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An introduction to optical spectroscopy, focusing on the study of atoms and molecules through their spectral characteristics. It explains how knowledge of spectral features can be used to deduce physical conditions in various regions and understand electronic structure. The document also presents a quantum-mechanical calculation of energy levels for a diatomic molecule and discusses the mechanics of making measurements and analyzing data.
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VIBRATION-ROTATION SPECTRUM OF CO
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.
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 μ. In our approximation the interatomic potential energy U = k ( r - r (^) e ) 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 = 2 r (^) e. This relationship is easy to solve for k , and μ 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 μm, in the far infrared region of the spectrum. The rotational energy levels can be immediately found from r (^) e 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.
n
Energy
m = ... (^) -4 -3-2-1 1 2 3 4 ...
n +
Fig 1. Energy-level diagram and resulting spectrum for ideal vibrator-rotator.
The deviations from a harmonic potential can be represented by expanding the interatomic potential U in a Taylor series about the equilibrium separation r (^) e :
U = f r (! r (^) e )^2! g r (! r (^) e )^3 (4)
Here g and f are constants, with g « f assumed. Applying perturbation theory to this potential yields energy eigenvalues
E vib = hc! e ( " + 12 ) # 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 transitions with! " = ±2, ±3 ... are also allowed, although only weakly. Both of these results, and the form of the energy, persist for other potential functions that might plausibly describe the interatomic force. The increase in separation with increasing J can be modeled by assuming that the connection between atoms is a simple spring of constant k. The rotational energy levels then become
E rot =!
2 2 I
2 Ire^2 k
This is more usually written with two constants as
E rot = hcBe J ( J + 1)! hcDe J^2 ( J + 1) 2 (7)
where the subscript e indicates the equilibrium value. Since the vibrational motion is very fast compared to the rotation, the increase in average separation with increasing " can be accounted for by using (^) 1 / r in B and D , rather than 1/ r (^) e. This
introduces two more constants to relate the vibrational-average quantities to the equilibrium values:
B! = Be "# e (! + 12 ) and D! = De + $ e (! + 12 ) (8)
The final expression for the rotational energy is then
% D $ (^) (^) # m^2 ( m + 1) 2 + D 0 m^2 ( m % 1)^2
If, as is often true, the! e correction term is negligible then D " ' = D 0 = D (^) e and the expression
simplifies somewhat. Rewriting the simplified form as a power series in m yields the equation needed to analyze the spectra
Note that we can obtain values for Be , # e and D (^) e from the several rotational transitions within a given " '. With two or more " ' we can also obtain $ e and $ e xe.
The goal is to measure the wavelengths of as many of the vibration-rotation lines of CO as are accessible to our instrument. We will do this by measuring the attenuation of light from a calibrated monochromator as it passes through a gaseous sample. The wavelengths of maximum absorption can then be statistically fit to determine the various constants in Eq. 14 and hence the properties of the molecule. A sample of gaseous CO at about 30 psig (4 atmospheres) is contained in a plastic cell with glass windows. The instructor will assist you in refilling the cell if it appears that the contents have leaked out. Note that carbon monoxide is a poisonous, flammable gas. If you open the valve on the sample cell or otherwise release the gas you will, therefore, create a modest chemical hazard as well as losing your sample material. The Cary Model 17 spectrophotometer is an integrated instrument which contains a light source, a monochromator, sample cell holder, detectors and chart recorder. It is a very flexible and somewhat fragile research device, so it is imperative that you read the manufacturer's instructions carefully and then consult with the instructor before attempting to operate it. We will be using the instrument in the infrared (IR) region, so in your reading you should concentrate on the instructions for operating the VIS-IR light source and the IR detector. Purging is not required in the wavelength range of interest here. Our instrument has the "Universal Absorbance - %T" slidewire installed, and it will usually be used in absorbance mode. Previous workers have found absorption patterns resembling Fig. 1 at wavelengths around 4670 nm and, with rapidly decreasing intensity, 2350 nm and 1575 nm. On the basis of our previous estimates, these are presumably the! = 0! 1, 0! 2 and 0! 3 transitions, respectively. Only the latter two transitions are within the range of our spectrophotometer, and the 0! 3 group is already too weak to observe, so you should concentrate on measuring as many lines as possible for the 0! 2 transition. Be sure also to repeat your measurement so that you can estimate errors.
Table I Vacuum wavenumbers of the 0! 1 vibration-rotation transitions for CO. From J. Molec. Spect. 4,518 (1960). A computer-readable file containing these data can be found at
