Dissociation energy of iodine by absorption spectroscopy

William Harvey

8446595

School of Physics and Astronomy

The University of Manchester

Third Year Laboratory Report

March 2015

This experiment was performed in collaboration with Alex Fortnam.

Abstract

The dissociation energy of Iodine was calculated using a Birge-Sponer extrapolation on data

obtained from absorption spectra, using a white light bulb and an LED. It was found to be

4283.52 11.371 for the excited state and 13246 22.11 for the ground state. The excited state equilibrium separation was calculated as 2.9227 0.127. Morse potential curves were plotted for the ground and excited states.

1. Introduction

When matter is exposed to some form of radiated energy, a spectrum may be produced.

Spectroscopy is the study of this phenomenon. Originally, the interaction between matter and

electromagnetic waves in the optical region was studied, but many different forms of

spectroscopy have since arisen. The radiated energy applied need not be limited to the

optical part of the electromagnetic spectrum, nor even to electromagnetic waves. Nearly any

particle, such as the electron or the muon, may be the source of energy in the form of de

Broglie waves.

In this experiment, the iodine molecule is studied using optical spectroscopy, giving rise to a

diatomic spectrum. Photons with energies equal to the energy of a transition may be absorbed

by the iodine, which will undergo this transition then re-emit the photon and transition back

down to its original state. From this data, the dissociation energy of the Iodine molecule may

be calculated.

2. Experimental Method

2.1. The experimental setup

Our setup consisted of a source emitting light through a collimator lens into a gas tube

containing iodine molecules. Light emitted from the gas tube passed through a focusing lens,

then through the entrance slit of the spectrometer. The light was diffracted, before passing

through an exit slit identical to the entrance slit and entering a photomultiplier tube. This

photomultiplier tube was connected to a computer running data acquisition.

Figure 1 [1]. A diagram of our experimental setup.

2.2 The source

Two light sources were used: A white light bulb and an LED. To ensure that light

passing through the entry slit was in the optical axis of the spectrometer, the source position

was carefully adjusted, with the image formed in the spectrometer being observed through the

exit slit. When the light was found to be in the centre of the exit slit, the source was in the

optical axis of the system. The light was then focused by adjusting the focusing lens until a

small, intense dot of light was observed on the centre of the entry slit.

The apparent spectral power distribution of the white light bulb was recorded:

Figure 2

The apparent spectral power distribution of the LED was also recorded:

Figure 3

2.3 The entry and exit slits

To calibrate the slit widths, the entry slit was fully closed, with the exit slit fully opened. The

exit slit was then slowly opened until a small photocurrent was produced, with the value on

the slit micrometer being recorded. This procedure was repeated, but with the exit slit being

fully opened and the entrance slit initially closed. The slit widths were increased until a

suitable signal to noise ratio was obtained. This ratio was 100:1.

2.4 The diffraction grating

The diffraction grating used was a blazed reflection diffraction grating and was

connected to a motor, rotating it with respect to the optical axis of the system, changing the

angle of incidence of incoming light. The position of the diffraction grating was measured

from the screw-gauge micrometer. The wavelength of light selected by the grating is related

to . To find the relationship, a cadmium bulb was manually observed through the exit slit, with the value being recorded when a reference emission line was found. This was plotted, assuming a linear relationship.

Figure 4

A cadmium spectrum was then recorded electronically. Spectra recorded by the software are

produced with values in terms of , necessitating finding the wavelength- relationship. Using the linear relationship the values of peaks in the calibration spectra were converted to wavelength values and matched to reference values for cadmium. There is a small quadratic

offset in the relationship between wavelength and , and thus a quadratic fit was applied to the data (see figure 5), using the lsfr26.m script [2].

Figure 5

3. Theory

3.1. Blazed reflection diffraction grating

The diffraction grating reflected incident light in a direction dependent on wavelength, so

only one wavelength would be able to propagate through the spectrometer (figure 6). A

blazed grating has teeth ruled in its surface, minimising the intensity of any diffraction that is

not first order. The grating equation is:

+ = (1)

where is the incident angle of the light, is the angle of reflection, is the number of teeth ruled per millimetre, m is the diffraction order and is the wavelength of the light.

As the diffraction grating is rotated, a range of different wavelengths will be able to

propagate through the spectrometer.

3.2. Energy of the diatomic molecule

The Born-Oppenheimer approximation states that:

= + + (2)

where is the total internal energy, is the electronic energy, is the vibration energy and is the rotational energy, where:

103 10

6

Within each electronic state, there are many vibrational energy levels and within each

vibrational state there are many rotational energy levels, giving coarse vibrational and fine

rotational structure in spectra.

Figure 6. Diagrammatical representation of equation

(1) [3].

3.3. Vibration

Vibration in the iodine molecule occurs about the equilibrium internuclear separation

distance - the point for which the potential energy is minimised. The vibrational energy is

quantised, with a vibrational quantum number of . Molecular vibration may be approximated as simple harmonic, but this approximation is poor. Anharmonicity in real

molecules is not negligible, causing higher vibrational levels to crowd together.

Dividing (2) by with in units of 1 gives term-values with units 1:

= + + (3)

where is the total energy term, is the vibrational energy term and the rotational energy term [4]. Working in these units allows the energy of a transition to be expressed in

terms of the wavenumber of the transition (the reciprocal of the photon wavelength). All

parameters corresponding to the excited electronic state are primed () and all parameters corresponding to the lower electronic state will be double primed () from this point onwards. An electronic-vibrational transition neglecting the rotational energy will therefore be, from

(3):

= (

) + ( ) + ( ) (4)

where is the wavenumber of the transition. The Boltzmann distribution of our iodine molecules is:

=1

(5)

where is an energy level with quantum number i, is the probability of finding a particle in this energy level, is the Boltzmann constant, is the temperature and is the partition function. Low vibrational levels with less anharmonicity may be approximated as simple harmonic motion:

= ( +1

2) (6)

where is the reduced Plancks constant and is the angular frequency. The partition function for (6) simplifies to:

=

12

1

(7)

Inserting (7) into (5) gives:

=

(1

) (8) where is the probability of finding a molecule in the th vibrational level [5]. From this, it is found that, at room temperature, almost all iodine molecules are in the = 0 state, and

the electronic transitions are between the ground state (X) and the first excited state (B), = 0 and

= 0. The vibrational energy term for a level is:

() = ( +

1

2)

(

+1

2)2 (9)

where is the frequency of infinitesimal amplitude vibrations between

= 0 and the zero of the potential well (see figure 7) and

is an anharmonicity constant. The energy

of a transition between and is (from (9)):

= + (

+1

2)

(

+1

2)

2

. (10)

The Birge-Sponer extrapolation is a plot of the energy level spacings (in 1) between subsequent vibrational levels against + 1, where values are applied to bands in absorption spectra using a Deslandres Table [6]. The equation for the energy level spacing,

from equations (4), (9) and (10) is:

= 2

(

+ 1) (11)

Figure 7 1 Morse potential plot calculated using parameters from our data. Diagrammatical representation of molecular

constants.

thus is the y-intercept and 2

is the gradient of this plot. Integrating to obtain the

area of this plot is identical to summing all of the energy level spacings, giving 0, the dissociation energy of the B state:

0 =

0 (12)

where is the convergence vibrational quantum number, or the x-intercept. , the

convergence limit, is:

= +

(13)

where is the wavenumber of the transition corresponding to , the highest measured.

The dissociation energy of the ground state, 0 is:

0 = () (14)

where () is the energy difference between a ground state atom and an atom in the first excited state with a value of 75891 [7].

3.4. Rotation, moment of inertia, equilibrium separation and the potential curves

A diatomic molecule may rotate around an axis passing through the centre of and

perpendicular to the bond joining them. , the moment of inertia, is:

= 2 (15)

where is the reduced mass of the molecule and is the equilibrium separation. The Morse potential energy curve is a good approximation of the molecular potential curve [8]:

( ) = (() 1)

2 (16)

where is the separation and is:

= 2

(17)

where is Plancks constant. may be obtained [9] using a literature value for

:

=

+1

(ln (1 +

U()

) (18)

From this, the and potential curves and may be found.

4. Results and discussion

4.1. White light results

Scanning over the widest possible range of wavelengths with the white light bulb gave Figure

8, from which the wavelength range of vibrational bands was found.

Figure 8.

Table 1 is the Deslandres:

() 27 0 543.47

28 0 541.18

29 0 539.89

30 0 536.87 Table 1.

A white light spectrum at room temperature, scanned across the wavelength range of interest,

was labelled from bands corresponding to = 28 up to = 40:

Figure 9. Not all of the band heads labelled for the Birge-Sponer extrapolation are labelled on this figure.

A Birge-Sponer extrapolation was plotted:

Figure 10

4.2. LED Results

Using the LED as the source gave clearer spectra with better signal to noise ratios. 5 LED

spectra were averaged, reducing noise 5 times. The spectrum was labelled:

Figure 11

18 data points were collected for the Birge-Sponer extrapolation, with peaks matching up

closely to the reference values in Table 1.

From figure 12 (and using a literature value for of 2.66 [10]) we extracted molecular

parameters:

Molecular parameter Value

127.51 1.25 1

0.935 0.162 1 0

4283.52 11.371 20835 22.11 0

13246 22.11

2.9227 0.127 8.995 0.0872 1.894 0.0721 1.143 0.1221

Figure 2

4.3. Errors and conclusion

The main source of error was in the measurement of , (0.04) on the screw-gauge, from which every other error in the experiment was calculated. The random error associated with

the noise was 0.01 volts, and was essentially negligible.

Figure 12

Our results are close to literature values, with the exception of . The spectra produced by the LED source are obviously superior to the white light spectra. This suggests that laser

spectroscopy would be a logical place to continue, building on the results of this experiment.

Word count: 1980.

5. References

[1]

https://www.teaching.physics.manchester.ac.uk/lab/scripts/year3/pdfs/Iodine_Absorption/201

3_Iodine_Absorption.pdf. Accessed on 17/03/15

[2] Lsfr26.m matlab script available at http://teachweb.ph.man.ac.uk/COURSES/lsq/lsfr26.m.

Accessed on 17/03/15

[3] http://www.shimadzu.com/products/opt/oh80jt0000001uz0.html. Accessed on 17/03/15 [4], [5] http://www.tau.ac.il/~phchlab/experiments_new/LIF/theory.html/ Accessed on 17/03/15 [6] ROSEN, B., editor, "Tables de Constantes et Donnks Numkriques, 4, Donnkes

Spectroscopiques," Hemann and Co., Paris V, 1951.

[7] Gaydon, A. G., Dissociation Energies, Chapman and Hall, London, 2nd Ed., Rev.,

1953.

[8] Morse, P.M., Phys. Rev., 34, 57 (1929)

[9], [10] I.J.McNaught, J.Chem.Edu., 1980, 57, 2, 101-105.

https://www.teaching.physics.manchester.ac.uk/lab/scripts/year3/pdfs/Iodine_Absorption/2013_Iodine_Absorption.pdfhttps://www.teaching.physics.manchester.ac.uk/lab/scripts/year3/pdfs/Iodine_Absorption/2013_Iodine_Absorption.pdfhttp://teachweb.ph.man.ac.uk/COURSES/lsq/lsfr26.mhttp://www.shimadzu.com/products/opt/oh80jt0000001uz0.htmlhttp://www.tau.ac.il/~phchlab/experiments_new/LIF/theory.html/