MOS: Molecular Spectroscopy

Leland Breedlove, Andrew Hartford, Roman Hodson, and Kandyss Najjar

Abstract

This set of experiments uses Fourier-Transform infrared spectroscopy (FTIR) to

determine the molecular characteristics of various molecules. The data from these experiments

provides good insight into the rovibrational levels of carbon monoxide, the effectiveness of the

greenhouse gases NO2 and CH4, and the X-State and B-State of molecular iodine. The obtained

results from the carbon monoxide experiment are close to the literature values, and provide

evidence that carbon monoxide acts more like a harmonic oscillator than an anharmonic

oscillator because its anharmonicity constant is small compared to the other obtained constants.

In addition, while the obtained values are all less than the literature values, the global warming

potentials of the greenhouse gases NO2 and CH4 indicate that NO2 is a more effective greenhouse

gas than CH4, as expected from theory, due to NO2’s time horizon in the atmosphere. Lastly, the

results from the absorption and emission of molecular iodine provide molecular constants for the

X-State and B-State. These values are close to the literature values, excluding the anharmonicity

constants due to extrapolation error. While the calculated equilibrium bond length for the X-State

is less than the literature value, the results show that the X-State has a smaller equilibrium bond

length than the B-State, which is expected from theory as the equilibrium bond length increases

with increasing vibrational energy. In essence, all three experiments provide the expected trends

from the theory.

1

Introduction

This set of experiments is concerned with the determination of structural features of

certain molecules, as well as global warming potentials of various greenhouse gases, through

molecular spectroscopy. Molecular spectroscopy studies the response of molecular structure to

electromagnetic radiation in the form of absorption and emission as well as any energy level

transitions that occur during these processes.1 In addition, it depends on nuclear and electronic

configurations as well as molecular behavior to distinguish molecules.

Central to molecular spectroscopy are rules pertaining to the energy of movement of

nuclei and electrons, as well as their respective frequencies.2 These rules adhere to the quantum

mechanical basis of energy quantization. For all particles, the kinetic and potential energy is

dependent on their motion. In the x, y, z space realm, the number of degrees of freedom

associated with n particles is 3n. When examining a molecule, the reference is its center of mass

outlined by Equation 1

r0=1M ∑

i

mi ri (1)

where M is the total mass of the system, mi is the mass of a particle, ri is the distance of the

particle from the center, and ro is the center of mass. For the nucleus, the vibrational, rotational,

and translational aspects of motion are all carried out with respect to this center of mass.

Electronic motion is spatially arranged with respect to molecular orbitals as electrons are

significantly smaller than nuclei resulting in a fixed configuration about them. Energy

quantization notes the discrete energy levels associated with different wave functions in the form

2

v=0, v=1, v=2, etc. for vibrational motion, J = 0, J = 1, J = 2, etc. for rotational motion, and S 0,

S1, S2 etc. for electronic motion as shown in Figure 1.

Figure 1. Energy Levels of Electronic, Vibration, and Rotational Energy 2

The strength of transition between two energy levels is dependent on the dipole moment of a

molecule dependent on Equation 2

Pi → f∨∫ψ final¿ µψ intial dr ¿2 (2)

where µ=∑i

qi ri

where the ψ terms are the wavefunctions of the particles, q is the charge of the particles, and r is

the length of the bond. The transitions that occur between these states are governed by selection

rules that determine whether a particular absorption transition is permitted. When an electron is

3

excited rotationally between energy levels such as v=0 and v=1, the excited state values must

follow that ΔJ = ±1.2 For vibrations, the selection rule states that Δv=±1.2

Rovibrational spectroscopy consists of analyzing the coupled rotational and vibrational

aspects of molecules using infrared radiation in the form of light. Infrared radiation (IR) has

enough energy to cause molecules to rotate and vibrate with rotations and vibrations represented

in Equations 3 and 4

F(J) = BJ(J+1) (3)

G(v) = (v + ½)νe (4)

where F(J) represents the rotational energy and G(v) represents the vibrational energy.3 In

addition, J and v represent the quantum numbers of the rotational and vibrational states

respectively, νe represents the frequency constant in wavenumbers (cm-1), and B represents the

rotational constant in wavenumbers. The equations for constants νe and B are represented in

Equations 5 and 6, with the moment of inertia represented in Equation 7

ve=1

2 πc √ kμ

❑

(5 )

B= h

8 π2cI(6)

I=μ Re2(7)

where c is the speed of light ( m s-1), k is the force constant (N m-1), μ is the reduced mass (kg), h

is Planck’s constant (J s), I is the moment of inertia (kg m2), and Re is the equilibrium bond

length (m).3 The force constant is proportional to the strength of the covalent bond, as it shows

how stiff the bond is.4 Stiffer bonds are more difficult to stretch and compress, and therefore

require a greater amount of energy to do so. As a result, stiffer bonds vibrate faster and absorb at

4

higher wavenumbers.4 The equilibrium bond length is the internuclear distance when the

internuclear potential energy is at a minimum, as shown by the Lennard-Jones potential in Figure

2. It is the thermal motion of the molecule that causes the iodine atoms to move around this

equilibrium position.5

Figure 2. Lennard-Jones Potential Diagram 5

The negative derivative of potential energy is force, as shown in Equation 21.

−dUdr

=F ( r )(21)

Therefore, on the Lennard-Jones potential diagram, the area to the left of the minimum is the

repulsive force the atoms feel, and the area to the right of the minimum is the attractive force the

atoms feel.5 The equilibrium bond distance is at the minimum of the curve, where the repulsive

and attractive forces cancel.5 Therefore, a small equilibrium distance corresponds to a larger

force constant. In addition, following Equations 5 and 7, the moment of inertia is directly

5

proportional to the size of the force constant. The terms I and Re combine to make the reduced

mass term, shown in Equation 22

μ=m1m2

m1+m2 (22)

Because μ is directly proportional to the force constant, diatomic molecules with larger masses

will therefore have larger force constants. We determined the molecular constants for carbon

monoxide in this experiment using rovibrational spectroscopy.

While Equations 3 and 4 provide a good model for rotations and vibrations alone, when

coupled they create interferences which need to be assessed. During a vibrational state transition,

the molecule experiences a force which causes the average bond length to increase.3 This

increase in bond length affects the rotational constant B, and therefore needs the terms Be and αe

to account for this bond length increase, as represented in Equation 8.

Bv=Be−αe (v+ 12 )(8)

In addition, Equation 8 implies that rotations are not based on a rigid rotor, so as the value of J

increases, the centrifugal distortion will cause the bond length to increase as well. This increase

in bond length due to centrifugal distortion, represented by the constant De, and is provided by

Equation 9.

F ( J )=Bv J (J +1 )−D e J 2 ( J+1 )2(9)

So far, the vibrational transitions have been based on the harmonic oscillator model, as shown in

Equation 3 and in Figure 3.

6

Figure 3. Simple Harmonic Oscillator Model 6

However, the harmonic oscillator model is only useful for low quantum numbers, as this

model does not account for bond dissociation or repulsive effects. In addition, the simple

harmonic oscillator forbids vibrational transitions which do not follow a change in vibrational

level of Δv = ± 1. However, such transitions can occur when enough energy is presented in the

system, such as the first overtone which corresponds to a molecule’s being excited from the

ground vibrational state to the second excited vibrational state.6 Therefore, another model known

as the anharmonic oscillator (shown in Figure 4) is used which accounts for these deviations

from the simple model.

7

Figure 4. The Anharmonic Oscillator 6

The anharmonic oscillator model shows the average bond length to change with

increasing quantum numbers, as well as that the vibrational energy levels are no longer equally

spaced for a molecule.6 The anharmonic oscillator demonstrates an increasing average bond

length for increasing quantum numbers. In addition, the anharmonic oscillator shows a

decreasing width of spacing of energy levels at higher excitation, as the curve provides less

constraint than the harmonic oscillator parabola.6 This is an effective model to use in

rovibrational spectroscopy, as it provides another constant, the anharmonicity constant (χe)

shown in Equation 10, which accounts for the deviations in bond length due to increasing

vibrational levels.

G (v )=(v+ 12 ) ve+xe ve (v+

12)

2

(10)

Rovibrational spectroscopy characterizes the structure of molecules by their rotational

energy levels corresponding to specific vibrational levels.7 We will identify rovibrational

characteristics of carbon monoxide through use of a FTIR spectrometer to develop an IR

8

spectrum. An FTIR spectrometer is of use in both inorganic and organic chemistry realms as it is

capable of determining structural characteristics from IR exposure. A major part of the

spectrometer is the Michelson interferometer which handles both the radiation exposure of a

sample and the Fourier Transformation required to develop a spectrum. From the source, a beam

of light is split and reflected off a motorized mirror, subsequently recombining to run through a

sample. A detector obtains the interferogram and Fourier transforms it into a spectrum. A block

diagram of a Michelson interferometer in Figure 5 outlines the major components.

Figure 5. Block Diagram of a Michelson Interferometer 8

We will use the rovibrational spectra to determine the fundamental transition and first

overtone of carbon monoxide between the ranges of 1950-2275 cm-1 and 4100-4400 cm-1,

respectively.9 A fundamental transition corresponds to Δv = +1, whereas the first overtone

corresponds to Δv = +2 for the CO molecule. Overtones correspond to Δv = ± n transitions, but

the probability of overtone transitions decreases as n increases.6 The anharmonic model shows

the overtones to be usually less than a multiple of the fundamental frequency.6 While the first

overtone corresponds to a higher energy, it is expected that its intensity will be less than that of

the fundamental.

9

We will determine the band center frequency (v0) to calculate rovibrational characteristics

like the anharmonicity constant ( χe ¿ and equilibrium frequency ¿) by plotting m vs wavelength

of the fundamental and overtone spectra as well as using Equation 11

ν0=ν e−2 νe χe (11)

We will also quantify the band force constant k by assessing the transition of the ground

state to first excited state as if it were a harmonic oscillator and using Equation 12 where ħ is

Planck’s constant divided by 2π, µ is the reduced mass, ω is the angular frequency, and k is the

band force constant, shown in Equation 12.

ħω=ħ√ kµ

(12)

The internuclear distance, or the bond length between atoms is when the systematic

potential energy is at its lowest level. The bond length is assumed to be identical for both the

ground and first excited energy state, and therefore we will use the transition frequency

difference for this calculation. Diatomic molecules such as CO and HCl have a center frequency

spectrum shown in Figure 6.

10

Figure 6. Center Frequency Spectrum of HCl 10

Using the internuclear distance, we calculated the moment of inertia for a diatomic

molecule using Equation 7. This formula corresponds to a singular point mass; for a defined

space containing multiple point masses, the moment of inertia is the summation of these terms.

We also determined the global warming potentials (GWP) of various greenhouse gases

including N2O and CH4 using IR absorbance. GWP represents the amount of heat trapped by

greenhouse gases when they are exposed to IR radiation emitted from the earth’s surface. The

GWP for a molecule is determined with respect to quantity, strength, and location of IR

absorption bands of the molecule with respect to the earth’s emitted IR radiation. GWP has been

of interest among researchers and political activists alike as it is a way of quantifying the adverse

effects and levels of harm these gases have on climate change. For example, the 1997 Kyoto

Climate Conference aimed to reduce emissions of six common greenhouse gases determined to

have high GWP’s to levels around 5.2% below 1990 levels by 2012.9 Radiation forcing capacity

is the summation of the IR spectrum and the emission of blackbody radiation from earth. It is

equivalent to the GWP in proportion to the time of residence the gases have in the atmosphere.

11

In order to obtain IR spectra, we will fill the gas cell of the FTIR spectrometer with samples at

pressures compatible with Beer’s Law (60 Torr for both N2O and CH4). Once obtained, we

integrated the spectra at 10 cm-1 intervals between 500-1500 cm-1 per the Pinnock et. Al model.11

The radiation forcing capacity of a sample can be determined relative to a reference gas. The

reference gas is normally CO2. The radiation forcing capacity is given by Equation 13 where RFA

is the radiation forcing capacity per 1 kg increase of sample, A(t) is time decay of the sample

pulse sample, and RFR and R(t) are that of the reference.

GWP =

RF A∗∫0

TH

A (t ) dt

RFR∗∫0

TH

R (t ) dt

(13)

In order to determine GWP in terms of mass as opposed to per molecule as in Equation

13, we used Equation 14 as shown below where τ is the atmospheric lifetime and MW is

molecular mass.

GWP =

RF A∗( 1000MW A )∗∫

0

TH

e−t / τ A

dt

RFR∗( 1000MW R )∗∫

0

TH

e−t / τR

dt

(14)

Absorbance spectroscopy is another aspect of this experiment, which works by measuring

the transmittance of light after it passes through the analyte. This transmittance relates to the

energy level transition from ground to an excited state. Transmittance is related to absorbance by

Equation 15, where I0 is the initial intensity of light and I is the transmitted intensity.

12

A = -log(II o

)

(15)

Absorbance uses a broad spectrum of visible light to raise the electrons to a range of

higher vibronic energy levels.14 Vibronic modes describe the simultaneous vibrational and

electronic transitions of a molecule.14 The broad spectrum allows for observation of multiple

excited states. An important aspect of absorbance is the population of the ground and excited

states. The population of these excited states is further described by the Boltzmann distribution

in Equation 16

NN o

=e−EkB T (16)

where N is the population of the excited state, No is the population of the ground state, E is the

energy (J), kB is the Boltzmann constant (J K-1), and T is temperature (K). The distribution states

that at higher temperatures, the populations of the ground and excited states become more equal.

The absorbance spectrum shows the vibrational level of the B-State. The B-state

describes the potential energy of the excited mode, which is a low-lying bound excited state. 12

The other state observed is the X-state, which describes the potential energy of the ground state.12

In this experiment, we observed the B-state of molecular iodine through the use of its absorbance

spectrum. An example of an absorbance spectrum for iodine is provided in Figure 7, with the B-

state and X-states shown in Figure 8.

13

Figure 7. I2 Absorbance Spectrum at 40oC 12

Figure 8. B-state and X-state of I212

As indicated in Figure 7, the absorbance spectrum consists of cold and hot bands. A cold

band is a transition from the lowest vibrational level of the ground electronic state to a certain

vibrational level in the B-state.12 On the other hand, a hot band is a vibrational transition between

two excited states.12 By taking the absorption spectrum and plotting wavenumber vs. v’ + ½, we

determined the spectroscopic constants for the B-state from a fourth order polynomial fit. The

14

spectroscopic constants are Te – G”(0), ve, vex’e, vey’e, E*, and D’e.12 The constant Te – G”(0)

corresponds to the energy offset between the two potential wells, where T’e is the spacing

between the bottoms of the two potential wells, and G”(0) is the vibrational energy in the ground

state.12 In addition, ve represents the fundamental vibrational number of the B-State, and x’e and

y’e are anharmonicity constants.12 The other constants, E* and D’e, correspond to the energy it

takes to move molecular iodine from the lowest vibrational energy level of the X-state to the

dissociation limit of the B-state, and the well depth of the B-state, respectively.12

Emission is similar to absorbance except the molecule is subjected a single wavelength of

light, in this case 514.5 nm. This selected wavelength excites the molecule to a singular excited

state. From this state, the molecule then relaxes back to various vibronic levels in the X-state. 12

These relaxations are measured and reveal the nature of the ground states. An example of an I2

emission spectrum is shown in Figure 9.

Figure 9. I2 Absorbance Spectrum 12

The peaks of the emission spectrum are referred to as bandheads. The bandhead

represents the highest energy point in the spectrum reached by the R branch.13 For molecular

15

iodine, the bandhead and band origin are close together. The band origin can be assumed equal to

the peak bandhead because at room temperature the rotational levels of I2 are largely populated.12

This large population means that the band maximum is at lower energy than the band origin.13

The numbers above of the bandheads correspond to the vibrational level which the molecular

iodine relaxes to after it is excited. As shown in Figure 9, the larger numbers correspond to a

higher wavelength, which means that the vibrational energy levels at these numbers correspond

to a higher energy.12 By taking the emission spectrum and plotting Δv vs. v” + ½, we obtained

the spectroscopic constants for the X-state. The value of Δv is obtained by subtracting the

wavenumber corresponding to the v” value from the wavenumber of the laser (19429.7694

cm-1).12 The spectroscopic constants of interest for molecular iodine are G”(0), v”e, vex”e, vey”e,

D”e, and E(I*).12 As stated earlier, G”(0) corresponds to the vibrational energy in the ground

state.12 Also, similar to the B-state, the v”e, vex”e, and vey”e constants refer to the fundamental

vibrational number of the X-state, and the two anharmonicity constants of the X-state,

respectively.12 Lastly, the D”e and E(I*) constants correspond to the well depth of the X-state and

the excitation energy corresponding to the lowest 2P1/2 ← 2P3/2 atomic transition of iodine,

respectively.12

We examined the emission and absorbance properties of I2 for its ground state and

excited state when exposed to an argon laser. When our I2 sample is exposed to the argon laser at

a short wavelength, the electrons will temporarily excite vibrationally and electronically before

relaxing back to the ground state. The emission in this experiment occured in the form of

fluorescence, as depicted in Figure 10.

16

Figure 10. Types of Emission 14

The emission levels for the spectra will be developed in relation to wavelength of the laser. The

expected peak of the absorption spectrum of iodine is at a peak of 500-600 cm -1. According to

Stokes Law, the emission peak is typically lower in intensity than the absorbance peak as the

general trend follows for a loss of vibrational energy when going from excited to ground state.14

This process is known as the Stokes Shift depicted in Figure 11.

Figure 11. Excitation and Emission spectrum of a Common Fluorochrome 14

The Franck-Condon Principle is a means of describing the intensity of a vibronic

transition within a molecule. It states that when a molecule undergoes an electronic transition,

17

there is not a major change in its nuclear configuration due to the inability of the nucleus to react

vibrationally before the transition ends, due to the massive size of the nucleus compared to the

electrons.15 The Born-Oppenheimer approximation accounts for this inability and quantifies the

vibrational and rotational motion separately as shown in Equation 17, where each value of E

corresponds to the energy of the type of transition.1

E = Eelec + Evib + Erot (17)

The cause of the vibrational state of the nucleus is the Coulombic forces that arise after

the transition. Integrating the wavefunctions for the ground and excited states determines their

overlap. Squaring the overlap terms provides the Franck-Condon factor as shown in Equation 18.

FC = ∑v '=0

∞

∑v=0

∞

Sv' , v

2 (18)

Although the summation terms range to infinity, the finite nature of overlap creates limitation

attributed to the finite number of absorption states of a molecule.17

Experimental

The first week of experimentation consisted of obtaining the rovibrational spectrum of

carbon monoxide (CO) using the FTIR spectrophotometer. Any time we were not using the gas

sample cell, we kept it in a desiccator because of its moisture sensitivity. Using the gas manifold,

we evacuated the gas sample cell using the vacuum, using the digital gauge to monitor the

pressure of any gases which could have been left inside it. We then placed the evacuated cell in

the FTIR spectrophotometer and collected a background spectrum. After collecting the

background spectrum, we then used the gas manifold to fill the gas sample cell with 100 mmHg

of CO. We collected CO spectra at resolutions of 4, 2, 1, 0.5, and 0.25 cm -1, collecting a new

18

background after each run. After collecting an adequate spectra showing the fundamental and

first overtones, we then shut down the program we used to obtain the spectra, and evacuated the

gas sample cell, storing it in the desiccator.

During the second week of experimentation we used the gas manifold to fill the gas

sample cell with N2O and CH4, using the FTIR spectrophotometer to determine their GWPs. In

order to prevent N2O from entering the vacuum pump, which could cause an explosion, we made

a gas trap to collect the N2O and any other contaminant gases. This process consisted of filling

the trap dewar with liquid nitrogen and placing it under the gas trap. After setting up the gas trap,

we then checked for leaks in the gas manifold, using the digital pressure gauge. After confirming

the absence of leaks, we then filled the gas sample cell with 60.1 mmHg N2O and took a

spectrum of it using the FTIR spectrophotometer, after taking the appropriate background. After

collecting an adequate N2O spectrum, we evacuated the gas sample cell, filled it with 60.0

mmHg CH4, and repeated the spectrum collecting steps. After collecting both spectra, we

concluded the procedure by evacuating the gas sample cell and placing it back in the dessicator.

The third week’s procedure consisted of taking the absorption and emission spectra of

molecular iodine. After adjusting the sample holders in an appropriate manner for absorption, we

took a reference background, adjusting the integration time to an appropriate value. After taking

the reference background, we then collected a dark background, and then took an absorbance

spectrum of the molecular iodine sample using a halogen light source, adjusting the signal to

noise ratio by increasing the number of scans until we obtained an appropriate spectrum. We

then set up the detector perpendicular to the laser beam in order to prepare for the collection of

the emission spectrum. After properly aligning the laser, we then covered the sample with a dark

cloth to prevent fluorescence, and then we took a dark background of the sample. After taking

19

the dark background, we then placed a filter in front of the path of the laser to maximize the

region of interest as well as to minimize the laser signal. We then turned on the laser and

collected the emission spectrum, adjusting the signal to noise ratio by increasing the number of

scans. After collecting an adequate spectrum, we ended the experiment by closing the shutter,

turning off the laser, and placing the iodine samples back in their containers.

Results and Discussion

The first week of experimentation consisted of obtaining the rovibrational spectrum of

CO, from which we obtained spectroscopic constants. We used the FTIR to determine the

fundamental and first overtones of CO, as shown in Figures 12 and 13.

2000 2050 2100 2150 2200 22500

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

2728-28

26

-27

25

-26

2423

-25

-24

2221

-23

20

-22

1918

-21-20

1716

-19

15

-18

1113

-17

14 12

-16

-15-14

-13

8

-12

10

-11

9

-10

7

-9

-8

6

-7

5

-6

-5

4 3

-4-3

2 1

-2

-1 1

0

2

1

3 4

2

5

43

6

5

7

6

8

7

9

8

10

9

11

10

12

11

13

12

14

13

15

14

16

17

1516

18

19

17

20

18

21

19

22

22

23

20

24

21

25

23

2627

252426

28

27

2928

Wavelength (cm-1)

Abso

rban

ce (A

U)

P-Branch R-BranchP-Branch R-Branch

J values

20

Figure 12. CO Fundamental Absorbance Spectra

4125 4175 4225 4275 43250.08

0.1

0.12

0.14

0.16

0.18

0.2

24-24

23

-23

22

-222120

-21-20

19

-19

18

-18

1716

-17

1514

-16

12

-15

13

-14-13-12

11

-11

10

-10

9 8

-9-8

7

-7

6 4

-6

5

-5

3

-4

2

-3

1

-2

-1

0

1

2

1

3

2

4

3

5 6

4 5

7

6

89

7

10

9

11

810

1213

14

1211

15

13

16

1415

1718

1920

1618

21

1719

2223

202221

24252324

Wavelength (cm-1)

Abso

rban

ce (A

U)

P-Branch R-Branch

J-values

Figure 13. CO First Overtone Spectra

As shown in Figures 12 and 13, the fundamental and first overtones are located in the

literature value ranges, from 1950-2275 cm-1 for the fundamental and 4100-4400 cm-1 for the first

overtone. This data makes sense as the first overtone corresponds to the second excited

vibrational state of the molecule, which is at a higher energy than the fundamental. In addition,

we located the P and R branches on these spectra, which allowed us to find the m values (located

above the peaks in Figures 12 and 13) corresponding to each J value. The R branch corresponds

to ΔJ = +1, and therefore has positive m values starting at 0.18 On the other hand, the P branch

corresponds to ΔJ = -1, and therefore has negative m values.18 However, its m values cannot start

at 0 because the value of J’ (the excited rotational state) cannot be -1.18 We then plotted the

wavelength versus the m values for the fundamental and first overtones, obtaining cubic,

quadratic, and linear fits, shown in Figures 14 and 15.

21

-30 -20 -10 0 10 20 301900

1950

2000

2050

2100

2150

2200

2250f(x) = − 5.71059552062594E-05 x³ − 0.0142504848197964 x² + 3.76634079344824 x + 2142.8071931871R² = 0.999693385573565

f(x) = − 0.0143372950823478 x² + 3.73758863579973 x + 2142.82216461854R² = 0.999682062419989f(x) = 3.72337699623958 x + 2138.73722924753R² = 0.996445029356287

m values

Wav

elen

gth

(cm

-1) Cubic

Quadratic

Linear

Figure 14. CO Fundamental Absorbance

-30 -20 -10 0 10 20 304000

4050

4100

4150

4200

4250

4300

4350f(x) = 3.76728929831439 x + 4252.2003626029R² = 0.98623969853831f(x) = − 0.0349102489872732 x² + 3.80184373919751 x + 4259.61006637168R² = 0.999973319486877

f(x) = − 1.05406482050996E-05 x³ − 0.0348941899747725 x² + 3.80578473873728 x + 4259.60800019687R² = 0.999973525196163

m values

Wav

elen

gth

(cm

-1)

Linear

Quadratic

Cubic

Figure 15. CO First Overtone

22

As shown in Figures 14 and 15, the graphs both have R2 values close or equal 1, which

shows the reliability of the obtained functions. We used the values obtained from the cubic

functions, and Equation 19 (shown below) to determine the spectroscopic and molecular

constants, shown in Tables 1-3. The calculations for these constants are provided in the appendix

section.

ν(m) = νo + (2Be - 2αe)m – αem2 – 4Dem3 (19)

Table 1. Fundamental and First Overtone Wavenumbers

Fundamental (cm-1) First Overtone (cm-1)2142.9 4259.6

Table 2. Rovibrational Spectra Constants

Equilibrium Frequency

(cm-1)αe (cm-1) Be (cm-1) De (cm-1) χe (cm-1)

Experimental Value

2168.8 0.0143 1.90 1.5 x 10-5 0.00599

LiteratureValue19 2169.8 0.0175 1.9313 6.2 x 10-6 0.00612

Percent error 0.0461% 18.3% 1.62% 142% 2.12%

Table 3. Molecular Constants

Moment of Inertia (kg m2)

Equilibrium bond (Å)

Force Constant (N/m)

Experimental Value 1.47 x 10-46 1.14 1903Literature Value19 1.4490 x 10-46 1.1281 1902

Percent error 1.45% 1.05% 0.0526%

As shown in Table 2, the experimental values are close to the literature values for the

larger spectroscopic constants, such as the equilibrium frequency. On the other hand, for smaller

constants, such as De and αe, the percent errors are large. This large percent error means that these

23

constants are not as well defined as the larger values. In addition, the larger values contribute

more to the rovibrational frequency than the smaller values. For instance, as shown in Table 2,

the rotational constants Be and αe contribute more to the rovibrational transitions than the

anharmonicity contant χe. A smaller anharmonicity constant means that the molecule acts more

like the ideal harmonic oscillator. In addition, the centrifugal distortion constant, De, is roughly

1000 times smaller than the anharmonicity constant. This previous statement means that the

centrifugal distortion caused by rotation contributes the least amount to the rovibrational states.

The experimental molecular constants shown in Table 3 are all close to those found in the

literature. These close values show that FTIR is an effective method of determining

spectroscopic and molecular constants.

To calculate the global warming potentials (GWPs) during this experiment, we took the

absorbance spectrum of CH4 and N2O, which provided the frequencies where CH4 and N2O

absorb Earth’s blackbody radiation. The absorbance spectra obtained in the lab as well as where

CH4 and N2O absorb Earth’s blackbody radiation are provided in Figures 16-19.

24

495.00 695.00 895.00 1095.00 1295.00 1495.00-0.05

0.15

0.35

0.55

0.75

0.95

Wavenumber (cm-1)

Ab

sorb

ance

(A

U)

Figure 16. CH4 Absorbance Spectrum

500 600 700 800 900 1000 1100 1200 1300 1400 1500-1

0

1

2

3

4

Blackbody Radiation of Earth

GHG Spectrum

Frequency (cm-1)

* 1

018

s

orf

Figure 17. Absorbance Spectrum of Methane and Blackbody Radiation of Earth

25

495.00 695.00 895.00 1095.00 1295.00 1495.00-0.10

0.10

0.30

0.50

0.70

0.90

Wavenumber (cm-1)

Ab

sorb

ance

(A

U)

Figure 18. N2O Absorbance Spectrum

500 600 700 800 900 1000 1100 1200 1300 1400 1500-1

0

1

2

3

4

Blackbody Radiation of Earth

GHG Spectrum

Frequency (cm-1)

* 101

8

sorf

Figure 19. Absorbance Spectrum of N2O and Blackbody Radiation of Earth

As shown in Figure 16, CH4 has a peak near 1300 cm-1, which means it absorbs at that

wavenumber value. This data correlates with Figure 17 as the CH4 peak has overlap with the

blackbody radiation of earth near 1300 cm-1. In addition, as shown in Figure 18, N2O has

absorbance peaks near 600 cm-1, 1200 cm-1, and 1300 cm-1. However, as shown in Figure 19,

26

N2O only has overlap with the blackbody radiation of earth near 1300 cm-1. The lack of peaks in

Figure 19 is due to the pressure of N2O used in the experiment. At higher pressures, the GHG

and blackbody spectra will overlap.

We used the data collected from the absorbance spectra to calculate the GWPs of CH 4

and N2O. GWP is a measurement of the ability for a gas to trap heat in the atmosphere. The

calculated GWP values are provided in Table 4.

Table 4. Greenhouse Gas (GHG) GWP Values

GHGLifetime (Years)

Time Horizon (Years)

Calculated GWP Literature GWPPercent

Difference (%)

N2O 12020 73.3 93 21.1100 69.3 88 21.3500 60.9 77 20.9

CH4 1520 33.3 37 10.0100 11.6 13 10.9500 5.9 6 2.46

Looking at Table 4, the GWPs for N2O are larger than those of CH4, which means that

N2O traps more heat in the atmosphere and therefore is a more effective GHG than CH4. This

data correlates well with the literature values, as the literature GWPs for N2O are greater than

those of CH4. Nitrous oxide has larger GWPs because it has a larger atmospheric lifetime than

CH4, and therefore decays less rapidly than CH4. However, as shown in Table 4, the calculated

GWPs are all less than the literature values, even though the pressures of the gases were 60.0

Torr for CH4 and 60.1 Torr for N2O, which correspond to the linear range of Beer’s Law. Smaller

GWPs mean that a GHG traps less heat in the atmosphere than gases with larger GWPs. These

smaller GWPs for N2O and CH4 portray that they are not as effective as holding in heat as the

literature states, which cannot be trusted. The smaller calculated GWPs could be due to some

possible reasons. One reason for this difference could be difficulties retaining a vacuum during

27

the filling of the IR cell as well as possible contamination. However, the effects of contamination

were minimal due to the precautions of the experimental set-up. In addition, we could have

found more accurate data by using smaller cm-1 intervals instead of using 10 cm-1 intervals, as the

literature values were taken at 2 cm-1 intervals.9 Also, the windows of the IR cell were different

than that used in the literature, which would affect the transmission limits. In addition, the

literature does not state exactly what pressures were used during experimentation, which leads to

uncertainty in its values. Variances in pressures also have the ability to change the GWP by a

significant amount. For example when using the GWP model, pressure values of 65 Torr and

60.1 Torr for N2O differ in values by 10%. However, the literature also states that a major source

of uncertainty in the GWP is the determination of the atmospheric lifetime of the GHG.9 Elrod

et. al states that GHGs with longer lifetimes are more accurately modeled by the Pinnock et. al

model because the gases are more well mixed globally.9 However, for our data the gas with the

larger atmospheric lifetime, N2O, was less accurately represented by this model. The biggest

source of error is that the model is an exponential decay.9 In reality CO2 follows three different

rates of decay, which means that this simplified model does not accurately portray the GWP for

both CH4 and N2O as well. Using a model which accurately portrays the decay of the molecules

would provide more accurate results. In essence, while the GWP model shows N2O to be the

more effective GHG than CH4, which agrees with the literature, the effectiveness of N2O and

CH4 as GHGs are underestimated, as their GWPs are less than the literature values.

During the third week of experimentation we determined the B-State and X-State

constants for molecular iodine using absorption and emission, respectively. Figures 20 and 21

provide the absorbance spectrum for molecular iodine and the bandhead energy versus v’ + ½,

respectively, with Table 5 providing the B-State spectroscopic constants.

28

Figure 20. I2 Absorbance Spectrum

5 10 15 20 25 30 35 40 4515500

16000

16500

17000

17500

18000

18500

19000

19500f(x) = − 0.0075886056938 x³ − 0.4380668461184 x² + 119.22591592413 x + 15689.93570027R² = 0.999860764677182

v' + 1/2

Wav

enu

mb

ers

(cm

-1)

Figure 21. I2 Bandhead Energy Versus v’ + ½.

29

Table 5. B-State Spectroscopic Constants

Spectroscopic Constants

Experimental Values (cm-1)

Literature Values12

(cm-1)Percent Error (%)

T’e – G”(0) 15690 15661.99 0.501v’e 119.23 125.67 5.12

vex’e 0.4381 0.7504 41.6vey’e -0.0076 -0.00414 83.6E* 19658.58 20043.2 1.92De 3968.578 4381.2 9.42

The bandheads in Figure 20 show the vibronic transitions from the ground state to

varying excited states. These bandheads are where the unresolved vibrational-electronic lines are

the strongest.15 We then plotted the bandhead energy versus v’ + ½ (Figure 21) which provided

us with a cubic fit, which we then used to calculate the spectroscopic constants for the B-State,

shown in Table 5. Looking at Table 5, the spectroscopic constants are relatively close to the

literature values, besides the anharmonicity constants, vex’e and vey’e. The error in this part of the

data analysis is due to the extrapolation of the cubic fit in order to find the maximum of the

function, which corresponds to De. In addition, even though the R2 is practically equal to 1, there

is always estimation error associated with extrapolation, which can lead to erroneous results,

particularly in the case of the anharmonicity constants.12

In addition to finding the B-State spectroscopic constants, we found the X-State

spectroscopic constants by taking the emission spectrum of molecular iodine. Figures 22 and 23

show aspects of the emission spectrum of molecular iodine, whereas Figure 24 shows the

bandhead energy versus v” + ½ and Table 6 provides the spectroscopic constants of the X-State.

30

500 550 600 650 700 750 8000

5000

10000

15000

20000

25000

1

2

3 4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

2223

2425 26

27

28

2930

31

3234

36

Wavelength (nm)

Inte

nsi

ty

Figure 22. I2 Emission Spectrum

508 508.2 508.4 508.6 508.8 509 509.2 509.4 509.6 509.8 510-20

0

20

40

60

80

100

120

140508.54 nm 509.14 nm

Wavelength (nm)

Inte

nsi

ty

Figure 23. I2 Emission Spectrum Doublet at v” = 0 Used to Find B”

31

0 5 10 15 20 25 30 35 400

1000

2000

3000

4000

5000

6000

7000

8000

f(x) = − 0.00122088 x⁴ + 0.07542046 x³ − 2.14035247 x² + 224.925209 x − 138.153683R² = 0.999964332382799

v" + 1/2

Wav

enu

mb

er (

cm-1

)

Figure 24. I2 Bandhead Energy vs v” + ½

Table 6. X-State Spectroscopic Constants

Spectroscopic constants

Experimental values (cm-1)

Literature Values12

(cm-1)Percent Error (%)

G”(0) 138.15 107.11 29.0v”e 224.93 214.53 4.85

vex”e 2.14 0.6130 249vey”e -0.5177 0.0754 73303.88D”o 11731.229 12440.2 5.70

E(I*) 7927.4 7602.98 4.27

As shown in Figures 22 and 23, the bandheads provide the vibronic transitions from the

excited state (v’ = 43) to varying ground states, where larger values of v” correspond to higher

energy vibronic levels. We plotted the bandhead energies versus v” + ½ to find the spectroscopic

constants of the X-State, as shown in Table 6. Like the B-State spectroscopic constants, the X-

State spectroscopic constants are close to the literature values, except for the anharmonicity

constants. Extrapolation of the fourth order fit is the cause of this error, even though the R 2 value

32

is equal to 1. Looking at Figure 22, the largest Franck-Condon factor is at v” = 5 because it has

the greatest intensity, and therefore corresponds to the greatest vibrational-electronic overlap.15

We then used these spectroscopic constants to find the Morse Potentials for both the X-

State and B-State. In order to generate these potentials from the spectroscopic constants, we used

Equations 20 and 21

E=Te+De¿ (20)

β=νe π √ 2 μch De

(21)

where R is bond length (Å), Re is equilibrium bond length (Å), μ is the reduced mass of

molecular iodine (g), νe is the equilibrium frequency (cm-1), h is Planck’s constant (J s), and c is

the speed of light (m/s). After resolving the bandheads corresponding to the v” = 0 transition

(Figure 23), we calculated the value of Re using Equation 6. We plotted these equations for both

the X-State and B-State, which provided the Morse Potentials, shown in Figure 25, with the

spectroscopic constants for the Morse Potentials provided in Tables 7 and 8.

33

2 2.5 3 3.5 4 4.5 5 5.5 60

5000

10000

15000

20000

25000

30000

35000

40000

X-State

B-State

Bond Length (Å)

En

ergy

(cm

-1)

T'e

E(I*)

D"e

E*

D'e

D"o

Figure 25. X-State and B-State Morse Potential Curves

Table 7. Morse Potential Spectroscopic Constants For X-State

Spectroscopic constants Experimental Values Literature Values12 Percent Error (%)

D”e Used Lit. Value 12547.3 cm-1 n/a

R”e 151 pm 266.64 pm 43.4Te 15828.15 cm-1 15769.1 cm-1 0.374v”e 224.93 cm-1 214.53 cm-1 4.85

Table 8. Morse Potential Spectroscopic Constants For B-State

Spectroscopic constants Experimental Values Literature Values12 Percent Error (%)

D’e 3968.578 cm-1 4381.2 cm-1 9.42

R’e Used Lit. Value 3.0267 Å n/aTe 15828.15 cm-1 15769.1 cm-1 0.374v’e 84.603 cm-1 125.67 cm-1 32.7

In determining the Morse Potentials, we used the literature value of D”e because

extrapolation of our curve in Figure 24 provided a negative value. The Morse Potentials in

Figure 25 cannot entirely be trusted even though they follow the theoretical shape of Morse

34

Potentials, as the value for R”e is smaller than the literature value. However, this value is still

smaller than R’e, which follows the theory, because bond lengths at higher vibrational levels will

increase due to the higher energy stretching the length of the bond. Other than the small value of

R”e, the other constants are relatively close to the literature values. In comparison with Figure 8,

the equilibrium bond length of the X-State should be larger. Our data shows smaller overlap

between the two potential wells. According to the Franck-Condon principle, the only vibronic

transitions occur within the overlap swathe which contains the potential wells of the X-State and

B-State. A larger swathe allows for more transitions to occur. Therefore, our data shows less

transitions than are actually possible. After plotting these Morse Potentials, we then plotted the

Morse Potentials using the program FCIntensity, by using our spectroscopic constants. Figures

26 and 27 show the Morse Potentials and Franck-Condon factor intensities, respectively, for the

X-State and B-State.

Figure 26. Morse Potential From FCIntensity Program

35

Figure 27. FC Intensities From FCIntensity Program

Looking at Figure 26, the Morse Potentials are not accurate as the equilibrium bond

length for the X-State is larger than that of the B-State, because higher vibrational energies

stretch the equilibrium bond length. In addition, the FC Intensities graph (Figure 27) from the

FCIntensity program show that the largest Franck-Condon factor is located near 800 nm.

However, we determined the largest Franck-Condon factor from the emission spectrum at v” = 5,

corresponding to a wavelength of 544 nm. The results from our calculations are more reliable

than those of the FCIntensity program because the equilibrium bond length of the X-State is less

than that of the B-State.

Conclusion

The results from the CO experiment show the accuracy of the FTIR, as the experimental

values are close to the literature values. The data shows the Be and αe constants contribute most

to the rovibrational frequency, with the smaller constants such as χe contributing the least. In

addition there is also an increase in percent error for the smaller constants because the smaller

36

constants are not as well defined as the larger ones. In addition, the findings show that CO acts

more like a harmonic oscillator as its anharmonicity constant is small compared to the other

constants.

The GHG data shows N2O to be a more effective greenhouse gas than CH4, because N2O

has a larger GWP than CH4. A larger GWP means that a greenhouse gas is more efficient at

trapping heat within the atmosphere. While all the experimental GWP values are less than the

literature values, the results show the expected result that N2O is a more efficient greenhouse gas

than CH4, which is due to N2O’s large time horizon in the atmosphere. The deviations from

literature values are mainly due to the inaccuracy of the Pinnock et. al model, which uses a

simple exponential decay model. In reality, molecules can have different decay models, such as

CO2 which uses three different rates of decay.9

The results from the third week provide insight into the different vibronic levels of the X-

State and B-State, as well as provide a model of the Morse Potentials for each state. The

constants obtained from the absorption and emission data are all relatively close to the literature

values, besides the anharmonicity constants, due to error stemming from extrapolation. However,

the most important aspect of this data analysis is the Morse Potentials. The experimental Morse

Potentials show the correct anharmonic oscillator curves, shown in Figure 8. However, the

experimental Morse Potentials show the R”e to be less than the literature value, which ultimately

leads to less overlap swathe between the X-State and B-State. This smaller swathe leads to a

smaller Franck-Condon factor than expected from the theory.

37

References:

1. Molecular Spectroscopy. Seton Hall University Chemistry Department. 

https://hplc.chem.shu.edu/NEW/Undergrad/Molec_Spectr/molec.spectr.general.html (accessed

Mar 8, 2015).

2. MIT. (n.d.). Principles of Molecular Spectroscopy. Retrieved March 23, 2015, from

http://web.mit.edu/ 5.33/www/lec/spec4.pdf.

3. McQuarrie, Donald A.; Simon, John D. Physical Chemistry: a Molecular Approach;

University Science Books: United States of America, 1997; 495-506.

4. MSU. (n.d.). The Nature of Vibrational Spectroscopy. Retrieved March 27, 2015, from

https://www2.chemistry.msu.edu/faculty/reusch/virttxtjml/Spectrpy/InfraRed/irspec1.htm.

5. UC Davis. (n.d.). Lennard-Jones Potential. Retrieved March 27, 2015, from

http://chemwiki.ucdavis.edu/Physical_Chemistry/Intermolecular_Forces/Lennard-

Jones_Potential.

6. University of Liverpool. (n.d.). Vibrational Spectroscopy. Retrieved March 13, 2015, from

http://osxs.ch.liv.ac.uk/java/spectrovibcd1-CE-final.html.

7. Tipler, Paul A. and Llewellyn, Ralph A., Modern Physics, 3rd Ed., W.H. Freeman, 1999.

8. University of California at Davis Chemistry Department. FTIR Block Diagram [Image].

9. Elrod, M. J. J. Chem. Ed. 1999, 76, 1702-05.

10. Georgia State University. Center frequency spectrum of HCl [Image].

11. Pinnock, S.; Hurley, M. D.; Shine, K. P.; Wallington, T. J.; Smyth, T. J. J. Geophys. Res.

Atmos. 1995, 100, 23227–23238.

12. Williamson, J. C. (2007). Teaching the Rovibronic Spectroscopy of Molecular Iodine.

Journal of Chemical Education, 84(8), 1355-1359.

38

13. University of Colorado. (n.d.). Band Spectra and Dissociation Energies. Retrieved March 27,

2015, from http://chem.colorado.edu/chem4581_91/images/stories/BS.pdf.

14. Microscopy Resource Center. Jablonski Energy Diagram; Excitation and Emission Spectrum

[Image].

15. Franck-Condon Principle. University of California at Davis Chemistry Department. 

http://chemwiki.ucdavis.edu/Physical_Chemistry/Spectroscopy/Electronic_Spectroscopy/Franck-

Condon_Principle (accessed Mar 8, 2015)

16. Properties of Iodine Molecules. University of Hannover.

https://hplc.chem.shu.edu/NEW/Undergrad/Molec_Spectr/molec.spectr.general.html (accessed

Mar 8, 2015).

17. Dunbrack, R.L. J. Chem Ed. 1986, 63, 953-55.

18. Konstanz. (n.d.). Electronic Spectroscopy of Molecules I - Absorption Spectroscopy.

Retrieved March 27, 2015, from http://www.uni-konstanz.de/FuF/Bio/folding/3-Electronic

%20Spectroscopy%20r.pdf.

19. Mina-Camilde, N., & Manzanares, C. (1996). Molecular Constants of Carbon Monoxide at v

= 0, 1, 2, and 3. Journal of Physical Chemistry, 73(8), 804-807.

39

Appendix: Sample Calculations

1. Moment of inertia

Be=h

8 π2 cI

I= h

8 π 2c Be

I= 6.62606957 x10−34 Js

(8 π2 29979245800cms)1.90 cm−1

I=1.47 x10−46 kgm2

2. Equilibrium Bond Length

I=μ r2

r=√ Iμ

r=√(1.47 x 10−46kg m2)(1000gkg

)(6.02214129 x 1023mol−1)

(12.0107g

mol)(15.9994

gmol

)

12.0107g

mol+15.9994

gmol

r=1.14 A

40

3. Vibrational Force Constant

ve=1

2 πc √ kμ

❑

k=μ(2 πcv e)2

μ=(12.0107

gmol

)(15.9994g

mol)

12.0107g

mol+15.9994

gmol

=6.86052081g

mol

k=(μ)¿

k=1903Nm

NOTE: Values were taken from cubic fits from the fundamental and first overtone graphs.

4. Calculating αe

α e m2=(0.0143 cm−1)m2

α e=0.0143 cm−1

5. Calculating Be

(2Be−2αe )m=(3.7663 cm−1)m

α e=0.0143 cm−1

Be=3.7663 cm−1+2 (0.0143 cm−1 )

2

41

Be=1.90 cm−1

6. Calculating De

4 De m3=(6 x10−5 cm−1)m3

De=1.5 x10−5 cm−1

7. Calculating equilibrium frequency νe

υo=υe−2υe χe

χe=υe−υo

2 υe

υoovertone=2 υe−6 υe

υe−υo

2υe

υoovertone=4259.6 cm−1

υo=2142.8 cm−1

υe=2168.8 cm−1

8. Calculating χe

χe=υe−υo

2 υe

υe=2168.8 cm−1

υo=2142.8 cm−1

χe=0.00599 cm−1

9. Percent error of moment of inertia

%error=absolute value(accepted−experimental)

acceptedx 100 %

42

% error=absolute value (1.47 x10−46 kgm2−1.449 x10−46kg m2 )

1.449 x10−46 kgm2 x100 %

% error=1.45 %

10. Calculating D”o

D} o=19429.7694 {cm} ^ {-1} -peak energ¿

peak energy=7698.54 cm−1

D} o=11731.2 {cm} ^ {-1¿

11. Calculating E(I*)

E ( I ¿)=E¿−D } ¿

E¿=19658.58 cm−1 (obtained¿extrapolation¿maximum )

E ( I ¿)=19658.58 cm−1−11731.2cm−1

E ( I ¿)=7927.4 cm−1

12. Calculating spectroscopic constant R”e

R} e = sqrt {{h} over {8 {π} ^ {2} c μ {B} ^ { ¿¿

B} = {(19664.13655-19640.96391) {cm} ^ {-1}} over {398} = ¿0.05822451 cm-1

R} e = sqrt {{6.626 x {10} ^ {-34} {kg {m} ^ {2}} over {{s} ^ {2}} s} over {8 {π} ^ {2} left (2.99 x {10} ^ {10} {cm} over {s} right ) left (126.9045 {g} over {mol} x {mol} over {6.022 x {10} ^ {23}} x {kg} over {1000 g} right ) 0.05822451 {cm} ^ {-1`}}} ¿

R} e=1.51 x {10} ^ {-10} m=1.51 angstrom ¿

13. Calculating β (for X-State)

β=νe π √ 2μch De

43

β=224.93 cm−1 π √ 2(126.9045g

mol )(3 x 108 ms ) kg

1000 gmol

6.022 x1023

(6.626 x 10−34 kgm2

s2 s)(12547.31

cm)( 100cm

m)

β=1.94 x 107 cm−1

14. Calculating Te

T e−G} left (0 right ) =15690 {cm} ^ {-1¿

E¿=19658.58 cm−1 (obtained¿extrapolation¿maximum )

G} left (0 right ) =138.15 {cm} ^ {-1} left (obtained from emission data right ¿

De=39678.578 cm−1 ¿

T e=−¿

T e=15828.15 cm−1

44