Dissociation Energy of Iodine by Absorption SpectroscopyFINAL

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 −  for the excited state and −  for the ground state. The excited state equilibrium separation was calculated as . Morse potential curves were plotted for the ground and excited states.

4283.52±11.37

13246±22.1 2.9227 2.9227 ±0.1 ± 0.127Å 27Å

1. Introduction Introduction

When matter is exposed to some form of radiated energy, a spectrum may be produced. Spectroscopy is the study study of this phenomenon. Originally, the interaction between matter matter and electromagnetic waves in the optical region was studied, but many different forms of spectroscopy have since arisen. arisen. The radiated energy energy applied need need not be limited to the optical part of the electromagnetic electromagnetic spectrum, nor even to electromagnetic 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. 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]. [1]. A diagram of of our experimental 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 rel ationship between wavelength 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 i s not first order. The grating equation is:

 + =   (1) where  is the incident angle of the light, li ght,  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.

 Figure 6. Diagrammatica Diagrammaticall representation representation of equation equation (1) [3].

As the diffraction grating is rotated, a range of different dif ferent wavelengths will be able to  propagate through through the spectrometer. 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:  ≈  ∙ 10 ≈  ∙ 10 Within each electronic state, there are many vibrational energy levels and within each vibrational state there are many rotational energy levels, l evels, giving coarse vibrational and fine rotational structure in spectra.

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.



ℎ with  in units of  − gives ‘term-values’ with units −:  =  +  +   (3) where   is the total energy term,   is the vibrational energy term and  the rotational

Dividing (2) by

energy term [4]. Working in these units allows the energy of a transition to t o be expressed in terms of the wavenumber of the transition (the reciprocal of the photon wavelength). All  parameters corresponding corresponding to the excited 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):

′′

ṽ =      + ′′′ +   

 

′

(4)

ṽ

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

 =   

 

(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:





 =  +  ℏ 



 

ℏ

where is the reduced Planck’s constant and function for (6) simplifies to:

(6)  is the angular frequency. The partition

ℏ   = −ℏ  

(7)

Inserting (7) into (5) gives:

 = −ℏ  1  1  −ℏ  

 

(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  state, and

 = 0

the electronic transitions are between between the ground state (X) and the first excited state (B),  and . The vibrational energy term for a level  is:

′′ = 0  = 0 ′  =   +     +         ′′ ′ ṽ = ′ +    +     + .  

(9)

 = 0

where  is the frequency of infinitesimal amplitude vibrations between  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)):  

(10)

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



−) between The Birge-Sponer extrapolation is a plot of the energy level spacings (in subsequent vibrational levels against  , 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:

  + 1

∆ṽ =   2  + 1 

′

(11)



2 

   is the gradient of this plot. Integrating to obtain the thus    is the y-intercept and area of this plot is identical i dentical to summing all of the energy level spacings, giving  , the dissociation energy of the B state:

′

 = ∫ ∆ṽ  

(12)



where   is the convergence vibrational vibrational quantum number, or the x-intercept. convergence convergence limit, is:

 ∆ṽ   ∗ = ṽ + ∫

(13)

ṽ

where   is the wavenumber of the transition corresponding to measured.

 is:  = ∗   ∗ 

∗, the

, the highest ′

The dissociation energy of the ground state,

 

(14)

∗  is the energy difference between a ground state atom and an atom in the first where excited state with a value of 7589 − [7].



3.4. Rotation, moment of inertia, i nertia, equilibrium separation and the potential curves A diatomic molecule may rotate around an axis passing through the centre of and  perpendicular to the the bond joining them. them. , the moment of inertia, is:

  =  



(15)



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

   = (−−  1)  (16) where  is the separation and   is:  =    (17) where ℎ is Planck’s constant.   may be obtained [9] using a literature value for  : ) ( U         =   +  ln1 ln1++    (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 28 29 30

′

0 0 0 0

′′

543.47 541.18 539.89 536.87



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 Not all of the band band heads labelled labelled for the the Birge-Sponer Birge-Sponer extrapolation extrapolation are labelled 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  times. The spectrum was labelled:

√ 5

 Figure 11

18 data points were collected for the Birge-Sponer extrapolation, with peaks matching up closely to the reference values in Table 1.

 Figure 12

From figure 12 (and using a literature value for  parameters: Molecular parameter

     ∗      

4.3. Errors and conclusion

 of 2.66Å [10]) we extracted molecular Value

 Figure 2

127.51± 127. 51 ± 1.25 25 −− 0.935 935 ± 0.162 162  − 4283.52±11.37− 20835 ±22.1− 13246 ±22.1 2.9227±0.127Å  8.995±0.087− 1.894±0.072Å− 1.143±0.122Å

 ±0.04

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

±0.01



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.manc https://www.teaching .physics.manchester.ac.uk/lab/sc hester.ac.uk/lab/scripts/year3/pdfs/Iodine_ ripts/year3/pdfs/Iodine_Absorption/201 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 http://teachweb.ph.man.ac.uk/COURSES/lsq/lsfr26.m.. Accessed on 17/03/15 [3] http://www.shimadzu.c http://www.shimadzu.com/products/opt/oh80 om/products/opt/oh80jt0000001uz0.html jt0000001uz0.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," Spectroscopiques," Hemann and Co., Paris V, 1951. [7] Gaydon, A. G., “Dissociation Energies,” Chapman  and Hall, London, London, 2nd Ed., Rev., 1953. [8] Morse, P.M., Phys. Rev., 34, 57 (1929) [9], [10] I.J.McNaught, I.J.McNaught, J.Chem.Edu., 1980, 57, 2, 101-105.