Quantitative Spectroscopy: Theory and Practice

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QUANTITATIVE SPECTROSCOPY: THEORY AND PRACTICE

This Page Intentionally Left Blank


BRIAN C. SMITH Spectros Associates Shrewsbury, Massachusetts

/ ^ ACADEMIC PRESS V _ ^ An imprint of Elsevier Science Amsterdam • Boston • London • New York • Oxford Paris • San Diego * San Francisco • Singapore • Sydney • Tokyo

This book is printed on acid-free paper. @ © 2002 Elsevier Science (USA) All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the Publisher. The appearance of the code at the bottom of the first page of a chapter in this book indicates the Publisher's consent that copies of the chapter may be made for personal or internal use of specific cHents. This consent is given on the condition, however, that the copier pay the stated per copy fee through the Copyright Clearance Center, Inc. (222 Rosewood Drive, Danvers, Massachusetts 01923), for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. Copy fees for pre-2002 chapters are as shown on the title pages. If no fee code appears on the title page, the copy fee is the same as for current chapters. $35.00 ExpHcit permission from Academic Press is not required to reproduce a maximum of two figures or tables from an Academic Press chapter in another scientific or research pubhcation provided that the material has not been credited to another source and that full credit to the Academic Press chapter is given. Academic Press An Elsevier Science Imprint 525 B Street, Suite 1900, San Diego, California 92101-4495, USA http://www.academicpress.com International Standard Book Number: 0 12 650358 3 Typeset by Keyword Pubhshing Services Ltd, Barking, Essex Printed in Great Britain by MPG Books Ltd, Bodmin, Cornwall 02

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CONTENTS

Preface

vii

Acknowledgments Chapter 1

xi

FUNDAMENTALS OF MOLECULAR ABSORPTION SPECTROSCOPY

1

Chapter 2

SINGLE ANALYTE ANALYSIS

43

Chapter 3

MULTIPLE COMPONENTS I: LEAST SQUARES METHODS

85

Chapter 4

MULTIPLE COMPONENTS II: CHEMOMETRIC METHODS

Chapter 5

AND FACTOR ANALYSIS

125

IMPLEMENTING, MAINTAINING, AND FIXING CALIBRATIONS

181

GLOSSARY

189

INDEX

195


PREFACE

The quantitation of the concentrations of molecules in samples has long been an important application of spectroscopy. In the last two decades, advances in algorithms, computers, instruments, and software have led to a burgeoning of interest in this field. These developments mean samples and analytes that were once considered intractable are increasingly yielding usable calibrations. The purpose of this book is to give readers a thorough grounding in the theory and practice of modern quantitative spectroscopic analysis. This book is geared towards anyone using spectroscopic absorbance measurements to determine concentrations in unknown samples, and should appeal to users of infrared, near-infrared, and UV-Vis instruments. There are several aspects of this book that make it, I beheve, a valuable contribution to this field. 1. The balance between theory and practice. Relevant theory is interspersed with practical discussions. Equations are explained at length to make difficult concepts comprehensible. I have attempted to strike a balance between the mathematical and experimental aspects of this field. 2. Example caHbrations using real world data. In many chapters actual experimental data are used to generate example calibrations. I believe

viii

PREFACE

there is no substitute for real data to make understandable the important concepts of this field. 3. One stop shopping. There are many aspects of quantitative analysis, including instrumentation, software, mathematics, and experimental technique. This book pulls together information from all these fields in one place. Everything you need to know to obtain a fundamental understanding of quantitative spectroscopy is here. 4. This book is written at an introductory level. Increasingly, cahbrations are being developed and implemented by people without math degrees. This book is written for them. All technical terms appear in italics, and are defined in the glossary at the end of the book. Much effort has been expended to make difficult concepts understandable to novices. 5. A spectroscopist's perspective. Spectra comprise half the information in a cahbration. How the spectra are measured, their quality, and what they mean is important. Being a spectroscopist, I have included topics on the spectroscopic part of quantitative spectroscopy to emphasize its importance. The book begins with a theory and background chapter. The properties of Hght and the interaction of Hght with matter are the first topics introduced. Next, Beer's Law is derived, and the complexities behind the absorptivity {s) are discussed. At its root, the absorption of Hght by matter is a quantum mechanical process, and understanding this process is key to understanding cahbrations. This is discussed in the appendix to Chapter 1. The appendix shows that many of the features of spectroscopy, including the existence of quantized energy levels and selection rules, are a natural outgrowth of microscopic bound systems. Do not let this scare you off, the appendix is comprehensible even if you have never taken a course in physical chemistry. The second chapter of the book focuses on single analyte determinations. It shows how Beer's Law is used in practice, and provides a detailed look at how linear regression and the least squares algorithm are used to generate calibration hues. Calculating statistics that give the accuracy and robustness of a calibration is discussed. The chapter concludes with a discussion of standard methods, and practical tips on how to avoid experimental error in analyses. A simple single analyte example cahbration is used throughout to illustrate important concepts. The third chapter covers the least squares methods (inverse and classical) of obtaining multi-analyte cahbrations. A review of matrix algebra is followed by the matrix algebra behind least squares techniques. The ideas of linear regression are extended to multiple analytes. The chapter includes a discussion on the strengths and weaknesses of each of these methods.

PREFACE

ix

A multicomponent system is used to generate example calibrations to illustrate important concepts. A large part of the rest of the book is devoted to factor analysis (chemometrics). The mathematics of factor analysis algorithms are discussed in words and equations. Then, readers are presented with a process that they should follow to obtain successful chemometric caHbrations. The process starts with ascertaining the quahty of the raw data used in the calibration, proceeds to checking a calibration for predictive ability, and ends with using independent data to vahdate a caUbration. A multicomponent system is used throughout to generate example calibrations to illustrate important concepts. Finally, the last chapter discusses the practical aspects of implementing and supporting a calibration, and the theoretical and practical limits of quantitative analysis in general. There are those who may prefer to treat the instruments and calibration methods of quantitative spectroscopy as a black box. My goal here is to open that box. You might be able to obtain a calibration without the knowledge in this book. However, if you want a caUbration that is of high quaUty, is robust, and is implemented properly, then an understanding of the topics in this book is necessary. I will be the first to admit that no book is perfect, and that most certainly this book can be improved. Therefore, I welcome your comments and criticisms, and can be contacted at the e-mail address below.

BRIAN C . SMITH, P H . D .

Shrewsbury, MA bcsmith@spectrosl .com September 2002


ACKNOWLEDGMENTS

The writing of a book, even by a single author, is never a soHtary endeavor. The support, advice, and love of a wide group of people is necessary to bring such creations to fruition. I would first like to thank my reviewers, John Winn, Howard Mark, and Rich Kramer. Their input and constructive criticism have made this a much better book. This book was developed as part of the Quantitative Spectroscopic Analysis course I teach for my company, Spectros Associates. It has been made better by comments from many of my course attendees. I would like to thank my students for their honesty, and for helping me earn a living. Software is an overriding aspect of quantitative spectroscopy these days. Obtaining and learning how to use quantitative software packages was key to generating the real world cahbrations found in this book. The factor analysis cahbrations in this book were generated using the PLSPlus/IQ software package from Galactic Industries of Salem, NH. The people there, including Jamie Duckworth, Sue Kaplan, Tony Nip, and Brie Gervin were most helpful in answering my questions, and in solving my problems with the software. The P-matrix calibration discussed in Chapter 3 was generated using the Optical Solutions Multiple Linear Regression Software Package from Optical Solutions of Folsom, CA. Don Goldman of that company was very helpful in obtaining me a copy of the program, and helping me troubleshoot it. Thank you Don.

xii

ACKNOWLEDGMENTS

There has been a parade of editors in my Hfe during the writing of this book, but they have all contributed to its completion. Emma Roberts was smart enough to see the need for a book such as this, Carla Kinney provided support and encouragement, and Derek Coleman made this book a reality by setting a deadhne for me. Thank you all. Above all, it takes the love, support, and encouragement of one's family to make a book a possibility. I would Hke to thank the lovely ladies in my hfe, Marian, Eleanor, and Isabel.

Dedicated to... all my parents

1 FUNDAMENTALS OF MOLECULAR ABSORPTION SPECTROSCOPY

I. Terms and Definitions This book is devoted to quantitative molecular absorption spectroscopy, one of the most important and common applications of spectroscopy. The word quantitative means we are measuring the concentrations of the chemical species in a sample. The word molecular means that we are interested in the molecules, not the atoms or elements, in a sample. The word absorption means we will use the amount of Ught absorbed by a sample to quantify the amount of substance present. Thus, the purpose of spectroscopic quantitative analysis in this context is to determine the concentration of a molecule or molecules in a sample using absorbance spectra. The molecule whose concentration is measured is called the analyte. The peak heights or areas in an analyte's absorbance spectrum are directly proportional to the concentration of the analyte. To establish the correlation

2


between absorbance and concentration, we must take the spectra of standards, samples that contain known concentrations of the analyte. Through a process called calibration, a mathematical model is generated that correlates the absorbances to the known concentrations in the standard samples. Once an accurate cahbration is in hand, the spectrum of a sample with an unknown concentration of the analyte, the unknown, is measured. The concentration of the analyte in the unknown sample is predicted by applying the calibration to the absorbances in the unknown spectrum. Spectroscopic quantitative analyses are based on the assumption that caHbrations model the standard and unknown samples equally well. Much of the work involved in obtaining a calibration is done to assure that this is true. Since we can never prove with 100% certainty that a given calibration gives a completely accurate description of an unknown sample, it is proper to say that we predict unknown concentrations rather than calculate them. Concentrations always contain a certain amount of error, and to say that they are calculated means we know their exact value, which is impossible.

II. The Properties of Light Spectroscopy is the study of the interaction of light with matter. The fundamental measurement obtained in spectroscopy is a spectrum, which is a plot of measured Hght intensity versus some property of Hght. An example of a spectrum, the mid-infrared absorbance spectrum of polystyrene, is seen in Figure 1.1. The F-axis of this spectrum is in absorbance, which is a measure of how much light a sample absorbs. The J^-axis is in wavenumbers (cm~^), a property of hght discussed later. An instrument used to measure a spectrum electronically is called a spectrophotometer, or sometimes simply a spectrometer. There are many different kinds of spectrophotometers in the world, and they use many different types of hght to obtain spectra. The details of how these instruments work are beyond the scope of this book. Several of the books hsted in the bibliography of this chapter are excellent references on how different spectrophotometers work. To understand how a molecule absorbs hght, we must first understand something about the properties of hght. Light can be thought of as being a wave or a particle, depending upon the particular property of light under consideration. This wave-particle duality is an inherent feature of light. For now, we will consider hght as a wave. Light beams are composed of electric and magnetic waves that undulate in planes perpendicular to each

1. FUNDAMENTALS OF MOLECULAR ABSORPTION SPECTROSCOPY

.3H .25

.2 .15 .1

wV

.05 4000

3500

3000

2500 2000 Wavenumber (cm-1)

1500

1000

500

Figure 1.1 The mid-infrared spectrum of polystyrene. Note that the Z-axis units are wavenumber (cm~^), and that the 7-axis units are absorbance.

other. Light is properly called electromagnetic radiation because it contains electric and magnetic waves. The light wave traverses through space in a direction defined by the line where the two planes containing the waves intersect. The interaction of the electric wave of hght, the electric vector, with matter is what is usually measured to obtain absorbance spectra. The amplitude of the electric vector changes over time and has the form of a sine wave, as shown in Figure 1.2. One of the properties used to distinguish between different types of Hght is the light's wavelength. A wavelength is the distance between adjacent crests or troughs of a wave, as seen in Figure 1.2. Wavelength is denoted by the lowercase Greek letter lambda (X). Different types of Hght have different wavelengths. For example, infrared radiation is longer in wavelength than visible Hght, which is longer in wavelength than X-rays. Another property of a Hght beam is its frequency. Frequency is denoted by the Greek letter nu (y), and equals the number of cycles a wave undergoes per second. A cycle is considered complete when a Hght wave starts at zero and then crosses the X-axis twice. The wave in Figure 1.2 undergoes almost three cycles. Frequency is measured in cycles/s or Hertz (Hz). Frequency is a measure of the number of cycles of a Hght wave per unit time. The frequency, wavelength, and speed of a Hght beam are related to each other via the following equation: c = vX

(1.1)


Time

Figure 1.2 A plot of the amplitude of the electric vector of a light wave versus time. The arrow denotes the distance between adjacent crests, and is called the wavelength, X. Note that the polarity of the wave, denoted by the + and — signs, changes over time. Note also the definition of a cycle.

where c = the speed of Hght ( 3 x 1 0 10 cm/s) V —frequency in Hertz (s~^) X — wavelength This equation shows that the product of frequency and wavelength for a Hght wave is a constant, the speed of Hght. Another property used to describe Hght is its wavenumber. A wavenumber is defined as the reciprocal of the wavelength as foHows: W =z l/k

(1.2)

where W = wavenumber X = wavelength If X is measured in cm, then W is reported as cm"^ or reciprocal centimeters. A wavenumber measures the number of cycles in a light beam per unit length. If we substitute Equation (1.2) into (1.1) and solve for c, we obtain c = v/W

(1.3)


5

which upon rearranging gives

v = cW

(1.4)

Equations (1.1)-(1.4) show that Hght waves may be described by their frequency, wavelength, or wavenumber. These equations also show that these three quantities are related to each other. Throughout this book, we will usually refer to hght waves by their wavenumber. However, at times it will be more convenient to refer to a hght beam's frequency or wavelength. As mentioned above, hght can also be thought of as a particle. A particle of light is called a photon. A photon has no mass, but it does have energy. The energy of a photon is directly related to frequency as follows: E = hv

(1.5)

where E = photon energy in Joules /z = Planck's constant (6.63 x 10"^"^ J s) y = frequency in Hertz (s~^) If we substitute Equation (1.4) into (1.5), we obtain E = hcW

(1.6)

which shows that photon energy also depends on the wavenumber. Note that high wavenumber light has more energy than low wavenumber light.

III. The Electromagnetic Spectrum There are types of hght in addition to the visible hght that we can detect with our naked eyes. All the different types of light are called the electromagnetic spectrum. Each different type of light can be characterized by a different frequency, wavelength, wavenumber, or energy. A section of the electromagnetic spectrum is illustrated in Figure 1.3. Note that in reading Figure 1.3 from right to left the frequency, the wavenumber, and the energy increase while wavelength decreases. When performing a quantitative absorption experiment, the first thing to decide upon is what type of hght to use in the analysis. The type of hght chosen for an analysis affects the types of samples that can be investigated, the sample preparation necessary, the type of instrument that


40000-14000 cm Visible& Ultraviolet

Higher Wavenumber Higher Frequency Higher Energy Shorter Wavelength

14000 to 4000 cm"' Near -Infrared

4000 to 400 cm' Mid-Infrared

400 to 4 cm"^ < 4 cm'* Far Infrared Microwaves

Lower Wavenumber Lower Frequency Lower Energy Longer Wavelength

Figure 1.3 The electromagnetic spectrum, showing the wavenumber, wavelength, frequency, and energy ranges for different types of light. Also illustrated is how wavelength, wavenumber, frequency, and energy change across the spectrum.

can be used, and ultimately impacts the calibration quality. The types of light listed in Figure 1.3 are the ones most commonly used in quantitative absorption spectroscopy. The lowest energy Ught seen in Figure 1.3 is microwaves, which appear below 4 cm~^ (this is the type of radiation used in microwave ovens). When this type of Ught is absorbed by a molecule, there is an increase in the rotational energy of the molecule. This is why microwave spectroscopy is sometimes called rotational spectroscopy. This technique is typically limited to gases because gas phase molecules are free to rotate, whereas soHd and Hquid phase molecules are not. Although microwave spectrometers exist, and have been used to perform quantitative analyses, their use is not widespread. Next in energy above microwaves is far-infrared radiation, found from 400 to 4 cm~^ When molecules absorb far-infrared light, the energy excites vibrational motion of the molecule's bonds. However, far-infrared absorbances are low in energy, and are typically found in heavy molecules such as inorganic and organometallic substances. Most organic molecules do not absorb in the far infrared, limiting the types of molecules this wavenumber range can be used to analyze. Mid-infrared radiation is found between 4000 and 400 c m ' ^ Chemical bonds vibrate when they absorb mid-infrared radiation, but with more energy than in the far infrared. Many of the molecules in the universe (of which there are more than 10 million) give usable mid-infrared spectra. Midinfrared absorbances are intense, and often times only a miUigram of material is needed to obtain a spectrum. Additionally, almost every type of material, including solids, liquids, gases, polymers, and semi-solids can have their mid-infrared spectra measured. A disadvantage of the strength of midinfrared absorbances is that it is easy for a sample to absorb all the hght impinging on it at a given wavenumber, making it difficult to record a spectrum. As a result, a good deal of time consuming manual sample preparation can be involved in making the sample dilute enough or thin


7

enough so that it absorbs the right amount of Hght. This problem with midinfrared samples is called the thickness problem. However, mid-infrared Hght is commonly used for quantitative analysis. Throughout the rest of this book, the terms mid-infrared and infrared will be used interchangeably. From 14,000 to 4000 cm~^ lies the near-infrared region of the electromagnetic spectrum. Like in the mid-infrared, molecules vibrate when they absorb near-infrared light, but with higher energy than in the mid-infrared. A disadvantage of near-infrared absorbances is that they are typically 10-100 times weaker than mid-infrared absorbances. This is bad if only a small amount of sample is available. However, near-infrared samples do not suffer from a thickness problem, so sample preparation can be faster and easier compared to mid-infrared spectroscopy. Because of this, and of the high performance of near-infrared spectrometers, there has been an explosion of quantitative applications of near-infrared spectroscopy in the last 30 years. The highest energy Hght we will consider in this book is the ultraviolet and visible (UV-Vis). This type of Hght falls from 40,000 to 14,000 cm"^ When this type of light is absorbed by a molecule, an electronic transition takes place and the Hght's energy promotes an electron from a lower energy level to a higher energy level. Although UV-Vis absorbances can be intense, only molecules with certain types of chemical bonds absorb UV-Vis light, somewhat limiting the types of molecules that can be analyzed. Despite this limitation, UV-Vis light was historically the first type of light widely used in quantitative analysis, and is still widely used today.

IV. Beer's Law The basis of most quantitative spectroscopic analyses is Beer's law. This law relates the amount of Hght absorbed by a sample, a spectroscopically observable property, to the concentration of absorbing species in the sample. This relationship is what aHows absorbance measurements to be used to predict concentrations. To derive Beer's law, we assume the experimental setup shown in Figure 1.4. We also assume that monochromatic Hght of wavelength A, impinges upon the sample perpendicular to one of its faces. Before the light enters the sample, it has intensity /Q. Light intensity is defined as the number of photons of light hitting a unit area per unit time. The thickness ox pathlength of the material is denoted by L. An infinitesimally thin slab of the absorbing material is denoted by dL. When the light leaves the sample it has intensity I < IQ due to absorbance of the light by the sample.


I05 A>^

r'

dL

Figure 1.4 A sample of absorbant material of pathlength L. Monochromatic light of wavelength X and initial intensity /Q impinges on the sample perpendicular to it. An infmitesimally thin slab of the material is denoted by dL. The light leaves the sample with a lower intensity / due to the sample absorption at wavelength X.

Before Collision

After Collision Photon Still With Energy Ep

\

Photon With Energy Ep

Methane Molecule at Rest

Methane Molecule Still at Rest

Figure 1.5 An elastic collision between a photon and a methane molecule. Ep is the energy of the photon.

One way to think about how molecules absorb light is to consider the interaction as a colUsion between two particles. Imagine photons of light and molecules as simple particles shaped like spheres. When two particles colHde (be they molecules or billiard balls), several different types of colHsion can occur. The particles may undergo an elastic collision, which is illustrated in Figure 1.5. Before the colHsion, the incoming photon has energy Ep = hcW as stated in Equation (1.6), and imagine the methane molecule has no kinetic or vibrational energy, and can be considered at rest. By definition, an elastic colHsion results in no net energy exchange between the molecules. After the colHsion, the photon still has energy hcW, and the methane molecule is still at rest. The only thing that has changed is the direction of travel of the photon, and to a lesser extent, the position of the methane molecule. Elastic colHsions between photons and molecules result in a phenomenon called Rayleigh scattering, in which the direction but not the energy of the photon is changed. The intensity of Rayleigh scattering is proportional to the fourth power of the wavenumber of the photon, W, involved in the scattering. The highest wavenumber Hght we can see is blue light, so blue light is


9

Rayleigh scattered more intensely than other colors of light. Molecules in the upper atmosphere preferentially scatter blue light, which is why the sky is blue. Another type of photon/molecule collision are inelastic collisions. In this type of collision, energy is exchanged between the particles, and they leave the colHsion with different energies than before the colHsion. This is illustrated in Figure 1.6. Before the coUision, the incoming photon has energy E^, and the methane molecule is at rest. After the colHsion, some of the photon's energy is deposited into the molecule as vibrational energy, E^. The photon's energy after the colHsion is Ei (i for inelastic). AU processes must follow the law of conservation of energy, thus, E^ = Ey-\-Ei. The concentric circles in Figure 1.6 indicate that the methane molecule is vibrationally excited. Also, note that the direction of the photon has changed after the colHsion. Inelastic collisions between molecules and photons give rise to a phenomenon called Raman scattering. The amount of energy lost by the photon, £'v, is characteristic of the molecule involved in the scattering. Thus, after the colHsion the inelastically scattered photon carries chemical information. When the intensity and the wavenumber of these photons are analyzed and plotted, one obtains a Raman spectrum. Raman spectra are similar to infrared spectra in that they measure the vibrational energy levels of a molecule. Raman spectra can be used to quantitate chemical species, but is beyond the scope of this book. A third thing that can occur when a photon encounters a molecule is a totally inelastic collision. This is illustrated in Figure 1.7. Before the coHision, the photon has energy E^, and the methane molecule is at Before Collision

After Collision Photon With Energy ""l \tlip - tLv) — tLi

Photon With ^ Energy Ep

\


^--"irr--'' Vibrationally Excited Methane Molecule With Energy Ev

Figure 1.6 An inelastic collision between a photon and a methane molecule. Energy from the photon is deposited into the molecule as vibrational energy, E^. The photon leaves the collision with energy E^.

10


Before Collision

After Collision Photon With Energy Ep


^--"JriExcited Methane Molecule With Energy Ep

Figure 1.7 An inelastic collision between a photon and a molecule, known as absorbance.

rest. After the collision the photon has disappeared; all its energy has been absorbed by the molecule leaving it excited. This phenomenon is known as absorbance. The wavenumber of the light absorbed and the intensity with which it is absorbed depends upon the molecule involved in the collision. Thus, chemical information can be derived from a plot of absorbance intensity versus wavenumber, called an absorbance spectrum. Such a spectrum is seen in Figure 1.1. A totally inelastic colUsion between a photon and a molecule results in the disappearance of the photon, and the transfer of all the photon's energy into the molecule. The total amount of light absorbed by a sample is simply equal to the total number of photons that undergo totally inelastic colHsions with molecules. The decrease in the number of photons leaving the sample will give rise to an absorbance feature in the spectrum of the sample. Now, photons may also be scattered by macroscopic-size particles such as dust grains, leading to a decrease in photons exiting the sample as well. Experimentally, this will look hke absorbance but it is not. Thus, we are tacitly assuming that our sample has no other species present other than molecules that will interact with the light beam. In Figure 1.4, the change in Ught intensity across slab dL of the sample will be denoted by d/. The more photons and molecules there are in dL, the more coUisions will occur. The number of molecules in dL is simply determined by the analyte concentration, c. The number of photons present is given by the intensity of Hght in dL, which is simply L Additionally, the thicker the slab dL, the more photons will be absorbed because there are more molecules encountered. Thus, we can write a proportionality for


11

the light absorbed in dL as -dIcxIcdL

(1.7)

where d/=: amount of light absorbed in dL / = intensity of light impinging on dL c = concentration of absorbing species in dL dL = thickness of an infmitesimally thin slab of sample and the negative sign of d/ means that the intensity of the light beam decreases as it traverses dL. This proportionaHty assumes that the change in /across the width of dL is negUgible because dL is infmitesimally thin. Thus, we assume that / is the same at all points in dL. Note that the amount of hght lost simply depends on the number of analyte molecules present (c), the number of photons present (/), and the sample thickness (dL). Equation (1.7) tells us what parameters determine the total number of photon-molecule colhsions. Remember that molecules and photons can undergo several different types of collisions. Only some percentage of these colhsions will be totally inelastic. It is the number of totally inelastic collisions that determines the amount of hght absorbed. To calculate the number of these colhsions, we must multiply the right-hand side of Equation (1.7) by the fraction of colhsions that are totally inelastic. This number is called the absorptivity, and is denoted with the Greek letter epsilon, s. The absorptivity can also be thought of as a probabihty, it is the probabihty that a photon-molecule collision wiU be totally inelastic. The absorptivity depends upon the identity of the molecule absorbing the hght, and the wavelength of hght being absorbed. By inserting the absorptivity into Equation (1.7), it functions as a proportionality constant making the formerly quahtative relationship between d/ and concentration quantitative. We can then remove the proportionahty sign from Equation (1.7) to obtain -dI = sIcdL

(1.8)

where s = the absorptivity of the analyte molecule at wavelength X and the other terms have the same meaning as before. We can combine terms and rewrite Equation (1.8) as follows: -dI/I

= 6cdL

(1.9)

12


Since both sides of this equation contain an infinitesimal, we can integrate both sides as such

/ AI/I = sc f dL Jlo

(1.10)

Jo

The left-hand integration limits in Equation (1.10) mean that we are integrating between the intensities before and after the light passed through the sample. The right-hand integration limits are for the entire thickness of the sample, L. Note that s and c have both been brought outside the integration sign. This means that we are assuming that the concentration and the absorptivity are not a function of pathlength. In essence, we are assuming that the concentration and absorptivity are the same everywhere in the sample, i.e. the sample is homogeneous. Remembering that the integral of any quantity 1/1^ is In X, we can apply this to Equation (1.10) and evaluate the integrals to obtain \n(Io/I) = 6cL

(1.11)

Equation (1.11) is one way of stating Beer's law. However, it is traditional to express Beer's law using base 10 logarithms rather than natural logarithms. Recall that ln(10) = 2.303. We divide the right-hand side of Equation (1.11) by 2.303, which we will wrap into the absorptivity and rewrite to obtain \og(Io/I) = scL

(1.12)

To simpUfy Equation (1.12), we define a new quantity called the absorbance, denoted by A and given by ^=log(/o//)

(1.13)

and this allows us to rewrite Beer's law one last time to obtain its final form A = 8lc where A = absorbance £ = absorptivity / = pathlength c = concentration

(1.14)


13

This is the form in which Beer's law is most commonly expressed, and will be the form used throughout this book. What this equation tells us is that the amount of light absorbed by a sample depends on the concentration of the analyte, the thickness of the sample, and the sample's absorptivity. Also, note that the relationships in Beer's law are Hnear. For example, doubling the pathlength or concentration of a sample doubles its absorbance. Many spectrometers are capable of measuring spectra with the F-axis in absorbance units. An example of a spectrum plotted in absorbance units is seen in Figure 1.1. Because the relationship between absorbance and concentration is Hnear, the peak height or area of an analyte's absorbance band will vary hnearly with concentration. Sometimes spectra are plotted with the F-axis units in transmittance, which is defined as follows T = {I/Io)

(1.15)

where r:= transmittance The 7-axis values in a transmittance spectrum can vary from 1, when I=Io, to 0 when 7 = 0 (this means no hght is being passed by the sample). Transmittance measures the fraction of light transmitted by the sample. Spectra are also sometimes plotted with F-axis units of percent transmission, which is defined as % r = Tx 100 where %T=Vo transmission The F-axis values in a % transmission spectrum can vary from 100%, when / = / o , to 0% when 7 = 0 (when no light is being passed by the sample). Percent transmission measures the percentage of light transmitted by a sample. An example of a spectrum plotted in % transmission units is seen in Figure 1.8. Note that the peaks point down. Rearranging Equation (1.15) shows that \/T = (Io/I)

(1.16)

Substituting Equation (1.16) into (1.13) yields ^-log(l/r)

(1.17)

14


— " V V T N y - W ^ — N #«/^

2500

Af\

2000

Wavammba

Figure 1.8 The mid-infrared % transmittance spectrum of polystyrene.

Equation (1.17) establishes the relationship between absorbance and transmittance. Most spectroscopic software packages allow you to switch between the two units, and Equation 1.17 is used to make the conversion. If we substitute Beer's law (Eq. (1.14)) into Equation (1.17) we obtain

£lc^\og{\/T)

(1.18)

Raising both sides of the equation to the power of 10 yields 10'^' = \/T

(1.19)

and rearranging yields the fmal result: T = 10-'^^"

(1.20)

Equation (1.20) shows that the relationship between transmittance and concentration is not linear, making transmittance spectra inappropriate for use in quantitative analysis. It is absolutely necessary that you use spectra plotted in absorbance units for quantitative analysis because of the linear relationship between absorbance and concentration. Some further discussion of the absorptivity is in order. The absorptivity is the proportionality constant between concentration and absorbance.


15

It depends upon the wavenumber of light being absorbed, and the identity of the molecule doing the absorbing. For example, we can write _1700 / _1600 ^HiO 7^ ^ H 2 0

where the subscript denotes molecule and the superscript denotes wavenumber. This inequaUty shows that the absorptivity of water depends on the wavenumber, and illustrates that the absorptivity for any molecule varies with the wavenumber. Additionally, we can write ^1700 / ^1700 H2O / Acetone

illustrating that the absorptivity varies with molecule, even for a fixed wavenumber. For a given molecule and wavenumber of light, the absorptivity is a fundamental physical constant of the pure molecule. For example, the absorptivity of pure acetone at 1700 cm~^ is a fundamental property of the molecule like its molecular weight or boihng point. The absorptivity can be thought of as an "inherent absorbance." It measures, in an absolute sense, how strongly a specific molecule absorbs light at a specific wavenumber. A quick look at Beer's law (Eq. (1.14)) shows that the absorbance is a unitless quantity, so the units on the right-hand side of Beer's law must cancel each other. The product cL gives units of (length x concentration), so the units of the absorptivity (s) must be (length x concentration)"^ for the units in Beer's law to cancel properly. This sometimes leads to the absorptivity being expressed in units such as Hter/mole-centimeter. These units may be hard to understand, but further analysis shows that these units make sense. Recall that the absorptivity represents the probability of a photonmolecule collision being totally inelastic. Do the units of the absorptivity express themselves as a probabihty? Physicists often express probabilities in units of area, and call the quantity a cross section. Imagine throwing a baseball at the broad side of a barn. The probabihty of your hitting the barn increases the larger its area, and decreases the smaller its area. It all depends upon the size of the target. Additionally, your probabihty of hitting the barn is higher the closer you are to it, and lower the farther you are from it. In this case, the absolute area of the barn does not change, but the apparent area, how big the barn appears to you, changes with your distance from it. This apparent area is called the apparent cross section of the barn. Simply stated, the apparent cross section measures how large the barn appears to be from where you are standing. The same concept can be applied to photon-molecule colHsions. Molecules that strongly absorb

16


light undergo totally inelastic collisions with a high probabiHty. From the point of view of a photon, the molecule appears "large," and the interaction between the two has a large apparent cross section. This is measured experimentally as a large absorptivity. Molecules that absorb light weakly have a low probabiHty of undergoing totally inelastic coUisions with photons. From the point of view of the photon, the molecule appears "small," and the interaction is characterized by a small cross section. This is measured experimentally as a small absorptivity. Are the units of the absorptivity consistent with this idea of apparent cross section? It is common to measure concentration in moles/volume. The units of volume are length^. The unit of pathlength are length, so the units of the absorptivity are length^/(mole x length). CanceUng gives units of lengths/mole, where length^ is the unit of area. Thus, the absorptivity has units of area/mole when measured in this fashion. Since probability can be expressed in area as well, what the absorptivity measures is the apparent section of a mole of analyte molecules with respect to totally inelastic photon collisions. Consequently, this quantity is called the molar absorptivity. The units of the absorptivity do make sense when thought of as a probability. The absorptivity for a single molecule can be calculated by dividing the molar absorptivity by Avogadro's number to obtain an apparent cross section in units of area/ molecule. V. Variables Affecting the Absorbance and Absorptivity For a calibration to be legitimately appHed to standard and unknown samples alike, the absorbance reading for a given concentration of the analyte must be reproducible. To achieve this, an understanding of the variables other than concentration that affect the measured absorbance is essential. The purpose of this section is to discuss the important variables that must be controlled when performing quantitative spectroscopic analyses. This will involve discussing the physical processes behind the absorbance of Hght by molecules. Though this discussion is mathematical by its very nature, the amount of complex math has been minimized. For the mathematical underpinnings of the concepts presented here, consult Appendix 1 at the back of this chapter, or any of the undergraduate texts on physical chemistry or quantum mechanics cited in the bibliography.

A. THE IMPACT OF TEMPERATURE ON ABSORBANCE

As we all know from personal experience, matter is "clumpy," it comes in discrete pieces, be they people, rocks, molecules, or quarks. It turns out


17

that energy is also "clumpy" and comes in discrete packets called quanta. Therefore, energy is quantized. A quantum of energy is extremely small, and the fact that energy comes in discrete but small packets is not important for the consideration of the physics of the macroscopic world. However, in the microscopic world of atoms and molecules energy quantization has a huge impact upon how matter and energy interact. The field of physics that deals with the behavior of atoms, molecules, nuclei, and quanta of energy is called quantum mechanics. Quantum mechanics has Uttle to tell us about physical systems that are unbound, i.e. the parts of a system that are free to move at will. Imagine an electron moving through a vacuum or a baseball being thrown by a pitcher as examples of unbound systems. However, any time microscopic particles are bound, such as the electrons and protons in a molecule, their energy levels become quantized. This means that the particles in the system cannot have just any energy, but that only certain specific energies are allowed. The rotational, vibrational, and electronic energies of molecules are quantized. The reason that molecules have discrete spectral bands and absorb light at discrete energies is that their energy levels are quantized. There are a number of things that determine the quantized energy levels of a molecule, including its mass, type of atoms present, the strength of the chemical bonds, and the arrangement of the atoms in space. Two hypothetical quantized energy levels for a molecule are shown in Figure 1.9, with energy E\ for the lower level and E^ for the upper level. When a molecule absorbs a photon of Ught, it is said to make a spectroscopic transition from the lower to the upper energy level, as indicated by the arrow in Figure 1.9. Equation (1.6) gives the energy of a photon of light. For a photon to be absorbed and give rise to the transition seen in Figure 1.9, the difference in energy between the two levels must equal the energy of the photon.

Figure 1.9 A spectroscopic transition from a lower energy level {E{) to an upper energy level (û) occurs when a molecule absorbs a photon of light, as indicated by the arrow, xj/x and ^^ stand for the wavefunctions of the lower and upper energy levels, respectively.

18


This idea can be summarized in an equation form as follows: AE = hcW

(1.21)

where AE = Eu - El /z = Planck's constant (6.63 x 10"^"^ Js) c = the speed of Hght (3 x 10^^ cm/s) P^=wavenumber in cm~^ Equation (1.21) holds for any two energy levels in a molecule. A photon can only be absorbed if its energy happens to match a AE in a molecule. If the energy of a photon does not match any of the AEs in a molecule, it will pass through unabsorbed. Thus, Equation (1.21) gives a necessary but not sufficient condition for molecular absorbance to take place. It tells us that absorbance can take place, but says nothing about the probability of absorbance taking place. For the transition seen in Figure 1.9 to occur, there must also be molecules present in a sample that have energy E\. If all the molecules in a sample were somehow excited to energy level E^ (via some process other than light absorption, such as molecular collisions), there would be no molecules available with energy Ei to make the transition from Ei to E^. As a result, no light would be absorbed. More specifically, it is the difference between the number of molecules in the upper and lower energy levels that determines the number of photons that can be absorbed. In equation form, we would say that Acx(Ni-N^)

(1.22)

\/A(x\/(Ni-N^)

(1.23)

or

where A = amount of hght absorbed Aû = the number of molecules with energy E^ Ni — the number of molecules with energy E\ In essence, the greater the number of molecules in E\, the more that are available to absorb light of energy A^" and make the transition to E^. The fewer the molecules with energy Eu the fewer are available to absorb light of energy AE and make the transition to Ey.


19

There is a law of physical chemistry, called the Boltzmann distribution, which gives the ratio NJNi, the ratio of the number of molecules in the upper state to the number of molecules in the lower state. The Boltzmann distribution assumes that energy levels are populated due to random thermal processes (collisions). The formula for the Boltzmann distribution is Nu/Ni = Q-^^/^^

(1.24)

where ^ = Boltzmann constant, 1.38 x 10~^^ J/K r = temperature in Kelvin (K) The Boltzmann distribution is derived assuming that there is no change in NJN\ due to the absorption of Hght. Note that as temperature is increased, the ratio NJN\ increases as more molecules populate the upper state. This leaves fewer molecules of energy Ei available to make the transition from El to E^ via hght absorption. Also, note that as A^" increases, the ratio NJNi increases as it becomes more difficult for thermal processes to populate high energy states. Solving Equation (1.24) for A^^, we obtain Nu = Nie-^^^^^

(1.25)

Examination of Equations (1.22) and (1.25) will show that each contains the quantity N^. By substituting Equation (1.25) into Equation (1.22) we obtain ^ a (Aî - NxQ-^^'^^) and rearrangement gives

There are two things in Equation (1.26) worth noting. First, note that A is proportional to Aî, the number of molecules in the lower state. The size of a peak in an absorbance spectrum responds not to the concentration of molecules in a sample, but to the concentration of molecules in state Aî. Any process that causes Aî to change will affect the absorbance independent of the total concentration of the analyte molecules in the sample. As described above, changes in the temperature affect the relative number of molecules with energies E\ and E^. This phenomenon is responsible for the

20


^-AE/kt ^^^^ ^^ Equation (1.26), which shows that as temperature goes up, absorbance goes down. Thus, changes in the temperature can affect the measured absorbance even at constant pathlength and concentration. This means that at a fundamental physical level, absorbance does depend upon the temperature, and that a caHbration obtained with samples at one temperature will not necessarily give accurate results for samples at a different temperature. Temperature is one of the most important variables to control when performing quantitative spectroscopic analysis. The practical impact of Equation (1.26) is typically small. At room temperature (298 K), each molecule in a sample has about 200 cm~^ of thermal energy. If Ei = 0 cm~^ and £"^ = 200 cm~^ then N^ will be appreciable because there is enough thermal energy present to populate E^. Thus, small changes in temperature will have a noticeable effect on Aî. On the other hand, if Ei = 0 cm~^ and E^ = 3000 cm~^ a small change in temperature will have little effect on Aî. It would take a large increase in temperature for there to be enough thermal energy present to populate E^ to any appreciable extent. Aî will be large and Aû will be effectively zero. So, although temperature is an important parameter to control in spectroscopic quantitative analysis, the types of caHbrations most sensitive to this effect will be those involving spectroscopic transitions that are low in energy. However, there are other ways in which changes in the temperature can affect caUbrations, as will be discussed below.

B. THE IMPACT OF ELECTRONIC STRUCTURE ON THE ABSORPTIVITY

What we have not discussed yet is why some absorbance bands are more intense than others. This phenomenon can be seen in the spectrum of polystyrene in Figure 1.1. Note that the band near 700 cm~^ is more intense than the peaks near 3000 or 1400 cm~^ This means that the spectroscopic transition that gives rise to the peak at 700 cm~^ absorbs photons with a higher probabiHty than the transition that gives rise to the other peaks. Alternatively, the absorptivity of polystyrene at 700 cm~^ is higher than its absorptivity at 3000 or 1400 cm~^ What fundamental physical property causes transitions in the same molecule to absorb hght with different intensities? More generally, why do some molecules absorb hght more strongly than others? Recall from above the discussion of quantum mechanics. One of the postulates of quantum mechanics is that there exists for energy levels in a molecule a wavefunction that contains all the information about that energy level. Wavefunctions are typically denoted with the Greek letter psi, x//. For the spectroscopic transition shown in Figure 1.9, we denote


21

the wavefunction of the lower energy level by Vî, and the wavefunction of the upper energy level by xj/u. If we could a priori know all the wavefunctions of a molecule, we would know everything there is to know about the molecule. In the real world, this is not usually the case. An important use of spectroscopy is to measure molecular energy levels, and use these to help formulate reaUstic wavefunctions for molecules. One of the important properties of wavefunctions is that their square gives a probability. For example, the probability of finding an electron at a specific place in a hydrogen atom is equal to the square of the electron's wavefunction at that point in space. Now, a spectroscopic transition involves two different wavefunctions, xj/i and x/r^- The quantity that gives the probability that a spectroscopic transition will take place is called the transition probability, and it is denoted by |R^^|^, where the superscripts 1 and u denote the two levels involved in the transition. The equation that gives the transition probabiUty is IR'»|2=|y

VÎMVÛ dr

(1.27)

where |R^"|^ = the transition probability in going from state 1 to state u Vî = the wavefunction for the lower state in the transition /JL = dipole moment operator Vû = the wavefunction for the upper state in the transition and the dr indicates that the integral is taken over all space. The quantity being squared on the left-hand side, R^" is called the transition moment for the transition between the states 1 and u. Note that the transition probability depends upon the product of the wavefunctions of the two energy levels involved in a spectroscopic transition. R^" also depends upon a quantity called the dipole moment operator, ii. A dipole is simply two charges separated by a distance, and a dipole moment is a measure of charge asymmetry. The dipole moment operator for a molecule is calculated from the following equation: M

where II = dipole moment operator q = charge r = distance

Y^^qtri

(1.28)

22


H—CI Figure 1.10 The chemical bond in a hydrogen chloride molecule. The 8'^ and 8~ represent the partial positive and negative charges on the hydrogen and chlorine atoms. The arrow represents the magnitude and direction of the dipole moment for the bond.

and the subscript / denotes that we calculate a dipole moment considering the position of all the charged particles in a molecule. The dipole moment is a vector quantity, having both a magnitude and a direction. Vector quantities can be represented by arrows, where the length of the arrow is proportional to the magnitude, and the arrow point gives the direction. The dipole moment vector for the H-Cl molecule is seen in Figure 1.10. Each bond in a molecule will typically have a dipole moment called the bond dipole. The overall, or net dipole for a molecule, is the vector sum of the bond dipoles. This is expressed in Equation (1.28). For hydrogen chloride, since there is only one bond, the bond dipole and net dipole are the same. For a spectroscopic transition like the one shown in Figure 1.9, the dipole moment operator is related to the change in dipole moment between the lower and upper states. Thus, a molecule's electronic structure, the distribution of electrons and nuclei in space, determines the molecule's net dipole moment, and ultimately impacts the spectroscopic transition probability. Recall from the derivation of Beer's Law that the absorptivity is simply a measure of the probability that a molecule will undergo a totally inelastic collision with a photon. The transition probability measures the probability with which a specific spectroscopic transition will be excited by absorption of a photon. These two quantities are measuring the same thing, the probability of photon absorption. Thus, they are related to each other and we can write £a|R^^|2

(1.29)

The equation that more precisely relates these two quantities can be found in standard spectroscopic textbooks, such as the ones listed in the bibliography. By making use of Equations (1.27) and (1.28) we can also write (1.30) and £ (x fi

(1-31)


23

and by applying Beer's Law (Eq. (1.14)) Aocjji

(1.32)

In the jargon of quantum mechanics, the "observable" for a spectroscopic transition, the thing we actually measure, is the absorptivity. The integrated intensity, or peak area, of a spectroscopic transition is called the dipole strength of the transition. When a spectrum with absorptivity on the 7-axis is plotted versus wavenumber on the Z-axis, the peak area (dipole strength) of the band is given by D=

i£/WdW

= / Vî/xiAudr

(1.33)

where D = dipole strength (peak area) W = wavenumber Equations (1.32) and (1.33) are two of the most important relationships in all of spectroscopy. Equation (1.32) connects the absorbance, a spectroscopically observable quantity, to the electronic structure of a molecule. Equation (1.33) relates peak area to the underlying quantum mechanics behind the absorption process. Thus, we now have a picture of how molecules and light interact to give rise to absorption spectra. A more thorough quantum mechanical treatment of the absorption of light by molecules is given in the appendix to this chapter. There it is shown that treating a simple model of chemical bonds naturally gives rise to the quantized energy levels and selection rules that are such a large part of the absorption spectroscopy. The appendix is recommended reading for anyone who truly wants to understand what gives rise to absorbance spectra. Given the impact of electronic structure on the absorptivity, what factors influence the electronic structure of a molecule? Neighboring molecules, particularly in the solid and liquid phases, weakly interact with each other, which can alter the electronic structure of an analyte molecule. A good example of a molecular interaction is the hydrogen bonding that takes place in liquid water, as illustrated in Figure 1.11. The partial negative charge of one oxygen molecule interacts with the partial positive charge on the hydrogen of a second molecule, forming a hydrogen bond. This type of intermolecular interaction affects the electronic structure of the molecules involved. The electronic structure of a lone water molecule is different from that of water molecules with one or

24


8-

§+ /

A

H

H

H

A

\

H

S"*^

y

A

H

/

Hydrogen

B""**

H

Figure 1.11 The interactions of the nearest neighbor molecules in liquid water. The oxygen atoms have a partial negative charge ((5~) and the hydrogen atoms have a partial positive charge ((5"^). This type of interaction is called hydrogen bonding, as indicated by the dotted Hnes.

two hydrogen bonds. These water molecules are said to Hve in different chemical environments. The strength and number of interactions between neighboring molecules partly determines a molecule's chemical environment. Molecules in different chemical environments will have different electronic structures. We showed above that the absorptivity of a molecule depends upon the electronic structure, so the absorptivity depends upon the chemical environment. The water molecules in different chemical environments in Figure 1.11 will have sHghtly different absorptivities, which is another way of saying that they will have slightly different spectra. The absorptivity that is measured for a sample is averaged over all the chemical environments of the molecules in the Ught beam. We call the overall chemical environment of a sample its matrix. Thus, a sample's matrix affects its absorptivity. What factors influence the matrix of a sample? 1. Composition—The identities of the molecules in a sample determine the molecular interactions present, and hence impact the electronic structure of an analyte molecule. An acetone molecule in pure acetone finds itself in a different chemical environment than if the acetone molecule is dissolved in water. Acetone would have different absorptivities in these two cases. 2. Concentration—Even if the type of molecules in a sample is controlled, their concentration impacts the chemical environment. In a concentrated solution of acetone in water, an acetone molecule will have on an average more acetone molecules surrounding it than in a dilute solution of acetone in water. The acetone molecules in these two solutions will then find themselves in different chemical environments, and consequently have different absorptivities. This change in absorptivity with concentration is why Beer's law plots sometimes show a nonlinear relationship between


25

absorbance and concentration; the absorptivity has changed over the concentration range being measured. 3. Temperature—lncvQ3.SQS in the sample temperature cause the average thermal energy of the molecules in a sample to increase. The molecules in a hot sample move faster and with more energy than in a cold sample. Increased thermal energy also increases the number and energy of molecular colHsions, and the consequent pushing and pulling alters the strength of the intermolecular interactions. Ultimately, the electronic structure and the absorptivity are affected. This is in addition to the effect temperature has on caHbrations discussed above. 4. Pressure—The pressure of a sample matters because it determines the spacing between the molecules. Molecules under high pressure are forced closer together than molecules under low pressure, impacting the strength of the intermolecular interactions. This ultimately means that electronic structures and absorptivities can change with pressure. For solids and Hquids, the impact of pressure on the electronic structure is typically small, and day-to-day atmospheric pressure variations are not normally of concern. For gases, however, pressure is an important variable and should be controlled when possible. In summary, the sample matrix affects the absorptivity, and ultimately the spectrum of a sample. For a caUbration to be truly applicable to both the standard and unknown samples, the sample matrix should be reproduced as well as possible. There is no way of knowing, a priori, how a specific change in the sample matrix will impact the absorptivity of a sample. It therefore makes sense to measure the spectra and perform calibrations under different conditions to get a feel for the range of conditions under which your caUbration is accurate. It is never safe to assume that your calibration is good over all the conditions under which you will obtain unknown samples and measure their spectra. It may be necessary to obtain several caHbrations, and apply the appropriate one for the circumstances of the unknown sample being examined.

C. SUMMARY

For many of us who perform quantitative spectroscopic analyses, there is a tendency to dismiss the absorptivity as just a proportionality constant. It is easy to believe that its value is determined by the absorbances and concentrations measured, and there is not much thought given to its physical meaning. The whole purpose of this section has been to firmly establish the fact that the absorptivity does have physical meaning, and that

26


its value can be affected by experimental parameters as summarized in Equations (1.29)-(1.31). When performing a calibration with standard samples, we normally know the pathlength and concentration of the sample, and we measure the absorbance. Thus, the only piece of Beer's law that we do not know for the standard samples is their absorptivity. The entire reason why we have to perform a caHbration in the first place is to obtain the absorptivity. Now, the absorptivity is a fundamental physical constant of a molecule. If that is the case, why cannot one look up in a table the absorptivities for specific molecules at specific wavenumbers, just hke we can look up the melting points of substances? The reason is Equations (1.29)-(1.31). The electronic structure of a molecule can be influenced by a number of variables in the sample and the environment as we have seen. In a nutshell, the absorptivity is matrix dependent. If the absorptivity were not matrix dependent, we would only have to measure the absorptivity of a pure chemical once at a specific wavenumber. These values could be tabulated, and it would be a matter of simply looking up the absorptivity, sparing us the work of calibration. Sadly, this is not the case. Since the absorptivity is matrix dependent, we must calibrate for each matrix in which the analyte may be found. This is why spectroscopic calibrations have to be performed in the first place, and why it is so important to get them right. It has been stated elsewhere in this chapter that the fundamental assumption of quantitative spectroscopic analysis is that a calibration describes the unknown samples as well as the standard samples. Since it is the absorptivity we seek when calibrating, we are assuming that the analyte absorptivity is the same in the standard and unknown samples. To ensure that this happens, the sample matrix must be reproducible. If you control all the variables that impact the absorptivity of your analyte, your calibration will be applicable to unknown samples. If you do not, you will be using the wrong calibration on your unknown samples, and predict erroneous concentrations. Often the failure to understand and control these variables leads to the failure of a calibration.

VI. Gas Phase Quantitative Spectroscopic Analysis Quantitative spectroscopy can be performed on soHds, liquids, and gases. In this book, the information presented is general enough to be apphed to almost any type of sample. This section, however, will be devoted to the theory behind the quantitative spectroscopy of gases. Gases present a unique case. For gases, a simple equation called the ideal gas law relates the physical properties of a gas to its concentration. This type of equation is called an equation of state because it can be used to calculate the state of


27

a sample given enough information. Solids and liquids also have equations of state, but they are nowhere as simple or easily understood as the ideal gas law. The purpose of this section is to combine the ideal gas law with Beer's law to obtain an equation that relates absorbance to the physical properties of a gas. The derivation of the ideal gas law can be found in any of the physical chemistry texts cited in the bibliography. The ideal gas law states PV = nRT

(1.34)

where P = pressure F = volume n = number of moles of gas R = universal gas constant (0.082 Hter-atmosphere/mole-deg) r = temperature (normally expressed in degrees Kelvin, K) What Equation (1.34) tells us is the following; that for a constant amount of gas at a given temperature, the product of the pressure and the volume is constant. The gas constant, R, has been found by experiment to be a constant for a wide variety of gases under a variety of conditions. It is, in essence, a proportionaUty constant. Rearranging the ideal gas law in various ways illustrates the relationship between the different parameters. For example, we can say that V=:nRT/P

(1.35)

which says that as pressure goes up, volume goes down. The more pressure you apply to a gas, the smaller it is going to get, and Equation (1.35) is just a mathematical way of saying the same thing. Also, note in Equation (1.35) that volume and temperature are proportional. This means when a gas is heated it expands. Equation (1.35) is just a mathematical way of expressing something we already know from everyday experience. Although we have not derived the ideal gas law, these ideas should help its form seem sensible. We can rearrange the ideal gas law yet again to obtain n/V = P/RT

(1.36)

which says that the moles per volume of a gas is proportional to pressure and inversely proportional to temperature. A convenient set of units to measure moles per volume is moles per liter, or moles/Uter. These are also the units of concentration commonly used by chemists. Thus, we can rewrite

28


Equation (1.36) c = P/RT

(1.36a)

where c = n/V= concentration The concentration of a gas can be calculated by simply knowing its pressure and temperature. Earlier in this chapter we derived Beer's law, namely that A = sic We can rearrange Beer's law to solve for the concentration such that c = A/8l

(1.37)

Both the ideal gas law and Beer's law contain concentration. We can then set Equation (1.36a) equal to Equation (1.37) and obtain A/sL = P/RT

(1.38)

A = PsL/RT

(1.39)

which on rearranging gives

Equation (1.39) relates the absorbance, a spectroscopically observable property, to the physical parameters characterizing a gas sample. Simply put, for gases the absorbance depends upon not only the pathlength and the absorptivity, but the pressure and the temperature of the gas as well. This makes sense if we think about it. Remember that concentration is a measure of the number of molecules per unit volume. Equation (1.39) says that absorbance is proportional to pressure. If pressure is increased, molecules are forced closer together, creating more molecules per unit volume or increasing the concentration. An increase in the concentration causes an increase in the absorbance. This means that the total pressure of the standard and unknown samples must be the same for gas phase quantitative analysis. Equation (1.39) also says that absorbance is inversely proportional to temperature. When a gas is heated, it expands. The molecules get further apart, the number of molecules per unit volume goes down, and so we would expect absorbance to go down.


29

Earlier in this chapter, we discussed at length the temperature dependence of the absorptivity. We noted that Beer's law is derived assuming that the absorptivity does not change with the temperature. However, we saw that the absorptivity does change with the temperature because of its dependence upon a molecule's electronic structure. The conclusion is that temperature needs to be controlled when performing quantitative spectroscopy. We would call the dependence of the absorbance upon temperature implicit, which is not obvious from looking at Beer's law. In Equation (1.39), the temperature dependence of the absorbance is explicit. Any change in the temperature will produce an immediate change in the absorbance even if the absorptivity does not change with the temperature. This makes using one calibration at multiple temperatures for gases even more inappropriate than it is for sohds and Uquids. We could call Equation (1.39) "Beer's law for the Gas Phase." It relates the spectroscopic and physical variables of a gas phase analysis to each other. It would be quite difficult to derive a similar expression for liquids or solids. This equation should be foremost in the mind of anyone performing quantitative spectroscopy in the gas phase. BIBLIOGRAPHY

B. C. Smith, Infrared Spectral Interpretation, CRC Press, Boca Raton, Florida, 1999. B.C. Smith, Fundamentals of FT IR, CRC Press, Boca Raton, Florida, 1996. N. Colthup, L. Daly, and S. Wiberley, Introduction to Infrared and Raman Spectroscopy, Academic Press, New York, 1990. Donald A. McQuarrie, Quantum Chemistry, University Science Books, Mill Valley, California, 1983. P. Griffiths and J. de Haseth, Fourier Transform Infrared Spectrometry, Wiley, New York, 1986. D. Skoog and D. West, Principles of Instrumental Analysis, Holt, Rinehart, & Winston, New York, 1971. J. Robinson, Undergraduate Instrumental Analysis, Marcel Dekker, New York, 1995. D. Peters, J. Hayes, and G. Hieftje, Chemical Separations and Measurements, WB Saunders, Philadelphia, 1974. M. Diem, Introduction to Modern Vibrational Spectroscopy, Wiley, New York, 1993. J.M. Hollas, Modern Spectroscopy, Wiley, New York, 1996. G. Castellan, Physical Chemistry, Addison-Wesley, Menlo Park, 1971. I. Levine, Physical Chemistry, McGraw-Hill, New York, 1983. John Winn, Physical Chemistry, Harper Collins, New York, 1995. T. Hirschfeld, Quantitative FTIR: A Detailed Look at the Problems Involved, Chapter 6 of FTIR: Applications to Chemical Systems, Vol. 2, edited by J. Ferraro and L. Basile. Academic Press, New York, 1979. D. Burns and E. Ciurczak, Handbook of Near Infrared Spectroscopy, Harcourt, New York, 1997.

30


Appendix: The Quantum Mechanics of Light Absorption The purpose of this appendix is to fully establish the theoretical basis behind the absorption of light by molecules. You do not need to understand the theory of hght absorption to run a spectrometer and measure a spectrum. However, to truly understand what a spectrum means, where it comes from, and what variables impact spectroscopic caUbrations, a familiarity with this theory is necessary. I have attempted to write this section assuming that the user is not famihar with quantum mechanics. This is, however, a daunting task given the complexity of the math in this field. Thus, a nodding familiarity with the basics of quantum chemistry will be very helpful in deriving maximum benefit from this section. This appendix will be very focused, only those quantum mechanical concepts that impact directly on the Hght absorption process will be discussed. For those wishing a more detailed discussion of quantum mechanics, consult any of the physical chemistry texts listed at the end of this section. The discussion in this section owes much to References [1, 2, 4]. A famiharity with the contents of Chapter 1 of this book will be assumed. We will begin the appendix with an introduction to wavefunctions and Schrodinger's equation. We will then proceed to apply the time independent Schrodinger's equation to a simple system, the particle in a box, and derive the energy levels for it. This system is an oversimpUfied, but useful model for systems such as electrons bound in chemical bonds. We will see that one of the most important aspects of quantum mechanical systems, the quantization of energy levels, is natural for bound microscopic systems. We will then look at doing spectroscopy on the particle in a box, ascertaining how the particle might make the transition from one energy level to another. We will apply the equation for the transition probability given in Chapter 1, and derive selection rules for the particle in the box very similar to those for vibrational and other kinds of spectroscopy. We will see that the selection rules are a natural outgrowth of the shape of the wavefunction for the particle in the box.

I. Wavefunctions and Schrodinger's Equation We all know from personal experience that matter is "clumpy." It comes in discrete pieces, be they molecules, atoms, protons, or quarks. It turns out that energy is also "clumpy" and comes in discrete packets called quanta, so energy is said to be quantized. Einstein's famous equation relating matter and energy, E = mc^, tells us that matter and energy are equivalent. Thus, if matter is clumpy, then energy should be clumpy as well. A quantum of energy is extremely small, and the fact that energy comes in discrete


31

packets is not important in the macroscopic world. However, in the microscopic world of atoms and molecules energy quantization has a huge impact upon how matter and energy interact. The field of physics that deals with the behavior of atoms, molecules, nuclei, and quanta of energy is called quantum mechanics. Quantum mechanics has little to tell us about physical systems that are unbound, i.e. the parts of a system that are free to move at will. Imagine an electron moving through a vacuum, or a baseball being thrown by a pitcher, as examples of unbound systems. However, any time microscopic particles are bound, such as the electrons and protons in a molecule, their energy levels become quantized. This means that the particles in a molecule cannot have just any energy, only certain specific energies are allowed. The rotational, vibrational, and electronic energies of molecules are quantized. The reason that molecules have discrete spectral bands is that they absorb light at discrete energies that match those of an energy level difference in the molecule. There are a number of things that determine the quantized energy levels in a molecule. These include the mass and type of atoms in a molecule, the strength of the chemical bonds, and the arrangement of the atoms in space. Two hypothetical energy levels for a molecule are shown in Figure A.l, with energy E\ for the lower level and E^ for the upper level. When a molecule absorbs a photon of light, it is said to make a spectroscopic transition from the lower to the upper energy level, as indicated by the arrow in Figure A. 1. One of the postulates of quantum mechanics is that there exists for energy levels in a molecule a wavefunction that contains all the information about that energy level. Wavefunctions are typically denoted with the Greek letter psi, i//. For the spectroscopic transition shown in Figure A.l, we denote the wavefunction of the lower energy level by Vî, and the

Figure A.l A spectroscopic transition from a lower energy level (£"1) to an upper energy level (£u) that occurs when a molecule absorbs a photon of light, î and 1/^^ stand for the wavefunctions of the lower and upper energy levels, respectively.

32


wavefunction of the upper energy level by Îr^. If we could a priori know all the wavefunctions of a molecule, we would know everything there is to know about the molecule. In the real world, this is often not the case. Frequently, the form of wavefunctions is assumed, or calculated from spectroscopic data. The simplest way of writing how to determine the energies of a bound system is the general formulation of Schr6dinger's equation, which says H^lr = Ef

(A.l)

where H = The Hamiltonian operator \lr = Wavefunction E = Energy This equation is not as simple as it looks. The symbol H is not a simple variable, but an operator. An operator is a symbol that tells you to do something to whatever follows the symbol. The Hamiltonian operates upon wavefunctions to give energies. For a given quantum mechanical system, it is not necessarily obvious what form the Hamiltonian operator should take. By definition, the Hamiltonian represents the total energy, kinetic plus potential, of a system. More succinctly, H = (K^P)

(A.2)

where ^ a k i n e t i c energy P = potential energy When possible, expressions for the kinetic and potential energy of a system are incorporated into the Hamiltonian. For a one-dimensional system whose states are stationary, i.e. not changing over time, the Schrodinger equation is [1] -fi/2md^/dx^ + P(xM where fi = {hl2ii) h = Planck's constant m = mass X = distance P = potential energy

= E^lr

(A.3)


33

The equivalence of Equations (A.l) and (A.3) can be seen by recognizing that the term in curly brackets on the left-hand side of Equation (A.3) is the Hamiltonian operator, H.

11. The Particle in a Box Perhaps one of the most simple quantum mechanical systems is the one-dimensional particle in a box. This is illustrated in Figure A.2. The box has length L. The walls of the box are infinitely high so the particle cannot escape. This means that the potential energy of the particle is infinity for X not between 0 and L. We will also assume that the potential energy of the particle inside the box is zero. Thus, P = 0

forO<x L

Since the potential energy inside the box is zero, Schrodinger's equation reduces to (A.5)

-fi/2md^/dx^}xl/=Ejl/

X =0

X=L

Figure A.2 A particle in a box of length L with infinitely high potential energy walls on each side at positions X=0 and X=L.

34


It may not be obvious at this point, but this simple model has some utiHty. The particle could be an electron, and the infinitely high walls represent the chemical bond force holding the electron in its orbital. A discussion of the nature of the wavefunction, i/^, is in order. In the macroscopic world, the square of the wavefunction describes the amplitude of a wave as a function of position and time. For example, imagine tying a piece of string to a doorknob and shaking it. If we knew the wavefunction for the motion, we could calculate the ampUtude of the string at any point at any moment in time. The French physicist de Broglie discovered that when any type of matter moves, it describes a wave. The wavelength of a "matter wave" is given by A = h/p

(A.6)

where X = matter wavelength h = Planck's constant p = momentum For macroscopic systems matter waves are small enough to be irrelevant. For example, a 5 oz baseball hurled by Pedro Martinez at 90 mph has a de BrogHe wavelength of 1.2 x 10~^^ m. A trivial size compared to that of a baseball, and a phenomenon that American League batters need not worry about. For a microscopic particle moving within a bound system, the square of the wavefunction can be thought of as the "ampUtude" or "intensity" of its matter wave [2]. It may seem nonsensical for matter to have intensity, but one interpretation of the square of the wavefunction is that it represents the probabiHty of finding the particle at a given location. The "intensity" of a matter wave is simply a probabiHty. For the particle in the box, the probability of finding the particle outside the box is zero, while the probability of finding the particle inside the box is 1 (the particle must be in the box). Thus,

/

}lf^dx=l

for 0 < jc < L

(A.7)

Another condition we impose upon the wavefunctions is that they be continuous. A continuous function has lvalues that vary smoothly with X; there are no "jumps" in a continuous function. An inexact, but useful


35

picture of a continuous function is that it can be drawn without Hfting your pencil off the paper you are drawing on [3]. In the case of our particle in the box wavefunctions, since the value of i/^^ outside the box is zero, the value of V^^ at the walls of the box must equal zero for the wavefunction to be continuous. Else, there would be a "jump" or discontinuity in the value of i/r^ in passing from inside the box to outside the box. In equation form this is expressed as follows: iA(0) = 0

(A.8)

and f{L) ^ 0 The conditions in Equations (A.7) and (A.8) are called the boundary conditions for the particle in the box problem. In many problems of this type, the boundary conditions play a large part in determining the form of the solution. Our job now is to solve Equation (A.5), the Schrodinger's equation for the particle in a box. Note that for any function V^ to be a solution to this equation its second derivative (on the left-hand side of the equation) must equal the function times a constant (the right-hand side of the equation). Combinations of sin and cosine waves fit this description. Thus, wavefunctions \lf that solve Equation (A. 5) have the form f = A sin(Cx) + B cos(Cx)

(A.9)

where A = constant B = constant C = (2mE)^^^/fi We now need to impose the boundary conditions on these wavefunctions. One boundary condition is that T//^(0) = 0. Recall that cos(0)=l. For the cosine term to follow the boundary condition, B = 0, which simplifies our solution to ilf = Asm(Cx)

(A. 10)

Another boundary condition is that \lf(L) = 0, thus V^(L) = ^ sin(CL) = 0

(A. 11)

36


The sin function has the value of zero at 180°, 360°, 540° etc, in other words at multiples of 180°. It is convenient to represent angles in units of radians. A 180° angle has n radians, a 360° angle contains 2n radians, etc. Thus, Equation (A. 11) is true if CL = nn

(A. 12)

where n =

0,1,2,3...

which simply depends on the fact that the sin of multiples of n radians is zero. Combining Equation (A. 12) with C = {2mE)^^^/fi gives mi = {2mE)^'^L/fi

(A. 13)

We next remember that fi = h/2n, substitute this into Equation (A. 13), and rearrange to solve for E as follows: E = h^n^/SmL^

(A. 14)

where n = 0,1,2,3...

the quantum number

This is the equation we have been seeking; it gives the quantized energy levels for the particle in a box. In this equation, n is referred to as a quantum number. Note that the quantization of the energy occurs as a natural outcome of restraining the particle to reside in the box, and assuming that the wavefunction is continuous. No other assumptions or constraints are needed. Now we can also derive the form of the wavefunctions for the particle in the box. We have already stated this form in Equation (A. 10). However, if we rearrange Equation (A. 12) we obtain C = nn/L

(A. 15)

-^ = A sm(nnx/L)

(A. 16)

substituting into (A. 10) yields


37

¥4

V3 E2 = 4h78m]

¥2

Vi

Figure A.3 The first four wavefunctions and energy values for the particle in a box.

H 2 C—CH—CH—CH 2 Figure A.4 The chemical structure and bonding pattern of 1,3-butadiene.

It can be shown via normalization that ^ = (2/L)^/^ [1]. The first four wavefunctions (n—l, 2, 3, and 4) and the energy levels for the particle in a box are shown in Figure A.3. Note that the wavefunctions look similar to some of the simple sin wave patterns that can be induced by wiggling a rope or string anchored at one end. The particle in a box models, at least crudely, the electrons in a chemical bond. Consider the structure of the butadiene molecule in Figure A.4. This molecule consists of two C=C bonds connected by a C C bond. The two double bonds interact with each other via the dumbbell shaped (p) orbitals shown on the right-hand side of Figure A.4. As a result, conjugation takes place and the four electrons in these bonds are free to travel the length of the molecule. Consider this bonding pattern as a particle in a box. The length of the box would be the length of the molecule. Assuming that the molecule is linear (which it is not), the box length would be equal to twice the bond length of a C=C bond plus the bond length of a C-C bond. Typical values for these quantities are 1.35 A and 1.54 A respectively, which would give a box length of 5.78 A.

38


n=3

n=2

n=l Figure A.5 The first three energy levels of the conjugated electrons in the butadiene molecule modeled as a particle in a box. The 4 electrons populate the n=\ and n = 2 levels. A spectroscopic transition, indicated by the arrow, can take place promoting an electron from the n = 2 to the n = 3 level.

Now, there are in actuality four electrons in this box. There is a rule of quantum mechanics called the Pauli Exclusion Principle that says no more than two electrons can occupy the same orbital. This means that two of the electrons are in the n = 1 energy level, and two of them are in the n = 2 energy level as shown in Figure A.5. The energy levels of the particle in a box are given by Equation (A. 14). An electron in the n = 2 level can in theory absorb a photon of light and undergo a spectroscopic transition to the n = 3 level. Thus, the resulting absorbance feature would appear at an energy that is the difference between n = 2 and n = 3 levels. Using Equation (A. 14), this gives A ^ = (3^ - 2^)h^/%mL^ AE = 5h^/^mL^

(A.17)

where A£' = energy difference Using L = 5.78 A and an electron mass of 9.11 x 10~^^ kg gives AE = 45,466 cm~^ In reaUty, the UV-Vis spectrum of 1,3-butadiene has an absorbance feature at 46,083 cm~^ (217 nm), an error of less than 1.5%. This is not a bad agreement given the simplicity of this model.


39

III. Transition Probabilities for the Particle in a Box Recall from Chapter 1 that the absorptivity of a spectral feature depends, ultimately, upon the transition probabiUty which is given by

|R'"l' = 1/

1

(A. 18)

where |R^"|^ = the transition probability in going from state 1 to state u Vî = the wavefunction for the lower state in the transition /x = dipole moment operator Vû = the wavefunction for the upper state in the transition If we take the square root of both sides of Equation (A. 18), we get an expression for the transition moment, R^^ as such

" ' " = / •VîMiAudT

(A. 19)

Recall also from Chapter 1 that the dipole moment operator is given by /x = ^ . ^ / r ,

(A.20)

where /x = dipole moment operator q = charge r = distance / = particle number in a molecule For the particle in a box problem, there is only one particle so i=l in Equation (A.20), and /x will depend upon the charge and position of the particle as such fi = eX

(A.21)

where e = charge A'= distance The wavefunctions for the particle in a box are given by Equation (A. 16). We can then insert Equation (A.21) and (A. 16) into Equation (A. 19) to

40


come up with an equation for the transition moment for the particle in a box as such R^^ = {2e/L) j sm(lnX/L)X

sm{unX/L) dx

(A.22)

where / = quantum number for the lower energy state u = quantum number for the higher energy state The charge on the particle e and the constant A have been brought outside the integral because they do not depend upon x. Equation (A.22) can be solved by applying known properties of the products and integrals of trigonometric functions [4] to obtain the following: ^ 1 ^ ^ eL {cos[(u -1)K]-1

COS[(U + / )7r] -

1}

We could use Equation (A.23) to calculate the transition moment for any two energy levels for the particle in a box. However, we seek to obtain a more general description of what transitions are allowed by noting when R^" = 0. It is a property of cosine waves that cos(^7i) = — 1 for n = odd

(A.24)

and cos(/77i) = 1 for n = even We can see from Equation (A.23) that the transition moment for the particle in a box depends upon the quantities [cos(w — /)7r — 1] and [cos(w + /);! — 1]. For any transition where cos(w — /)TC and cos(w + /)7r equal 1 will mean both cosine terms in the numerators in Equation (A.23) would equal 1, and after subtracting 1 both terms go to zero, giving a transition moment of zero. It turns out that if u and / are both even, or are both odd, this condition is met (i.e. (odd + odd) = even, (odd — odd) = even, (evenH- even) = even, (even —even) = even). For example, for 1 = 2 and w = 4 (w-h/) and (u — l) are both even, and R^^ = 0. This type of transition is said to be a. forbidden transition because its transition moment is zero. Alternatively, combinations with u odd and / even, or u even and / odd give cosine terms in the numerators in Equation (A.23) of - 1 (i.e. (odd + even) = odd, (odd even) = odd, (even - odd) = odd). The numerators then each reduce to - 2 ,


41

and the total value for R^" is nonzero. For example, for 1 = 2 and w = 3 (the transition seen in Figure A.5) both (/+w) and {l-u) are odd, making R^" nonzero. This type of transition is said to be an allowed transition because the transition moment is nonzero. If An = {u — l), transitions are forbidden for An = even and allowed for An = odd. We can summarize all of this in equation form as follows: R^^ = 0

for A« = ±2, ± 4, ± 6 . . .

(A.25)

and R^^TÔ for A/i = ± l , i b 3 , i b 5 Equations (A.25) are an example of what are called selection rules. These rules determine what types of spectroscopic transitions are allowed in bound systems. Selection rules like those in Equations (A.25) exist for mid-infrared, near-infrared, and UV-Vis spectroscopy. The point of this derivation is that selection rules, like energy quantization, are a natural consequence of bound systems. Many of the important features of absorption spectra are determined by the energy levels of molecules, and the selection rules imposed upon those energy levels. Examination of the particle in the box shows that energy quantization and selection rules arise naturally out of binding microscopic particles, no other conditions or assumptions are necessary. Although this treatment will not accurately predict the actual absorption spectra of real world molecules, it at least gives a feel for what gives rise to important aspects of absorption spectra. REFERENCES

[1] Max Diem, Introduction to Modern Vibrational Spectroscopy, Wiley, New York, 1993. [2] Donald McQuarrie, Quantum Chemistry, University Science Books, Mill Valley California, 1983. [3] Sherman Stein, Calculus and Analytic Geometry, McGraw-Hill, New York, 1977. [4] Ira Levine, Physical Chemistry, 2nd Edition, McGraw-Hill, New York, 1983.



The purpose of this chapter is to provide theoretical and practical information in determining the concentration of a single analyte in a sample. The field of multiple analyte analysis is covered in later chapters. Even if you do not plan to perform single analyte analyses, you should read this chapter. Many of the important concepts of quantitative spectroscopy are best learned using a single analyte as an example. This chapter begins with a discussion of the basics of precision and accuracy, and then moves on to how Beer's law is used to achieve cahbrations. Since this usually involves plotting a straight line, a discussion of how to plot and determine the equation for a straight Hne is given. This leads to a presentation of several important statistics, and how they are used to determine cahbration quahty and robustness. The chapter concludes with sections on standard methods, experimental protocols, measuring absorbances properly, and avoiding experimental errors.

43

44


I. Precision and Accuracy Before we proceed to a discussion of how spectroscopic calibrations are generated, and how their quaUty is measured, the ideas of precision and accuracy need to be discussed. These two quantities measure data quahty, and are often confused. All measurements contain some level of error, regardless of how well we perform an experiment, or how well our instruments work. For example, we may know the concentration of the analyte in a standard sample to be 60 ± 5 % , or the position of an absorbance band to be 1700 ± 2 cm~^ (the number after the ± represents the amount of error in the measurement). Another name for experimental error is noise. Noise is what we try to minimize to maximize the quality of a measurement. Signal is the measurement itself, which is the number before the it. The best data have maximum signal and minimum noise. A quantity used to measure data quahty is the signal-to-noise ratio (SNR) SNR = Signal/Noise where SNR = Signal to Noise Ratio Note that the higher the SNR, the better the data. Two types of error contribute to uncertainties in measurements. Random error is caused by variables over which we have no control, such as pressure, temperature, fluctuations in the electrical voltage powering an instrument, and sloppiness in the experimental technique of the analyst. A digital clock whose battery is dying and whose display flashes random times would give time measurements with a great deal of random error. The sign of random error is random; it can be either positive or negative. Hence, the di symbol is used to denote the size of random error. Performing multiple measurements and adding the measurements together can reduce the random error. This is why we frequently perform an experiment multiple times and report the average of the data as the final result. For example, the reason we add scans together in Fourier Transform spectroscopy is to reduce the noise level of the spectrum. Random error can be reduced, but never eliminated. We can never have complete control over every variable that affects a measurement. Systematic error is error that always has the same sign and magnitude. A digital clock that is 1 hour slow because it was not set for daylight savings time is an example of a systematic error. The clock will always be

2.

45


in error by the same amount (1 hour) and in the same direction (slow). In order to calculate systematic error, we must have a true reference value to compare to the measured value. For the digital clock example, the actual time is the reference value. Systematic error is also called bias. Unlike random error, systematic error can be eliminated. In the case of the digital clock, its systematic error is removed by setting the clock ahead by 1 hour. Precision is a measure of the size of the random error in a measurement. Precision is often determined by measuring the same quantity several times, then looking at the scatter in the data. Accuracy is a measure of how far away a given measurement is from its true value. Like systematic error, determining accuracy requires the existence of a true value to which to compare the measured value. Accuracy includes the effects of random and systematic errors, and is a more complete description of the quahty of a measurement than precision. Precision and accuracy are not the same, as can be visualized by the targets seen in Figure 2.1. Consider the center of the bull's eye to be the true value of a measurement. The left-hand target in Figure 2.1 is the worst case scenario. The data are imprecise, scattered all over the place, and are inaccurate, falling far away from the center of the target. The data in the middle target are somewhat better. The data are precise, forming a tight cluster with httle scatter. However, the data are inaccurate because the cluster is centered far from the bull's eye. The distance from the center of the data cluster to the center of the target is a measure of the systematic error in the data. Note that precision is a measure of random noise only, and that data devoid of random noise can still be wrong because of bias. The rightmost target in Figure 2.1 is the ideal situation. The data are precise, forming a tight cluster, and the data are accurate, falling near the bull's eye. Note that good data have small amounts of both types of error, and that accuracy measures both the systematic and random errors.

Imprecise & Inaccurate

Precise & Inaccurate

Precise & Accurate

Figure 2.1 A visualization of the concepts of precision and accuracy using a target.

46


II. Calibration and Prediction with Beer^s Law Beer's law was derived in Chapter 1. As a review, we will restate its form here A=:6lc

(2.1)

where A = absorbance 6 = absorptivity / = pathlength c = concentration To obtain a cahbration using Beer's law, we must first observe that its form is analogous to the formula for a straight line. The formula for a straight line is Y = mX^b

(2.2)

where Y= 7-axis value of a data point m = slope of line X= X-axis value of a data point b — F-intercept of the Hne (where the Hne crosses the F-axis) Comparison of Equations (2.1) and (2.2) shows that if absorbance is plotted on the 7-axis, and concentration is plotted on the Z-axis, a straight hne should be obtained whose slope is given by m = sl

(2.3)

The slope of a Une can be measured by choosing two (or preferably more) data points on the line and calculating the difference between their 7-axis values (AY), then calculating the difference between their X-axis values (AX). Then, we divide A F b y AX (or put more simply, calculate rise/run) to obtain the slope of the Une. To obtain the data needed to develop a calibration the absorbances of a series of standard samples are measured. The peak height at a specific wavenumber, or an integrated peak area calculated with fixed limits, is used (see below for more on measuring peak heights and areas). The concentrations are determined by some method other than the one used to obtain the calibration. Frequently, standards are made with measured

47

2. SINGLE ANALYTE ANALYSIS

Peak Area Calibration for IPA y = 0.165X - 0.6834 R = 0.999 12

^

10 8 6 4 2 0

^

"¥

20

40 IPA

60

80

Figure 2.2 A calibration line for the analysis of isopropanol (IPA) in water. The 7-axis is the area of the IPA peak centered at 2973 cm^^ The peak area was integrated between 3008 and 2947 cm~^ The I'-axis is volume percent IPA in water. The data are contained in Table 2-1.

masses or volumes of analytes. Alternatively, a method such as titration or chromatography can be used to determine the concentration of the analyte in standard samples. This points out that quantitative spectroscopy is a secondary method of determining concentrations. We cannot caHbrate a spectrometer until some primary method of analysis is used to determine the analyte concentration in the standards. An example of a calibration obtained using Beer's law is seen in Figure 2.2. This figure is a Beer's law plot or calibration line for the determination of isopropanol (IPA, also known as rubbing alcohol) mixed with water. The 7-axis is the area of the IPA peak centered at 2973 cm~^ (the methyl group asymmetric C-H stretch, integration limits 3008-2947 cm~^). The X-axis is the volume percent of IPA in water. This hne is the mathematical model relating absorbance to concentration. The data used in making this plot are seen in Table 2-1. An overlay of the peaks analyzed is seen in Figure 2.3. The experimental details of how the spectra were measured are contained in the appendix to this chapter. If a plot of absorbance versus concentration is linear, a system is said to follow Beer's law. Beer's law plots may be nonhnear for a number of reasons as discussed later. Once a cahbration is in hand, the next step is to predict the concentration of the analyte in an unknown sample. We first need to write Beer's law for the unknown sample *unk = skyunk

(2.4)

48


TABLE 2-1 Volume % and peak area for calibration line seen in Figure 2.2. %IPA

IPA

Area

9 18 35 53 70

1 2.2 4.9 8 11

2973 cm-1

^

T 2980 2970 2960 Wavenumber (cm-1)

2950

2940

Figure 2.3 An overlay of the five peaks whose areas are given in Table 2-1. These data are used in the calibration line seen in Figure 2.2. The dashed vertical line shows how the peak maximum changes with the concentration. This is an example of absorptivity drift as discussed in Chapter 1.

where ûnk = absorbance of the unknown sample el = thQ product of the absorptivity and the pathlength (= the slope of the calibration line) Cunk = the analyte concentration in the unknown sample Ultimately, it is the concentration of the analyte in the unknown sample, Cunk, ill which we are interested. Rearranging Equation (2.4) gives ûnk — ^ u n k / ^ t


49

From Equation (2.3) m = £l giving Cunk = Aunk/fn

(2.5)

According to Equation (2.5), predicting the concentration of an analyte in an unknown sample is a matter of dividing the measured absorbance of the unknown sample by the slope of the caHbration Une. Note that to use Equation (2.5) you do not need to know the absorptivity and the pathlength expHcitly, just their product. In comparing Equations (2.1) and (2.2), note that Beer's law does not have a second term. This means that b, the F-intercept of the hne, should equal zero. In other words, the Une should pass through the origin. Sometimes plots of absorbance versus concentration have nonzero F-intercepts. At the origin the analyte concentration is zero, so should not the absorbance be zero as well? Not necessarily; part of the reason is absorbance by interferents. For example, solvent or impurity molecules may absorb at the same wavenumber as the analyte. Noise is also a culprit; the line may not pass exactly through zero due to the noise in the absorbance data. In general. Beer's law plots should not be forced through the origin. In doing so, it is assumed that the absorbance at zero concentration is exactly zero, which is not necessarily the case. In Chapter 1, there was a discussion of how concentration affects the absorptivity. An example is seen in Figure 2.3. Looking at the figure from top to bottom, the concentration of IPA goes from 70 to 9%. The vertical dashed line in the figure passes through the peak in the top spectrum at 2973 cm~^ At lower IPA concentrations, the peak maximum moves to a higher wavenumber and the peak becomes broader, making it increasingly difficult to determine the peak edges needed to perform an accurate integration. Ultimately, these changes in the spectrum of IPA with concentration were not enough to greatly affect the quality of the caHbration. However, Figure 2.3 illustrates that changes in absorptivity with concentration are real, and if the changes are large enough, inaccurate caUbrations may result.

III. Plotting and Analyzing Lines In single analyte analysis the calibration model is typically a straight line. To understand these cahbrations we must learn more about straight Hues and their properties. This section will cover how equations for straight Hues are calculated from a set of data, and what statistics can be calculated to assess the quality and robustness of a caHbration.

50


A. LINEAR REGRESSION: LEAST SQUARES FITTING

Regression is the process by which a mathematical model is generated relating sets of data. Linear regression assumes that two sets of data have a linear relationship. One method of using Hnear equations to regress two sets of data and calculate the line relating them is called the least squares equations. The resulting Hne is sometimes called a least squares fit [1-5]. You have probably heard of the least squares fitting routine, and may have used it to plot lines using various computer programs such as Microsoft Excel®. It is easy to treat least squares fitting routines as black boxes, simply stuffing the data in and getting out caHbration lines. However, least squares routines have certain attributes and assumptions associated with them that must be understood to use them properly. It is time we opened the black box.

1. Properties of the Least Squares Fit

It can be shown [1] that the slopes and F-intercepts calculated using the least squares methods are unbiased. This means no systematic error is introduced into the results by the calculation. This may seem an obvious requirement for a fitting method, but there are fitting methods that introduce bias into the calculated results [1]. The least squares fit will not remove bias already present in a set of data, it will simply not add any bias of its own. Another property of least squares methods is that they provide the best fit possible to the data. In Figure 2.2, you can see that all of the data points fall a Httle below or above the calculated Une. The distances of the data points from the hne represent the error in the least squares fit. It can be shown [1, 2] that a least squares fit provides a line that minimizes the distance between the data and the calculated hne. Thus, a least squares fit provides a hne with the least possible error in it. The proof of this idea [3], while beyond the scope of this book, is recommended reading because it is relatively easy to follow and provides an interesting insight into how least squares methods work. Thus, two important properties of least squares fits are that they provide unbiased, minimum error results for the slope and the intercept of lines being fit to the data. It is no wonder then that this method is widely used for regression analysis.

2. Assumptions of the Least Squares Method

There are a number of assumptions made when applying least squares fits to data. A complete understanding of these assumptions is necessary to


51

insure successful calibrations. Here are some of the assumptions behind the least squares method [1]. 1. Perfect Model—In using the least squares method, it is assumed that the equation being used perfectly describes the data. In other words, there are no variables affecting the data outside those represented in the equation describing the data. This puts a huge burden on the analyst. It is up to you to figure out which equation best describes the data in front of you. For single analyte quantitative spectroscopic analysis, the equation being fit is Beer's law. By using Beer's law in a least squares fit, we are assuming that the system follows Beer's law, and that there are no parameters other than the absorptivity, pathlength, and concentration that affect the absorbance. We already know from Chapter 1 that many variables affect the absorbance besides those in Beer's law. If those variables are not controlled, not only will it be inappropriate to use Beer's law, it will be inappropriate to use least squares fitting on the data as well. 2. Linear Model—When performing linear regression, the equation being fit must be linear. Specifically, this means that the coefficients must be linear. In the case of Beer's law, A = elc, the coefficients are the absorptivity and the pathlength. It is assumed that these quantities vary Hnearly with absorbance. If, for example, absorbance happened to be proportional to F, it would be inappropriate to use linear regression to analyze the data. 3. No Systematic Error—This is just another way of saying that all the error in the data is random error. Since we are fitting plots of absorbance versus concentration, we are assuming that these data have no systematic error in them. Any bias in the data will produce bias in the line calculated from the data. In reality, it may be impossible to assure that there is absolutely no systematic error in the data. A more realistic assumption is that the systematic error is significantly less than the random error. If this is the case, the least squares methods can usually be appUed legitimately. 4. Error in the Y-Axis Data Must Have Certain Properties—This concept is best illustrated with an example. Imagine for the moment that the F-axis data is concentration, and that you have five standard samples, hence five F-axis data points. Also, imagine that we use titration to analyze the concentration of each standard, and that we perform nine titrations of each standard under controlled conditions. For each standard, we will determine nine concentrations, and because of random error, there will be some scatter of these concentrations around the average for each standard. Strictly speaking [3], the shape and extent of this scatter has to be the same for each sample to legitimately apply the least squares fitting to the data. For those familiar with statistics, we are saying that the errors for each

52


sample follow a Gaussian distribution, and that the standard deviation for each sample is the same (these concepts are explained later in this chapter). 5. There is no Error in the X-Axis Data—This is perhaps the most important assumption from a practical point of view. Error free data do not exist, thus this assumption is violated practically any time least squares fitting is performed. When fitting absorbance versus concentration data, we are assuming that there is no error in the concentration measurements, which is patently false. What then should we do? We know that both the absorbances and concentrations have error in them. However, the concentration error is often an order of magnitude greater than the absorbance error. The best way to minimize the damage caused by violating this assumption is to put the data with the least error in it on the Z-axis. This means putting absorbance on the 1^-axis, concentration on the 7-axis, and fitting plots of concentration versus absorbance. This is actually the recommended practice, as will be discussed in more detail later in this chapter. To summarize, to apply the least squares equations to a problem, conditions are placed upon the equation to be fit, and the error in the data. If these assumptions are met, least squares fitting can be apphed legitimately to your data.

3. Calculating a Least Squares Line

Table 2-1 contains two sets of data, a series of concentrations and a series of absorbances. There are a number of computer programs readily available today (including Microsoft Excel®) that can be used to calculate the least squares line relating two sets of data. However, it is useful to examine the equations used in this process to understand where caUbrations come from. Assuming two sets of data, X and 7, are related to each other by the following equation: Y ^mX

+b

Then the least squares slope of a Hne fitting this data is given by [1]:

"^ = Hi (^' -^)(Yi-Y)/

E / (^' - ^)'

^^'^^

and the least squares 7-intercept is given by b=Y-mX

(2.7)


53

where m = slope of the Hne / = data point index b = 7-intercept of the hne X = the average of the Z-axis values Y = the average of the T-axis values The calculations involved in determining the cahbration hne include subtracting the average X from all X values, the average Y from all Y values, then performing the appropriate summing, multiplication, squaring, and division. Recall from the previous section that one of the assumptions of the least squares method is that there is no error in the Z-axis data. This assumption is best met by plotting concentration versus absorbance. To do this we use what is known as the inverse form of Beer's Law. Rather than writing absorbance as a function of concentration as seen in Equation (2.1), we write concentration as a function of absorbance as such c = A/sl

(2.8)

Equation (2.8) means that a calibration hne plotted using the inverse form of Beer's law would have concentration on the F-axis, absorbance on the X-axis, and a slope of I/si. We would call this graph an inverse Beer's Law plot. Inverse Beer's law plots are convenient because the unknown variable, the concentration, has already been solved for in Equation (2.8) and can be determined directly. There are other advantages of using the inverse form of Beer's law [3] discussed in Chapter 3. An inverse Beer's law plot for the IPA data in Table 2-1 is seen in Figure 2.4. Using the data in Table 2-1 and applying Equation (2.6), we can calculate the slope of the line seen in Figure 2.4. The data needed to determine the slope are listed in Table 2-2. Using the sums in the bottom row of Table 2-2, the slope is 411/68 or 6.04. To calculate the 7-intercept we use Equation (2.7) and find that /? = 37-(6.04 x 5.4) or 4.39. The final equation for the line plotted in Figure 2.4 is %IPA = 6.04(Peak Area) + 4.39

(2.9)

This equation would be used to predict the concentration of IPA in an unknown sample given the unknown sample's peak area. Note that this equation was calculated manually. The equation for this hne calculated using Microsoft Excel is seen at the top of Figure 2.4. While the two calculations did not produce identical results, they are very similar, and

54


Inverse Beer*s Law Plot for IPA y = 6.0528X + 4.1936 R = 0.999

10 Peak Area Figure 2.4 The Inverse Beer's Law plot for the data contained in Table 2-1. This line also acts as a calibration line, relating volume percent IPA to peak area. TABLE 2-2

Least squares slope calculation for the calibration line seen in Figure 2.4. Zs and Ys from Table 2-1. X = 5.4, Y = 37. Xi

Yi

(Peak Areas) 1 2.2 4.9 8 11

(Xi - X)

(X, -

Xf

(Y^ - Y)

(% IPA) 9 18 35 53 70

{Xi -

X)

x ( r , - Y) -4.4 -3.2 -0.5 2.6 5.6

19.4 10.2 0.25 6.76 31.4 E = 68.0

-28 -19 -2 16 33

123 60.8 1 41.6 185 E = 411

the differences can probably be explained by the rounding of numbers in the manual calculation.

B. STATISTICS FOR DETERMINING CALIBRATION QUALITY AND ROBUSTNESS

One way to think about the problem of calculating cahbration models is that the variance in the data must be modeled. For our purposes, the variance in a data set is determined by the difference between each individual data point and the average for that set of data. For example, the variance in a set of concentration data would be determined by the difference between each individual concentration and the average concentration. The greater the difference between a set of data points and their average, the greater the variance in the data. Ultimately, variance is a measure of the scatter, or spread, in a data set.

55


Variance Modeled (SSR, inner circle)

\ / /^^^r^m\ / /^^^•^v\ [ 4» X • k 1

Total Variance ^^^^^ ^^^^^ circle)

Unmodeled Variance (SSE, torus) Figure 2.5 An example of variance in a set of data. The "X" represents the average of the data. The outer circle represents the total variance, the inner circle the variance accounted for by the model. The area between the two circles (torus) is the unmodeled variance, the variance due to error.

If a calibration model were perfect, it would completely model all the variance in a set of data. For example, if all the data points in Figure 2.4 fell exactly on the caHbration line, we would have a perfect model with no unmodeled variance. In reahty, the data points in Figure 2.4 do not fall exactly on the Hne, which means the hne has not modeled all the variance in the data. The unmodeled variance can be attributed to error either in the data or in the model. The idea of variance and how a caHbration models it can be visualized by looking at Figure 2.5. This figure is a hypothetical plot of a set of data points. The large bold X in the middle of Figure 2.5 represents the average of the data. There is obviously some scatter in this data. It can be shown [3, 4] that the total variance in a data set, the variance explained by the mathematical model, and the variance due to error are given by the following equations: SST = ^ . ( r , - Y)^

(2.10)

SSR-^.(r-

f)2

(2.11)

SSE = ^ . ( r -

Yf

(2.12)

where r = actual (measured) 7-axis values Y = the average value of Y Y' = the value of Y predicted by the calibration model SST = sum of squares total (the total variance)

56


SSR = sum of squares explained by regression (the variance modeled) SSE = sum of squares due to error (variance not modeled) Note that SST is the total variance and depends only upon the data and its average. The area of the outer circle in Figure 2.5 represents the total variance. SSR is the amount of the variance explained by the calibration model (the R stands for regression), and depends upon the predicted values of Y and the average Y. The area of the inner circle in Figure 2.5 represents SSR. SSE is the amount of variance not explained by the caHbration model, or the amount of variance due to error. SSE depends on the actual and predicted Y values. It is represented by the area between the two circles in Figure 2.5. The area of this torus (doughnut shape) is what the least squares fits attempt to minimize. A perfect model would have outer and inner circles of equal areas, all variance would be modeled, and SSE equal zero. Since a caHbration model explains some but not all the variance, the total variance must depend on the sum of the variance modeled and the variance due to error. It can be shown [3, 4] that SST = SSR + SSE

(2.13)

Equation (2.13) is another way of saying that the area of the outer circle in Figure 2.5 is equal to the sum of the areas of the inner circle and the torus. Using the data in Tables 2-1 and 2-2, we can calculate SST, SSR, and SSE for the caUbration Une seen in Figure 2.4. In this example, Y^ values are calculated by substituting the measured peak area values for each sample into Equation (2.9). The data used to calculate these values is seen in Table 2-3. TABLE 2-3

Data used to calculate SST, SR, and SSE for IPA% calibration line seen in Figure 2.4. f = 37.

Y

r

9 18 35 53 70

10.4 17.7

Sums

34 52.8 70.9

(7, - Yf

(r - Yf

784 361 4 256

708 372 9 250

1089

1149 J2i = 2488 (SSR)

Ei = 2494 (SST)

(F- rf 1.96 0.09

1 0.04 0.81

J2i = 3.9 (SSE)


57

For this calibration model, SST = 2494, SSR = 2488, and SSE = 3.9. Ideally, this data should satisfy Equation (2.13). However, 2488 + 3.9 = 2491.9 or ^2492 2494 7^ 2492 The data do not satisfy Equation (2.13) exactly, but do so within 0.1%. The probable explanation is the accumulation of the round off error in manually performed calculations. There are a number of very useful statistics that can be calculated from SST, SSR, and SSE that describe the quahty and robustness of a cahbration model. The best known of these is called the standard deviation, and is calculated from the following equation [1, 3]: a = [SSE/(« - l)]^/2

(2.14)

where a = standard deviation n == number of data points Note that the standard deviation depends only on SSE, the unmodeled variance, and does not depend upon SST or SSR. The term (n—l) term in the denominator is called the degrees of freedom. This term appears instead of n, the number of data points. The degrees of freedom take account of the fact that it only takes (n—\) data points to describe a data set containing n points. Here is how to think about it. Imagine we have the following simple algebraic equation: l + 2 + 3 + x = : 10 It is easy enough to solve this equation to discover that x = 4. Now, we only need three of four data points on the left-hand side of the equation to solve this equation, so this equation has three degrees of freedom. The same idea apphes to data sets being fit by a model. The standard deviation, then, is a standardized way of expressing the amount of error per data point. From the data in Table 2-3, we can calculate that a = [3.9/(5 - 1)]^^^ or 0.987% IPA (we will round this value up to one from here on out). Note that the standard deviation has the same units as the 7-axis data. The standard deviation is commonly used to refer to the accuracy of the predicted values. For example, if Equation (2.9) were used to predict the volume percent of IPA in an unknown sample, it would be legitimate to say that the predicted value had an accuracy of ± 1 % .

58


Another statistic that describes how well a caUbration model fits a data set is the correlation coefficient. It is given by [1-4] 7? = (SSR/SST)^/2

(2.15)

where R = correlation coefficient The term "correlation coefficient" is used because this quantity describes how well the model fits, or correlates to, the data. R depends upon the fracfion of variance in the data described by the model, SSR. If the model perfectly describes the data, SSR = SST (SSE is 0) and 7^==1, all of the variance is modeled. If the model describes none of the variance, SSR = 0 and R = 0. Thus, the correlation coefficient varies between 0 and 1. A value of 1 is for a perfect model; a value of 0 is for the worst possible model. In terms of Figure 2.5, the correlation coefficient depends upon the ratio of the area of the inner circle to the area of the outer circle. The greater the fraction of the total area modeled, the bigger the inner circle, the smaller the torus, and the better the model. From the data in Table 2-3, the correlafion coefficient for the %IPA calibrafion is (2488/2494)^/^ or 0.998. This indicates that the model fits the data very well. Analysts occasionally refer to the quality of their cahbrations by the number of "9s" in the correlation coefficient. For example, a calibration with "one 9" would have 7^>0.9, a caHbration with two "9s" would have 7? > 0.99, etc. This cahbration has R = 0.998, so it has two 9s, but is close to three 9s. Another useful statistic that can be derived from SST, SSR, and SSE calculations describes the robustness of a caHbration. For our purposes, robustness is a measure of how sensitive a caHbration is to random error. If a small increase in the random error has a small effect upon the predicted values, a calibration is robust. On the other hand, if a small increase in the random error gives a large effect upon predicted values, a caHbration is not robust. The statistic used to measure the caHbration robustness is called the Ffor Regression and is given by the following formula [2]:

(SSE) where i^=:F for regression n = number of data points m = number of independent variables

m


59

Note that the F for regression depends upon the amount of variance described by the model (SSR) divided by the amount of variance due to error (SSE). You can think of this calculation as a signal-to-noise ratio (SNR). SSR is the amount of variance that can be explained, the signal. SSE is the error, or noise, in the determination. By taking their ratio, we measure how much variance the model explains compared to what it does not explain. The reason the F for regression is a measure of robustness is as follows. Imagine a cahbration where F = 10, or in other words the SNR ==10. If a problem with the spectrometer causes the absolute error in measured absorbances to increase by 1 unit, SSE would increase 1/10 or 1%. On the other hand, if F for a calibration is 1000, an increase in error of 1 unit changes F b y 1/1000 or 0.1%. Simply stated, 1 is a smaller percentage of 1000 than it is of 10. A robust calibration with a large F is less sensitive than a fragile calibration with a low F to the same increase in error. In terms of Figure 2.5, the F for Regression depends upon the area of the inner circle divided by the area of the torus. A robust model will have a small torus; a fragile model will have a large torus. Using the data in Table 2-3, the F for regression for the IPA calibration in Figure 2.4 is [2488/(3.9/3)] or 1914. The variable n in Equation (2.16) is determined by the number of standard samples. As n goes up, so does F. This means that the more standard samples included in a calibration, the higher F becomes and the more robust the calibration will be. The variable m in Equation (2.16), the number of independent variables, is for our purposes a measure of how many different absorbance values in each spectrum are used in a cahbration. For the calibration in Figure 2.4, one absorbance band's area was plotted giving a value of m = 1. Since a Beer's law equation can be written for each absorbance in a spectrum used, m also determines the number of equations in a calibration. Note in Equation (2.16) that as m goes up, F goes down. This means the more equations used in a calibration, all else being equal, the less robust the cahbration becomes. F contains the quantity SSR/m, a measure of how much variance per equation is modeled. A robust calibration will model the most variance with the fewest equations. A fragile cahbration will need many equations to model the same amount of variance. The measures of cahbration quahty for IPA cahbration discussed in this section are summarized in Table 2-4. Overall, this is an excellent calibration. Other absorbance bands, including the one at 948 cm~^ were used in trial calibrations. These typically gave standard deviations of two or greater. The key to the success of this cahbration was that the peak at 2973 cm~^ was not very intense. This peak's intensity for the most concentrated sample is 0.76. As will be discussed below, ideally only absorbance bands

60


TABLE 2-4

Calibration metrics for the IP A calibration. Metric

Value

Standard Deviation (a) Correlation Coefficient (R) F for Regression

1.0 0.998 1914

whose intensity is 0.8 or less should be used in caUbrations. At higher absorbances, the relationship between absorbance and concentration tends to be nonHnear. One of the reasons Beer's law fails at high concentrations is absorptivity drift. As was stated in Chapter 1, the absorptivity is a function of the chemical environment of a molecule. The greater the concentration range modeled in a calibration, the more the chemical environment changes, and the more probable a measurable absorptivity change will be observed.

IV. Methods for Making Standards and Measuring Spectra It is important when making standards to give some thought to the concentrations of the analytes in the standards. First, the concentration range should be broad enough to bracket the concentration that any unknown example might exhibit. This prevents applying the calibration to an unknown that falls outside the concentration range of the caUbration. Second, the concentration range should be significantly broader than the error in the concentration measurements. For example, if the error in a series of concentration determinations is ± 3 % , and if standards are prepared with nominal concentrations of 10, 12, 14, and 16%, the concentration error bars for all these standards would overlap and there would be no unique data points. The solution to this problem is to space out the concentrations, so that they are farther apart than the error bars. In this case, standard concentrations of 10, 20, 30, and 40% would work. A third thing to consider about the analyte concentrations is their distribution. A caUbration Hne is based on data points. The more data points in a specific concentration region, the more accurate the line is in that region. The concentrations plotted in Figure 2.4 are about evenly spaced. This is ideal; it means that the cahbration is equally accurate throughout its range. On the other hand, if the standards contained four samples with IP A concentrations below 40%, and one sample at 70%, the caUbration would be very accurate below 40% and less accurate at higher concentrations. There are situations where you may not have control over the


61

concentrations in the standard samples, and have to Hve with whatever concentrations the standards contain. However, if you do have control over the concentrations in the standards, try to space them out. The next two sections discuss two different standard methods. These are different ways of using standards in creating a caHbration.

A.

EXTERNAL STANDARDS

The method of external standards can be summarized as calibrate now, predict later. It follows the same steps outlined earlier in this chapter for obtaining a caHbration. Standards are made, their spectra obtained, and their absorbances measured. The absorbances and concentrations are plotted to give a caHbration curve such as Figure 2.4. The concentration of unknown samples is determined by obtaining their spectrum, measuring their absorbance, and then using the caHbration Hne to predict the concentration in the unknown sample. The method of external standards is so called because the calibration and prediction steps are performed at different points in time. The caHbration plotted in Figure 2.4 was obtained using this method. The external standards method works fine for many samples, and is relatively simple to perform. However, it does have some drawbacks. It does not take into account fluctuations in random variables over time, such as changes in the instrument performance or sampHng. These variables can have an unknown effect on the measured absorbances, contributing an unknown amount of error to the predicted concentrations. When using the external standards method (in the absence of calibration checking), one can never be 100% sure if a calibration obtained in the past is stiH working weH. A way of insuring the validity of a calibration over time is to check the calibration with a new standard sample. This is done by simply making up a new standard with a known concentration of the analyte, measuring its spectrum, predicting its concentration, then comparing the known and predicted concentrations for the sample. If the concentrations agree reasonably weU, the caHbration is stiH good. However, if the predicted and known concentrations are not in reasonable agreement, the caHbration may no longer be accurate, and the source of variability must be eliminated, or a completely new caHbration may need to be obtained. Some users of the external standards method create a new standard sample every time they analyze an unknown sample, and check the calibration at the same time they are predicting the concentration in the unknown sample. This is an excellent way of insuring the caHbration is stiH accurate.

62


B. INTERNAL STANDARDS

The method of internal standards was developed to try to get around the problems of the external standards method. It involves adding a known amount of a known material, called the internal standard, to each standard sample and to each unknown sample. The internal standard should be chemically stable, not react or interact with other molecules in the sample, not interfere with the analyte absorbance, and have a unique absorbance band free of interferences. Once the spectrum is measured the absorbance of the analyte is divided by the absorbance of the internal standard. The caUbration curve is generated by plotting this absorbance ratio versus analyte concentration. For unknown samples, the ratio of the analyte and internal standard absorbances is calculated, and then used with the calibration line to predict the concentration of the analyte in the unknown. The internal standards method assumes that any unknown source of error will affect the internal standard and analyte absorbances equally. When these absorbances are ratioed, the effects of the unknown variables are canceled, yielding a more accurate analysis. As a result, internal standards analyses are generally more accurate than external standards analyses. However, the internal standards method involves more work and calculations than the external standards method. The internal standards method is useful for any type of sample where the pathlength is unknown or difficult to reproduce (such as in KBr pellets in FTIR). This is illustrated by first writing Beer's law equations for the analyte and the internal standard. (In this discussion, the subscript "a" stands for the analyte, and the subscript 'is' stands for the internal standard.)

When the two absorbances are ratioed, we obtain the following:

The pathlength for the analyte and the internal standard are the same because they are in the same sample. When the ratio is calculated the pathlength cancels and the result is a pathlength independent quantity as such: ^ a / ^ i s — ââ/îsîs

v^-^ ')


63

A calibration obtained using Equation (2.17) would be pathlength independent, and is an example of how the internal standards method compensates for random variations in experimental parameters. C. A N EXPERIMENTAL PROTOCOL FOR SINGLE ANALYTE ANALYSES

The following is an experimental protocol that, if followed, will help in the attainment of accurate single analyte caHbrations. Analyzing for a Single Analyte 1. Prepare the standards. Be as accurate as possible since any concentration error in the standards is carried through to generate error in the predicted concentrations of the unknown samples. 2. Take two aUquots of each sample and obtain separate spectra of them. In theory, these two spectra should be identical because they are obtained with the same sample, instrument, and sampling accessory. 3. Subtract the two spectra of each sample using a subtraction factor of one, giving a subtraction result or residual. Look closely at the residual. Two spectra of the same sample should be identical, and should give a residual that is a reasonably flat hne containing only noise. Such a residual is seen in the top of Figure 2.6. The only features seen in this residual are noise and a CO2 peak at 2350 cm~^ The bottom residual in Figure 2.6 shows what happens when absorbance does vary between two spectra of the same sample. This spectrum shows analyte peaks, meaning something caused the absorbances to change. Since these are spectra of the same sample, the problem may be caused by the instrument or sampling variability. At a minimum, the two spectra of the standard should be re-measured and a new subtraction performed. It may take a good deal of experimentation and instrument adjustment to control the variable(s) causing the problem. 4. Examine the spectra and choose a band whose height or area changes with concentration (more on this below). 5. Measure the appropriate absorbance and plot the caHbration hne (see above). Examine the plot to insure linearity, and then calculate its slope. Use the statistics discussed above to check the quality and robustness of the cahbration. 6. Obtain the spectra of the unknown samples, and calculate the unknown concentrations using the cahbration curve's slope and the absorbance of the unknown. If these simple steps are followed, an accurate single analyte calibration can often be obtained.

64


Noise Only "~X

LM,,^.**^-^^

Lw--->--J 1 3500

1 3000

1 2500

1 2000

\ 1500

1 1000

Wavenumber (cm-1) Figure 2.6 Spectral residuals obtained by subtracting two spectra of the same sample. Top: No spectral features are seen, just noise and a CO2 peak. This indicates that there are no unusual sources of absorbance variability in the experiment. Bottom: Real spectral features, pointing up or down, indicate a source of absorbance variability exists in the experiment.

V. Measuring Absorbances Properly A. PEAK AREAS VERSUS PEAK HEIGHTS

Correctly measuring the absorbance of a spectroscopic feature is more difficult than might first be imagined. You must first insure that the spectral feature being measured is due to the analyte, and that there are no interfering absorbances from other components. It may appear that reading the absolute peak height directly off a spectrum will give an accurate absorbance. However, the baselines of spectra can drift up and down on the 7-axis, due to things Hke instrumental problems. As a result, spectra of the same sample measured at different points in time may give different absolute absorbances. To avoid this problem, it is common to draw a baseline connecting the beginning and ending points of a peak, and then measuring the peak height or area with respect to that baseline. This is shown in Figure 2.7 for the C = N stretching band in the mid-infrared spectrum of benzonitrile. Note the vertical lines in Figure 2.7 denoting the beginning and endpoints of the basehne. It is critical that these endpoints be chosen such that they include the entire analyte peak, but no spectral features due to other molecules present. The peak height is measured from the baseline to the top of the peak.

65

2. SINGLE ANALY TE ANALYSIS

2228

Height = 0.445 Area = 6.58 .4-

.3-

o

.2-

.12270 0- 1

1

^

2195

> ^ ,

2260

p-

2240

1

2220

-L

1 2200

1—

2180

Wavenumber (cm-1) Figure 2.7 An example of a spectroscopic absorbance band, the C=N stretch of benzonitrile. Note the basehne drawn between the peak edges, used to remove basehne drift. Also, note the integration limits of 2270 and 2195 c m ' \

The peak area is measured by integrating the peak between the basehne endpoints. In Figure 2.7 the basehne endpoints were 2270 and 2195 cm~^ By drawing the basehne such that the endpoints both touch the spectrum, spectral drift on the F-axis is subtracted from the peak height or area calculation. This ultimately gives a more accurate and robust calibration. Many of us instinctively choose to measure the peak heights at the peak top. Why is it done this way? The peak seen in Figure 2.7 consists of 40 data points. In theory, the absorbance at any of these points could be used to create a cahbration since all of these points respond to changes in the analyte concentration. We use the data point at the top of the peak for two reasons. First, assuming the noise level across the peak is constant, the maximum absorbance at the peak top will have the best SNR of all the data points in the peak. Second, the slope of the spectrum at the top of a peak is close to or equal to zero, so a small error in the X-axis (AX), such as a shift in the measured wavenumber or the wavelength, will produce a small shift in the measured absorbance (AY). However, on the steep sides of an absorbance band where the slope is large, a small change in the wavenumber or the wavelength means a large change in the absorbance. This is why absorbances measured on the side of the peak are more sensitive to X-axis error than absorbances measured near the top of a peak. There is an ongoing debate as to whether peak areas or the peak heights provide better calibrations. Here is one author's opinion. When a peak height is measured, only one of the many data points in a peak whose

66


absorbance is sensitive to the analyte concentration is used in the analysis. This is like putting all your eggs in one basket. If anything causes the accuracy of this one data point to be compromised (an interferent in the unknown sample for example), the entire calibration is thrown off. This cahbration would not be robust with respect to these types of absorbance errors. Peak areas are measured by adding the absorbances of many different data points together. For the band seen in Figure 2.7, the peak area is calculated by adding together the absorbance at 40 different wavenumbers. If one of these data points is inaccurate, it is averaged over the other 39 data points that are accurate. By using more data points, the cahbration is less sensitive to error in any specific data point. Additionally, by adding the absorbances together, the SNR of the area increases with respect to that of the peak height. In general, peak areas produce more robust and accurate caUbrations because they are not as dependent on a single data point, and the raw data has a better SNR. However, there are real world examples of chemical systems where peak height caUbrations work better than peak area caUbrations. Therefore, the author recommends that when establishing a cahbration on a new chemical system, that peak areas be tried first. If you have difficulty in getting a good cahbration with peak areas, then try peak heights. There is no guarantee that this will fix your cahbration, but it might. It is now trivial to measure the peak heights, the peak areas, and generate caUbrations using computers. Therefore, it is straightforward to try both types of cahbration and choose the one that works best. It would be useful in this discussion to have some measure of the quality of the set of peak areas or peak heights used in a calibration. So far, we have discussed signals, noise, and SNRs as measures of spectral quality. Typically, we use the absolute absorbance of a feature in a spectrum as a measure of its signal. How should we characterize the "signal" in a set of peak areas or heights? It is the change in absorbance with concentration, not individual absorbances that we are interested in when calibrating. Thus, the average Av4 (peak area or peak height) for a spectral set characterizes the signal in the data. The noise in the data is measured as the average of the noise for spectra in the set. We can calculate the SNR for the spectral set plotted in Figure 2.3 using the data seen in Table 2-5. The noise levels in these spectra were measured as the peak-to-peak noise (maximum minus minimum noise value) in the 2600-2500 cm~^ spectral region. This region was chosen because the basehne was fiat and free of absorbance bands. The AA values were calculated by subtracting sequential values from each other. The averages for APeak Area, APeak Height, and noise are in the bottom row of Table 2-5. The SNR we desire is the average AA divided by the average noise level. For the peak area data,


67

TABLE 2-5

Data used to calculate SNR for the IPA spectral set used in the calibration seen in Figure 2.4. % IPA

9 18 35 53 70 Average

A Peak Peak Area (3004-2947 cm"^;) Area

1 2.2 4.9 8 11

— 1.1 2.7 3.1 3 2.5

Peak Height (2973 cm-^)

A Peak Height

Peak-to-Peak Noise (2600-2500 cm-^)

0.05 0.1 0.22 0.37 0.51

_ 0.05 0.12 0.15 0.14 0.11

0.003 0.002 0.0015 0.0012 0.001 0.0017

SNR = 2.5/0.0017 or 1471. For the peak height data, SNR = 0.11/0.0017 or 65. Thus, the peak area data have a 23 times better SNR than the peak height data. This is to be expected since peak areas involve adding multiple absorbances together, whereas peak heights are a single absorbance. The point is that the higher SNR for the peak area data means calibrations based on this data are more robust with respect to absorbance error than the peak height data. B. DEALING WITH OVERLAPPED PEAKS

In an ideal world, we measure the absorbance of an analyte band that is baseline resolved, and has no interference from bands due to other molecules. The peak in Figure 2.7 is a good example of this. However, there are samples where analyte bands are overlapped with bands due to other molecules. The peak at 2973 cm~^ in Figure 2.8 is an example. It is an IPA peak superimposed upon a much broader and stronger absorbance band due to water. Are usable calibrations possible in such circumstances? The solution to overlapped peaks is judiciously drawing an appropriate baseline for the spectral feature being measured. Only data points whose absorbance is sensitive to the concentration of the analyte should be included in a peak height or area measurement. The advantage of using the peak heights or areas measured with an appropriate baseline is seen in the data in Table 2-6. The table contains nonbaseline corrected peak areas for the IPA peak at 2973 cm~^ The spectra from which these data are taken are seen in Figure 2.3. The calibration Hne obtained from the data in Table 2-6 is seen in Figure 2.9. We can compare the cahbration hne in Figure 2.9 to that in

68


1.2H

l-[

.2H

1

1

3400

3300

1

1

3200

1

1

1—

3100 3000 2900 2800 w avenumD er (^cm-1;

2700

—I— 2600

Figure 2.8 An example of a shoulder peak, and how to draw a baseline to accurately measure its peak height or area. TABLE 2-6 Nonbaseline corrected peak area and %IPA data. Nonbaseline Corrected Area

%IPA

16.3 19.9 22.9 23.2 26.2

9 18 35 53 70

IPA Uncorrected Peak Area Calibration y = 6.3099X • 99.925 R = 0.948 80 •

60 40 20 0 15

20

25

30

Peak Area Figure 2.9 The calibration line obtained using uncorrected IPA peak areas, based on data in Table 2-6.


69

i

1260

1220 1200 Wavenumber (cm-1)

Figure 2.10 A series of overlapped peaks that share a common baseUne, and how to draw basehne segments to measure their absorbances.

Figure 2.4. Both the cahbrations were obtained using the same integration Umits and the same spectra, the difference is that the data in Table 2-1 were baseline corrected; the data in Table 2-6 are not. Recall from earher in this chapter that the correlation coefficient, R, is a measure of model quahty, and that R=\ is a perfect model. The calibration using basehne corrected peak areas has 7^ = 0.998 as seen in Table 2.4. The model using nonbaseline corrected peak areas has 7^ = 0.948, significantly worse than the basehne corrected model. By judiciously choosing the basehne endpoints that include only the IPA feature, the effect of the interferents and the basehne drift on the absorbance of the analyte has been reduced. Another instance when it can be difficult to measure the absorbances properly is when a group of peaks overlap and are not basehne resolved, as illustrated in Figure 2.10. To deal with this situation, use the minima between the overlapped peaks as baseline endpoints, as shown in Figure 2.10. Used properly, this approach isolates the absorbances of the analytes, and removes the effects of baseline drift from the measurement. C. CORRECTING FOR INTERFERENTS

When dealing with the determination of single analytes in a mixture, the analyte band may be overlapped with those of interferent molecules. As long as the interferent band appears as a shoulder on the analyte band and can be somewhat resolved, judicious drawing of basehnes can

70


Polystyrene/Lexan Mixture

' Lexan

A / Pure Polystyrene

Wavenumber (cm-l) Figure 2.11 An example of an interferent band totally overlapping with an analyte band. The bottom spectrum is of polystryene, the interferent. The middle spectrum is of the polycarbonate Lexan®, the analyte. The top spectrum is of a mixture of the two polymers. The dashed lines show that the polystrene features are almost totally subsumed by the polycarbonate features in the mixture spectrum.

compensate for the interferent absorbance as shown above. However, there are times when the interferent bands totally overlap the analyte bands, making it difficult to draw proper basehnes. An example of this is seen in Figure 2.11. The spectra in Figure 2.11 are of polystyrene, the polycarbonate Lexan®, and a mixture of the two. The dashed Unes show that the polystyrene bands are almost totally subsumed by the polycarbonate bands. This is why we cannot draw a baseline to remove their absorbance from the Lexan absorbance. How then could we obtain a calibration for Lexan using these features without the polystyrene interference? A technique called interferent correction helps get around this problem. Here is how it works. First, assume the analyte feature whose absorbance we want to use in a cahbration is overlapped with a peak of one interferent molecule. We will call this spectral feature the analytical hand. We will further assume that no other molecules but the analyte and the interferent contribute to the absorbance of the analytical band. Next, assume that the interferent has an absorbance band away from the analytical band that is isolated and due only to the interferent, called the interferent hand. For this situation.

2.


71

we can write the following equations: ^ t - A + î

(2.18)

Ct = Ca + C,

(2.19)

where the indices t, a, and i represent total, analyte, and interferent respectively. Equation (2.18) states that the total absorbance of the analytical band is equal to the absorbance of the analyte plus the absorbance of the interferent. We are taking advantage here of a property of the absorbances called their additivity. This simply means that the total absorbance at any wavenumber is the sum of all the absorbances of all the species present at that wavenumber. Concentrations are also additive, which allows us to write Equation (2.19). Like any other quantitative analysis, we seek Ca, the concentration of the analyte. We rewrite Equation (2.19) to solve for c^ to obtain c^ = a-

Ci

(2.20)

which shows that the analyte concentration is simply the total concentration minus the interferent concentration. Using Beer's law we can write the following for the total absorbance: At = Bile,

(2.21)

c, = A/8,1

(2.22)

and rearrange to obtain

We can apply Beer's law to the interferent concentration similarly to obtain A; = Silci

(2.23)

Ci = Ai/sil

(2.24)

and rearrange to obtain

Substituting Equations (2.24) and (2.22) into Equation (2.20) gives

c. = A/eJ - A/sJ

(2.25)

72


Equation (2.25) gives us a means of correcting for the interferents even if the analyte and interferent features are completely overlapped. Note that the right-hand side of Equation (2.25) has two terms. The first term involves parameters for the analyte and interferent together, the second just the interferent. How would we use Equation (2.25) to predict the analyte concentrations? Recall from earUer in this chapter that the slope of a single analyte cahbration line is equal to si. What we need to do is make up a series of standards with known concentrations of the analyte and the interferent. Once the spectra of these standards are obtained, the absorbance of the analytical band and the interferent band are measured. Then, two independent calibrations can be performed. Using the analytical band and Equation (2.21), the slope of the cahbration hne is found to be e^l. Using the interferent band and Equation (2.23), the slope of the cahbration line is found to be £{1. Prediction would simply be a matter of measuring A^ and Ai in the unknown spectrum, then using the slopes of the cahbration lines and Equation (2.25) to obtain the analyte concentration. Note from Figure 2.11 that if we did not have spectra of the pure analyte and interferent, we would not have been able to see how the polystyrene spectrum interfered with that of the Lexan spectrum. It is generally a requirement when performing single analyte analyses in mixtures to have pure spectra of all the different molecules present in the unknowns. You then need to closely examine these spectra to look for isolated analyte bands. If none are found, then either judicious drawing of baselines, or the interferent technique discussed here need to be used. If none of these techniques work, or if there are multiple analytes needing to be determined with multiple interferents, the more powerful calibration algorithms discussed in Chapters 3 and 4 should be used.

VI. Avoiding Experimental Errors Avoiding errors is the key to obtaining good calibrations. It is convenient to divide error into different categories. Since for a quantitative spectroscopic cahbration the two pieces of data are concentrations and absorbances, it is appropriate to speak of concentration errors and absorbance errors. This section will cover how to measure and avoid these types of errors, in addition to discussing general sources of experimental and instrumental errors.

A. SPOTTING OUTLIERS

The concentration error in each sample in a set of standards should be approximately the same. This happens if standards are made up following


73

an exact, reproducible experimental technique. The errors in the absorbances, which are based on the SNR of the spectrum, should also be approximately the same if careful experimental technique is followed. However, from time to time human or instrumental errors can cause a specific concentration or absorbance to have much more error than average. We will call those data points that have significantly more error than the rest of the data in a model outliers. Outliers can be spotted by subtracting the predicted data from the real data to obtain a residual. The size of the residual measures how well the model predicts a specific data point, such as a standard sample's concentration or absorbance. If the error across a set of data is approximately the same, then the model should predict the properties of each sample with approximately equal accuracy. If a specific sample has an unusually large residual compared to the rest of the data in a set, it may be an outlier. If all the other data points are well predicted, it probably means that the problem is with the data point rather than the model itself Once an outlier is spotted, the data should be examined to try to determine why the outher is present. This can possibly lead to removing the data from the calibration. The next two sections discuss how to spot outliers in the concentration and absorbance data. 1. Concentration Outliers

There are many reasons why analyte concentrations contain error, including mistakes in making up standards. Another problem is transcription errors. If the concentration value is recorded incorrectly, or if it is entered incorrectly into the caHbration software, it will throw off a caHbration. Lastly, reactions between the sample components, evaporation, or sample precipitation can also cause the predicted analyte concentration to be different from the actual. It is therefore important to pay attention to the preparation and condition of every standard sample. A standard whose actual and predicted concentrations lie far apart is called a concentration outlier. A quantity called the concentration residual can be used to spot concentration outUers. The concentration residual for a sample is calculated as follows: R^ = c-c' where Re = concentration residual c = actual concentration c' = predicted concentration

(2.26)

74


TABLE 2-7

Actual, predicted, and residual concentration data for the IPA calibration in Figure 2.4. Sample Number

c

c

Re

1 2 3 4 5

9 18 35 53 70

10.4 17.7 34 52.8 70.9

-1.4 0.3 1.0 0.2 -0.9

Concentration Residuals for IPA Calibration -5 2 9

S

1

^

0

o -1

-2

I 0.3

-M10.2 • -0.9 6

• -1.4 Sample #

Figure 2.12 A plot of the concentration residuals from the IPA calibration, based on data from Table 2-7.

Table 2-7 shows the concentration residuals and the actual concentrations for the IPA calibration seen in Figure 2.4. The equation hsted in Figure 2.4 was used to predict the concentrations. A scatter plot of the concentration residuals of Table 2-7 is seen in Figure 2.12. Note that the X-axis on this plot is the sample number, and that the F-axis is in concentration units. The residuals in Figure 2.12 fall in a "clump" around zero. This is normal if all the sources of errors in the concentration values are random. If the clump is centered on a value other than zero, this may indicate a systematic error in the concentration data. The clump should look like a "random scatter;" it should have no structure. If the points fell on a line, or had a recognizable shape to them, it would again indicate that the error sources are not random, and that a systematic error may be present. If one of the residuals falls far outside the clump, it might be an outlier. An example of a residual plot containing a suspected outlier is seen in Figure 2.13. Note the suspected outlier in Figure 2.13 has a residual of

2.

75


IPA Concentration Residuals

«

0

' 0.8. This may explain why the cahbration with a less intense peak is better than a calibration with a bigger peak. Now, the concentration range for this cahbration, from 9% to 70% IPA is rather broad, varying by greater than a factor of seven. The physical reason behind the 2970 cm~^ peak following Beer's law better than the 948 cm~^ peak may be that the absorptivity at 2970 cm~^ is less sensitive to changes in IPA concentration than the absorptivity at 948 cm~^ Additionally, using summed peak areas improves the signal-to-noise ratio of the absorbance data compared to using a single peak area. Because using many absorbances can improve a cahbration, it is common to use many absorbances in multicomponent calibrations. However, there are some limits. First, using hundreds of absorbances makes the cahbration data set large, and it can be time consuming to perform the calculations to obtain a calibration. However, this is less of an issue as time goes by because of improvements in computer performance. Second, the calibration algorithm plays a role in determining the number of absorbances that can be used. These details are explained below and in Chapter 4. Third, more absorbances are usually better as long as one or more components absorb at the wavenumbers selected. Inclusion of wavenumbers whose absorbance is pure noise may add error to the calibration and decrease its accuracy. Also, indiscriminately adding wavenumbers to a cahbration is not a good idea. The threat is overfitting, generating a calibration with too many equations, some of which only model noise.

106


E. MULTICOMPONENT CALIBRATIONS: GENERALITIES

The discussion of multicomponent calibration methods that follows will focus on their mathematical and practical differences. The types of cahbration techniques available to you will depend on the software you have. In general, the calculations described below are transparent to the user when using a computer; the choice of which technique to use is simply a matter of setting a software parameter. However, an understanding of what the computer is doing to your data is necessary to truly understand how to develop an accurate quantitative method. Be aware that the same high quaHty lab technique needed for a single component analysis is just as important for multicomponent analysis. The awesome mathematical power of these algorithms will not compensate for poorly made standards, sloppy technique, or lousy spectra. The focus for the rest of this chapter will be on two "least squares" cahbration algorithms, the mathematics behind them, and their relative advantages and disadvantages.

IV. The Classical Least Squares (K Matrix) Method One of the methods of simultaneously determining multiple components in an unknown sample is the Classical Least Squares (CLS) method. Like all multicomponent calibration algorithms, CLS starts with Equation (3.11). Then we write a new equation combining the pathlength and absorptivity matrices into a single matrix called K as follows: K = EL

(3.12)

where K = the absorptivity-pathlength matrix E = the absorptivity matrix L = the pathlength matrix (or vector) The classical least squares method is often called the "K matrix" method because of Equation (3.12). Using Equation (3.12), we can rewrite Equation (3.11) as A = KC

(3.13)

Note that in Equation (3.13) the absorbance is written as a function of concentration, or absorbance is the dependent variable and concentration is the independent variable. To obtain a calibration using the CLS method,

3.

MULTIPLE COMPONENTS I: LEAST SQUARES METHODS

107

the values in the K matrix must be determined. This means measuring the absorbances of standards to obtain the data for the A and C matrices. From what we have learned about matrix algebra, it would be tempting to solve Equation (3.13) for K by writing the following K = AC-^

(3.14)

where C~^ = the inverse of the concentration matrix The problem with Equation (3.14) is that it ignores some of the known properties of matrix inverses. A given matrix may have one inverse, many inverses, or no inverse at all [1]. This makes it difficult to use Equation (3.14) as written. Square matrices are better behaved when it comes to inversion. A square matrix possesses either a unique inverse or no inverse at all. There is a way of "squaring" the concentration matrix, then using its inverse to calculate K. It makes use of the transpose of C, C^ as follows: K = AC^(CC'^)-^

(3.15)

The matrix operation shown in Equation (3.15) does more than square the concentration matrix so its inverse can be found. Equation (3.15) is the matrix definition of a least squares fit. The properties, assumptions, and mathematics of the least squares fit for one component were discussed in Chapter 2. Here we are simply extending its use to multiple components and the solving of simultaneous equations. Everything that was said about the least squares algorithm in Chapter 2 will hold true here for multiple components. Recall that a least squares fit, by definition, provides the best model relating two sets of data to each other. Once K is known, we can predict concentrations in unknown samples using the following equation C,nk= (K'rK)-^KTA,nk

(3.16)

where Cunk = component concentrafions in the unknown sample Aunk = absorbances of unknown sample Equation (3.16) is also a least squares fit, and will produce the best possible prediction of C. Note that by calculating C this way, all the unknown concentrations are determined at once.

108


A limitation in using CLS is that the number of absorbances measured must equal or exceed the number of chemical species present in the samples. This is simply another way of stating that there must be at least one equation containing the concentration term for each species. A species whose concentration does not appear in any of the equations in the caHbration cannot be modeled. A. ADVANTAGES AND DISADVANTAGES OF CLS

The K matrix method is useful because the mathematics are straightforward and relatively easy to understand. Additionally, as long as the number of absorbances exceeds the number of species present, you can use as much of the spectrum in the caHbration as you wish. Using many absorbances gives an averaging effect that improves the calibration quality as seen above. If many wavenumbers are used in a calibration, the absorptivity matrix E will contain the predicted absorbance spectra of the components. This allows comparison of predicted and actual standard spectra, which can be useful in ascertaining calibration quahty. Despite these advantages, there are a number of problems with the CLS method that prevents its widespread use. Most of the problems stem from the form of the equations themselves. Equations (3.17) show the first three rows of the matrices used in CLS, assuming that there are three components called d, e, and f, and we are measuring absorbances at wavenumbers 1 and 2 Ai = kidCd + kiQCe + kifCf ^2 = k2dCd + ^2e

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