 18
6 DETERMINATION OF Zn(II) AND Cu(II) IN A DRINKING WATER
SAMPLE .................................................................................................. 119
6.1. EXPERIMENTAL .................................................................................. 120
6.1.]. Solution Preparation ........................................................................... 120
6.1.2. Kinetic-Spectrophotometric Data Collection ............................................... 122
6.1.3. Data Processing ................................................................................. 122
6.2. DETERMINATION OF ZINC(II) AND COPPER(II) ........................................ 122
6.3. REFERENCES ...................................................................................... 127
7 CONCLUSIONS AND FUTURE PERSPECTIVES ................................................ 130
7.1. CONCLUSIONS AND SUMMARY ............................................................ 130
7.2. REFLECTIONS ON FUTURE DIRECTIONS ................................................ 133
7.3. REFERENCES ..................................................................................... 136
APPENDD(: MATLAB CODE FOR GENERATING SIMULATED
KINETIC-SPECTROPHOTOMETRIC DATA .................................... 138
MULGEN_A ............................................................................................. 140
KINETIC .................................................................................................. 146
VARY_K ................................................................................................. 147

ix

LIST OF TABLES

Table 4-1
Single component calibration and unknown sets for the
equilibrium determination of Ni(II)

Table 4-2
Single component calibration and unknown sets for the
equilibrium determination of Ga(III)

Table 4-3
Spectrophotometric (equilibrium) determination of N i(II)

Table 4-4
Spectrophotometric (equilibrium) determination of Ga(III)

Table 4-5
Multicomponent spectrophotometric (equilibrium)
determination of Ni(II) and Ga(III)

Table 4—6
Multicomponent kinetic determination of Ni(II) and Ga(III)

Table 4-7
Multicomponent kinetic-spectrophotometric determination of Ni(lI) and Ga(III)

Table 5-1
Kinetic-spectrophotometric determination of Ga(III) and Ni(II) with two different
values of reagent excess

Table 6-1
Certified concentrations (ppb) of metal cations in a drinking water sample

Table 6-2
Concentrations of analytes and interferents in the unknown samples in M units

Table 6-3
Results of the determination of Cu(II) and Zn(H) in a series of unknown samples

83

83

89

9O

91

94

95

115

120

123

123

LIST OF FIGURES

Figure 1-1: Multivariate calibration as a “black box”.
Figure 1-2: Decomposition of data into principal components.

Figure 1-3: Determination of the number of principal components using a scree
graph.

Figure 1-4: Relationship between principal component regression, partial least
squares regression, multiple linear regression and continuum regression.

Figure 1-5: Schematic diagram of a neuron.
Figure 2-1: Timing diagram of signals sent to and from the diode array.

Figure 2-2: Schematic diagram of the peak track-and-hold circuit. 112V power
connections to the LT1363 operational amplifiers and +5V power connections to
the 741.8123 monostable multivibrators are omitted in the interest of clarity.

Figure 2-3: Schematic of the observation cell.

Figure 2—4: Plot showing dependence of relative error in absorbance to
absorbance.

Figure 2-5: Dependence of S/N on absorbance.
Figure 2-6: Dependence of SIN on intensifier gain.
Figure 2-7: Dependence of S/N on integration time.

Figure 2-8: Linearity of measured absorbance with the known absorbance of a
series of neutral density filters. The linear regression shown was calculated from
all absorbances up to and including one absorbance unit.

Figure 3-1: Kinetic angle as a function of rate constant ratio. The fraction of the
slower reaction observed was held constant at 90%, and the number of data points
was held at 100.

Figure 3-2: Kinetic net analyte signal for the slower (squares) and the faster
(circles) reactants in a two-component mixture. The fraction of the slower reaction

observed was held constant at 90%, and the number of data points was held at 100.

Figure 3-3: Synthetic spectra used for simulation studies. Spectral angles range
between 77.7 and 0.

xi

l3

14
35
37

39
43

45
46
47

54

55

57

Figure 3-4: Spectral net analyte signal as a function of spectral angle. The spectra
used to generate this plot are shown in Figure 3-3.

Figure 3-5: Concentrations of calibration (stars) and unknown (open circles)
samples.

Figure 3-6: Relative standard error of a PLS prediction as a function of the kinetic
and spectral angles.

Figure 3-7: Relative standard error of a PCR prediction as a function of the kinetic
and spectral angles.

Figure 3-8: Relative standard error of a CR prediction as a function of the kinetic
and spectral angles.

Figure 3-9: Relative standard error of an nPLS prediction as a function of the
kinetic and spectral angles.

Figure 3-10: Relative stande error of an MLR prediction as a function of the
kinetic and spectral angles.

Figure 3-11: Relative standard error of a PARAFAC prediction as a function of
the kinetic and spectral angles.

Figure 3-12: Kinetic angle as a function of rate constant ratio and fraction of the
slower reaction observed.

Figure 3-13: Relative standard error of a CR prediction as a function of the kinetic
angle and instrumental noise.

Figure 3-14: Relative standard error of an nPLS prediction as a function of the
kinetic angle and instrumental noise.

Figure 3-15: Relative standard error of a CR prediction as a function of the
spectral angle and instrumental noise.

Figure 3-16: Relative standard error of an nPLS prediction as a function of the
spectral angle and instrumental noise.

Figure 3-17: Relative standard error of a CR prediction as a function of the kinetic
angle and rate constant fluctuation.

Figure 3-18: Relative standard error of an nPLS prediction as a function of the
kinetic angle and rate constant fluctuation.

xii

58

59

62

62

63

63

65

67

69

70

71

72

74

75

Figure 3-19: Absorption spectra of PAR and its Ni(H) and Ga(III) complexes used
for simulation studies.

Figure 3-20: Relative standard error of a CR prediction as a function of the rate
constant ratio and the fraction of the slower reaction observed. The Ga(III)/Ni(H)
spectra shown in Figure 3-19 were used. The spectral angle was 6.3.

Figure 3-21: Relative standard error of a CR prediction as a function of the rate
constant ratio and the fraction of the slower reaction observed. The Ga(III)/Ni(H)
spectra shown in Figure 3-19 were used. The spectral angle was 6.3.

Figure 4-1: Acid dissociation of PAR.

Figure 4-2: Concentrations of calibration (stars) and unknown (open circles)
samples used for the two-component equilibrium determinations.

Figure 4-3: Concentrations of calibration (stars) and unknown (open circles)
samples used for the ph 8.5 kinetic and kinetic-spectrophotometric determinations
of Ga(III) and Ni(II).

Figure 4-4: Concentrations of calibration (stars) and unknown (open circles)
samples used for the ph 7.0 kinetic and kinetic-spectrophotometric determinations
of Ga(III) and Ni(II).

Figure 4-5: Plot showing the predicted vs. actual concentrations of Ga(III) for a
two-component equilibrium determination. The circles show the predicted
concentrations. The solid line has a SIOpe of unity and represents a prediction with
no error. The dashed lines show :l:10% error tolerances.

Figure 4—6: Plot showing the predicted vs. actual concentrations of Ni(II) for a
two-component equilibrium determination. The circles show the predicted
concentrations. The solid line has a slope of unity and represents a prediction with
no error. The dashed lines show i10% error tolerances.

Figure 5-1: Example of a linear (pseudo-first-order) kinetic system. This data was
generated with a rate constant ratio of 1.7 and a kinetic angle of 10.2. The reagent
was in 281 fold excess The angle from linearity was 0.017.

Figure 5-2: Example of a nonlinear kinetic system. This data was generated with a
rate constant ratio of 1.7 and a kinetic angle of 10.2. The reagent was in 1.4 fold
excess. The angle from linearity was 6.449.

Figure 5-3: Concentrations of calibration (stars) and unknown (open circles)

samples used for the kinetic-spectrophotometric determinations of Ga(III) and
Ni(II) under linear and nonlinear kinetic conditions.

xiii

76

78

79

81
84

85

86

92

93

104

105

106

Figure 54: Relative standard error of a CR prediction as a function of the kinetic
angle and angle from linearity.

Figure 5-5: Relative standard error of an nPLS prediction as a function of the
kinetic angle and angle from linearity.

Figure 5-6: Relative standard error of a CR prediction as a function of the spectral
angle and angle from linearity. The kinetic angle for this data was ~10°.

Figure 5-7: Relative standard error of an nPLS prediction as a function of the
spectral angle and angle from linearity. The kinetic angle for this data was ~10°.

Figure 5-8: Relative standard error of a CR prediction as a function of the kinetic
angle and reagent excess. The gallium/nickel/PAR spectra (with the PAR
absorbing) were used to generate the data. The spectral angle for this data was
~11°.

Figure 5-9: Relative standard error of an nPLS prediction as a function of the
kinetic angle and reagent excess. The gallium/nickel/PAR spectra (with the PAR
absorbing) were used to generate the data. The spectral angle for this data was
~l 1°.

Figure 5- 10: CR determination of Ni(II) by reaction with 1mM PAR (circles) and
40 11M PAR (plus signs). At the PAR concentration of 1mM the average excess is
28 fold; at 40 11M it is 1.2 fold.

Figure 5-11: CR determination of Ga(III) by reaction with 1mM PAR (circles) and
40 M PAR (plus signs). At the PAR concentration of 1mM the average excess is
28 fold; at 40 M it is 1.2 fold.

Figure 6-1: Calibration set (stars) used in the determination of Zn(II) and Cu(II).
The unknown sample is depicted by an open circle.

Figure 6-2: Spectra of the PAR complexes of Zn(II) and Cu(II). The zinc complex
is shown as a solid line, the copper complex as a dotted line.

Figure 6-3: Kinetic profiles of the formation of the PAR complexes of Zn(II) and

Cu(II). The zinc complex is shown as a solid line, the copper complex as a dotted
line.

xiv

108

109

110

111

113

114

116

117

121

125

126

CR
GRAmA

nPLS

PAR
PARAFAC
PCR

PLSR
RSEP

SIN

LIST OF ABBREVIATIONS

Artificial Neural Network

Continuum Regression

Generalized Rank Annihilation Method
Multiple Linear Regression

N-way Partial Least Squares Regression
4-(2-Pyridylazo)-resorcinol

Parallel Factor Analysis

Principal Component Regression

Partial Least Squares Regression
Relative Standard Error of Prediction

Signal to Noise

XV

CHAPTER 1

INTRODUCTION

Beware the man of one book.
--St. Thomas Aquinas

The subject of this thesis is the application of chemometric methods to
kinetic data. It is important, therefore, to understand other work that has been done
in both the fields of chemometrics and of kinetic determinations.

First, chemometrics will be discussed and some commonly used
chemometric techniques will be described. Particular attention will be paid to
multivariate calibration techniques. Kinetic determinations will then be reviewed,
and the application of chemometric techniques to kinetic data will be highlighted.

1.1. CHEMOMETRICS

Chemometrics has been defined as “the science of relating measurements
made on a chemical system to the state of the system via application of
mathematical or statistical methods and of designing optimal experiments for
investigating chemical systems.”1 As highlighted by several recent texts}.3

chemometrics as a field of study has been steadily growing in recent years.

1.1.1. Nomenclature and Conventions

In this document, matrices, vectors and scalars are written as follows.
Matrix A, vector a, and scalar a are written as shown. The transpose of matrix A
and vector a are written as AT and aT, respectively. All vectors are assumed to be
column vectors unless written as a transpose. In data matrices it is assumed that
rows represent samples or observations and that columns represent variables.
Often, for simplicity, the examples shown presume that each row is a sample and
each column is a measurement at a discreet time point.
1.1.2. Multivariate calibration

To use terminology developed by Booksh and Kowalski in their Theory of
Analytical Chemistry,4 zeroth order data is data where only one data point is
collected for each sample. Examples of zeroth order data include absorbance at a
single wavelength, a single pH measurement, or a single temperature
measurement. First order data consists of an array of data for each sample. First
order data is extremely common, and examples include absorption spectra,
emission spectra, chromatograms, and kinetic profiles. Second order data is made
up of a matrix of data for each sample. Many hyphenated chemical analysis
techniques produce second order data. Examples include fluorescence excitation-
emission spectra, chromatograms with array detection, and kinetic profiles
collected with array detectors. When the data consists of an array of absorbance
measurements taken at every point in the kinetic profile, we will call the result

kinetic-spectrophotometric data.

Calibration is the process of relating the known state of a system to
measured data collected from the system. Multivariate calibration, then, is the
process of relating the known state of a system to a series of measured variables
describing the system. From the above definitions, it is clear that multivariate
calibration is calibration performed on first or higher order data. Since much of the
data collected by analytical scientists is at least first order, the use of multivariate
calibration is becoming routine.

In spite of the above (strict) definition of multivariate calibration, the term
is often used to describe the combination of two distinct processes. After
multivariate calibration has been performed, the relationship found between the
measured variables and the state of the measured system can be used to predict the
state of another system, given the values of variables measured on that system.
The term multivariate calibration is thus often used to describe the process of
performing a calibration and then using the calibration relationship (or model) to
perform a prediction. This combined process can be seen graphically in Figure 1.
Here the two steps are distinctly separated. In the calibration step, the multivariate
calibration algorithm is provided with data for several samples and the system
states (in many cases, analyte concentrations) whose relationship to the data is to
be discerned. The algorithm finds an empirical relationship (or model). In the
second step, data from an unknown system is provided. Using the model built in
the calibration step, the state of the unknown system (e.g., the concentration of the

analytes in the unknown samples) is predicted.
3

m Algorithm

 

Calibration Set |
Data Matrix

      
 

Calibration
Model

 

 

Calibration Set
Concentration
Matrix

 

 

Unknown

Data Matrix Regressron

 

 

 

 

Figure 1-1: Multivariate calibration as a “black box”.

1.1.2.1. Multiple Linear Regression

Multiple linear regression (MLR) is perhaps the simplest and most
straightforward of the multivariate calibration techniques. The mathematics have
been well described in the literaturel-3v5'6, and so only a brief discussion will be
presented here.

Multiple linear regression assumes a linear relationship between the
observed data (e.g., kinetic profiles) and a matrix or vector of weighting factors

describing the state of the system (e.g., initial concentrations)

4

x = [)5
where x is the vector of measured data, S is the matrix of calculated pure
component responses (kinetic profiles) and p is the vector of weights (initial

concentrations). If S has already been calculated from known x and p
T '1 T
S = (p p) p x
An unknown punk can be determined from a vector of data xunk

punk = xunks+
where S+ is the pseudo inverse of S and is defined as:
3+ = sT (SST)l

The major limitation of this technique is that the pure component responses must
be linearly independent for S+ to be defined.
1.1.2.2. Principal Component Regression

Principal component regression (PCR) has also been well described in the
literature.1'3v5'9 PCR is, like other multivariate calibration methods, a two step
process. In the first step, principal component analysis (PCA), a data matrix is
decomposed into a set of abstract factors (or principal components). These
principal components are linear combinations of the measured variables. The
variables are combined and weighted such that the first principal component

explains the largest fraction of the variance in the data. The manner in which this

is accomplished is as follows.

Given a data matrix X of size m x n (m samples and n variables) the

covariance matrix can be defined as

T
cov(X) = £25-
m—l

PCA decomposes X as
X = tlpir + tng + + tka + E
where k is less than or equal to the smaller of the number of variables (11) and
number of samples (m) and E is the residuals (error) matrix. The orthogonal t.
vectors (the scores) contain information about intersample relationships. The pi
vectors (the loadings) are the orthonormal eigenvectors of the covariance matrix,
cov(X)pi = kipi
where It, is the eigenvalue corresponding to eigenvector p. . Thus, the scores are

the projections of the data matrix onto the loadings vector.

As already mentioned, the eigenvalues are arranged in order of magnitude. The
first eigenvalue, Al , is the largest and is associated with the pair (t1 , 1),). This first
principal component contains more information about the system than any other.
By examining the eigenvectors it is possible to determine how many principal
components must be used to describe the data adequately. Most often, the number
of principal components is much smaller than the number of variables. Indeed, one

of the main advantages of PCA/PCR is this reduction in dimensionality. In

addition, principal components generated by PCA are often useful as descriptors
of a chemical system. They are often more robust than measured experimental
variables because of the averaging inherent in PCA. Some artificial neural network

applications use PCA scores rather than experimental data as inputs“).

Experimental Variable 3

 

Figure 1-2: Decomposition of data into principal components.

Figure 1-2 illustrates the application of PCA to a system described by three
experimental measurements. Plotting the data derived from these measurements
reveals that all the data points lie in a plane. Using PCA the three variables can be
consolidated into two principal components (PCs) that correspond to two axes in
this plane and so the dimensionality of the system can be reduced. The first PC
describes the main source of variation. The second PC corresponds to the next
greatest source of variation in the data. Used in this context, a component that is a

major source of variation in the data is one that has a large effect on the measured

data (X). Again, only PCs that have significant effects on the data are used in
modeling the system.

The second step of principal component regression involves using the
principal components calculated with PCA to create a calibration matrix. In a
manner similar to that used in multiple linear regression, the pseudo inverse, X+,

can be calculated as
-1
x“ = P(TTT) TT
such that
Punk = xunitX+

The major difference between the two methods is that in PCR the data are
regressed on the scores of principal components rather than on measured values.
This reduction in dimensionality serves to eliminate some noise and provides well
conditioned (orthogonal) data for regression. If all the available PCs are used,
there is no reduction in dimensionality and PCR converges to MLR. The proper
number of PCs to use in the regression can be determined in a variety of ways.
The most obvious, and therefore the most common, criterion for choosing the
number of principal components is the percentage of the total variation that is
described by a set of selected PCs. Generally, the minimum number of PCs that

combine to describe a desired fraction (usually 80-90%) of the variation in the

data set is chosen.

Eigenvalue vs. PC Number

800

mac.»

500..
.3 50.0 i“
a

 

.3 30.0 J-
11.1
ZOO-e
1QO~~
QO-

 

 

 

12

 

Figure 1-3: Determination of the number of principal components using a
scree graph.

Graphical methods for determining the number of principal components are
also very popular. A scree graph, shown in Figure 1-3, is a plot of the eigenvalue
associated with each PC. The reading of a scree graph is not an exact science, but
rather relies heavily on the common sense and intuition of the analyst. In general,
the “elbow point” where the graph begins to have a nearly zero slope is the last
significant PC. In Figure 1-3, it occurs at the second component, and so two PCs
should be included in the model of the system. Often, the location of this elbow
point is not as clear as that shown and is thus subject to interpretation.

Cross-validation methods for determining the number of PCs can be more
computationally intense than the two methods mentioned above. They usually
involve splitting the calibration data into two parts. The first is decomposed into

PCs. These PCs are then used to perform a prediction on the other set. The number
9

of PCs that provide the best prediction is then chosen. Most often the process is
repeated numerous times so that each sample is used in both a calibration and a
validation set. An excellent discussion of various methods for choosing the proper
number of PCs can be found in several references.2»3"5’8
1.1.2.3. Partial Least Squares Regression

Partial least squares regression (PLSR)l-3"5~1 1-12 can perhaps best be thought
of as a compromise between MLR and PCR. Multiple linear regression finds a
single factor that correlates data (e.g., kinetic profiles) with weightings (initial
concentrations). Principal component regression finds factors that best describe the
trends (variance) in the data. PLS attempts to find factors that describe the
variance in the data and correlate weightings to the data. PLS is thus less
susceptible to error arising from variables that fluctuate significantly, but are
unrelated to the weights.

Given a data matrix (e.g., kinetic profiles), X, and a matrix of predicted

variables (e.g., initial concentrations) Y, PLS decomposes X and Y as:
X = TPT
Y = UQT
where T is the matrix of scores for the data, P is the matrix containing the loadings

for the data, U is the matrix of scores for the dependent variables, and Q is the

matrix containing the loadings for the dependent variables. In addition, a vector of

10

weights w that relate U to X and a vector b that relates U and T are created. The

pseudo inverse used in calibration is then defined as:
-r -1
x” = w(PTw) (TTT) TT
Then as in MLR and PCR,
Punk = Xunkx+

Again, if all the available latent variables are used in the prediction PLSR
converges to MLR. Similar cross validation techniques are used in both PCR and
PLSR.
1.1.2.4. Continuum Regression

Continuum regression (CR) is a hybrid technique that has been described in
several recent publicationsJ’U'lé From at least one perspective, it can be argued

that MLR, PCR, and PLSR are all special cases of continuum regression. In a

manner similar to PLS and PCR, the observed data is decomposed

x = usvT
where U and V are orthonormal matrices and S is a diagonal matrix. This
decomposition is similar to that performed in PCR. In fact, the scores, V, are
identical to the scores produced by PCR (P). The PCR loadings are related to the
other matrices produced by the CR decomposition by
US = T
In continuum regression the diagonal matrix S is then raised to a power and a new

data or X matrix is formed as
11

xm = us'r'r'vT

After suitable manipulations and the use of PLSR on the new data matrix, a
properly scaled regression vector can be obtained. The value of the exponent m
determines the extent to which the algorithm considers the contribution of
variables to the overall variance in the data when forming its set of abstract
factors. At high (above unity) powers, the data set is deformed to exaggerate the
variance, leading the PLSR algorithm to place inordinate weight on those variables
that contribute variance to the data. At extremely high powers (nominally near
infinity; practically, above 8) CR converges to PCR.

At low (below unity) powers, the data matrix is deformed to compress the
data and to reduce the apparent variance. In this case, with little variance available,
the PLSR algorithm places inordinate weight on those variables that are strongly
correlated to the concentrations (or the state of the system). At extremely low
powers (near zero) CR converges to MLR.

At a power of unity, the data set is undeforrned, and CR converges to

PLSR. This is summarized in Figure 1-4.

12

 

   

@®®

oo 1 O

 

m

Figure 1-4: Relationship between principal component regression, partial
least squares regression, multiple linear regression and continuum regression.

1.1.2.5. Nonlinear Partial Least Squares Regression

Several varieties of nonlinear PLSR exist. The algorithms for these
methods are available in the literature.‘ In general, these techniques are
functionally identical to PLSR, except that some sort of nonlinear (polynomial or
spline, most commonly) fit is used to relate the scores to the concentrations.
1.1.3. Factor Analysis

Factor analysis is a well established technique that is well represented in the
literaturez'3’fr9v17-l3. It, like PCA and PLS, involves decomposing the data into
abstract factors. The factors, in this case, are not necessarily aligned to capture
maximum variance or covariance, but rather are rotated so as to have physical

meaning. In a spectrophotometric application, the abstract factors often are good

13

estimates of the absorption spectra of the analytes they describe. Factor analysis
techniques do, however, usually require pure component spectra or other
information to use in target testing steps.
1.1.4. Artificial Neural Networks

Artificial neural networks (ANNs) are a powerful new tool in the arsenal of
analytical chemistsl9'20. The primary element of an ANN is the neuron. These
neurons are arranged in input and output layers sandwiching one or more “hidden”
processing layers. Neurons can be thought of as weighted transfer functions.
Neurons can have single or multiple inputs. The processing neurons apply a
weighted sum of their inputs and transfer the result to the output. Often the

transfer function is non-linear (sigmoidal functions have been most often used).

Ir \ r’f neuron \

I2\l\ n a

.W2\‘£___.F___'

. MH— —

 

 

 

 

 

Figure 1-5: Schematic diagram of a neuron.

A diagram of a neuron is shown in Figure 1-5. Here I. is an input, w, is the
weighting associated with input I, b is the bias introduced into the summation, n is
the output of the weighted sum and a is the output of the transfer function (F); i.e.,

=F(wI+b)

l4

During the training or calibration phase the weightings are adjusted to accurately
fit the calibration data. Often PCs are used as inputs to the network instead of
experimental variables. This reduces the necessary number of neurons immensely.
1.1.5. Multiway Methods

All of the aforementioned chemometric techniques are designed to operate
on first order data. If second or higher order data is to be used with these
algorithms it must first be “unfolded” such that it appears as first order data. This
results in a loss of information. Some more recently popular techniques are
capable of handling second and higher order data directly without unfolding.
These are briefly described below.
1.1.5.1. GRAM

Generalized rank annihilation factor analysis, also known as the generalized
rank annihilation method (GRAM)21-24, is one of the most popular multiway
methods used by chemists. In GRAM a single calibration sample is used to predict
concentrations in an unknown sample. An advantage of GRAM is that interferents
are normally handled well, and that pure component responses of the analytes can
be recovered. GRAM’s major weakness, however, is its restriction to only one
calibration sample. This makes the algorithm sensitive to collinearity in the data
set.
1.1.5.2. PARAFAC

Parallel factor analysis (PARAFAC)25'27 is a technique that is quite

powerful, but also quite limited. PARAFAC, unlike GRAM, is not limited to one
15

calibration sample. It is generally possible to recover pure component responses
for each component in each measured dimension. PARAFAC is susceptible,
however, to nonlinearity in the data. If the measured data in each dimension is not
linearly related to the concentrations being predicted PARAFAC fails to converge
to meaningful solutions.
1.1.5.3. Multiway Partial Least Squares Regression

A multiway version of the PLSR algorithm has recently been developed.28
This N-way PLSR (or nPLSR) algorithm has been shown to be superior to PLSR
acting on unfolded data. It is known to be quite tolerant of noise.

1.2. KINETIC DETERMINATIONS

In recent years, kinetic methods for multicomponent determinations have
become more popular. Several books and reviews covering the principles and
applications of kinetic methods can be found29'36. Several other papers have
reviewed the application of chemometric techniques to kinetic determinations3642.

Multicomponent kinetic methods involve similar species reacting with a
common reagent or undergoing a common process. Differences in the reaction or
process kinetics are used to distinguish among the components without any
physical separation. The major limitation of many conventional techniques for
processing kinetic data is their reliance on an accurate model of the kinetics of the
system under study. These techniques can, as a group, be referred to as hard

modeling techniques. Such techniques require that the analyst have knowledge of

16

at least the reaction order and usually of the rate constants for each of the reactions
in the chemical system.

It is becoming increasingly clear that perhaps the most useful chemometric
techniques for handling kinetic data are those that do not assume a kinetic model.
In particular, multivariate calibration techniques have shown great promise.
Recent publications have demonstrated the use of many of these techniques.

1.2.1. Kinetic Determinations using Hard Modeling Techniques

Hard modeling methods have some inherent advantages. By presuming a
model for the chemical system they are often able to perform determinations using
small calibration sets, or by not using calibration sets at all (except, perhaps, for
determining the parameters of the model).
1.2.1.1. Kinetic Determinations using Reaction-Rate Methods

Several workers have reviewed the use of reaction rate based methods for
kinetic determinations. The papers by Love and Pardue434’t5 do an excellent job of
comparing most of the common (and some more uncommon) techniques. Other
excellent reviews of these methods can also be found37’46.
1.2.1.2. Kinetic Determinations using Other Miscellaneous Methods

A host of techniques are available for effecting kinetic determinations. The
H-point standard additions method“ is an excellent technique that is designed for
the simultaneous kinetic determination of the components of a binary mixture. The
continuous addition of reagent technique48 is one that has found some application

in systems with complex kinetics”. The kinetic wavelength-pair method“)-51
17

involves measurements using an array detector or fast-scanning spectrophotometer
and relies on measurements at two pre-selected pairs of wavelengths for the
determination of the two components in a binary mixture. Both the continuous
addition of reagent technique and the wavelength pair method were developed in
the laboratory of Professor Dolores Perez-Bendito at the University of Cordoba.

Schechter 52'55has developed several techniques that are error compensated
or error tolerant. One technique that does not require prior knowledge of the rate
constants is based upon detecting the intermediate species in a system of
consecutive reactions”. Other techniques developed in his laboratory include an
algorithm that can effect determinations of systems of mixed first and second
order reactions (A-)C & A+B—>C)53, a recursive algorithm for extrapolating slow
kinetic profiles and instrumental responses“, and an error-compensating algorithm
capable of determining the reaction order, rate constants, and concentrations of an
system of general-order reactions55. This latter algorithm, however, requires
modification for use with a first order system.
1.2.1.3. Kinetic Determinations using Nonlinear Regression

In cases where the reaction order and rate constants of a reaction system are
well known, nonlinear regression methods can be used to determine the
concentration of one or more analytes. These regression methods have been
applied to a variety of chemical systemsS‘r“. Some workers have also described

the use of multidimensional nonlinear least squares fits of kinetic-

18

spectrophotometric data in cases where both the rate constants and absorptivities
of the analytes are known61v65v66.
1.2.1.4. Kinetic Determinations using the Kalman Filter

The Kalman filter67 is a recursive algorithm well suited to use with kinetic
data“. Its use for such data has been reviewed”. It has also been found to be
useful for enzymatic kinetic determinations"). Use of the Kalman filter requires
knowledge (or at least a good estimate) of the rate constants. It presumes a
reaction order.

The application of the extended Kalman filter to kinetic-spectrophotometric
data has been described by Quencer and Crouch“. It has been used in binary
systems with first order kinetics72 as well as systems of consecutive reactions73.

Parallel Kalman filter networks have been applied to kinetic determinations
by Wentzell74. The Kalman filter and extended Kalman filter have been compared
to other data processing techniques in a number of reviews37»33»4‘.

1.2.2. Kinetic Determinations using Soft Modeling Techniques

Soft modeling techniques have the obvious advantage that they do not
presume a model that may not fit the system being studied. They do, however,
have the disadvantage that models they generate are often simply empirical and so
have no physical meaning beyond their predictive ability. Several of these soft

modeling techniques are described below.

19

1.2.2.1. Kinetic Determinations using Factor Analysis

While not strictly a soft modeling technique (first order kinetics are usually
assumed), factor analysis is closely related to some other multivariate calibration
techniques, and so in included in this section. In systems where there is at least a
moderate degree of spectral and kinetic resolution between the analytes, factor
analysis has been used to determine the rate constants, absorbance spectra, and
concentrations of mixtures of analytes75'31.
1.2.2.2. Kinetic Determinations using MLR, PCR and PLS

Multivariate calibration techniques are finding broad application for kinetic
determinations. This use has been reviewed in several papers36o33’39’41’42. Several of
these applications are summarized here.

Gallium and aluminum react with 4-(2-pyridylaxo) resorcinol (PAR) to
produce products with very similar spectra. The ratio of the rate constants is
kN/kca = 3.67. Using a stopped-flow, flow injection (FI) system with diode array
detection, Blanco, et a1.82 determined mixtures of Ga and A1 with an error of less
than 10%.

In other work, O-O’-bis-(2-aminoethyl) ethylene glycol-N,N,N’,N’
tetraacetic acid (EGTA) complexes of Fe(II), Co(II), and Zn(II) were reacted with
PAR"). These metal ions react with similar kinetics to form products with very
similar kinetic profiles. This experiment was performed in a stopped-flow FI
system with diode array detection. PCR and PLSR were used to determine Fe, Co

and Zn successfully. The kinetics can be complicated by performing the
20

experiment in two steps in a flow system. If Co, Fe, and Zn are directly injected
into the flow system, where they first react with EGTA and then with PAR, the
kinetics of the Co and Zn are essentially the same as for the case where the EGTA
complexes are directly injected. Iron(II), however, reacts slowly with EGTA and
thus the kinetics associated with the formation of the Fe-PAR complex are
significantly altered (in a non-linear fashion). Both methods (PCR and PLSR)
were used to determine Co, Fe, and Zn using data collected in this second manner.
Almost identical results were obtained with PCR and PLSR. They predicted the
concentration of Zn and Co with good accuracy, but performed less well in
determining Fe.

In other work, Havel and coworkers determined vanadium and cobalt by
PLSR using kinetic data”. The reaction studied was that of V and Co with the
TrAMeR reagent (4-(1’H—l’,2’,4’-triazolyl-3’-azo)-2-methylresorcinol). The
reaction was monitored at 60 3 intervals for 30 minutes at five wavelengths
between 500 and 540 nm. The average relative percent error was 4%. In the same
paper, a stopped Fl determination of Zn, Co, and Fe was described. The average
error associated with this determination was also about 4%.

Lopez-Cueto and coworkers84 have described the determination of
aminophenol isomers. These authors used PLSR with kinetic-spectrophotometric
data that were acquired with a diode array detector. The reaction studied was one

that required that the reagent not be present in excess. Also, the concentration of

21

each isomer influenced the reaction rate of the others. In spite of the inherent
kinetic non-linearity, acceptable results were obtained.

Havel and coworkers85 reported on the kinetic-spectrophotometric
determination of europium, terbium and lanthanum using PLSR. Binary mixtures
of the metal ions reacted with Xylenol Orange to produce similar spectra.
Acceptable errors were obtained (0.2-4%). The authors noted that the PLSR
algorithm required at least four latent variables for a satisfactory fit. They also
reported that, while excellent results were obtained with binary mixtures, ternary
mixtures could not be resolved with acceptable error levels.

A variety of other kinetic determinations have been achieved using
multivariate calibration techniqueslo’ll‘“'32-‘06.
1.2.2.3. Kinetic Determinations using Artificial Neural Networks

Artificial neural networks (ANNs) have been used for a variety of kinetic
determinationslo’l3959940910741°. In general, it has been found that ANNs are
similar to PLSR and PCR in their predictive ability in most cases. Artificial neural
networks require more rigorous and lengthy calculations. They also require larger
calibration sets. In most cases, the advantage (if any) of using an ANN is more
than outweighed by these drawbacks. ANNs do, however, achieve superior results
in cases where the data is nonlinear or in other ways ill-behaved10-13’95.
1.2.2.4. Kinetic Determinations using Multiway techniques

Multiway techniques have been applied to kinetic determinations in only a

handful of papers. The use of GRAM for kinetic-spectrophotometric data111 has
22

been investigated. Several papers have described initial studies of the application

of nPLS to kinetic-spectrophotometric datam’m. A number of papers have

reported the use of multivariate curve resolution for kinetic-spectrophotometric

determinations1 14" 16.

1.3.

(1)

(2)

(3)

(4)

(5)

(6)
(7)

(8)

(9)

(10)

(11)

REFERENCES

Wise, B. M.; Gallagher, N. B. PLS_Toolbox 2.0 User Manual ; Eigenvector
Research, Inc.: Manson, WA, 1998.

Kramer, R. Chemometric Techniques for Quantitative Analysis ; Marcel
Dekker, Inc.: New York, 1998.

Beebe, K. R.; Pell, R. J .; Seasholtz, M. B. Chemometrics: A practical guide
; John Wiley and Sons: New York, 1998.

Booksh, K. S.; Kowalski, B. R. "Theory of analytical-chemistry" Anal.
Chem. 1994, 66, A782—A79l.

Sanchez, E. S.; Kowalski, B. R. "Tensorial calibration: I. First-order ..

calibration" J. Chemometr. 1988, 2, 247-264.
Martens, H.; Naes, T. Multivariate Calibration ; Wiley: New York, 1989.

Malinowski, E. R.; Howery, D. G. Factor Analysis in Chemistry ; John
Wiley and Sons: New York, 1980.

Jolliffe, I. T. Principal Component AnalysisSpringer Series in Statistics, ; /

Springer-Verlag: New York, 1986.

Brereton, R. G. "Deconvolution of mixtures by factor-analysis" Analyst
1995, 120, 2313-2336.

Blanco, M.; Coello, J.; Iturriaga, H.; Maspoch, S.; Redon, M. "Artificial
neural networks for multicomponent kinetic determinations" Anal. Chem.
1995, 67, 4477-4483.

/

/

/

Geladi, P.; Kowalski, B. R. "Partial least-squares regression - a tutorial" /

Anal. Chim. Acta 1986, 185, 1-17.

23

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(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

Geladi, P.; Kowalski, B. R. "An example of 2-block predictive partial least-
squares regression with simulated data" Anal. Chim. Acta 1986, 185, 19-32.

Blanco, M.; Coello, J .; Iturriaga, H.; Maspoch, S.; Redon, M.; Villegas, N.
"Artificial neural networks and partial least squares regression for pseudo-
first-order with respect to the reagent multicomponent kinetic-
spectrophotometric determinations" Analyst 1996, 121, 395-400.

Wise, B. M.; Ricker, N. L. "Identification of finite impulse-response
models with continuum regression" J. Chemometr. 1993, 7, 1-14.

Stone, M.; Brooks, R. J. "Continuum regression - cross-validated 4

sequentially constructed prediction embracing ordinary least-squares,
partial least- squares and principal components regression" J. R. Stat. Soc.
Ser. B-Methodol. 1990, 52, 237-269.

Kalivas, J. H. "Interrelationships of multivariate regression methods using
eigenvector basis sets" J. Chemometr. 1999, 13, 111-132.

Malinowski, E. R. "Automatic window factor analysis - a more efficient
method for determining concentration profiles from evolutionary spectra" J.
Chemometr. 1996, 10, 273-279.

Harvey, D. T.; Bowman, A. "Factor-analysis of multicomponent samples"
J. Chem. Educ. 1990, 67, 470-472.

Zupan, J.; Gasteiger, J. "Neural networks: A new method for solving
chemical problems or just a passing phase?" Anal. Chim. Acta 1991, 248, l-
30.

Despagne, F.; Massart, D. L. "Neural networks in multivariate calibration"
Analyst 1998, 123, 157R-l78R.

Sanchez, E.; Kowalski, B. R. "Generalized rank annihilation factor-
analysis" Anal. Chem. 1986, 58, 496-499.

Sanchez, E.; Ramos, L. S.; Kowalski, B. R. "Generalized rank annihilation
method .1. Application to liquid- chromatography diode-array ultraviolet
detection data" Journal of Chromatography 1987, 385, 151-164.

Sanchez, E. S.; Kowalski, B. R. "Tensorial calibration: II. Second-order
calibration" J. Chemometr. 1988, 2, 265-280.

24

L/

(24)

(25)

(26)

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(34)

(35)

(36)

(37)

Ramos, L. S.; Sanchez, E.; Kowalski, B. R. "Generalized rank annihilation ’

method .2. Analysis of bimodal chromatographic data" Journal of
Chromatography 1987, 385, 165-180.

Bro, R. "PARAFAC. Tutorial and applications" Chemometrics Intell. Lab.
Syst. 1997, 38, 149-171.

Bro, R. "The N-way on-line course on PARAFAC and PLS"
http://newton/foodsci.kvl.dk/n-way 1998.

Harshman, R. A. "Foundations of the PARAFAC procedure: Model and
conditions for an ’explanatory’ multi-mode factor analysis" UCLA Working
Papers in Phonetics 1970, I6, 1.

Bro, R. "Multiway calibration. Multilinear PLS" J. Chemometr. 1996, 10,
47-61.

Perez-Bendito, D.; Silva, M. Kinetic Methods in Analytical Chemistry ;
Ellis Horwood: Chichester, 1988.

Perez-Bendito, D. "Inorganic differential kinetic-analysis - a review"
Analyst 1984, 109, 891-899.

Perez-Bendito, D. "Approaches to differential reaction-rate methods"
Analyst 1990, 115, 689-698.

Mottola, H. A. Some Kinetic Aspects of Analytical Chemistry ; Wiley: New
York, 1988.

Crouch, S. R. "Kinetic methods for intelligent automation" Chemometrics
Intell. Lab. Syst. 1990, 8, 259-273.

Crouch, S. R. "Trends in kinetic methods of analysis" Anal. Chim. Acta
1993, 283, 453-470.

Crouch, S. R. "Kinetic methods of analysis - how do they rate" J. Chin.
Chem. Soc. 1994, 41, 221-229.

Crouch, S. R.; Cullen, T. F.; Scheeline, A.; Kirkor, E. S. "Kinetic
determinations and some kinetic aspects of analytical chemistry" Anal.
Chem 1998, 70, 53R-106R.

Quencer, B. M.; Crouch, S. R. "Multicomponent kinetic methods" Crit.
Rev. Anal. Chem. 1993, 24, 243-262.

25

(38)

(39)

(40)

(41)

(42)

(43)

(44)

(45)

(46)

(47)

(48)

Perez-Bendito, D.; Silva, M. "Recent advances in kinetometrics" Trac-
Trends Anal. Chem. 1996, 15, 232-240.

Otto, M. "Chemometrics in kinetic-analysis" Analyst 1990, 115, 685-688.

Cerda, V.; Cladera, A.; Estela, J. M. "Multicomponent techniques as
applied to kinetic analysis" Quim. Anal. 1996, 15, 341-350.

Xie, Y. L.; BaezaBaeza, J. J.; RamisRamos, G. "A comparative study of
several chemometric methods applied to the treatment of two-way kinetic-
spectral data for mixture resolution" Anal. Chim. Acta 1996, 321 , 75-95.

Cullen, T. F.; Crouch, S. R. "Multicomponent kinetic determinations using
multivariate calibration techniques" Mikrochim. Acta 1997, 126, 1-9.

Love, M. D.; Pardue, H. L. "Systematic comparison of data-processing
options for kinetic- based, single-component determinations of
noncatalysts: Effects of random noise" Anal. Chim. Acta 1996, 326, 95-106.

Love, M. D.; Pardue, H. L. "Systematic comparison of data-processing
options for kinetic- based single-component determinations of non-catalysts
.1. Review, systematic classification, mathematical descriptions,
performance-characteristics and perspectives" Anal. Chim. Acta 1994, 299,
195-208.

Love, M. D.; Pardue, H. L. "Systematic comparison of data-processing
options for kinetic- based single-component determinations of non-catalysts
.2. One- rate, 2-point/fixed-time, 2-rate, 3-point/fixed-time options" Anal.
Chim. Acta 1994, 299, 209-218.

Wentzell, P. D.; Crouch, S. R. "Comparison of reaction-rate methods of
analysis for systems following lst—order kinetics" Anal. Chem. 1986, 58,
2855-2858.

Boschreig, F.; Campinsfalco, P.; Sevillanocabeza, A.; Herraezhemandez,
R.; Molinslegua, C. "Development of the h-point standard-additions
method for ultraviolet visible spectroscopic kinetic-analysis of 2-
component systems" Anal. Chem. 1991, 63, 2424—2429.

Marquez, M.; Silva, M.; Perez-Bendito, D. "Continuous addition of reagent
technique - a new approach to differential reaction-rate methods" Anal.
Chim. Acta 1990, 239, 221-227.

26

(49)

(50)

(51)

(52)

(53)

(54)

(55)

(56)

(57)

(58)

(59)

Velasco, A.; Silva, M.; Perez-Bendito, D. "Processing analytical data

obtained from 2nd-order reactions by using continuous reagent addition"
Anal. Chem. 1992, 64, 2359-2365.

Carreto, M. L.; Lunar, L.; Rubio, S.; Perez-Bendito, D. "Simultaneous

spectrophotometric determination of o-cresol and m-cresol in urine by use
of the kinetic wavelength-pair method" Analyst 1996, 121, 1647-1652.

Carreto, M. L.; Lunar, L.; Rubio, S.; Perez-Bendito, D. "Simultaneous
spectrophotometric determination of chlorpromazine, perphenazine and

acetopromazine by use of the kinetic wavelength pair-method" Anal. Chim.
Acta 1997, 349, 33-42.

Schechter, I. "Error-compensated kinetic determinations by detecting the
intermediate product in successive reactions, without prior knowledge of
reaction constants" Anal. Chem. 1991, 63, 1303-1307.

Schechter, I.; Schroder, H. "Error-compensated kinetic determinations in
systems of mixed 1st-order and 2nd-order reactions, without prior
knowledge of reaction constants" Anal. Chem. 1992, 64, 325-329.

Schechter, I. "Predictive mode of kinetic-analysis and transient detection
responses - evaluation of a recursive algorithm" Anal. Chem. 1992, 64,
2610-2614.

Schechter, I. "Simultaneous determination of mixtures by kinetic-analysis
of general-order reactions" Anal. Chem. 1992, 64, 729-737.

Cost, J. R. "Non-linear regression least-squares method for determining
relaxation-time spectra for processes with lst-order kinetics" J. Appl. Phys.
1983, 54, 2137-2146.

Gomez, E.; Cladera, A.; Estela, J. M.; Cerda, V. "Multidata treatment
applied to the simultaneous resolution of catechol resorcinol mixtures by
kinetic enzymatic processes" Talanta 1993, 40, 1601-1607.

Diaz, A. N.; Garcia, J. A. G. "Nonlinear multicomponent kinetic-analysis
for the simultaneous stopped-flow determination of chemiluminescence
enhancers" Anal. Chem. 1994, 66, 988-993.

Cai, Q. Y.; Wang, R. G.; Wu, L. Y.; Nie, L. H.; Yao, S. Z. "A nonlinear

regression model applied to kinetic studies of ester hydrolysis with a
surface acoustic wave sensor" Talanta 1996, 43, 699-705.

27

(60)

(61)

(62)

(63)

(64)

(65)

(66)

(67)

(68)

(69)

(70)

Gill, A.; Leatherbarrow, R. J.; Hoare, M.; PollardKnight, D. V.; Lowe, P.
A.; Fortune, D. H. "Analysis of kinetic data of antibody-antigen interaction
from an optical biosensor by exponential curve fitting" J. Biotechnol. 1996,
48, 117-127.

Havel, J.; Cuesta, F.; Gonzalez, J. L.; Canedo, M. D. "Multicomponent
kinetic analysis and kinetic studies .1. Multipurpose KILET(94) non-linear
least squares program" J. Anal. Chem. 1996, 51, 100-105.

Kuroda, Y.; Kawashima, A.; Ogoshi, H. "Computer analyses of complex
kinetics containing equilibrium processes. Example of application for
unusual atropisomerization of a tetraphenylporphyrin derivative" Chem.
Lett. 1996, 57-58.

Roecker, L.; Lee, S. M.; Liu, L. W.; Sun, L. J.; Zaerpoor, K. "Analysis of
first-order kinetic data by a differential technique" Int. J. Chem. Kinet.
1997, 29, 333-338.

Shultz, L. L.; Nieman, T. A. "Stopped-flow analysis of Ru(bpy)(3)(3+)
cherrriluminescent reactions" J. Biolumin. Chemilumin. 1998, 13, 85—90.

Chau, F. T.; Mok, K. W. "Multiwavelength analysis for a 1st-order
consecutive reaction" Comput. Chem. 1992, 16, 239-242.

Endo, M.; Sasaki, 1.; Abe, S. "Kinetic spectrophotometric determination of

iron oxidation- states in geological-materials" F resenius J. Anal. Chem.
1992, 343, 366-369.

Brown, S. D. "The Kalman filter in analytical-chemistry" Anal. Chim. Acta
1986, 181, 1-29.

Wentzell, P. D.; Karayannis, M. 1.; Crouch, S. R. "Simultaneous kinetic
determinations with the Kahnan filter" Anal. Chim. Acta 1989, 224, 263-
274.

Xiong, R.; Velasco, A.; Silva, M.; Perez-Bendito, D. "Performance of the
kalman filter algorithm in differential reaction-rate methods" Anal. Chim.
Acta 1991, 251, 313-319.

Lin, D. Z.; Tan, H. W.; Bao, L. L.; Nie, L. H.; Yao, S. Z. "Study on a

modified algorithm - iterative Kalman filter and fits application in
enzymatic kinetics" Analyst 1995, 120, 2573-2578.

28

(71)

(72)

(73)

(74)

(75)

(76)

(77)

(73)

(79)

(80)

(81)

Quencer, B. M.; Crouch, S. R. "Extended Kalman filter for
multiwavelength, multicomponent kinetic determinations" Analyst 1993,
118, 695-701.

Quencer, B. M.; Crouch, S. R. "Multicomponent kinetic determination of
lanthanides with stopped-flow, diode-array spectrophotometry and the
extended Kalman filter" Anal. Chem. 1994, 66, 458-463.

Mok, K. W.; Chau, F. T. "Application of the extended Kalman filter for
analysis of a consecutive first-order reaction" Trac-Trends Anal. Chem.
1996, 15, 170-174.

Wentzell, P. D.; Vanslyke, S. J. "Parallel Kalman filter networks for kinetic
methods of analysis" Anal. Chim. Acta 1992, 257, 173-181.

Cladera, A.; Gomez, E.; Estela, J. M.; Cerda, V. "Resolution of
simultaneous kinetic spectrophotometric processes by factor-analysis"
Anal. Chem. 1993, 65, 707-715.

Tam, K. Y.; Chau, F. T. "Simultaneous multiwavelength study of the
reduction reaction of chromic-acid with oxalic-acid using a photodiode-
array spectrophotometer" Spectr. Lett. 1993, 26, 1195-1212.

Tam, K. Y.; Chau, F. T. "Multivariate study of kinetic data for a 2-step
consecutive reaction using target factor-analysis" Chemometrics Intell. Lab.
Syst. 1994, 25, 25-42.

Liang, X. H.; Andrews, J. E.; deHaseth, J. A. "Resolution of mixture
components by target transformation factor analysis and determinant
analysis for the selection of targets" Anal. Chem. 1996, 68, 378-385.

Garland, M.; Visser, E.; Terwiesch, P.; Rippin, D. W. T. "On the number of
observable species, observable reactions and observable fluxes in

chemometric studies and the role of multichannel integration" Anal. Chim.
Acta 1997, 351, 337-358.

Fister, J. C.; Harris, J. M. "Resolving component overlap in
multiwavelength kinetic spectroscopy: Application to Raman scattering
from intermediates in triplet-state photoreactions" Photochem. Photobiol.
1997, 65, 47-56.

Amrhein, M.; Srinivasan, B.; Bonvin, D. "Target factor analysis of reaction
data: Use of data pre- treatment and reaction-invariant relationships" Chem.
Eng. Sci. 1999, 54, 579-591.

29

(82)

(83)

(84)

(85)

(86)

(87)

(88)

(89)

(90)

Blanco, M.; Coello, J.; Iturriaga, H.; Maspoch, S.; Riba, J.; Rovira, E.
"Kinetic spectrophotometric determination of Ga(III)-Al(III) mixtures by
stopped-flow injection-analysis using principal component regression"
Talanta 1993, 40, 261-267.

Havel, J.; Jimenez, F.; Bautista, R. D.; Leon, 1. J. A. "Evaluation of
multicomponent kinetic-analysis data by a partial least-squares calibration
method" Analyst 1993, 118, 1355-1360.

Lopez-Cueto, G.; Maspoch, S.; Rodriguez-Medina, J. F.; Ubide, C.
"Simultaneous kinetic spectrophotometric determination of o-, m- and p-
aminophenol using partial least squares calibration" Analyst 1996, 121,
407-412.

Fraga, J. M. G.; Abizanda, A. I. J.; Moreno, F. J.; Leon, J. J. A.; Havel, 1.
"Kinetic spectrophotometric resolution of mixtures of europium, terbium,
and lanthanum in a micellar medium using multivariate calibration"

Microchem J. 1996, 54, 32-40.

Havel, J .; Cuesta, F.; Jancar, L. "PLS self-calibration method for the
evaluation of rate constants and or initial concentrations in multicomponent
kinetic-analysis" React. Kinet. Catal. Lett. 1993, 49, 189-195.

Blanco, M.; Coello, J.; Iturriaga, H.; Maspoch, S.; Riba, J. "Kinetic
spectrophotometric method for analyzing mixtures of metal-ions by
sopped-flow injection-analysis using partial least-squares regression" Anal.
Chem. 1994, 66, 2905-2911.

Baxter, P. J.; Christian, G. D.; Ruzicka, J. "Multivariate approach for the
simultaneous determination of total biomass and glucose from a yeast
fermentation by sequential injection-analysis" Chem. Anal. 1995, 40, 455-
471.

Blanco, M.; Coello, J .; Iturriaga, H.; Maspoch, S.; Redon, M. "Partial least-
squares regression for multicomponent kinetic determinations in linear and
nonlinear-systems" Anal. Chim. Acta 1995, 303, 309-320.

Garcia, J. M.; Jimenez, A. 1.; Arias, J. J.; Khalaf, K. D.; Moralesrubio, A.;
Delaguardia, M. "Application of the partial least-squares calibration method

to the simultaneous kinetic determination of propoxur, carbaryl,
ethiofencarb and formetanate" Analyst 1995, 120, 313-317.

30

/

/-.

(91)

(92)

(93)

(94)

(95)

(96)

(97)

(98)

(99)

On, Z. G; Wang, X. D. "The application of principal component regression
on simultaneous multicomponent determinations through a single catalytic
kinetic run" Talanta 1995, 42, 205-210.

Lupu, M.; Todor, D. "A singular-value decomposition based algorithm for
multicomponent exponential fitting of NMR relaxation signals"
Chemometrics Intell. Lab. Syst. 1995, 29, 11-17.

Meras, I. D.; Mansilla, A. E.; Lopez, F. S. "Simultaneous kinetic
spectrophotometric determination of 2- furfuraldehyde and 5-
hydroxymethyl-Z-furfuraldehyde by application of a modified Winkler’s
method and partial least- aquares calibration" Analyst 1995, 120, 2567-
2571.

Wang, X. D.; Gu, Z. C. "Generalized-method for simultaneous
multicomponent determinations through a single catalytic kinetic run by
using the rate spectrum" Analyst 1995, 120, 1839-1842.

Blanco, M.; Coello, J .; Iturriaga, H.; Maspoch, S.; Redon, M.; Rodriguez, J.
F. "Artificial neutral networks and partial least-squares regression for
second-order multicomponent kinetic determinations" Quim. Anal. 1996,
15, 266-275.

de la Guardia, M.; Khalaf, K. D.; Hasan, B. A.; Morales-Rubio, A.; Arias,
J. J .; Garcia-Fraga, J. M.; Jimenez, A. I.; Jemenez, F. "Simultaneous kinetic
spectrophotometric determination of five phenolic compounds by reaction

with p-aminophenol, using partial least squares data treatment" Analyst
1996, 121, 1321-1326.

Khalaf, K. D.; MoralesRubio, A.; DelaGuardia, M.; Garcia, J. M.; Jimenez,
F .; Arias, J. J. "Simultaneous kinetic determination of carbamate pesticides
after derivatization with p-aminophenol by using partial least squares"
Microchem J. 1996, 53, 461-471.

Lopez-Cueto, G.; Rodriguez-Medina, J. F.; Ubide, C. "Individual kinetic
determinations using partial least squares calibration" Analyst 1997, 122,
519-523.

Xing, W. L.; He, X. W. "Kinetic determination of organic vapor mixtures
with single piezoelectric quartz crystal sensor using artificial neural
networks and partial least squares" Chem. Lett. 1996, 1065-1066.

31

(100)

(101)

(102)

(103)

(104)

(105)

(106)

(107)

(108)

(109)

Xing, W. L.; He, X. W. "Kinetic determination of organic vapor mixtures
with single piezoelectric quartz crystal sensor using artificial neural
networks" Talanta 1997, 44, 959-965.

Blasco, F.; Medina-Hemandez, M. J .; Sagrado, S. "Use of pH gradients in
continuous-flow systems and multivariate regression techniques applied to
the determination of methionine and cysteine in pharmaceuticals" Anal.
Chim. Acta 1997, 348, 151-159.

Blasco, F.; Medina-Hernandez, M. J .; Sagrado, S.; Fernandez, F. M.
"Simultaneous spectrophotometric determination of calcium and
magnesium in mineral waters by means of multivariate partial least-squares
regression" Analyst 1997, 122, 639-643.

Sans, D.; Nomen, R.; Sempere, J. "Interactive self-modelling of chemical

reaction systems using multivariate data analysis" Comput. Chem. Eng.
1997, 21, $631-$636.

Izquierdo, A.; Lopez-Cueto, G.; Medina, J. F. R.; Ubide, C. "Simultaneous
determination of niobium and tantalum with 4-(2- pyridylazo) resorcinol
using partial least squares regression and artificial neural networks" Quim.

Anal. 1998, 17, 67-74.

Kappes, T.; Lopez-Cueto, G.; Rodriguez-Medina, J. F.; Ubide, C.
"Improved selectivity in multicomponent determinations through

interference modelling by applying partial least squares regression to
kinetic profiles" Analyst 1998, 123, 2071-2077.

Peralta-Zamora, P.; Kunz, A.; Nagata, N.; Poppi, R. J. "Spectrophotometric
determination of organic dye mixtures by using multivariate calibration"
Talanta 1998, 47, 77-84.

Ventura, S.; Silva, M.; Perez-Bendito, D.; Hervas, C. "Multicomponent
kinetic determinations using artificial neural networks" Anal. Chem. 1995,
67, 4458-4461.

Galvan, I. M.; Zaldivar, J. M.; Hernandez, H.; Molga, E. "The use of neural
networks for fitting complex kinetic data" Comput. Chem. Eng. 1996, 20,
1451-1465.

Ventura, 8.; Silva, M.; Perez-Bendito, D.; Hervas, C. "Estimation of

parameters of kinetic compartrnental models by use of computational
neural networks" J. Chem. Inf Comput. Sci. 1997, 37, 517-521.

32

i.‘\

(110)

(111)

(112)

(113)

(114)

(115)

(116)

Ventura, 8.; Silva, M.; Perez-Bendito, D.; Hervas, C. "Computational
neural networks in conjunction with principal component analysis for
resolving highly nonlinear kinetics" J. Chem. Inf. Comput. Sci. 1997, 37,
287-291.

Xie, Y. L.; BaezaBaeza, J. J.; RamisRamos, G. "Second-order tensorial
calibration for kinetic spectrophotometric deterrrrination" Chemometrics
Intell. Lab. Syst. 1996, 32, 215-232.

Pettersson, A. K.; Karlberg, B. "Simultaneous determination of
orthophosphate and arsenate based on multi-way spectroscopic-kinetic data
evaluation" Anal. Chim. Acta 1997, 354, 241-248.

Xie, Y. L.; Baezabaeza, J. J .; RamisRamos, G. "Kinetic spectrophotometric
resolution of binary-mixtures using 3-way partial least-squares"
Chemometrics Intell. Lab. Syst. 1995, 27, 211-220.

Tauler, R.; Smilde, A. K.; Henshaw, J. M.; Burgess, L. W.; Kowalski, B. R.
"Multicomponent determination of chlorinated hydrocarbons using a

reaction-based chemical sensor .2. Chemical speciation using multivariate
curve resolution" Anal. Chem. 1994, 66, 3337-3344.

Saurina, J .; Hemandez-Cassou, S.; Tauler, R.; Izquierdo-Ridorsa, A.
"Multivariate resolution of rank-deficient spectrophotometric data from
first-order kinetic decomposition reactions" J. Chemometr. 1998, 12, 183-
203.

Saurina, J.; Hemandez-Cassou, S.; Tauler, R. "Multivariate curve resolution
and trilinear decomposition methods in the analysis of stopped-flow kinetic
data for binary amino acid mixtures" Anal. Chem. 1997, 69, 2329-2336.

33

CHAPTER 2
INSTRUMENT DESIGN AND CHARACTERIZATION

Never worry about theory as long as the
machinery does what it’s supposed to do.
--Robert A. Heinlein

The data acquisition system used to collect the kinetic-spectrophotometric
data discussed in this document consisted of a home-built stopped-flow apparatusl
interfaced to a Tracor Northern (Model TN-6123) 512 element intensified diode
array (Tracor Northern, Philadelphia, PA). Both the stopped-flow apparatus and
the computerized interface between the stopped-flow and the diode array were
modified for this research. These modifications and the subsequent
characterization of the entire instrument are the subject of this chapter.

2.1. DESIGN OF DIODE ARRAY INTERFACE

A new computerized interface between the diode array and the stopped-
flow was designed and constructed. This interface controls the diode array and is
responsible for sending timing signals to the array and acquiring the array’s
output. The interface is composed of a peak track-and-hold circuit and a computer
program that controls National Instruments LabPC+ and PC-TIO-lO data

acquisition and timing boards.

34

A timing diagram describing the signals sent to and from the array is shown

in Figure 2-1.

 

 

Trigger

 

Scan

 

 

 

 

 

START

 

 

BEOS

 

 

 

 

STB I J

 

Pulse Out A

P

 

Peak Track] | [—

 

 

Read 1

Reset I L.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 2 4 6 8 10 12118

Figure 2-1: Timing diagram of signals sent to and from the diode array.

The Trigger signal is sent by an opto-interrupter circuit attached to the stopped-
flow when a push has occurred and flow is stopped. When the computer sees
Trigger go high, it sets Scan high. The length of time Scan stays high is
determined by the length of the planned acquisition, i.e., on the number of scans

required and on the scan frequency. START is gated off of Sean, and has a
35

frequency related to the desired scan rate. Upon receiving the START pulse, the
diode array sets BEOS (Buffered End Of Scan) high. The STB (Scan Time Base)
timing signal is gated off of BEOS and begins to send out pulses when BEOS
goes high. The frequency of the STB pulses is related to the desired integration
time for each diode. Each STB pulse results, after a short delay, in the appearance
of a ~100ns pulse on the Pulse Out line. The pulses are the result of each diode’s
accumulated charge being converted into a voltage pulse in turn. The magnitude of
a pulse is related to the number of photons that impacted its diode during the
integration time. The Peak Track line is the output of a peak track-and-hold
circuit which is described in detail later in this section. This circuit tracks the pulse
to its maximum and then holds at that level until reset. Six microseconds after an
STB pulse, the Read line goes low, triggering the computer’s data acquisition
board to initiate an acquisition of the Peak Track line. After a short delay, the
Reset signal goes low and resets the peak track-and-hold circuit into tracking
mode. Both the Read and the Reset signals are initialized by the START pulse.

In reality, each scan of the array requires two START pulses and two STB
pulse trains. The first “scan” is not read into memory, but is merely used to clear
the array. The STB frequency is used to control the period between the clearing

and reading of each diode.

36

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37

Figure 2-2 is a schematic of the peak track-and-hold circuit. The Pulse Out
signal is inverted by an inverting amplifier. The second op amp has gain and
inverts the signal again. The track-and-hold functionality is provided by a fast
signal diode, the third op amp, and a high quality capacitor. The capacitor charges
as the rising edge of a pulse comes down the Pulse Out line. As the pulse hits its
maximum and begins to fall, the diode prevents the capacitor from discharging.
The input impedence of the voltage follower immediately after the capacitor
prevents discharge in that direction. The capacitor thus charges while a pulse is
rising and the holds at the peak value after the pulse begins to fall. The capacitor is
discharged by closing a switch that provides a low resistance path to ground. The
switch is closed by a high on the Reset line. Reset is the output of a monostable
multivibrator that clocks off of the STB pulse train. Shortly before the Reset goes
high, a second monostable causes Read to go low, triggering the computer’s data
acquisition board to initiate a sample-and-hold operation on the Peak Track
signal.

The LT1363 operational amplifiers used for the inverting amplifier, the
inverting amplifier with gain, and the voltage follower are 70 MHz amplifiers with
slew rates of 1000V/us in order to be able to respond to the very short duration
pulses on the Pulse Out signal. The switching is done with a 4066 high speed

quad bilateral switch.

38

2.2. REDESIGN OF STOPPED-FLOW OBSERVATION CELL AND
OPTICS

The optical path of the stopped-flow apparatus begins with a fiber optic
bundle carrying light from a tungsten lamp to the observation cell. The light passes
through the observation cell, where some is absorbed by the sample, and is
collected by a second fiber optic bundle which carries it to the diode array

detector.

From mixer

 

    

Fiber Optic Bundle
to Diode Array

 

Fiber Optic Bundle
from Tungsten Lamp

To stop
syfinge
Figure 2-3: Schematic of the observation cell.

As seen in Figure 2-3, the fiber optic bundles are inserted into the block containing
the observation cell and are butted against spherical sapphire ball lenses. These

lenses are fitted against the openings at the ends of the observation cell and serve

39

both to seal the cell and to collimate and collect the light passed through the
observation cell. They are held in place by custom-designed Delrin screws.

In previous incarnations of the instrument, the fiber optic bundles did not
extend into the block itself. Rather, a three centimeter quartz rod was inserted into
the block and butted against a flat window. The rod was screwed into place with
sufficient pressure to force the window to seal the cell. The fiber optics were
touched to the quartz rod. Thus, the light had to pass from the fiber optic bundle,
through the rod, through the window, and into the cell. This resulted in several
interfaces where light could be lost due to reflection.

The current scheme, in which the quartz rods have been eliminated and the
fragile quartz windows replaced by extremely durable sapphire balls, is
significantly simpler to operate and maintain. The light throughput is at least as
good as the previous arrangement.

2.3. CHARACTERIZATION OF DATA ACQUISITION SYSTEM

After the computerized interface was in place and the redesigned optical
path had been implemented, a detailed characterization study of the instrument as
a whole was performed. The delay time and maximum scan rate were determined.
The mixing time was not determined for reasons discussed below. The signal-to-
noise ratio of the instrument was measured, and the linear range was found.

2.3.1. Determination of Delay Time and Maximum Acquisition Rate

Literature procedures2 were used in an attempt to determine the dead and

mixing times of the instrument. An attempt was made to experimentally determine
40

the dead time of the stopped flow by observing the reaction of iron (III) with
thiocyanate to form a colored productz. Due to instrumental and electronic
limitations, the dead time was not determined; rather another parameter we shall
call the delay time was calculated by extrapolating the reaction curve back to the
initial absorbance of the reactant mixture. Delay time, then, is defined as the time
for which the reactants are mixing in the observation cell but are unobserved, i.e.,
the time between the mixing of the sample and reagent and the collection of the
first data point.

If the first data point is defined to be acquired at time zero, the delay time
is the negative of the time at which the extrapolated reaction curve reaches the
absorbance of the reactants. For this system, the delay time was determined to be
75 ms. Previous work using this same stopped-flow apparatus1 had found the dead
time of the stopped-flow itself be approximately 5 ms. The difference between the
two values can be attributed to several factors. Those that contribute most to the
difference are the delay between stoppage of flow and the triggering of the opto-
interrupter circuit, the delay between the arrival of the Trigger signal and the
generation of the first START pulse, and the need for one complete scan of all
512 diodes (to clear the array) before any data can be acquired.

The mixing time of the stopped-flow apparatus was not accurately
determined because it was shorter than the delay time of the instrument as a
whole, and so not measurable using the experimental setup used for the other

experiments.
41

The maximum rate at which spectra could be collected was determined, and
found to be a function of the desired integration time. The integration time
determines the frequency of the STB pulse train, and therefore the length of time
necessary to complete one full scan of the array. Due to the maximum STB
frequency (80 kHz) supported by the track-and-hold circuit and the lag between
the end of the clearing scan of the array and the beginning of the data acquisition
scan (determined in part by the speed of the PC running the interface program, in
our case a 33 MHz 486SX), the minimum practical integration time is 13 ms. At
this minimum integration time spectra can be acquired at 10.5 Hz. Many studies
described in this work were performed with 35 ms integration times. At that
integration time, 5.5 spectra can be acquired per second..

2.3.2. Determination of signal/noise ratio

The signal/noise ratio (S/N) for each wavelength was measured as the range
of recorded intensity measurements for a series of spectra acquired from a static
system. The average S/N was computed as the average of the S/Ns of all
wavelengths between 500 and 700 nm (the range used in these experiments).
These S/N measurements were carried out under a variety of circumstances. The
S/N was found to be a function of percent of the radiation transmitted through the
observation cell, intensifier gain, and integration time.

The dark noise (the range of intensities measured when no light impinges
on the array) is a major source of noise. Accordingly, attempts were made to

minimize it by cooling the array. A Peltier thermoelectric cooler was used to bring
42

the array temperature to —5.9°C. The cooler decreased the dark signal by only 2%,
but decreased the amplitude of the dark noise by 30%. For all measurements, the
average dark signal at each wavelength was removed in a background subtraction

step.

0.0

 

-O.5 -

-1.0 -

-1.5 -

109(0 IA)

-2.0 -

-2.5 -

 

 

 

'3.0 T i I T r 1
O 0.5 1 1 .5 2 2.5 3 3.5

Absorbance

Figure 2-4: Plot showing dependence of relative error in absorbance to
absorbance.

Figure 2-4 shows the relationship between absorbance (in this case the
absorber was a neutral density filter that attenuates the light) and the relative error
in absorbance. As is expected, log(oA/A) goes through a minimum at an
absorbance of ~0.5. It can also be seen that log(oA/A) is fairly constant between
absorbances of 0.2 and 1.0. For this reason all experiments were run under

conditions where no absorbances greater than 1.0 were measured.

43

 

    

 

 

 

2.5 4
Iog(S/N) = (0.91 *log(is)) + 2.2
A 2 d
z R2=O.999
\
SP, 15 -
O)
.9
1 _
0.5 -
0 t l r I j t
-2.5 -2 -1 .5 -1 -0.5 0 0.5 1

109(is)
Figure 2-5: Dependence of SIN on absorbance.

Figure 2-5 shows a linear relationship between the logarithm of the signal
to noise ratio and the logarithm of the measured signal (is), where the signal level
was varied by varying the light intensity The different measured intensities are due
to the attenuation of the light with a series of neutral density filters. The measured
slope was 0.91. For a shot noise limited system a slope of 0.5 is anticipated; when
blank noise dominates a slope of unity is expected3. It can thus be inferred that the
measurement is not shot noise limited, but over this region a mixture of shot noise
and blank noise predominates. We can speculate that blank noise is dominant at
the low end of the region and shot noise is dominant when light levels are at the

high end of the region shown in Figure 2-53.

2.2

 

2.1 4 log(S/N)= (0.64*|og(is)) + 2.1
R2 = 0.994

 
   

log (S/N)

 

 

 

.5 _L _l —L —L
in b) K1 in to N
1 1 1 1 1

db
on
I
o
a:
I
9
rs
('3
N
o

0.2
1090s)
Figure 2-6: Dependence of SIN on intensifier gain.

Figure 2-6 shows a linear relationship between the logarithm of the signal
to noise ratio and the logarithm of the measured signal (is) where the measured
signal was varied by changing the intensifier gain at a constant integration time
(13 ms, the minimum integration time) and constant light intensity. As gain
increases, so does the measured signal amplitude. At this minimum integration
time, the slope of the linear plot is 0.64. Blank noise contributes less and the

system is more nearly shot noise limited when intensifier gain is increased.

45

 

2
1.95 -
1.9 ~ log(S/N)= (0.93*log(is)) +2.26
1.85 i R2=0.998
1.8 —
1.75 -
1.7 ~
1.65 ~
1.6 -
1.55 -
1.5 T i . .
-O.8 -o.7 -0.6 -0.5 -o.4 -o.3

1090s)
Figure 2-7: Dependence of SIN on integration time.

 
    

Iog(S/N)

 

 

 

Figure 2-7 shows the relationship between the logarithm of the signal to
noise ratio and the logarithm of the measured signal (is) where the measured signal
is a function of integration time at a constant intensifier gain (0.2, a very low gain)
and constant light intenisty. As integration time increases, so does the signal. Here
the measured slope is close (0.93) to unity. At low gains blank noise is important.
2.3.3. Determination of linear range

The range over which absorbance measurements are linearly related to the
absorption properties of samples in the optical path was determined in two ways.
First, a series of neutral density filters with known absorbances across a broad
region of the spectrum were inserted into the optical path. Absorbance as
measured by the diode array at all wavelengths between 500 and 700 nm was

compared to the known absorbance of the filters.
46

 

  
   

y = 0.98x - 0.0026
R2 = 0.997

 

Measured Absorbance
N

 

 

 

0 I A i 1
0 1 2 3 4

Nominal Absorbance
Figure 2-8: Linearity of measured absorbance with the known absorbance of

a series of neutral density filters. The linear regression shown was calculated
from all absorbances up to and including one absorbance unit.

As shown in Figure 2-8, the array shows linearity for absorbances at least
up to one absorbance unit. Above this point stray light causes a negative deviation
from Beers law. As a second check of linearity, a series of chromium (III)
solutions were passed through the observation cell of the stopped-flow and their
spectra recorded. A calibration plot of absorbance at 569 nm versus concentration
was generated and found to be linear to absorbances of greater than 1.50 (the

absorbance of the solution with the highest concentration).

47

2.4.

(1)

(2)

(3)

REFERENCES

Beckwith, P. M.; Crouch, S. R. "An automated stopped-flow
spectrophotometer with digital sequencing for millisecond analyses" Anal.
Chem. 1972, 44, 221-227.

Stewart, J. E. "Flow deadtime in stopped-flow measurements" Dionex
Application Notes , 1-4.

Ingle, J. D., Jr.; Crouch, S. R. Spectrochemical Analysis ; Prentice Hall:
Englewood Cliffs, NJ, 1988.

48

CHAPTER 3

INITIAL SIMULATION STUDIES:

STUDY OF THE EFFECT OF EXPERIMENTAL VARIABLES

The sciences do not try to explain, they hardly even try
to interpret, they mainly make models. By a model is
meant a mathematical construct which, with the
addition of certain verbal instructions, describes
observed phenomena. The justification of such a
mathematical construct is solely and precisely that it is
expected to work
--John Von Neumann

I do not fear computers. I fear the lack of them.
--Isaac Asimov

In order to understand the factors that limit the accuracy with which an
analyte can be determined in a sample using kinetic-spectrophotometric data, a
systematic study of the effect of an array of experimental parameters on the
accuracy of a kinetic-spectrophotometric determination was performed. Because
of the impracticality of performing the number of experiments needed to complete
the study, simulated experiments were used. In these simulations experimental
parameters were varied and the effect of these variations on the accuracy of a
multicomponent (in this case, a two-component) kinetic-spectrophotometric

determination was noted.

49

3.1. EXPERIMENTAL

3.1.1. Generation of Simulated Data

Multicomponent kinetic-spectrophotometric data were simulated using a
program written in MATLAB (The Math Works, Natick, Mass). The algorithm
generates kinetic profiles for parallel second order reactions (first order in analyte
and first order in reagent) by numerically solving the appropriate differential
equations. The program mulgen_a which was used for this purpose is given in the
appendix.

For most of the simulation studies, synthetic spectra and rate constants were
used to generate kinetic-spectrophotometric data. The absorbance data were
generated by assuming that the analytes and reagent do not absorb in the spectral
region of interest and that the absorption spectra of the reaction products can be
modeled as overlapped Gaussian-shaped profiles. These synthetic spectra are
shown later in this chapter.

In all cases, adherence to Beer’s law was presumed for each component,
and the total absorbance at each wavelength was assumed to be the sum of the
absorbances of the components.

3.1.2. Data processing

Time dependent spectra were collected in triplicate and averaged, i.e., each metal
ion solution was reacted with PAR three times. The resulting three sets of kinetic-
spectrophotometric data were averaged. Data were mean-centered (i.e., the mean

of each variable vector was subtracted from each of its elements) before being
50

input to the appropriate algorithms. Multivariate calibration algorithms provided in
the PLS_TOOLBOX (Eigenvector Technologies, Manson, WA.) and run in
MATLAB were used to perform determinations.

3.2. METHODS FOR QUANTIFYING KINETIC AND SPECTRAL

DIFFERENCES FOR TWO COMPONENTS

In a kinetic-spectrophotometric determination, the accuracy with which an
analyte can be determined in unknown samples depends on the degree of kinetic
and spectral differentiation between the analyte and the other components of the
sample. In order to examine the effect of an array of experimental parameters on
both degree of kinetic and spectral differentiation and on the accuracy of a kinetic-
spectrophotometric determination, it was necessary to develop means for
quantifying the degree of kinetic and spectral differentiation between the analytes
in a mixture.

3.2.1. Methods for quantifying kinetic differences

In the following discussion, and indeed in most of this document, it is
assumed that in kinetic or kinetic-spectrophotometric determinations analytes react
with a common reagent to form different products at different rates. The plot of
concentration of product vs. time (or of absorbance vs. time, if either the analyte
or its product absorbs in the wavelength region of interest) is referred to as a

kinetic profile.

51

The amount of kinetic differentiation between two components can be
described by the ratio of the rate constants for the reactions of the analytes with
the reagent. As the ratio of the rate constants increases, so does the difference
between the rate constants and thus the difference in the kinetic profiles of the
reactions of the analytes.

The angle (6K) between the kinetic profiles of the two reaction products
was used as a measure of the degree of kinetic differentiation between the
analytes, and was calculated as the arccosine of the correlation between the kinetic
concentration profiles of the reaction products.

6K = arccos [1%.]
0' K1 ' 0K2
where K and K2 are the two kinetic profiles, 0x1 and 0112 are their standard
deviations, and (7sz is the covariance of K1 and K2. A kinetic angle of 0°
indicates that the kinetic profiles are completely correlated, while an angle of 90°
indicates complete independence. Practically, as the kinetic angle decreases, the
degree of kinetic differentiation decreases and so does the amount of kinetic
information that can be used to differentiate the analytes.

The calculation of figures of merit for multivariate data has been
described”. Using these methods, which state that the net analyte signal for the
kth analyte in a mixture can be calculated as the portion of the data orthogonal to

the data due to all of the other components of the mixture, net analyte signal and

52

selectivity can be easily calculated from the mixture data if the pure component

response of the components of the mixture are known:
v = ( I - XX“ )1:

where v is the portion of data vector (pure component response) u that is
orthogonal to data matrix X (which consists of the pure component responses of
all of the other components), and X+ is the pseudoinverse of X. The norm of v,
Ilvll, is the net analyte signal of the analyte. The selectivity, s, is calculated as the
ratio of the norms of the orthogonal part of the pure component response and the
pure component response,

_|lvl|
llull

The kinetic net analyte signal and kinetic selectivity for each analyte were
calculated from the concentration kinetic profile for the analyte and the sum of the
concentration kinetic profiles of all components of the mixture.

The rate constant ratio is a measure of the similarity of the reaction rates,
and is insensitive to factors other than the rate constant which affect the kinetic
profile. The kinetic angle is sensitive to factors other than the rate constant; indeed
the length of time for which the reaction is observed has a large effect on the
kinetic angle. A plot of kinetic angle vs. rate constant ratio (with other factors that

contribute to the kinetic angle held constant) is shown in Figure 3-1.

53

 

50.0
45.0 4
40.0 4
35.0 a

le

5’ 30.0 —
25.0 —
20.0 ~

Kinetic An

15.0 _
10.0 ~
5.0 ~

 

 

 

0.0 r T 1 r
0 2 4 6 8 10

Rate Constant Ratio
Figure 3-1: Kinetic angle as a function of rate constant ratio. The fraction of

the slower reaction observed was held constant at 90%, and the number of
data points was held at 100.

Neither the rate constant ratio nor the kinetic angle provide information about
which analyte is more resolved. The kinetic net analyte signal does provide this
information. For the case where the rate constant ratio is varied (by varying the
faster rate constant) and the length of time the reaction is observed is held constant
(at the time at which the slower reaction has reached 90% completion), the plot of

kinetic net analyte signal vs. rate constant ratio is as shown in Figure 3-2.

54

 

5.0E-06
4.5E-06 ~
4.0E-06 ~
3.55-06 -
3.0E-06 i
2.5E-06 _
2.0E-06 r
1.5E-06 *

Kinetic Net Analyte Signa

1.0E-06 a
5.0E-07 4

 

 

 

 

0.0E+00 1 . , g

Rate Constant Ratio

Figure 3-2: Kinetic net analyte signal for the slower (squares) and the faster
(circles) reactants in a two-component mixture. The fraction of the slower
reaction observed was held constant at 90%, and the number of data points
was held at 100.

More information is available for the slower analyte than the faster analyte. The
analytical signal of the slower component is larger than that of the faster
component since it reaches a higher equilibrium concentration. When all other
variables are equal and the concentration sets shown later in Figure 3-3 are used,
the slower analyte can be more accurately determined than the faster analyte

because of its higher net analyte signal.

55

3.2.2. Methods for quantifying spectral differences

The degree of spectral overlap (or more precisely, the amount of spectral
differentiation) between the analytes was quantified by calculating the angle
between the two spectra. This angle, referred to in this work as the spectral angle
(05), is defined as the arccosine of the correlation between the spectra (represented
by their respective molar absorptivities (81) so that the spectral angle is

independent of concentration, and so therefore also of time):

ass
65 =arccos ———'——2—
0511752

where 81 and 52 are the two spectral profiles, 0'51 and 0'52 are their standard
deviations, and 05152 is the covariance of S1 and S2, A spectral angle of 0°
indicates that the spectra are completely correlated, while an angle of 90° indicates
complete independence. Practically, as the spectral angle decreases, the degree of
spectral differentiation decreases and so does the amount of spectral information
that can be used to differentiate the analytes. The synthetic spectra used in the

simulations are shown with their spectral angles in Figure 3-3.

56

Spectral Angle=77.7 Spectral Angle=40.7

 

 

  

 

 

 

 

 

 

 

Spectral Angle=20.0 Spectral Angle=13.5

 

 

 

 

 

 

 

Spectral Angle=10.3 Spectral Angle=8.6

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

Spectral Angle=7.6 Spectral Angle=0.0
Spectral Channel Spectral Channel

Figure 3-3: Synthetic spectra used for simulation studies. Spectral angles
range between 77.7 and 0.

A spectral net analyte signal can be calculated in a manner similar to the
method for calculating the kinetic net analyte signal. It, like the kinetic net analyte
signal, has the advantage of proving information that reveals which analyte is
more resolved, and so should be more accurately determined. Figure 3-4, the
spectral net analyte signal for each analyte is plotted vs. the spectral analytes
between the analytes. The dashed line corresponds to the analyte whose spectrum

is depicted with a dashed line in Figure 3-3.

57

 

9.0E+05

8.0E+05 —

7.0E+05 *

6.0E+05 a

5.0E-1-05 r

4.0E+05 -

3.0E-1-05 a

Net Analyte Signal

2.0E+05 4

 

1.0E-1-05 r

 

 

 

ODE-+00 I 1 1 1
O 20 40 60 80 1 00

Spectral Angle

Figure 3-4: Spectral net analyte signal as a function of spectral angle. The
spectra used to generate this plot are shown in Figure 3-3.

The more strongly absorbing analyte (depicted with the dashed line) has a higher
net analyte signal. As the spectra become more similar, the difference between the
net analyte signals decreases as well. We would thus expect the more strongly
absorbing analyte to be determined with greater accuracy.

3.3. EFFECT OF EXPERIMENTAL VARIABLES

Data were generated for a collection of 12 standard calibration mixtures and
four unknown mixtures. The arrangement of these mixtures in the concentration
space of the two analytes is shown below. In all cases, the simulated data were

58

generated using 30 spectral channels (wavelengths); in most cases 100 time points

were generated (unless the number of time points was being varied).

 

 

 

 

5 T T I I I 1' T 1 fir
4.5 - -
A 4 ~ 111 a .
2
:1.
c 3.5 ~ 0 r
.2
on
g 3 - 111 111 111 111 .
c
m
g 4
O 2 5 O O
0
g 2 - 111 a a a -l
at
c
g 1.5 r- o _
.2
(D 1 b 111 111 -
0.5 '- -
O I 1 1 l 1 l 1 L 1
o 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Fast Analyte Concentration (11M)

Figure 3-5: Concentrations of calibration (stars) and unknown (open circles)
samples.

These data were used to generate calibration models with each of the
multivariate calibration techniques being investigated. These models were applied
to data from the four unknown mixtures, and the error of prediction for each

analyte was calculated as the relative standard error of prediction (RSEP)

59

 

 

% RSEP = i=1 .100

where C, is the true concentration of the analyte in sample i, C,- is the predicted
concentration of the analyte in sample i, and h is the number of samples.
3.3.1. Effect of kinetic and spectral angles

The kinetic resolution of the analytes was varied by changing the rate
constant of the slower analyte and by varying the neamess to completion of the
slower reacting analyte at the end of data collection (fraction of the reaction
observed). These two experimental parameters were found to influence the
accuracy of the determinations through their influence on the kinetic angle, and so
on the amount of kinetic differentiation observed between the analytes.

In all cases, the slower analyte was more accurately determined. Examining
the kinetic net analyte signal (Figure 3-2) reveals that this is expected. The slower
reacting analyte was also the more strongly absorbing analyte, and so it has a
higher spectral net analyte signal.

For all cases studied, principal component regression and partial least
squares regression produced predicted concentrations of similar accuracy.
Determinations using continuum regression were slightly more accurate. In every
case, multiple linear regression produced inferior predictions. PARAFAC
generally produced poor predictions, and multiway PLS (nPLS) in most cases

produced the most accurate predictions. The following plots illustrate the accuracy
60

of the predicted concentrations produced by each algorithm (expressed as the
%RSEP) as a function of kinetic and spectral angles.

Figures 3-6 through 3-9 shows the effect of varying kinetic and spectral
angles on the accuracy of a PLS, PCR, CR, and nPLS prediction, respectively.
PLS, PCR and CR all behave similarly. The accuracy of the nPLS prediction is
generally higher, but has the same dependence on the kinetic and spectral angles
as do the PLS, PCR and CR determinations.

The accuracy with which the faster analyte can be predicted decreases as
the spectral angle decreases. The slower analyte is mostly unaffected by a decrease
in spectral angle. At any given kinetic angle, the slower analyte has a larger kinetic
net analyte signal (Figure 3-2) and so its prediction is less affected by a decrease
in spectral differentiation because it is can rely more on kinetic information than
can the prediction of the faster analyte. Changes in the kinetic angle have little
effect at all but the lowest spectral angles, revealing that the regression algorithms
rely more heavily on spectral than kinetic differences. At very low spectral angles
(<10°), the amount of spectral information available becomes negligibly small,
and the prediction of both analytes depend on the kinetic information. In this case
decreasing kinetic angle produces marked increases in the error of both
predictions. The error of the prediction of the faster analyte increases more quickly
since less kinetic information about it is available. At very low spectral and kinetic

angles an accurate determination can not be performed.

61

Fast Analyte Slow Analyte

  
  

%RSEP (PLS)
.~
%RSEP (PLS)

M
0| 0

.5
(II

N .
I I l J

  

 

 

SpectralAngIo 4° Spectrum
KIneticAflab

 

Figure 3-6: Relative standard error of a PLS prediction as a function of the
kinetic and spectral angles.

Fast Analyte Slow Analyte

%RSEP (PCR)
1"
%RSEP (PCR)

 

 

   

30

‘0 Spectral Angle 4° Spectral Angle
Kinetic Angle Kinetic Angle

Figure 3-7: Relative standard error of a PCR prediction as a function of the
kinetic and spectral angles.

62

Fast Analyte Slow Analyte

%RSEP (CR)
1"

 

 

 

4° Spectral Angle

 

Kinetic Angle Kinetic Angle

Figure 3-8: Relative standard error of a CR prediction as a function of the

kinetic. and snectral angles.
Fest Analyte Slow Analyte

 

 

   

4.5 \ '
3.
9 9
a! 11
5 5
n. n. 2.
in ur
in m
a: a:
a? a9
1.5 \ '
0.5 4 ~
Spectral Angle 4° Spectral Angle
Kinetic Angle Kinetic Angle

Figure 3-9: Relative standard error of an nPLS prediction as a function of the
kinetic and spectral angles.

63

The effects of varying kinetic and spectral angles on an MLR determination
are shown in Figure 3-10. The same general trends observed for PLS, PCR, CR,
and nPLS can be seen here as well, though the magnitude of the errors is much

larger.

Feet Analyte Slow Analyte

xnssp (MLR)
%nsep (MLR)

 

 

 

KineticAnge

 

Figure 3-10: Relative standard error of an MLR prediction as a function of
the kinetic and spectral angles.

Figure 3-11 shows the effect of varying kinetic and spectral angles on the
accuracy of a PARAFAC determination. At kinetic or spectral angles of less than
10° neither analyte can be accurately determined, though when both angles are
high the accuracy of the predictions rivals that of CR and nPLS. PARAFAC’s
flaw, then, is its need for resolution in both, rather than one, dimension.

anwme fimkflm

a a
g a
$25 :2
n. n.
Lu Lu
U) U)
I: a:
x a

2
1.5
1

_‘
- in

O

or
9
or
1

 

 

Figure 3-11: Relative standard error of a PARAFAC prediction as a function
of the kinetic and spectral angles.

In the following sections, the results shown will be those produced by
continuum regression and multiway partial least squares regression.
3.3.1.1. Contributors to kinetic angle

The effect of decreasing kinetic information is not evenly distributed over

the two analytes. As already mentioned and seen in Figures 3-6 to 3-11, the

65

prediction of the faster-reacting of the two components is affected to a much larger
degree than is the prediction of the slower-reacting analyte. Using CR or nPLS
(the techniques to which it can be assumed that most assertions in the next several
sections refer) the slower-reacting analyte can be successfully determined in a
two-component mixture if either angle is greater than 8°; the faster-reacting
component requires that at least one angle be 15° for a determination to succeed.
The faster component is also more sensitive to small spectral angles than is the
faster component.

The fraction of the slowest reaction observed (defined as the fraction of the
slower analyte molecules that have reacted at the time observation of the reaction
ceases) was changed by varying the length of time over which the reaction was
monitored. It was found that kinetic angle is a function of both the ratio of the rate

constants and the fraction of the slower reaction observed.

66

 

 

Fraction Reaction Observed "8‘9 00mm “360

Figure 3-12: Kinetic angle as a function of rate constant ratio and fraction of
the slower reaction observed.

The relationship between kinetic angle and these two parameters is shown in
Figure 3-12. Generally, kinetic angle increases as either the fraction of the slower
reaction observed increases or the ratio of the rate constants increases. The effect
of increasing rate constant ratio at a constant 90% reaction observed was shown in
Figure 3-1. The same relationship observed there can be seen in Figure 3-12. At
low fractions of the reaction observed, the relationship is much the same, though
the total change is kinetic angle is smaller. At any constant rate constant ratio, an
increase in the fraction of the slower reaction observed results in a fairly linear

increase in kinetic angle.
67

3.3.2. Effect of number of data points acquired (time points)

The number of time points acquired was varied while the time for which
data was taken was held constant (i.e., the rate at which spectra were acquired was
varied). In general it was found that varying the number of spectra between 10 and
100 produced no appreciable change in the accuracy of the determination.

3.3.3. Effect of instrumental noise

Heteroschedastic (i.e., not uniform across the spectrum) instrumental noise
proportional to absorbance at each wavelength was added to all simulated data. In
most cases, 1% noise was added; in a few cases this value was varied, and the
effect of varying levels of noise on the accuracy of kinetic-spectrophotometric
determinations was explored. It was found that as the instrumental noise was
varied with kinetic angle (at a low spectral angle) the accuracy of the
determination using the multivariate calibration techniques decreased uniformly at

all kinetic angles as the noise level increased (see Figure 3-13).

68

Fast Analyte Slow Analyte

%RSEP (on)
%RSEP (on)

 

 

 

Figure 3-13: Relative standard error of a CR prediction as a function of the
kinetic angle and instrumental noise.

Determinations using multiway PLS were more tolerant of instrumental noise at

high kinetic angles as can be seen from Figure 3-14 below.

69

Fast Analyte Slow Analyte

%RSEP (nPLS)
%nsep (nPLS)

 

 

 

4o lfinetic Angle

 

Figure 3-14: Relative standard error of an nPLS prediction as a function of
the kinetic angle and instrumental noise.

When the level of the instrumental noise was varied concurrently with the spectral
angle (at a constant low kinetic angle), it was found that the effect of the
instrumental noise was less pronounced at high spectral angles. Figure 3-15 shows

this effect.

70

Fast Analyte Slow Analyte

 

 

E? E‘
e 9
a a
3: (n
a: a:
a! 32

Spectral Angle

 

Figure 3-15: Relative standard error of a CR prediction as a function of the
spectral angle and instrumental noise.

71

This same phenomenon was observed with nPLS, though the magnitude of the

errors was lower. One can see from Figure 3-16 that errors are somewhat lower

than from Figure 3-15.

Fast Analyte Slow Analyte

%nsep (nPLS)
%nsep (nPLS)

 

 

  

4O

 

Spectral Angle Spectral Angle
% Noise 96 Noise

Figure 3-16: Relative standard error of an nPLS prediction as a function of
the spectral angle and instrumental noise.

It is apparent that instrumental noise (as defined and added to the data) has
more influence on the spectral differentiation between the analytes than on the
kinetic differentiation, i.e., noise tends to blur spectral differences, especially
when the spectra are similar. Techniques that use unfolded data are unable to

ignore the spectral data and simply focus on the kinetic data; they are thus

72

unaffected by increased kinetic differentiation. Multiway PLS is able to more
exclusively use the kinetic data and so is able to take advantage of increasing
angles. At constant kinetic angles, all techniques perform better with high noise
levels if the spectral angle is large, though multiway PLS is (as usual) more
accurate.
. 3.3.4. Effect of rate constant fluctuations

Rate constant fluctuations during the course of the reaction (which could be
caused by temperature fluctuations or other external perturbations) were simulated
by allowing the rate constant to vary with a Gaussian distribution centered on the
true value and having a standard deviation proportional to a percentage of the rate
constant.

Large rate constant fluctuations produce acceptable errors if either the
spectral or kinetic angles are moderately large. Indeed, at spectral angles of 20° or
larger, rate constant fluctuations up to 15% had no noticeable effects, as can be

seen from Figures 3-17 and 3-18.

73

Fast Analyte Slow Analyte

 

 

   

10 ~ -
E E
9, e
n. o.
m u:
w w
c: at
32 a2
5
4o Kinetic Angle
‘36 Rate Constant Fluctuation % Rate Constant Fluctuation

Figure 3-17: Relative standard error of a CR prediction as a function of the
kinetic angle and rate constant fluctuation.

74

Fast Analyte Slow Analyte

 

 

   

1 5 \ 15 a
10 q ' i
E E
9, 9,
LL a.
tu Lu
«n 0:
cc 1:
a? a!
5
Spectral Angle Spectral Angle
96 Rate Constant Fluctuation % Rate Constant Fluctuation

Figure 3-18: Relative standard error of an nPLS prediction as a function of
the kinetic angle and rate constant fluctuation.

From comparing Figures 3-13 through 3-16 with Figures 3—17 and 3-18,
we can see that the addition of instrumental noise (as defined above) has a much
larger effect on the accuracy of determinations than do rate constant fluctuations.
For this reason, it was decided to focus more effort on decreasing the instrumental
noise level of the experimental system than on connolling factors that could lead

to rate constant fluctuations.

75

3.4. EXPLORATION OF EFFECT OF EXPERIMENTAL
VARIABLES ON SIMULATIONS OF THE GA(III) -- NI(II)
SPECTRAL SYSTEM

For these studies, experimentally determined spectra (Figure 3-19) of 4-(2-
pyridylazo)-resorcinol (PAR) and its complexes with gallium (III) and nickel (II)
were used to generate the simulated data. The procedure by which these spectra

were acquired is given in chapter four.

 

2.5- ..... ,

.0
.0
o

1.5— ‘.".. -4

Absorptivity (M cm)‘1

 

 

 

 

Wavelength (nm)

Figure 3-19: Absorption spectra of PAR and its Ni(II) and Ga(III) complexes
used for simulation studies.

76

The spectral angle between the analytes is 6.3; the spectral net analyte signal for
the Ni(H)-PAR complex is 26% higher than that for the Ga(IH)-PAR complex.
This indicates that it should be much more accurately predicted. Ni(II) is the faster
reacting analyte, and so always has a lower kinetic net analyte signal. In practice,
the much higher spectral net analyte signal overwhelms the slightly lower kinetic
net analyte signal. It is thus expected that Ni(II) should be determined more
accurately than Ga(III).

Using the measured spectra, the remaining experimental parameters were
varied. The simulations were carried out in the same manner as the previously
described studies.

3.4.1. Effect of Kinetic Angle

In this highly spectrally overlapped system, it is not unexpected that the
kinetic angle should affect the accuracy of kinetic-spectrophotometric
determinations (see Figures 3-6 through 3-11). The multivariate calibration
techniques are adversely affected by factors that lower the kinetic angle. This is
seen in Figure 3-20 which shows that errors below 5% can be achieved for the
faster (less spectrally overlapped) analyte under most kinetic conditions, i.e., with
all but the lowest kinetic angles (or as shown here, with all combinations of rate
constant ratios and fractions of the slower reaction observed save those with low
(<3) rate constant ratios and less than 40% of the slower reaction observed). The

slower(more spectrally overlapped) analyte is less accurately determined; errors

77

near 5% can be determined only when at least 50% of the slower reaction is

observed.

Fast Analyte Slow Analyte

 

 

‘0 2°\ . ..............
E E
0- n- ...........
u’ i.“
m (D
m m
32 32
5 1o
0
2
Ram .......................
conmm giant

  

  

Ratio

   

6 0.8 05 0.4 0.2 s 0.5 03 0.4 0.2
Fraction Reaction Fraction Reaction
Observed Observed

Figure 3-20: Relative standard error of a CR prediction as a function of the
rate constant ratio and the fraction of the slower reaction observed. The
Ga(III)/Ni(H) spectra shown in Figure 3-19 were used. The spectral angle was
6.3.

Multiway PLS is more able to use the limited kinetic and spectral information
available in this system. As such, the accuracy of its predictions is largely
unaffected by the parameters that affect the kinetic angle. Looking back at Figure
3—1 1, it can be seen that a spectral angle of ~6, the kinetic angle has little effect.
Figure 3-21 shows that neither the rate constant ratio nor the fraction of the slower

78

reaction observed had an appreciable effect on the prediction of either analyte. The

faster (less spectrally overlapped) analyte is predicted with greater accuracy.

Fast Analyte Slow Analyte

_.

o 20“;

%RSEP (nPLS)
%RSEP (nPLS)

U!

.a

O
1

 

 

         

0
Rate 2 Rate ......................
Constant Constant “““
Ratio Ratio . -
6 0'8 05 0.4 0.2 6 0‘8 05 0.4 0.2
Fraction Reaction Fraction Reaction
Observed Observed

Figure 3-21: Relative standard error of a CR prediction as a function of the
rate constant ratio and the fraction of the slower reaction observed. The
Ga(III)/Ni(H) spectra shown in Figure 3-19 were used. The spectral angle was
6.3.

79

3.5.

(1)

(2)

(3)

REFERENCES

Faber, K.; Lorber, A.; Kowalski, B. R. "Analytical figures of merit for
tensorial calibration" J. Chemometr. 1997, 11, 419-461.

Lorber, A.; Faber, K.; Kowalski, B. R. "Net analyte signal calculation in
multivariate calibration" Anal. Chem. 1997, 69, 1620-1626.

Messick, N. J.; Kalivas, J. H.; Lang, P. M. "Selectivity and related
measures for nth order data" Anal. Chem. 1996, 68, 1572-1579.

80

CHAPTER 4

DETERMINATION OF GALLIUM (III) AND NICKEL (II)

No man ’5 knowledge here can go beyond

his experience.
«John A. Locke

The determination of nickel (II) and gallium (IH) using their reaction with
4-(2-Pyridylazo)-resorcinol (PAR) was carried out under a variety of experimental
conditions. Kinetic, spectrophotometric, and kinetic-spectrophotometric
determinations were performed. Several of chemometric methods were used in the
determinations.

PAR is a highly sensitive photometric reagent that reacts with a variety of

metal ionsl. It is a triprotic weak acid whose dissociation progresses as1:

H3L“ —,£_='—>H,L—.i’—+m; ;EE—HLZ'
pde pH 3~5.5 pH 6~12.5 pH>12.5

Q—N =N—IQ-OH
0

Figure 4-1: Acid dissociation of PAR

In all of the work described in this document, PAR exists in its singly protonated

form. PAR complexes with metal ions as either a bi- or tridentate ligand. The two

81

metal ions whose determination is discussed in this chapter, Ga(III) and Ni(II),

both form complexes where two PAR molecules react with one metal ionl.

Niz*+2 PAR -,2 Ni PAR,
Ga3+ +2 PAR -‘-—-‘ Ga PAR2

Gallium (III) and nickel (II) were chosen for several reasons. Both react
with PAR, and both are soluble in the buffer solutions used in these studies. They
also present a highly challenging system. The rates at which they react with PAR
are very close together, and the absorption spectra of their reaction products with
PAR are quite similar.

The use of PAR as a spectrophotometric reagent for the determination of
gallium and especially of nickel has been reported in the literature”. Other
workers have described the determination of nickel or gallium using a
chromatographic separation and either a pre-concentration step involving PAR or
a post-column reaction with PAR10'13. There have been a few reports of kinetic

determinations of nickel or gallium using PAR as a reagent14’l5.
4.1. EXPERIMENTAL
4.1.1. Solution Preparation.

All solutions were prepared with distilled water and reagent grade
chemicals. Buffers of pH 8.5 and 7.0 were prepared from sodium borate and
sodium phosphate, respectively, and adjusted with nitric acid. All working

solutions were prepared in one of these buffers.

82

The equilibrium studies were performed using a set of three calibration sets.
The single component calibration and unknown solution sets contained six and
four solutions, respectively, and are described by Tables 4-1 and 4-2.
Table 4-1

Single component calibration and unknown sets for
the equilibrium determination of Ni(II)

Ni(II) concentration (uM)

 

 

 

 

Solution # Calibration Set Unknown Set
1 0.06 2.00

2 1.60 4.00

3 2.60

4 3.60

5 4.60

6 5.60

Table 4-2

Single component calibration and unknown sets for
the eguilibrium determination of Ga(III)

Ga(III) concentration (1.1M)

 

 

 

Solution # Calibration Set Unknown Set
1 1.60 2.00

2 2.60 4.00

3 3.60

4 4.60

5 5.60

6 6.60

 

The equilibrium multicomponent determinations were performed using a solution
set containing eight calibration solutions and two unknown solutions. These
samples were reacted with 1mM PAR at a pH of 7.0. The position of these

samples in concentration space is shown in Figure 4-2.

83

 

 

 

 

5 I If r r T I I I I
4.5 L * 4
4 ' r
A are
33.5 » .
§ 3 - 4
Z
I.—
o o
c 2.5 .- * 'i
.9
iii
5
«E 2
8
o
S 1.5 r- * * -
O
1 - _
0.5 P .
0 1 r l i l l i- t r 1 i ‘1'
o 0.5 1 1.5 2 3 3.5 4 4.5 5

2.5
Concentration of Ga(III) (pM)

Figure 4-2: Concentrations of calibration (stars) and unknown (open circles)
samples used for the two-component equilibrium determinations.

For the pH 8.5 kinetics studies, thirteen calibration mixtures and four
unknown mixtures were made. These were mixed and reacted in the stopped flow
system with a solution of 10’3 M PAR. The calibration and unknown sets used for

the pH 8.5 kinetics studies are can be seen in Figure 4-3.

84

 

 

 

 

60 I I I I I I I T I
a a a
50’ i
O
3 4 4
340 - ° -
2 s 4
'5
c 30 '- ..
.9
:6: o a
E
c 4
c 2° 0
8 a at
10L J
I’ § §
0 l l J l l l P l l
20 25 3O 35 40 45 50 55 60 65 70

Concentration of Ga(lll) (uM)

Figure 4-3: Concentrations of calibration (stars) and unknown (open circles)
samples used for the ph 8.5 kinetic and kinetic-spectrophotometric
determinations of Ga(III) and Ni(H).

The pH 7.0 studies were performed using a similar, but slightly different set
of solutions. Sixteen calibration and four unknown solutions were mixed and
reacted in the stopped flow system with a solution of 10'3 M PAR. The location of

the mixtures in the concentration space of the analytes is given in Figure 4-4.

85

 

 

 

20 r I I T T I I I I
18— J
16- ~
A is a a it
V
%12- a a as a 1
._ o
O
c 10* O 'l
3% e a it a
l-
E 8
o
g 6" * fi 4.. * -(
4h— -r
2* .
O L 1 l l l 1 l l L
0 5 10 15 20 25 3O 35 4O 45 50

 

Concentration of Ga(III) (pM)

Figure 4-4: Concentrations of calibration (stars) and unknown (open circles)
samples used for the ph 7 .0 kinetic and kinetic-spectrophotometric
determinations of Ga(III) and Ni(II).

4.1.2. Spectrophotometric (Equilibrium) Data Collection

Gallium (III) and Nickel (II) were reacted with PAR in a pH 7.0 phosphate
buffer. The reactions were allowed to proceed to equilibrium, and spectra were
obtained. Data were collected in two wavelength ranges (500-550nm and 500-
600nm) using a Hitachi U-4001 UV-visible spectrophotometer. A one centimeter
quartz cuvctte was employed as a sample holder. A pH 7.0 phosphate buffer was
used as a blank. Spectra were collected at 201 wavelengths between 500 and 600

nm, and 101 wavelengths between 500 and 550 nm..
86

4.1.3. Kinetic-spectrophotometric Data Collection

Data were collected using a home-built stopped—flow apparatus interfaced
to a thermoelectrically cooled Tracor Northern (Model TN-6123) 512 element
intensified diode array (Tracor Northern, Philadelphia, PA) configured to acquire
spectra in the 400-800 nm range as described in chapter 2.

In the wavelength range of interest (520-560 nm), absorbance was
measured at 52 equally-spaced wavelengths. In cases where the entire 520-560 nm
spectral region was not used, a subset of the 52 wavelengths was employed.
Specifically, in the range 540-560 nm, 26 wavelengths were used.

At pH 8.5, kinetic information was obtained by acquiring 26 spectra at a
rate of 5.0 scans per second for a total acquisition time of 5.2 seconds. At pH 7.0,
100 spectra were acquired at a rate of 7.575 scans per second over a total
acquisition time of 13.07 seconds.

4.1.4. Data processing

Time dependent spectra were collected in triplicate and averaged, i.e., each
metal ion solution was reacted with PAR three times. The resulting three sets of
kinetic-spectrophotometric data were averaged. Data were mean-centered (i.e.,
the mean of each variable vector was subtracted from each of its elements) before
being input to the appropriate algorithms. Multivariate calibration algorithms
provided in the PLS_TOOLBOX (Eigenvector Technologies, Manson, WA.) and

run in MATLAB were used to perform determinations.

87

4.2. DETERMINATION OF RATE CONSTANTS AND

ABSORPTION SPECTRA

Experiments were performed to determine pure component spectra and rate
constants for the reaction products of the reaction of Ni(H) and Ga(III) with PAR.
The pure spectrum of PAR was subtracted from observed absorbance versus time
data resulting from the reaction of PAR with a single metal ion. These subtracted

data were fit to the equation16’17:
A. = A... -(A.. ‘A0)’e-k"t
The pseudo-first order rate constant, k’, was calculated from data at several
wavelengths, and an average value was computed. The second-order rate constant
was calculated from this average k’ and the known excess concentration of PAR as
k = k’[PAR]
From the fit values of A”, A0, and k’, the initial rate of the reaction was calculated

at each wavelength:
6 ,
A‘ at = k (AW _ A0)
The initial rate was plotted versus concentration. The molar absorptivities of the

reaction product at each wavelength were computed from the slope of this plot:

slope
kl

 

These values were then used in the simulations described in other chapters.

88

4.3. RESULTS OF THE DETERMINATION OF GALLIUM (IH)
AND NICKEL (II)

A spectrophotometric (equilibrium) determination of Ga(III) and Ni(II) was
performed, and the ability of the various chemometric algorithms to accurately
predict the concentration of the analytes in unknown mixtures was examined. Rate
constants and pure component spectra were obtained experimentally for the
reactants and products of the reactions of Ni(II) and Ga(III) with PAR. Kinetic-
spectrophotometric determinations of Ga(III) and Ni(II) were performed. These
determinations were then compared to the spectrophotometric (equilibrium)
determination and to a kinetic determination carried out at a single wavelength.
4.3.1. Spectrophotometric (Equilibrium) Determination of Gallium (III) and
Nickel (II)

Continuum regression and partial least squares regression both produced
acceptable results for single-component equilibrium determinations. The relative
standard errors of prediction (as defined in chapter two) for each determination are

shown in tables 4—3 and 4-4.

 

 

Table 4-3
Spectrophotometric (equilibrium) determination of N i(II)
Method % RSEP % RSEP
(500-550 nm) (500-600 nm)
CR 1.2 1.2
PCR 20.4 20.4
PLS 1.2 1.2
MLR 1.6 1.9

 

89

Table 4-4
Spectrophotometric (emiilibrium) determination of Ga(III)

 

 

Method % RSEP % RSEP
(500-550 nm) (500-600 nm)

CR 8.6 8.3

PCR 30.6 36.8

PLS 8.6 8.3

MLR 19.6 21.0

 

The determination of Ni(II) produced relative standard errors of prediction
of approximately 1.2% for both wavelength ranges when CR and PLS were used.
PCR produced inaccurate predictions for reasons which are not altogether clear.
Ga(IH) was less accurately predicted than was Ni(II). Because the molar
absorptivity of the Ni-PAR complex is much higher than that of the Ga-PAR
complex (see chapter 3), more accurate predictions of N i(II) were expected.

The determination of Ga(III) produced slightly lower errors for the larger
wavelength range (8.3%) than for the smaller wavelength range (8.6%) when CR
and PLS were used. The addition of more wavelengths decreases the error of the
Ga(III) determination more than it does the error of the Ni(H) determination
because although the additional wavelengths add much more noise than
information about the analytes, the algorithms used (CR and PLS) are able to
extract the small additional amount of information present in the added
wavelengths from the background noise. It is interesting to note that the same
effect is not seen when principal component regression or multiple linear
regression is employed. It is probable that the ability of continuum regression and
partial least squares regression to place lower weights on variables (wavelengths)

90

that vary substantially but are not correlated to the concentrations of the analytes is
responsible for their more accurate predictions. The Ga(III) determination is most
strongly impacted by this phenomenon, as the lower absorptivity of the Ga-PAR
complex results in smaller observed absorbances (and thus less information related
to the concentrations being determined).

The multicomponent equilibrium determination of Ga(III) and Ni(H)
showed significantly higher errors than the single-component determinations.

These results are summarized in table 4-5.

Table 4-5
Multicomponent spectrophotometric (equilibrium)
determination of Ni(II) and Ga(III)

 

 

500-550 nm 500-600 nm
Method % RSEP % RSEP % RSEP % RSEP
(Ga) (Ni) (Ga) (Ni)
CR 26 6 21 6
PCR 22 6 21 8
PLS 21 6 21 7
MLR 23 4 24 5

 

As expected, Ni(II) was determined with greater accuracy than was Ga(III). The
spectra are highly overlapped, and so there is little information present that can be
used to differentiate the two analytes. Thus, it is not surprising that the additional
information that is present in the larger wavelength range results in lowered error
for determinations using all algorithms. Figures 4-5 and 4-6 are plots of the

predicted vs. actual concentrations.

91

 

5.5 I I 1 I I I I

Predicted Concentration of Ga(III) ( pM)

 

 

 

l l

 

0.5 J— 1 1 1 1
1 1.5 2 2.5 3 3.5 4 4.5 5

Actual Concentration of Ga(III) (11M)

Figure 4-5: Plot showing the predicted vs. actual concentrations of Ga(III) for
a two-component equilibrium determination. The circles show the predicted
concentrations. The solid line has a slope of unity and represents a prediction
with no error. The dashed lines show i10% error tolerances.

92

 

N on b 01 O)
r r I 1 r
1 1 1 1

Predicted Concentration of Ni(ll) ( 11M)

_a
I
l

 

 

 

O A 41 l l l

o 1 2 a 4 s 6
Actual Concentration of Ni(ll) (11M)

Figure 4-6: Plot showing the predicted vs. actual concentrations of Ni(II) for
a two-component equilibrium determination. The circles show the predicted
concentrations. The solid line has a slope of unity and represents a prediction
with no error. The dashed lines show i10% error tolerances.

4.3.2. Kinetic Determination of Gallium (III) and Nickel (11)

Using the data collected during the kinetic-spectrophotometric
determinations, a single-wavelength (550 nm) kinetic determination of Ga(III) and
Ni(II) was performed at both pH 7.0 and pH 8.5. The results of these

determinations are summarized in table 4-6.

93

Table 4-6
Multicomponent kinetic determination of Ni(II) and Ga(III)

 

 

pH=7.0 pH=8.5
Method % RSEP % RSEP % RSEP % RSEP
(GA) (Ni) (Ga) (Ni)
CR 35 26 10 6
PCR 30 20 10 6
PLS 35 19 10 4
MLR 47 31 12 9

 

The results at pH 8.5 (where a good deal of kinetic differentiation exists
between the analytes) are quite comparable with those from the kinetic-
spectrophotometric determination (table 4-7). There is little spectral differentiation
present in this system, and so the addition of more wavelengths adds little
additional information about the analytes. At pH 7 .0 (where much less kinetic
differentiation exists) the kinetic determination is clearly inferior to the kinetic-
spectrophotometric determination. The additional differentiation between the
analytes afforded by the spectral information is, in this case, necessary.

4.3.3. Kinetic-spectrophotometric determination of Gallium(III) and
Nickel(II)

Two different pH values and two wavelength ranges were used to
determine Ga(III) and Ni(II). The results of these determinations are presented in
table 4-7. As can be seen, errors of prediction at pH 8.5 were generally half those
at pH 7 .0. This can be rationalized in several ways. First, the kinetic angle at pH

8.5 (40.1) is much larger than the angle at pH 7.0 (12.6). This represents a large

94

increase in the amount of kinetic differentiation between the analytes. The spectral
angles at pH 8.5 and pH 7.0 are similar, but since both are small (~63), kinetic
differentiation is the major source of selectivity; it is thus expected that a larger

kinetic angle will result in a smaller error of prediction. Indeed, this is what is

 

 

observed.
Table 4-7
Multicomponent kinetic-spectrophotometric determination of N 101) and Ga(III)
540-560 nm 520-560 nm
pH Method % RSEP % RSEP % RSEP % RSEP
(Ga) (Ni) (Ga) (NiL
7.0 CR 19 17 17 18
PCR 18 18 20 19
PLS 20 17 19 17
MLR 50 12 11 20
nPLS 21 15 29 16
8.5 CR 11 6 14 8
PCR 9 5 l 1 6
PLS 9 6 14 6
MLR 31 5 29 8
nPLS 9 5 ll 6

 

The results of these determinations can be compared to the results of the
equilibrium spectrophotometric determination described previously (table 4-5).
The results of the kinetic-spectrophotometric determination performed at pH 8.5
show a marked improvement over the equilibrium results for all techniques except
multiple linear regression.

In comparing the multivariate calibration techniques, continuum regression,
partial least squares regression, principal component regression and multiway

partial least squares regression all produce predictions of similar accuracy.

95

Multiple linear regression sometimes produced slightly more accurate predictions,
but, more often, was clearly inferior. For the type of data studied, continuum
regression and principal component regression proved the most stable of the
multivariate calibration techniques. While not always the best choice, they were
rarely worse than any others and often significantly better. Multiway PLS (nPLS)
was often superior to the one way techniques, but also performed poorly on
occasion. The reasons for these failures are not clear.

The spectral region between 540-560 nm was found to contain the majority
of the spectral differentiation between the analytes. Increasing the spectral window
greatly increased the data processing time, but did not appreciably affect the error
of prediction.

The results are similar to what is expected based on the simulation studies
done using the Ga(III) and Ni(H) spectra and on the relative kinetic and spectral
net analyte signals (as discussed in chapter 3). Ni(II) was more accurately

predicted, as expected.

96

4.4.

(D

(D

G)

M)

(5)

(O

0)

(&

(%

REFERENCES

Shibata, S., "2-Pyridylazo compounds in analytical chemistry", in Chelates
in Analytical Chemistry; Flaschka, H. A., Barnard, A. J ., Jr, Eds; Marcel
Dekker: New York, 1972; Vol. 4.

Bobrowska Grzesik, E.; Grossman, A. M. "Derivative spectrophotometry in
the determination of metal ions with 4-(pyridyl-2-azo) resorcinol (PAR)"
F resenius J. Anal. Chem. 1996, 354, 498-502.

Cladera, A.; Gomez, E.; Estela, J. M.; Cerda, V.; Cerda, J. L. "Computer
method for the simultaneous kinetic determination of compounds in

mixtures based on the use of diode-array spectrophotometry" Anal. Chim.
Acta 1993, 272, 339-344.

Gomez, 13.; Estela, J. M.; Cerda, V.; Blanco, M. "Simultaneous
spectrophotometric determination of metal-ions with 4-(Pyridyl-2-
Azo)Resorcinol (PAR)" F resenius J. Anal. Chem. 1992, 342, 318-321.

Kolomiets, L. L.; Pilipenko, L. A.; thud, I. M.; Panfilova, I. P.
"Application of derivative spectrophotometry to the selective determination
of nickel, cobalt, copper, and iron(III) with 4- (2-pyridylazo)resorcinol in
binary mixtures" J. Anal. Chem. 1999, 54, 28-30.

Ni, Y. N. "Trace-metal determinations by spectrophotometry with a double
chromogenic system and a chemometric approach" Anal. Chim. Acta 1993,
284, 199-205.

Ridder, C.; Norgaard, L. "Simultaneous determination of cobalt and nickel
by flow- injection analysis and partial least-squares regression with outlier
detection" Chemometrics Intell. Lab. Syst. 1992, 14, 297-303.

Mori, I.; Kawakatsu, T.; Fujita, Y.; Matsuo, T. "Selective
spectrophotometric determination of gallium(III) with 2-(5-bromo-2-
pyridylazo)-5-diethylaminophenol in the presence of sodium dodecylsulfate
and Brij 35" Anal. Lett. 1999, 32, 613-622.

Taljaard, R. E.; van Staden, J. F. "Simultaneous determination of cobalt(II)

and Ni(II) in water and soil samples with sequential injection analysis"
Anal. Chim. Acta 1998, 366, 177-186.

97

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

Khalaf, K. D.; MoralesRubio, A.; DelaGuardia, M.; Garcia, J. M.; Jimenez,
F.; Arias, J. J. "Simultaneous kinetic determination of carbamate pesticides

after derivatization with p-aminophenol by using partial least squares"
Microchem J. 1996, 53, 461-471.

Chakrapani, G.; Murty, D. S. R.; Mohanta, P. L.; Rangaswamy, R.
"Sorption of PAR-metal complexes on activated carbon as a rapid
preconcentration method for the determination of Cu, Co, Cd, Cr, Ni, Pb
and V in ground water" Journal of Geochemical Exploration 1998, 63 , 145-
152.

Karve, M. A.; Khopkar, S. M. "Liquid-liquid-extraction of gallium with
high-molecular-weight amines from ascorbate solutions" Chem. Anal. 1993,
38, 469-476.

Ming, X. Y.; Wu, Y. H.; Schwedt, G. "HPLC analysis of V, Co, Fe, and Ni
By 4-(2-Pyridylazo)- Resorcinol, PAR, and H202 and studies on complex
properties influencing retention" F resenius J. Anal. Chem. 1992, 342, 556-
559.

Arruda, M. A. Z.; Zagatto, E. A. G.; Maniasso, N. "Kinetic determination
of cobalt and nickel byflow-injection spectrophotometry" Anal. Chim. Acta
1993, 283, 476-480.

Blanco, M.; Coello, J.; Iturriaga, H.; Maspoch, S.; Riba, J.; Rovira, E.
"Kinetic spectrophotometric determination of Ga(III)-Al(III) mixtures by
stopped-flow injection-analysis using principal component regression"
Talanta 1993, 40, 261-267.

Moore, J. W.; Pearson, R. G. Kinetics and Mechanisms ; Wiley: New York,
1991.

Mieling, G. E.; Pardue, H. L. "Kinetic method that is insensitive to
variables affecting rate constants" Anal. Chem. 1978, 50, 1611-1618.

98

CHAPTER 5

KINETIC-SPECTROPHOTOMETRIC DETERMINATIONS

IN SYSTEMS WITH NONLINEAR KINETICS

When you can measure what you are speaking about,
and express it in numbers, you know something about
it. But when you cannot- your knowledge is of a

meagre and unsatisfactory kind.
«Lord Kelvin

Most kinetic determinations are carried out under conditions such that the
kinetics of the reaction are pseudo first order in the analyte. In cases where
multiple analytes are determined, conditions are usually arranged such that the

analytes each react according to pseudo first order kinetics:

A+R -—) PA

B+R —-> PB

g:— = —kAAR = -k,’,A
k; = k AR

5%? = —kBBR = -k,’,B
k; = kBR

A7 = Ace-k}!

B, = B0 e4?"

Under these conditions, the concentration of the analytes (and of their products) at

any time t is linearly related to the initial concentration of the analytes. The

99

condition that must be met is that the concentration of the reagent be sufficiently
large that it effectively remains constant over the course of the reaction.

Several factors can cause kinetic non-linearity. Most obviously, a low
reagent concentration invalidates the assumptions necessary for pseudo first order
kinetics. In this case, the concentration of the analyte at any given time is no
longer linear with the initial concentration. For the case of a single analyte reacting

with the reagent:

 

A+R —> PA

1A- : —kAAR

dt

A, _ AOR, eprokAI-Roh!
R0

Here the concentration of the analyte at time t is a complex nonlinear function of
the initial concentration of the analyte and reagent. When two or more analytes are
present, a discreet solution for the concentration of an analyte at any time can not

be written; rather, the set of differential equations

%=—k,-A.R
%z—kB-B-R

dR
E=-(kr'A°R)-(ks-B°R)

must be solved numerically.
Some other conditions also result in nonlinear kinetics. Perhaps the most

common of these is the existence of synergistic effects. Here the rate of reaction of

100

one analyte is a function of the concentration of the other, even if the reagent is
present in large excess.

dA
—=—k ~AoR+ B
dt A f( )

In all of these cases, most traditional methods for performing kinetic
determinations fail.

Some work has been done in the area of performing kinetic determinations
under conditions where nonlinear kinetics prevail. Modified nonlinear regression
algorithms have been used].2 to determine analytes using second and third order
reaction schemes. Several workers have used artificial neural networks to process
kinetic data collected from reactions with nonlinear kinetics. Blanco and
coworkers3 used an artificial neural network to perform a determination of three
analytes. Two react with the reagent according to pseudo first order kinetics; the
third follows a complex multistep process that is nonlinear. The ANN produced
predictions of acceptable accuracy for all three analytes and outperformed both
PLS and PCR. In another study4, Blanco’s group used simulated data to perform a
detailed study of the use of PLS for nonlinear kinetic data. Here the nonlinearity
was added through the addition of a synergistic term (see above equation). In all
but the most nonlinear cases studied, the predictions returned by PLS were
acceptably accurate.

Blanco and coworkers5 have also explored the use of ANN s in situations

where the nonlinearity is introduced by having the analytes present in sufficiently

101

high concentrations that the reaction is pseudo first order in reagent. Benzlyarnine
and Butlyamine were determined through their reaction with salicylaldehyde. The
ANN outperformed PLS and predicted the concentration of the analytes with only
4% error.

Ventura et al.6 used an ANN whose inputs were PCA factors to perform
kinetic determinations in the presence of synergistic effects and the inherent
nonlinearity of the continuous addition of reagent technique. The results of the
experiments were quite impressive and the concentrations of the analytes were
predicted with good accuracy (errors of prediction of about 5%).

In this chapter a standardized way of reporting the degree of kinetic non-
linearity present in a system is developed. The results of simulations and
experiments performed under conditions of kinetic non-linearity are presented.

5.1. EXPERIMENTAL

5.1.1. Simulations
Simulation studies were performed in a manner similar to that described in
chapter 3. The degree of nonlinearity was controlled by varying the concentration
of the reagent, and was measured as an angle from linearity.
The angle from linearity (0,) was calculated by plotting the concentration
of the analyte at a fixed time vs. the initial concentration of the analyte for several
samples with different concentrations of the two analytes. The angle 0L is related

to the correlation between the initial and time-dependent concentrations.

102

0A A
HLA =arccos ° ‘

0A0 "5A.
where A0 and Al are the concentrations of analyte A for several samples at times 0
and t, respectively, 0A0 and 0A, are their standard deviations, and O'Aom is the
covariance of A0 and A.. An angle from linearity of 0° indicates complete
linearity, while an angle of 90° signifies complete nonlinearity. Practically, when
the system is linear, the plot is linear; when there is kinetic nonlinearity the
concentration of one analyte affects the concentration of the other at the fixed

time. The following figure shows this plot for a situation where linear kinetics

prevail.

103

 

 

 

 

m5
53-
g a
< s 2.5" "
n. O-'
° 2
C 0
go 2r- .1
g o
*-' E
C...
8" -l
C o 1.5"
o 3
0|-
3 1.- -4
0.5- -
0 1 1 1 1 1 1 1 1 1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Initial Analyte Concnetration (11M)

Figure 5-1: Example of a linear (pseudo-first-order) kinetic system. This data
was generated with a rate constant ratio of 1.7 and a kinetic angle of 10.2. The
reagent was in 281 fold excess The angle from linearity was 0.017.

Kinetic nonlinearity causes samples with the same initial concentration of an
analyte to have different concentrations of the analyte after a fixed time. Figure 5-
2 shows this effect. The effect of the concentration of the second analyte on the
kinetics of the first (because the second analyte uses up some reagent, thereby

changing the reaction rate for the analyte in question) can be clearly seen.

104

 

1.8 r T r r r r

1.6-

Analyte Concentration At Two Tune Constants (11M)

0.6 *-

 

 

 

l 1 l l J l

 

0.6 0.8 1.2 1.4 1.6 1.8

1
Initial Analyte Concentration (pM)

Figure 5-2: Example of a nonlinear kinetic system. This data was generated
with a rate constant ratio of 1.7 and a kinetic angle of 10.2. The reagent was
in 1.4 fold excess. The angle from linearity was 6.449.

5.1.2. Solution Preparation
Solutions of Ni(II) and Ga(III) were made in a pH 8.5 borate buffer as
described in chapter 4. Solutions of PAR at concentrations of 1 mM and 40 11M

were also made in the same buffer. The solutions prepared were positioned in the

concentration space of the analytes as shown in Figure 5-3

105

 

 

 

 

18 r r r r I r r r n
16“ "
* ‘I
14" O 4
A
2
:1.
v12l- * * ¥ *- ..
§ 0
Z
*10" O -
O
c ‘I ‘U * *
.9 8
H F 4
E
E O
8
C 6” 'I l- .(
O
U
4— d
2" .l
0 1 1 1 1 1 1 1 1 1
0 5 10 15 20 25 30 35 40 45 50

Concentration of Ga(lll) (11M)

Figure 5-3: Concentrations of calibration (stars) and unknown (open circles)
samples used for the kinetic-spectrophotometric determinations of Ga(III)
and Ni(II) under linear and nonlinear kinetic conditions.

5.1.3. Kinetic-spectrophotometric Data Collection

Data were collected using the stopped-flow apparatus and diode array
detector as described in chapter 4. One hundred spectra were acquired; the rate at
which spectra were acquired was dependent on the concentration of the PAR. For
the 1 mM PAR, 100 spectra were acquired over a period of 50 seconds; when the
40 11M PAR solution was used 100 spectra were acquired over a period of 200

seconds. For this study, a wavelength range of 500-550 nm was empirically

106

chosen. Absorbances were measured at 64 equally-spaced wavelengths in this
range.
5.1.4. Data processing

Time-dependent spectra were collected in triplicate and averaged, i.e., each
metal ion solution was reacted with PAR three times. The resulting three sets of
kinetic-spectrophotometric data were averaged. Data were mean-centered before
being input to the appropriate algorithms. Multivariate calibration algorithms
provided in the PLS_TOOLBOX (Eigenvector Technologies, Manson, WA.) and
run in MATLAB were used to perform determinations.

5.2. SIMULATION STUDIES INVOLVING SYSTEMS WITH

NONLINEAR KINETICS

Detailed simulation studies that explored the effect of nonlinear kinetics on
the accuracy of predictions generated by an array of chemometric techniques were
conducted. Two limiting cases were considered. In the first, the synthetic spectra
described in chapter 3 were used to generate data; in these spectra the reagent does
not absorb. The second case was one where the reagent does absorb. These used
the Ga(III)-PAR / Ni(H)-PAR spectra which were also used in the studies
described in chapter 3.

5.2.1. Systems where the reagent does not absorb

In the case where the reagent does not absorb, the only effect of kinetic

nonlinearity is the type of effect seen in figure 5-2. The slower analyte is slightly

more affected by the nonlinearity, as the faster analyte consumes some of the

107

reagent before the slower analyte has reacted to any appreciable extent. This effect
is exaggerated at high kinetic angles where the difference between the reaction
rates of the analytes is larger and the faster analyte consumes more of the reagent

before the slower analyte begins to react.
Fast Analyte Slow Analyte

I"
at
l

%RSEP (CR)
%RSEP (CR)

 

 

   

20
4° Kinetic Angle 4° Kinetic Angle
Angle From Unaarity Angle From Linearity

Figure 5-4: Relative standard error of a CR prediction as a function of the
kinetic angle and angle from linearity.

Figure 5-4 shows the effect of kinetic angle and angle from linearity on the
accuracy of a CR prediction. Error of prediction increases as the angle from
linearity increases and kinetic angle decreases. The concentration of the slower

analyte is predicted more accurately; this is expected, as when the determination is

108

limited by kinetic information the slower analyte is favored by the reaction scheme

(see chapter 3). The spectral angle is low (8.6), and so the effects of the kinetic

angle can easily be seen. The angle from linearity has a much larger effect at high

kinetic angle than low ones (for the reasons already discussed).

3.5—1.”.

u
L

96385? (nPLS)
to
or
1

to
1

   

i7.
1

 

-
.......

.
.....
.
-------
.......
.
......

0..
._..

2 1 20

4° Kinetic Angle

Angle From Linearity

 

Slow Analyte

 

‘ 4o
Kinetic Angle
Angle From Linearity

Figure 5-5: Relative standard error of an nPLS prediction as a function of the

kinetic angle and angle from linearity

Figure 5-5 shows the response of multiway PLS to the same conditions depicted in

figure 5-4. Here, it can be seen that the slower analyte is more affected by kinetic

non-linearity, especially at high kinetic angles. It can also be seen through

comparison with Figure 5-5 that multiway PLS produces more accurate

109

predictions than do the multivariate calibration techniques (of which CR is the
best).
At a constant (low) kinetic angle the interaction of spectral angle with

kinetic nonlinearity can be demonstrated.

    

 

 

 

 
     

Fast Analyte Slow Analyte
5 ~ 5 a
4.5 4 4.5 ~
4 a 4 ~
3.5 ‘1 3.5 ‘-
A 3 ~ A
a: a:
e 9
$ 2.5 ~ g 2.
w to
t: a:
a2 2 g 39
1.5 ~ 1
1 a
0 5 ~ 0.5 ~
0 J _ o o
2.5 2 1.5 1 50 2.5 2 1.5 1 ' 50
0.5 Spectral Angle 0.5 Spectral Angle
Angle From Linearity Angle From Linearity

Figure 5-6: Relative standard error of a CR prediction as a function of the
spectral angle and angle from linearity. The kinetic angle for this data was
~10°.

In figure 5-6, it can be clearly seen that at high spectral angles kinetic non-
linearity has little affect on the accuracy with which multivariate calibration
techniques can predict the concentration of the faster analyte. The prediction of

the slower analyte is more strongly impacted by the angle from linearity. At low

110

spectral angles, the kinetic information becomes important, and the slower analyte

is predicted with greater accuracy. The increase in error of prediction with

increasing angle from linearity can be seen for both analytes.

%RSEP (nPLS)
33 u

to
1

 

1

Fast Analyte

 

0-5 Spectral An 9
Ange From Unearity g

%RSEP (nPLS)
N
a: u
L l

N
J

1.5»“1

 

 

1 ' 5°
05 SpectralAngle

Slow Analyte

4“
fffff

.....
.....
~~~~~~
‘‘‘‘‘‘
........
........
.........
.......
..\.

 

 

Angle From Linearity

Figure 5-7: Relative standard error of an nPLS prediction as a function of the
spectral angle and angle from linearity. The kinetic angle for this data was

~10°.

Figure 5-7 depicts the response of multiway PLS to various spectral angles and

angles from linearity. The same general trends described above are visible in these

plots, but it is apparent that the errors of prediction are lower. Again, multiway

PLS proves superior to traditional multivariate calibration techniques.

11]

5.2.2. Systems where the reagent does absorb

When the reagent absorbs in the spectral region of interest the effect of
kinetic nonlinearity is complicated. In addition to a nonlinear relationship between
initial and fixed—time analyte concentrations, the non-constant reagent
concentration contributes a time-dependent rather than a constant spectral
background. This effect is becomes more pronounced as the reaction proceeds and
the reagent is consumed. This results in less accurate determinations of the slower
analyte, especially at high rate constant ratios where the faster reactant has used
more of the reagent before the slower analyte has reacted to any appreciable
degree. Figure 5-8 shows the results of a CR prediction for a system with the
gallium-nickel spectra described in chapter 3. The determination of the faster
analyte is not greatly affected by decreasing kinetic angle or reagent excess. The
determination of the slower analyte, however, suffers loss of accuracy as the

reagent excess decreases, especially at high rate constant ratios (kinetic angles).

112

Fast Analyte Slow Analyte

%RSEP (CR)
%RSEP (CR)

 

 

   

Reagent Excess

Figure 5-8: Relative standard error of a CR prediction as a function of the
kinetic angle and reagent excess. The gallium/nickel/PAR spectra (with the
PAR absorbing) were used to generate the data. The spectral angle for this

data was ~11°.

Figure 5-9 shows the results of applying nPLS to the same data used for Figure 5-

8. In general, nPLS seems more adversely affected by increasing non-linearity

than does CR.

113

Fast Analyte

%RSEP (nPLS)
%RSEP (nPLS)

 

 

   

Figure 5-9: Relative standard error of an nPLS prediction as a function of the
kinetic angle and reagent excess. The gallium/nickel/PAR spectra (with the
PAR absorbing) were used to generate the data. The spectral angle for this
data was ~11°.

5.3. DETERMINATION OF GALLIUM (III) AND NICKEL (11) IN A
SYSTEM WITH NONLINEAR KINETICS

A determination of Ga(III) and Ni(H) was carried out as described in
sections 5.1.2 through 5.1.4. The degree of nonlinearity was determined by
varying the concentration of PAR. Two PAR concentrations were used. At the
higher concentration (1.0 mM) there is an average reagent excess of 28.1 fold.

Here the excess of reagent is calculated by dividing the PAR concentration by the

114

average sum of the analyte concentrations. At the lower PAR concentration (40

M) the PAR is present in 1.2 fold excess.

Table 5-1: Kinetic-spectrophotometric determination of Ga(III) and
Ni(II) with two different values of reagent excess

 

 

1.0 mM PAR 40 11M PAR

Method % RSEP % RSEP % RSEP % RSEP
(Ga) (Ni) (Ga) (Ni)

CR 19 4 28 4
PCR 19 6 34 4
PLS 19 4 43 6
MLR 26 14 56 4
nPLS 21 5 49 4

 

The results of these determinations are presented in Table 5-1. Inspection of
the data reveals that, as is expected based on the results shown in Figures 5-8 and
5-9, the determination of Ni(II) is not affected by the change in PAR
concentration. The determination of Ga(III) is more strongly affected by the
kinetic nonlinearity present in the data due to the lower PAR concentration.

Figures 5-10 and 5-11 show the results of predictions using CR. As in
chapter 4, the solid center line in these plots represents a perfect prediction (a

slope of unity). The two dashed lines represent 10% error limits.

115

 

d .e d .e .5
o N & Q m 8

Predicted Concentration of Ni(ll) ( uM)
O

 

 

 

 

45 1.0 115 20
Actual Concentration of Ni(ll) (uM)

Figure 5-10: CR determination of Ni(II) by reaction with 1mM PAR (circles)
and 40 11M PAR (plus signs). At the PAR concentration of 1mM the average
excess is 28 fold; at 40 uM it is 1.2 fold.

Figure 5-10 depicts the prediction of nickel. The plot clearly shows that the

prediction is largely unaffected by the kinetic nonlinearity.

116

 

8

8

Predicted Concentration of Ga(lll) ( 1.1M)
23 8

d
o

 

 

 

 

o L I l l I l l l

5 1o 15 20 25 so 35 4o 45 so
Actual Concentration of Ga(III) (uM)

Figure 5-11: CR determination of Ga(III) by reaction with 1mM PAR
(circles) and 40 uM PAR (plus signs). At the PAR concentration of 1mM the
average excess is 28 fold; at 40 uM it is 1.2 fold.

In Figure 5-11 the results of the prediction of Ga(III) can be seen. The
overall error is large (28%, see table 5-1), but three of the four unknown samples
are predicted well. The fourth, with the highest concentration of Ga(IH), is
predicted quite poorly. The PAR is not in excess for this sample, and so the
prediction fails. This result is of extreme interest, as it shows that reagent excesses
of just a single fold are sufficient for accurate determinations and that only when

the reagent is the limiting reactant does the kinetic-spectrophotometric

determination fail. This is important, as it reveals that large excesses of extremely

117

high absorptivity reagents in order to ensure pseudo-first order conditions are not

necessary. This, then, is perhaps one of the most compelling and powerful

arguments for the use of multivariate calibration techniques for kinetic and

kinetic-spectrophotometric determinations.

5.4.

(1)

(2)

(3)

(4)

(5)

(6)

REFERENCES

Schechter, 1.; Schroder, H. "Error-compensated kinetic determinations in
systems of mixed lst-order and 2nd-order reactions, without prior
knowledge of reaction constants" Anal. Chem. 1992, 64, 325-329.

Schechter, I. "Simultaneous determination of mixtures by kinetic-analysis
of general-order reactions" Anal. Chem. 1992, 64, 729-737.

Blanco, M.; Coello, J.; Iturriaga, H.; Maspoch, S.; Redon, M. "Artificial
neural networks for multicomponent kinetic determinations" Anal. Chem.
1995, 67, 4477-4483.

Blanco, M.; Coello, J .; Iturriaga, H.; Maspoch, S.; Redon, M. "Partial least-
squares regression for multicomponent kinetic determinations in linear and
nonlinear-systems" Anal. Chim. Acta 1995, 303, 309-320.

Blanco, M.; Coello, J .; Iturriaga, H.; Maspoch, S.; Redon, M.; Villegas, N.
"Artificial neural networks and partial least squares regression for pseudo-
first-order with respect to the reagent multicomponent kinetic-
spectrophotometric determinations" Analyst 1996, 121, 395-400.

Ventura, S.; Silva, M.; Perez-Bendito, D.; Hervas, C. "Computational
neural networks in conjunction with principal component analysis for
resolving highly nonlinear kinetics" J. Chem. Inf. Comput. Sci. 1997, 37,
287-291.

118

CHAPTER 6

DETERMINATION OF Zn(II) AND Cu(II)

IN A DRINKING WATER SAMPLE

D0, or do not. There is no try.
--George Lucas: The Empire Strikes Back (Yoda)

The determination of zinc (II) and copper (II) in a standard drinking water
sample was carried out using their reaction with PAR. Good accuracy was
achieved in spite of the presence of interfering species.

Both Zn(II) and Cu(II) react with PAR according to a 1:2 stoichiometrylz

Zn“ +2PAR :2 Zn PAR2
Cu2+ +2PAR .-_\ Cu PAR,

The determination of zinc and copper in environmental and clinical samples
is of some importance? While both metals are essential in small concentrations,
they are toxic at higher concentrationsz. The EPA has set limits of 1.3 ppm for
copper in drinking water3 and 5 ppm for zinc in drinking water3.

The use of PAR as a spectrophotometric reagent for the
determination of copper and zinc has been reported in the literature“? Other
workers have described the determination of zinc or copper using a

chromatographic separation and either a pre-concentration step involving PAR or

119

a post-column reaction with PAR14-23. There have been a few reports of kinetic
determinations of copper or zinc using PAR as a reagent2034’25.

6.1. EXPERIMENTAL

The sample in which the Zn(II) and Cu(II) were determined was purchased
from NSI Solutions, Inc. (Research Triangle Park, NC). It contained a mixture of
11 metal cations in a standard drinking water matrix. These metals were certified

to be present in the following concentrations:

 

 

Table 6-1
Certified concentrations (ppb) of metal cations in a
drinking water sample
Certified Certified
Metal Concentration Concentration
(Ppb) 11M
Arsenic 113 1.5
Beryllium 5.17 0.6
Cadmium 15.4 0.1
Chromium 103 2.0
Copper 834 13.1
Lead 69.4 0.3
Manganese 222 4.0
Mercury 10.3 0.05
Nickel 246 4.2
Selenium 23.2 0.3
Zinc 1 130 17.3

 

Of these metals, all but arsenic, beryllium and selenium will react with PARl.
6.1.1. Solution Preparation.

All solutions were prepared with distilled water and reagent grade
chemicals. A buffer of pH 8.5 was prepared from sodium borate and adjusted with
nitric acid. All working solutions were prepared in this buffer, though some

concentrated stock solutions were made in water and then diluted with buffer.

120

Twelve calibration mixtures in the micromolar concentration range were
prepared from 1 mM stock solutions of the metal nitrates. The Cu(II)
concentration was varied between 11 and 17 M; the Zn(H) concentration ranged
from 15 to 21 M. These were mixed and reacted in the stopped-flow system with
a solution of 100 M PAR in pH 8.5 buffer. A plot of the calibration set in

concentration space is given in figure 6-1.

 

22 1 l r I r I r
21 F 111 a _
20+ -

19- e a a a -

Concentration of Zn(ll) (uM)

16'- -

 

 

 

15 - I a -
14 I l l L l I l
10 11 12 13 14 15 16 17 18

Concentration of Cu(II) (uM)

Figure 6-1: Calibration set (stars) used in the determination of Zn(II) and
Cu(II). The unknown sample is depicted by an open circle.

121

6.1.2. Kinetic-spectrophotometric Data Collection

Data were collected using a home-built stopped-flow apparatus interfaced
to a thermoelectrically cooled Tracor Northern (Model TN-6123) 512 element
intensified diode array (Tracor Northern, Philadelphia, PA) configured to acquire
spectra in the 400-800 nm range as described in chapter 2.

In the wavelength range of interest (500-550 nm), absorbance was
measured at 64 equally-spaced wavelengths. Kinetic information was obtained by
acquiring 22 spectra at a rate of 10.5 spectra per second for a total acquisition time
of 2.0 seconds. The spectra and kinetics of the analytes and their reaction products
are described in detail later in this chapter.

6.1.3. Data processing

Time dependent spectra were collected in quadruplicate and averaged, i.e.,
each metal ion solution was reacted with PAR four times. The resulting four sets
of kinetic-spectrophotometric data were averaged. Data were mean-centered (as
described in chapter 3) before being input to the appropriate algorithms.
Multivariate calibration algorithms provided in the PLS_TOOLBOX (Eigenvector
Technologies, Manson, WA) and run in MATLAB were used to perform

determinations.

6.2. DETERMINATION OF ZINC (II) AND COPPER (II)
Zn(H) and Cu(II) were determined in both a standard drinking water sample
and in a series of synthetic unknown samples. Ni(II) and Mn(II) were the major

interfering species in the drinking water unknown and so these were added to the

122

calibration set solutions, and to several of the synthetic unknowns. The calibration

samples were all 4.0 uM in Mn(II) and 4.2 11M in Ni(II). The concentrations of the

analytes and the levels of the interferents present in each unknown sample are

given in Table 6-2.

 

 

 

Table 6-2
Concentrations of analytes and interferents in the unknown sam les in uM units
Drinking Water Synthetic Unknow Pynthetic Unknowri Synthetic Unknown
#1 #2 #3
Cu 13.1 13.1 13.1 13.1
Zn 17.3 17.3 17.3 17.3
Mn 4.0 4.0 13.1 4.0
Ni 4.2 -- -- 4.2
Cd 0.1 -- -- --
Cr 2.0 -- -- --
Pb 0.3 -- -- --
fig 0.05 -- -- --

 

 

 

 

 

 

 

The results of the determination of Cu(II) and Zn(II) in each of these samples are

summarized in Table 6—3.

 

 

 

Table 6-3
Results of the determination of Cu(II) and Zn(II) in a series of unknown samples
Drinking Water Synthetic Unknow Synthetic Unknow Synthetic Unknown
#1 #2 #3
Method % RSEP % RSEP % RSEP % RSEP % RSEP % RSEP % RSEP % RSEP
(Cu) (Zn) (Cl!) (Zn) (Cu) (Zn) (Cu) (Zn)
PLS 6 6 ll 3 14 2 7 3
PCR 6 6 ll 3 13 2 6 3
MLR l 4 9 l 9 3 9 3
CR 5 1 ll 3 14 2 11 1
Parafac 5 3 21 2 27 2 19 5
nPLS 6 6 11 3 14 2 7 3

 

 

 

 

 

 

 

The EPA has designated acceptance limits for the determination of Cu(II)

and Zn(II) in the drinking water sample. The acceptable range for the

determination of copper is $9.8 % error. Acceptable determinations of zinc have

123

errors not larger than i7.98%. All of the chemometric techniques used produced
predicted analyte concentrations of acceptable accuracy. Copper was determined
with an error of approximately 5%; zinc with an error of 3-6%. Both are well
within the EPA guidelines.

The presence of the interferents had little effect on the determination. In the
second synthetic unknown, the manganese concentration is more than three times
its level in the calibration set. The accuracy with which the analytes were
predicted is only slightly worse (e.g., 14% vs. 11% error for the CR determination
of copper) than it is for the first synthetic unknown. In the third synthetic
unknown, which has the interfering species present in the same concentrations as
the calibration set, the analytes are determined with better accuracy (an average
decrease of 1-3% relative error compared to the first synthetic unknown). The
calibration step compensates for the analytical signal due to the interfering species,
and so the third unknown is least affected by the interferents. The other two
synthetic unknowns are missing Ni(II) (see Table 6-2) , and so are not well
described by the calibration set, which does contain nickel.

In all cases, zinc is determined more accurately than is copper; the error
with which Zn(II) is determined is generally half to one third that of Cu(II) (see
Table 6-3). This can be explained by examining the spectral and kinetic
contributions to the net analyte signal for each analyte. As a first step, the spectra
of the analytes (or specifically, of their reaction products with PAR) can be

compared.

124

 

0.016 I I I I I I I I I

0.014- _

I
I

0.012

0.01 -

0.008 -

Absorptivity ( uM cm)‘1

0.006 r-

0.004 *-

 

 

 

0 l l l I l l I l l

500 505 510 515 520 525 530 535 540 545 550
Wavelength (nm)

 

Figure 6-2: Spectra of the PAR complexes of Zn(II) and Cu(II). The zinc
complex is shown as a solid line, the copper complex as a dotted line.

The spectral angle can be computed to be 17.1, a sufficiently high value to suggest
that a reasonable determination might be attempted. The spectral net analyte
signals reveal that zinc is likely to be more accurately predicted than copper;
zinc’s spectral net analyte signal is 5.8 times larger than copper’s. This is true
because although the copper complex has a higher molar absorptivity, zinc is

present in a high concentration.

125

 

18 I I I I t r T j l

 

.5

5
f
L

.............
...................

d d
O N
r I
L 1

Product Concentration ( 11M)
0

 

 

 

0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 2

T111116 (s)

Figure 6-3: Kinetic profiles of the formation of the PAR complexes of Zn(H)
and Cu(II). The zinc complex is shown as a solid line, the copper complex as a
dotted line.

The kinetics of the reactions under the conditions in which the drinking
water sample were determined are shown in Figure 6-3. The rate constants for the
two reactions are very close; the ratio of the rate constants is 1.1, and the kinetic
angle is 2.4. The kinetic data provides slightly more information about the slower
reaction of zinc with PAR; zinc’s kinetic net analyte signal is 1.3 times larger than
copper’s. Since zinc has both a larger kinetic and spectral net analyte signal, it is
not surprising that it is determined with greater accuracy (usually between half and

a third the error of the copper determinations—see Table 6-3).

126

6.3.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

REFERENCES

Shibata, S., "2-Pyridylazo compounds in analytical chemistry", in Chelates
in Analytical Chemistry; Flaschka, H. A., Barnard, A. J., Jr, Eds; Marcel
Dekker: New York, 1972; Vol. 4.

Lobinski, R.; Marczenko, Z. Spectrochemical Trace Analysis for Metals
and MetalloidsWilson and Wilson’s Comprehensive Analytical Chemistry, ;
Elsevier: New York, 1996.

Environmental Protection Agency Office of Ground Water and Drinking
Water, "Technical Drinking Water and Health Contaminant Fact Sheets"
[Online] Available http://www.epa.gov/OGWDW/dwh/t-ioc/zinc.html,
1999

Bobrowska Grzesik, E.; Grossman, A. M. "Derivative spectrophotometry in
the determination of metal ions with 4-(pyridyl-2-azo) resorcinol (PAR)"
F resenius J. Anal. Chem. 1996, 354, 498-502.

Engstrom, E.; Jonebring, 1.; Karlberg, B. "Assessment of a screening
method for metals in seawater based on the non-selective reagent 4-(2-
pyridylazo)resorcinol (PAR)" Anal. Chim. Acta 1998, 371, 227-234.

Fernandez deCordova, M. L.; Molina Diaz, A.; Pascual Reguera, M. I.;
Capitan Vallvey, L. F. "Determination of Trace Amounts of Copper With
4-(2- Pyridylazo)Resorcinol By Solid-Phase Spectrophotometry" F resenius
J. Anal. Chem. 1994, 349, 722-727.

Gomez, E.; Estela, J. M.; Cerda, V.; Blanco, M. "Simultaneous
spectrophotometric determination of metal-ions with 4-(Pyl‘idyl-2-
Azo)Resorcinol (PAR)" F resenius J. Anal. Chem. 1992, 342, 318-321.

Kolomiets, L. L.; Pilipenko, L. A.; thud, I. M.; Panfilova, I. P.
"Application of derivative spectrophotometry to the selective determination
of nickel, cobalt, copper, and iron(III) with 4- (2-py1idylazo)resorcinol in
binary mixtures" J. Anal. Chem. 1999, 54, 28-30.

Manouri, O. C.; Papadimas, N. D.; Salta, S. E.; Ragos, G. C.; Demertzis,
M. A.; Issopoulos, P. B. "Three approaches to the analysis of zinc(II) in

pharmaceutical formulations by means of different spectrometric methods"
Farmaco 1998, 53, 563-569.

127

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

Molina, M. F.; Nechar, M.; Bosque-Sendra, M. "Determination of zinc in
environmental samples by solid phase spectrophotometry: Optimization
and validation study" Analytical Sciences 1998, 14, 791-797.

Ni, Y. N. "Trace-metal determinations by spectrophotometry with a double
chromogenic system and a chemometric approach" Anal. Chim. Acta 1993,
284, 199-205.

Pollak, M.; Kuban, V. "Comparison of spectrophotometric methods of
determination of Zinc(II) in biological-material and study of its complex-
formation reactions with 4-(2-Pyridylazo)Resorcinol" Collection of
Czechoslovak Chemical Communications 1979, 44, 725-741.

Ren, S. X.; Gao, L. "Simultaneous spectrophotometric determination of
copper(II), lead(II) and cadmium(II)" Journal of Automatic Chemistry
1995, 17,115-118.

Al-Shawi, A. W.; Dahl, R. "The determination of cadmium and six other
heavy metals in nitrate phosphate fertilizer solution by ion
chromatography" Anal. Chim. Acta 1999, 391, 35-42.

Cardellicchio, N.; Dell’Atti, A.; Giandomenico, S.; Di Leo, A.; Cavalli, S.
"Determination of transition metals in mineral waters by ion

chromatography and spectrophotometric detection" Annali Di Chimica
1998, 88, 819-827.

Cardellicchio, N.; Cavalli, S.; Ragone, P.; Riviello, J. M. "New strategies
for determination of transition metals by complexation ion-exchange

chromatography and post column reaction" J. Chromatogr. A 1999, 847,
251-259.

Chakrapani, G.; Murty, D. S. R.; Mohanta, P. L.; Rangaswamy, R.
"Sorption of PAR-metal complexes on activated carbon as a rapid
preconcentration method for the determination of Cu, Co, Cd, Cr, Ni, Pb
and V in ground water" Journal of Geochemical Exploration 1998, 63, 145-
152.

de Jesus, D. S.; Casella, R. J.; Ferreira, S. L. C.; Costa, A. C. S.; de
Carvalho, M. S.; Santelli, R. E. "Polyurethane foam as a sorbent for
continuous flow analysis: Preconcentration and spectrophotometric
determination of zinc in biological materials" Anal. Chim. Acta 1998, 366,
263-269.

128

(19)

(20)

(21)

(22)

(23)

(24)

(25)

Hardy, S.; Jones, R; Riviello, J. M.; Avdalovic, N. "Construction and

investigation of a post-capillary reactor for trace metal analysis by capillary
electrophoresis" J. Chromatogr. A 1999, 834, 309-320.

Lucy, C. A.; Dinh, H. N. "Kinetics and equilibria of the Zn-EDTA-PAR
postcolumn reaction detection system for the determination of alkaline-
earth metals" Anal. Chem. 1994, 66, 793-797.

Nagaosa, Y.; Tanizaki, M. "Simultaneous determination of zinc(II) and
iron(III) in human serum by liquid chromatography using post-column
derivatization with 4-(2-pyridylazo)-resorcinol" Journal of Liquid
Chromatography & Related Technologies 1997, 20, 2357-2366.

Shotyk, W.; Immenhauser Potthast, I. "Determination of Cd, Co, Cu, Fe,
Mn, Ni and Zn in coral skeletons by chelation ion chromatography" J.
Chromatogr. A 1995, 706, 167-173.

Vasconcelos, M. T.; Gomes, C. A. R. "Limitations of ion chromatography
with postcolumn reaction for determination of heavy-metals in waters
containing strong chelating-agents" J. Chromatogr. A 1995, 696, 227-234.

Cladera, A.; Gomez, E.; Estela, J. M.; Cerda, V.; Cerda, J. L. "Computer
method for the simultaneous kinetic determination of compounds in

mixtures based on the use of diode-array spectrophotometry" Anal. Chim.
Acta 1993, 272, 339-344.

Taljaard, R. E.; van Staden, J. F. "Simultaneous determination of cobalt(II)

and Ni(II) in water and soil samples with sequential injection analysis"
Anal. Chim. Acta 1998, 366, 177-186.

129

CHAPTER 7

CONCLUSIONS AND FUTURE PERSPECTIVES

Data without generalization is just gossip.
«Robert Pirsi g

The preceding chapters have described the application of chemometric data
processing techniques to kinetic-spectrophotometric data 'under a variety of
circumstances. In this chapter, the work as a whole is summarized, and thoughts
on future work are presented.

7.1. CONCLUSIONS AND SUMMARY

Chapter one presents a fairly detailed overview of the fields of
chemometrics and kinetics as applied to analytical chemistry and highlights the
application of chemometric techniques to kinetic data. Multivariate calibration
algorithms are described in detail; some attention is given to artificial neural
networks, multiway algorithms, and other techniques as well.

A new data acquisition system was designed and built. This redesign of the
existing system involved the fabrication of a new optical path for the stopped-flow
apparatus as well as the creation of a new computerized interface for the diode
array detection system. The new system was characterized; the results of this

characterization and the details of the new designs are found in chapter two. The

130

redesign of the system made possible the collection of the kinetic-
spectrophotometric data analyzed in the remaining chapters.

In chapter three the results of a series of simulated experiments are
described. In these simulation studies the effect of an array of experimental
variables on the accuracy of a kinetic-spectrophotometric determination of a two
component mixture are explored. Methods for quantifying the amount of kinetic
and spectral information present in kinetic-spectrophotometric data were
discussed. The kinetic and spectral angles were introduced and shown to be good
measures of the quantity of information available in each dimension. Kinetic and
spectral net analyte signals were also developed and were shown to be good
predictors of the relative accuracy with which analytes can be determined. Simply
put, the kinetic and spectral angles allow prediction of the general feasibility of a
determination and the dimension (kinetic or spectral) that will be relied upon most
heavily. From these data and the net analyte signals it is possible to infer which
analyte will be determined most accurately. As an example, the case of gallium
and nickel discussed in chapter four can be cited. The kinetic and spectral angles
make it clear that spectral information is more heavily relied upon than kinetic in
this case. Thus, although gallium has a higher kinetic net analyte signal, nickel’s
greater spectral net analyte signal indicates that it will be determined more
accurately. This indeed is the result of both the simulated and experimental

determinations.

131

The kinetic angle was shown to have several contributing factors. The ratio
of the analyte rate constants is the largest contributor, but the fraction of the
slower reaction for which data is acquired and the number of spectra acquired also
impact the kinetic angle.

Chapter four describes the determination of Ga(III) and Ni(II). In the
studies discussed, the effect of the kinetic angle was explored experimentally; the
determination was carried out at two different values of solution pH where the
ratios of the reaction rate constants are different. The expected results were
obtained; the determination was more accurate at the pH (8.5) where the kinetic
angle was largest. A comparison was drawn between the various chemometric
algorithms and it was found that continuum regression generally out-performed
the other multivariate calibration techniques, but that partial least squares
regression and principal component regression were nearly as good. PARAFAC
was found to be useful only in limited circumstances, and multiway PLS showed
an ability to compete with and often surpass the best of the multivariate calibration
techniques when dealing with multiway data.

In chapter five the effect of kinetic non-linearity was examined. Causes of
kinetic non-linearity were discussed and recent work in the area of kinetic
determinations in nonlinear systems was highlighted. The degree of kinetic non-
linearity was measured as an angle from linearity. This angle was shown to be a
good predictor of the accuracy with which a determination could be performed.

The degree of non-linearity was varied in a series of simulations. In general, it was

132

found that the techniques used were fairly tolerant of nonlinear kinetics, but that at
very large angles from linearity the determinations became inaccurate.
Experimental studies in which the degree of linearity was varied were also
described. In these studies Ga(III) and Ni(II) were determined under two sets of
experimental conditions with varying degrees of non-linearity. The results
obtained paralleled the simulations and were quite encouraging. Again, the various
chemometric algorithms were compared, and similar results to those found in
chapter three were obtained. Continuum regression and nPLS proved best suited to
handling the nonlinear kinetic data.

Chapter six presents the determination of Cu(II) and Zn(II) in a real sample
with clinical and environmental relevance. The accuracy of the determination was
within the EPA’s acceptance limits for both analytes. The kinetic and spectral
angles and net analyte signals were used to explain the relative accuracy with
which the analytes were determined.

7 .2. REFLECTIONS ON FUTURE DIRECTIONS

All of the work described in this document has focused on the
determination of two analytes. The extension to three or more analytes is certainly
logical, but also is highly challenging and will present many difficulties. Not least
of these will be the need to develop new methods for quantifying the amount of
kinetic and spectral information available. Ratios of rate constants and angles
between profiles lose meaning when more than two analytes are present. Net

analyte signals are better, but are still not capable of the necessary resolution. The

133

net analyte signals reveal the degree to which a profile (kinetic or spectral) is
overlapped by other profiles, but do not reveal which specific profiles overlap the
profile of interest; e.g., an analyte might be highly spectrally and kinetically
overlapped, but by different species. In this situation, the net analyte signals will
both be small, but the determination can still be accurately performed since the
analyte is not overlapped in both dimensions by the same species. Other problems
that will be encountered include the determination of an unoverlapped analyte in
the presence of two other highly overlapped analytes, and the economics of scale
associated with three-analyte calibration sets.

The trend in chemometrics is toward the wider use of multiway algorithms.
As they become more popular, their use with multiway data will supercede the use
of first order algorithms on unfolded multiway data. This has had and will
continue to have an impact on the way in which data are collected and used. The
use of multiway algorithms for kinetic-spectrophotometric data has been briefly
explored in this work, and it is expected that it will continue to be an active area of
study.

In 1993 Crouch1 identified seven trends in kinetic methods of analysis.
Several of these are relevant to the work discussed in this document. These trends
are:

1. Increasing use of “intelligent automation”

2. Growing utilization of multidimensional instrumentation

3. Continuing development of sophisticated data processing techniques

134

4. Additional progress in error-compensation techniques

5. Innovations in multicomponent kinetic procedures

6. Expanding applications of kinetic determinations

7. Enlarging the kinetic approach to include miscellaneous time-dependent

responses
The work in this thesis is testament to the continuation of trends 2, 3, 4, and 5. As
the field of chemometrics continues to advance and to become more widely
accessible and accepted, trend three will hold as new (at least to kinetics
researchers) chemometric techniques are applied to kinetic data.

It is the contention of this author that wider accessibility of and familiarity
with chemometric techniques and the proliferation of multidimensional
instrumentation will eventually result in an increase in the number of analytical
measurements employing kinetic or other time-dependent data. This is in
accordance with the sixth and seventh trends listed above. As Crouchl'2 and
Mottola3 have pointed out, most analytical methods involve some sort of kinetic or
transient response. Often, great pains are taken to ensure that these time-dependent
responses are avoided or that they are compensated for in the data processing.
Modern multiway chemometric data processing options are able to correctly
handle higher order data and can separate and make use of several data
dimensions. This implies that time-dependent data can be used in almost all
analytical measurements, resulting in increases in selectivity and (in many cases)

sensitivity'v3-5. Some of the areas in which the transient data is already being

135

applied include luminescence lifetime spectroscopy, fluorescence lifetime imaging
microscopy, sequential injection analysis and transient electroanalytical chemsitry.

In all, the future of analytical chemistry appears bright. Greater computing
power allows the acquisition of more data in more dimensions and grants the
ability to use these data to perform more and better determinations. The field of
kinetic methods will no doubt benefit from this trend, and the field of
chemometrics will continue to grow. This thesis is but one example, a simple
harbinger of this new direction and focus.

7 .3. REFERENCES

(l) Crouch, S. R. "Trends in kinetic methods of analysis" Anal. Chim. Acta
1993, 283, 453-470.

(2) Crouch, S. R. "Kinetic methods of analysis - how do they rate" J. Chin.
Chem. Soc. 1994, 41 , 221-229.

(3) Mottola, H. A. Some Kinetic Aspects of Analytical Chemistry ; Wiley: New
York, 1988.

(4) Perez-Bendito, D.; Silva, M. Kinetic Methods in Analytical Chemistry ;
Ellis Horwood: Chichester, 1988.

(5) Perez-Bendito, D.; Silva, M. "Recent advances in kinetometrics" Trac-
Trends Anal. Chem. 1996, 15, 232-240.

136

APPENDIX

137

APPENDIX

MATLAB CODE FOR GENERATING SIMULATED

KINETIC-SPECTROPHOTOMETRIC DATA

You think you know when you learn, are more sure
when you can write, even more when you can teach,
but certain when you can program.

--Alan J. Perlis

Anyone who considers arithmetical methods of
producing random digits is, of course, in a state of sin.
--John von Neumann

Code designed to generate simulated kinetic-spectrophotometric data was
written in MATLAB’s native programming language. The main program is
mulgen_a.m. This programs calls several others.

Mulgen_a begins by collecting the necessary information for the generation
of simulated data. The user provides a matrix of absorptivities for each reactant,
reagent, and product, a matrix of the initial concentrations of each reactant, the
fraction of the slower reaction to observe, the initial reagent concentration, the
desired levels of instrumental noise and rate constant fluctuation, and the number
of spectra to generate.

The program then uses the variable perrxn (the input fraction of the slower

reaction to observe), and calculates the time at which to cease “data acquisition.”

138

It does this by making the initial, worst-case assumption that the faster reaction
will have reached equilibrium by the time the slower reaction has had time to
begin. Using this assumption, the time at which the slower reaction will have
reached the desired fractional completion is calculated. Five hundred points are
then generated for the parallel reaction of the two reactants with the reagent
between zero and this calculated time. The time at which the concentration of the
slower product reaches the threshold determined by the fraction of the reaction to
be observed is noted, and 500 more points are generated between time zero and
this new time. Again, the time at which the product concentration reaches the
threshold level is found, and this time is used as the stopping time for the data
generation in all future calculations.

The program proceeds to calculate the concentration of all species at all
times in all samples. The absorptivities are then used to calculate the absorbance
of each species, and these absorbances are summed to generate the final data
matrix.

Calculations of the concentration of the reactants, reagent, and products at
any time during the reaction are performed by the program kinetic.m. This
program sets up and calls the MATLAB routines for numerically solving systems
of ordinary differential equations. These routines require a model of the system
that includes the differential equations to be solved. This model is supplied by the

function vary_k.m.

The code of mulgen_a.m. kinetic.m, and vary_k.m are found below.

139

MULGEN_A

function [signal,signa13,Dim_signa1,tfinal] =...
mulgen_a(absorbs,concentrations,perrxn,concrO,...
noisek,noisei,pts)
[signal,signa13,Dim_signa1,tfinal] =
mulgen_a(absorbs,concentrations,perrxn,concrO,.
noisek,noisei,pts);

generates absorbance data for a specified # of wavelengths
over a specified time period. It uses the kinetic.m routine,
and so can handle any kinetic system that can be modeled
by ode45.
Molar absorbtivities should be organized in columns:

epsl epsZ ... epsreagent epspl epsp2

Concentrations should be also be arranged in columns.

complsampl compZSampl
complsamp2 comp23amp2

%

%

%

%

%

%

%

%

%

%

%

%

%

%

%

%

%

%

%

%

%

% This version handles 2 components, any # of wavelengths, and
% any # of samples.

% It allows the user to input the amount of variation in the
% rate constants, the level of instrumental noise, and the
% percent completion of the slower reaction.

%

% It requires that k1 and k2 be declared as global variables
%

% The output “signal" is in the form:

%

% slwltl slw2t1 slw3t1 ... slw1t2 slw2t2

%
%
%
%
%
%
%
%
%
%
g
g
‘g
1%

sZwltl $2w2tl $2w3t1 ... sZwltZ sZw2t2
stltl s3w2tl s3w3tl ... s3w1t2 s3w2t2

The output "Dim_signa1" is in the form:

[samples waves pts]

The output I'signal3" has each row as a time, each column as a
wavelength, and each sample as a page.

140

Created 9/20/95 by Tom Cullen
Last Updated 6:04 PM 8/16/99 by Tom Cullen

096969690909

%*************************************************************‘k‘k

% Get info from user
%***************************************************************
[samples,comp]= size(concentrations);

[waves,xcomp] = size(absorbs);

global k1
global k2

%***************************************************************

% Determine stopping time
%***************************************************************

[lowk,which_one_low]=min([k1,k2]);
[highk,which_one_high]=max([k1,k2]);
if kl==k2
lowk=k2;
highk=k1
which_one_low=2;
which_one_high=1;
end

medianconcs=median(concentrations);
medianconc1=medianconcs(which_one_high);
medianconc2=medianconcs(which_one_low);
medianconcr=concr0-medianconc1-medianconc2;

concratio=medianconc1./medianconc2;
medianconcs=[medianconc1 medianconcZ];
reagtexcess=concr0 ./(medianconc1+medianconc2);

newtf=(-log(1-(perrxn)))./
(lowk*(concrO—medianconcs(which_one_high)));
[profile1,profile2,profiler] =
kinetic(medianconcs,concr0,0,newtf,’vary_k’,500);
if which_one_high==1
slow;profi1e=profile2;
else
slow;profile=profilel
«end
times=1inspace (0 , newtf, 500) ,-
for i=1:500

141

if slow;profile(i) < (l-perrxn)*medianconcs(which_one_low)
newtf=times(i);
break;
end;
end;

[profilel,profileZ,profiler] =
kinetic(medianconcs,concr0,0,newtf,’vary_k’,500);
if which_one_high==
slow;profile=profile2;
else
slowgprofi1e=profilel
end
times=linspace(0,newtf,500);
for i=1:500
if slow;profi1e(i) < (l-perrxn)*medianconcs(which_one_low)
newtf=times(i);
break;
end;
end;
tfinal=newtf;

%************************‘k*********'k‘k***************************

% Expand scalar data into matrices... prepare other matrices
%***************************************************************

c01 = ones(samp1es,pts);
c02 = ones(samp1es,pts);
for i=1:pts

c01(:,i) = concentrations(:,l);
c02(:,i) = concentrations(:,2);
end;

k1_array = k1*ones(samples,pts);
k2_array = k2*ones(samples,pts);
k1_noisy_array = noise(k1_array,noisek);
k2_noisy_array = noise(k2_array,noisek);
k1_noise = abs(k1_array - k1_noisy_array);
k2_noise = abs(k2_array — k2_noisy_array);
noisesign = sign(rand(samp1es,pts));
time = 1inspace(0,tfina1,pts);
times = ones(samp1es,pts);
for i=1:samples

times(i,:)=time;
end;

for s=1:samples
for c = lzcomp

eval(['con' int2str(c) ’_’ int25tr(s) ’ =...
ones(waves,pts);'])
eval(['conp' int23tr(c) ’_’ int2str(s) ' =...

142

ones(waves,pts);’])

eval(['abs’ int25tr(c) '_’ int23tr(s) ’ =...
ones(waves,pts);’])
eval([’absps’ int25tr(c) ’_’ int25tr(s) ' =...
ones(waves,pts);'])
end;
end;

absr = ones(waves,pts);

datapoints = waves*pts;
signal = ones(datapoints,samples);

%***************************************************************

% Calculate concentrations as a function of time
%************************‘k**************************************

[profilel,profile2,profiler] =...
kinetic(concentrations,concr0,0,tfinal,’vary_k’,pts);

k1t_noise = exp(-k1_noise .* times .* profiler) .* noisesign;

k2t_noise = exp(-k2_noise .* times .* profiler) .* noisesign;

profilel profilel .* k1t_noise;

profile2 profileZ .* k2t_noise;

profiler = profiler .* k1t_noise .* k2t_noise;

profilepl = c01 - profilel;

profilep2 = c02 - profi1e2;

for s=1:samples
for c = 1:comp
for i = 1:waves
eval(['con' int25tr(c) ’_’ int25tr(s) '([i],:) =...
profile’ intZStr(c) ’ (s,:);'])
eval(['conp’ int23tr(c) '_' int23tr(s) '([i],:) =...
profilep' int25tr(c) ' (s,:);’]);
eval([’conr_’ int23tr(s) ’([i],:) =...
profilerls,:);’]);
end;
end;
end;

%*************************‘k'k************************************

% Convert absorptivities into a useful fonm
%***************************************************************

for c = 1:comp
eval([’eps’ int25tr(c) ’ = absorbs(:,’ int23tr(c) ' );'])
eval(['epsp' int25tr(c) ' =...
absorbs(:,' int23tr(c+comp+1) ’ );’])
end;
eval([’epsr = absorbs(:,' int23tr(comp+1) ’ );’])

143

for C: 1:comp
eval([’extinct’ int25tr(c) ’ = ones(waves,pts);’])
eval([’extinctp’ int2str(c) ' = ones(waves,pts);’])
end;
extinctr = ones(waves,pts);

for C: 1:comp
for i = lzpts
eval(['extinct' int25tr(c) ' (:,[i]) =...
eps’ int25tr(c) ’ (:,1);’])
eval(['extinctp' int23tr(c) ' (:,[i]) =...
epsp' int23tr(c) ’ (:,1);’])

end;
end;
for i = lzpts

extinctr(:,[i]) = epsr(:,l);
end;

%************************'A’**************************************

% Calculate absorbances
%***************************************************************

for S: 1:samp1es
for C: 1:comp
eval(['abs’ int23tr(c) ’ ’ int23tr(s) ' = extinct’...

int25tr(c) ' .* con' int23tr(c) ' ’ int25tr(s) ';’])

eval(['absps’ int25tr(c) ’ ' int25tr(s) ' =extinctp’...

int2str(c) ' .* conp’ int23tr(c) ’_' int23tr(s) ’;'])
eval([’absr_’ int2str(s) ’ = conr_’ int25tr(s) ’...
.* extinctr;'])
end;

end;

for S: 1:samp1es
eval([’sig_’ int23tr(s) ’ = absr_’ int23tr(s) ';’])
for C: 1:comp
eval([’sig_’ int23tr(s) ’ =...
sig_' int25tr(s) ’ + abs' int2str(c) ’_' int25tr(s) '...
+ absps' int25tr(c) '_' int25tr(s) ’ ;’])

end;
end;

%***************************************************************

% Reshape data and add instrumental noise
%***************************************************************

for S: 1:samp1es
eval(['signa1_' int23tr(s) ' =...
reshape(sig_' int23tr(s) ',datapoints,1);'])
eval([’signal(:,s) = signa1_’ int2str(s) ’;'])
end;

144

signal noise(signal,noisei);
signal = signal’;

Dimhsignal=[samp1es waves pts];

signa13=reshape(signal,samples,waves,pts);
signal3=permute(signa13,[3 2 1]);

%**********************************‘k'k***************************

% All done!!!!!

%***************************************************************

145

KINETIC

function [profilel,profilez,profiler] =...

kinetic(concs,r0,t0,tfinal,model,pts)

% [profilel,profi1e2,profiler] =...

% kinetic(concs,r0,t0,tf,model,pts);

% is a general-purpose m-file that uses the ode45

% function to generate kinetic profiles. It must be given
% initial concentrations, beginning and ending times,

% and the filename of the ode45 function that describes the
% system. This m-file uses a cubic spline to interpolate
% as it generates a user-supplied number of data points

% per profile.

%

global k1

global k2

[mcon,ncon] = size(concs);

profilel=zeros(mcon,pts);
profile2=zeros(mcon,pts);
profiler=zeros(mcon,pts);
for 22: lzmcon

c0= [concs(zz,:),r0];
[t,c]=ode45(model,[t0 tfinal],c0);
ti=linspace(t0,tfinal,pts);
ci=interp1(t,c,ti,’spline’);
Ci=ci’;
profile1(zz,:)=ci(1,:)
profile2(zz,:)=ci(2,:),
profiler(zz,:)=ci(3,:)

I

I

end;

146

VARY_K

function cdot=varyk(t,c);

% function cp=varyk(t,c);

% m-file that returns state derivatives when given state and
% time valus for a second order kinetic process given by:
% A+R=PA

% B+R=PB

% Created 8/7/96 by Tom Cullen

global kl

global k2

% Define cl=A, c2=B, and c3=R

% cl’ = -k1 .* c1 .* c3

% c2’ = -k2 .* c2 .* c3

% c3’ = (—kl .* cl .* c3) + (-k2 .* c2 .* c3)

cdot=[(-k1)*c(l)*c(3);(-k2)*c(2)*c(3);((-k1)*c(1)*c(3))+...
(-k2*c(2)*c(3))];

147

MICHIGAN S

TATE UNIV. LIBRARIES
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93020582064

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