# Rice PHYS 332 - Vibration Rotation Spectrum of CO (10 pages)

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## Vibration Rotation Spectrum of CO

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- Phys 332 - Junior Physics Lab Ii

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Rice University Physics 332 VIBRATION ROTATION SPECTRUM OF CO I INTRODUCTION 2 II THEORETICAL CONSIDERATIONS 3 III MEASUREMENTS 8 IV ANALYSIS 9 April 2011 I Introduction Optical spectroscopy is one of the most valuable tools available for the study of atoms and molecules At the simplest level knowledge of spectral characteristics allows us to detect the presence of particular substances in stars planets comets the upper atmosphere and even interstellar space With a more detailed analysis of various spectral features it is often possible to deduce the physical conditions in these regions The role of atomic spectra in understanding the electronic structure of atoms is well known to students of quantum mechanics Molecular spectra can similarly be used to understand the motions of electrons in molecules and also the vibration and rotation of the nuclei The study of the electronic properties has led to a theoretical understanding of chemical valence and bonding From the vibrational frequencies we can deduce the forces between atoms while the rotational frequencies provide accurate information about bond lengths and other geometric features of molecules Knowledge of the physical properties of individual molecules in turn allows us to better understand properties of molecular gases and elementary chemical reaction processes In the present exercise we will be concerned with the spectrum of carbon monoxide particularly in the infrared region We will argue that the observed groups of lines can be understood as transitions between vibration rotation levels of the molecule Careful measurement of the transition frequencies will let us deduce the interatomic spacing and some characteristics of the interatomic potential Section II presents a quantum mechanical calculation of the energy levels of a diatomic molecule which can both vibrate and rotate in space From the energy levels we can calculate the frequencies at which the molecule should absorb radiation for comparison with observations Section III discusses the mechanics of making the measurements and Section IV presents some suggestions for analysis of the data 2 II Theoretical Considerations A Energy level calculation The chemist s ball and stick model of a molecule suggests that we need to consider both vibration along the bond between the atoms and rotation about that bond Because we will be concerned only with low energy irradiation we will neglect the possibility of exciting any of the electrons to higher states To a first approximation then we expect the allowed energies to be the sum of a vibrational and a rotational part E Erot Evib 1 The rotational levels are given by Erot 2 J J 1 2I 2 where I is the moment of inertia about an axis through the center of mass and perpendicular to the bond and J is an integer quantum number In terms of microscopic quantities I re2 where is the reduced mass and re is the equilibrium interatomic distance The vibrational levels are also familiar Evib k 12 hc e 12 3 where k is the effective spring constant for the interatomic potential and is the vibrational quantum number For compactness and later use the second equality expresses the energy in terms of the wavenumber e a reciprocal wavelength which we will later define carefully The observed spectrum is determined by the selection rules which specify the transitions allowed between energy levels In order for a potential transition to absorb light the electric dipole operator must have a non zero matrix element between the two states For vibrational states this requires that change by 1 while for the rotational states J must also change by 1 Figure 1 shows the vibration rotation energy levels with some of the allowed transitions marked The figure also shows the resulting idealized spectrum labeled in a way that will become convenient later 3 3 J 2 1 0 n 1 m 4 3 2 1 1 2 3 4 Energy J J 1 3 J 2 1 0 J J 1 n Fig 1 Energy level diagram and resulting spectrum for ideal vibrator rotator We can now combine the energy expressions with chemical data to estimate the transition energies for CO For the vibrational transitions we need to obtain k and In our approximation the interatomic potential energy U k r re 2 2 and we assume that the molecule is in equilibrium at the bottom of the potential well Thermal measurements can be used to deduce the energy needed to dissociate the molecule into constituent atoms about 11 eV for CO As a very rough approximation we assume that the dissociation occurs when the bond is stretched to twice its equilibrium length of about 0 1 nm so that U is equal to the dissociation energy when r 2re This relationship is easy to solve for k and is known from the atomic masses so we readily find a vibrational level spacing of about 0 1 eV Radiation of this energy has a wavelength of about 10 m in the far infrared region of the spectrum The rotational energy levels can be immediately found from re and the atomic masses giving E 0 7 meV for the J 0 1 transition The fact that this is much smaller than the vibrational energy indicates that the rotational levels will show up as fine structure on the vibrational transitions B Improved energy level calculation The idealized situation considered so far leaves out some important factors The interatomic potential is not exactly quadratic so there should be deviations from the simple harmonic oscillator energies If we view the connection between the atoms as a spring it is obvious that the spring will stretch as the molecule spins increasing the moment of inertia and decreasing the rotational level spacing Additionally in an anharmonic oscillator the average separation between atoms depends on the vibrational level which in turn affects the moment of inertia and effectively couples the vibrational and rotational motions We need to consider all these effects to get an accurate representation of the spectrum 4 The deviations from a harmonic potential can be represented by expanding the interatomic potential U in a Taylor series about the equilibrium separation re 2 U f r re g r re 3 4 Here g and f are constants with g f assumed Applying perturbation theory to this potential yields energy eigenvalues Evib hc e 1 2 hc e xe 12 2 5 where xe is a small constant related to g and e refers to the vibration energy at the equilibrium separation The anharmonicity has two main effects The level spacing decreases as increases sometimes called a softening of the potential The selection rule is relaxed so that

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