FUNDAMENTALS OF MOLECULAR ABSORPTION SPECTROSCOPY

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 QUANTITATIVE SPECTROSCOPY: THEORY AND PRACTICE

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

.05 wV 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 wave-number (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)

QUANTITATIVE SPECTROSCOPY: THEORY AND PRACTICE

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 (3x10 V —frequency in Hertz (s~^) X — wavelength

10 cm/s)

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)

1. FUNDAMENTALS OF MOLECULAR ABSORPTION SPECTROSCOPY 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

14000 to 4000 cm"' Near -Infrared

4000 to 400 cm' Mid-Infrared

400 to 4 cm" Far Infrared

< 4 cm'* Microwaves

Higher Wavenumber Higher Frequency Higher Energy Shorter Wavelength

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. Mid-infrared 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 mid-infrared 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


enough so that it absorbs the right amount of Hght. This problem with mid-infrared 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.


dL r'

I05 A>^

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 infmite-simally 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

Photon With Energy Ep

After Collision

Photon Still 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


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

Methane ^--"irr--'' Molecule at Rest 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^.


Before Collision

After Collision Photon With

Energy Ep

Methane ^--"Jri-Molecule at Rest Excited 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


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)


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

/ AI/I = sc f Jlo Jo

dL (1.10)

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

^ = l o g ( / o / / ) (1.13)

and this allows us to rewrite Beer's law one last time to obtain its final form

A = 8lc (1.14)

where

A = absorbance £ = absorptivity / = pathlength c = concentration


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

^ - l o g ( l / r ) (1.17)


— " V V T N y - W ^ — N #«/^ Af\

2500 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)

Raising both sides of the equation to the power of 10 yields

10'^' = \/T

and rearranging yields the fmal result:

T = 10-' "

(1.18)

(1.19)

(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.


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^ ^H20

where the subscript denotes molecule and the superscript denotes wave-number. 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 photon-molecule 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


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


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 (^u) 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.


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)

or

\/A(x\/(Ni-N^) (1.23)

where

A = amount of hght absorbed Au = 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.


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 (A i - NxQ-^^'^^)

and rearrangement gives

There are two things in Equation (1.26) worth noting. First, note that A is proportional to A i, 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 A i. Any process that causes A i 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


^-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 A i.

On the other hand, if Ei = 0 cm~^ and E^ = 3000 cm~^ a small change in temperature will have little effect on A i. It would take a large increase in temperature for there to be enough thermal energy present to populate E^ to any appreciable extent. A i will be large and Au 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


the wavefunction of the lower energy level by V i, and the wavefunction of the upper energy level by xj/u. If we could a priori know all the wave-functions 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 V IMV U dr (1.27)

where

|R^"| = the transition probability in going from state 1 to state u Vi = the wavefunction for the lower state in the transition /JL = dipole moment operator

Vu = 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 Y^^qtri (1.28)

where

II = dipole moment operator q = charge r = distance


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

and

(1.30)

£ (x fi (1-31)


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 = / V i/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


8-

A §+ / \ S"* y Hydrogen

H H / B""**

A A H H H 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


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


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


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


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)

which on rearranging gives

A = PsL/RT (1.39)

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.


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.


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


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 V i, 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, ^i and 1/ ^ stand for the wavefunctions of the lower and upper energy levels, respectively.


wavefunction of the upper energy level by ^Ir^. 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

^ak ine t ic 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 = E^lr (A.3)

where

fi = {hl2ii) h = Planck's constant

m = mass X = distance P = potential energy


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 f o r O < x < L (A.4)

and

P = OQ X < 0 and x > L

Since the potential energy inside the box is zero, Schrodinger's equation reduces to

-fi/2md^/dx^}xl/=Ejl/ (A.5)

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.


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


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, wave-functions \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)


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)

substituting into (A. 10) yields

-^ = A sm(nnx/L) (A. 16)


¥4

V3

¥2

Vi

E2 = 4h78m]

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.


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.


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 Vi = the wavefunction for the lower state in the transition /x = dipole moment operator

Vu = 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

" ' " = / • V iMiAudT (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


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 ,


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^ T^O for A/i = ± l , ib3 , ib5

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.

FUNDAMENTALS OF MOLECULAR ABSORPTION SPECTROSCOPY

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