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/