Lab Report

by My Name

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Vibrational Spectra of SO2 and CO2

Teaching Assistant: TA Name

Chem 375-003

Group #5


Performed: 3/6/97

Lab Partner: Name


Using vacuum and FTIR apparatus, vibrational spectra of gas phase SO2 and CO2 were obtained at moderate to low pressures and room temperatures. The spectra of SO2 were then used to calculate its heat capacity. In addition, the agreement of the valence force model is compared to the results.

Introduction & Theory

The infrared spectrum extends from 10 000 cm-1 down to 10 cm-1 (1 m to 1000 m). The range from 4000 to 400 cm-1 is of the most interest, as vibrational frequencies of most molecules lie within this region.

When a photon strikes a molecule, the photon may be absorbed if the energy of the photon (hv) corresponds to the energy difference from the molecules current state to a more excited state. This excited state may be manifested by moving electrons to higher shells or changes in vibration, rotation, or translational energy. This experiment investigates changes in vibrational energy, however the other effects may be seen at different wavelengths (energies).

Vibrational energy may be stored in a number of ways. A nonlinear molecule containing N atoms has 3N-6 degrees of freedom, so a molecule like SO2 has three different forms of vibration. These forms of vibration are called normal modes. Linear molecules, such as CO2, have one additional normal mode, the linear stretch.

Each vibrational mode has numerous rotation modes within it, giving rise to the fine structure found in vibrational spectra. The resolution of the apparatus used will determine whether the fine structure will be visible.

In CO2 molecules, the linear stretch is not infrared-active, so there will be no peak corresponding to it. In addition, the ranges of two peaks within the spectrum of CO2 overlap, so only one peak is seen. As a result, only two peaks are expected. On the other hand, all of the modes of SO2 are optically active and non-degenerate, so three peaks are expected. The bending frequency v2 is given by the lowest frequency band, the antisymmetric stretch frequency v3 is given by the highest frequency band, and the symmetric stretch frequency is given by the frequency of the band between the previous two.

At high pressure, much weaker bands may be observed due to overtones (2vi, 3vi, . . .) or combination bands (vi vj, 2vi vj, . . .).

Using SO2 as an example, the normal vibrational modes may be expressed using the atomic masses (mS and mO), the O-S-O bond angle (2), changes in the O-S-O bond angle (), changes in S-O distances (r1 and r2), and force constants ki. The potential energy of SO2 is then given by

From this, using classical mechanics for harmonic oscillators, one can obtain

By expressing v in wavenumber units (cm-1), mO=15.995 amu
, and mS=31.972 amu, one can replace 42 with 42c2/103N0=5.8918x10-5 to obtain k1 and k/l2 in units of N m-1. Sulfur dioxide has a bond angle 2 of 119.5, and l=0.1432 nm.

The heat capacity at constant volume for ideal gases is the sum of contributions from translational, rotational, and vibrational modes. For nonlinear polyatomic molecules, the molar heat capacity due to translation is equal to 3/2 R, and that due to rotation is 3/2 R as well. As the energy levels of a harmonic oscillator are represented by (v + )hv, the harmonic oscillator partition function for the ith normal mode qiHO is given by

The vibrational partition function qvib is well approximated by

This can then be used to calculate the vibrational contribution to the molar heat capacity at constant volume by

Where ui=hvi/kT=1.4388(wavenumberi)/T and the summation is made over all modes.


The procedure given in the text(1) was followed, with three changes. The spectrum of SO2 was not recorded at 900 Torr due to equipment limitations; additionally, two runs of CO2 at 300 and 9 Torr were included.


From the eqns. 2 and 4, the following relationships can be shown:

From the attached spectra, the average values of v1, v2, and v3 for SO2 are 1171.98, 538.67, and 1378.80 cm-1. Substituting into eqns. 8 and 9 yields:

Substitution into the left side of eq. 3 yields:

Substitution into the right side of eq. 3 yeilds:

The two sides of eq. 3 then differ by a factor of 2.28.

Rotational structure is evident in nearly all of the bands observed as multiply peaks centered around a point.

Using eq. 7, the molar heat capacity due to vibration at constant volume of SO2 can be calculated to be 6.2795 J mole-1 K-1 at 298 K and 12.8833 J mole-1 K-1 at 500 K. Translational and rotational considerations can be added to give a total molar heat capacity at constant volume of 31.2230 J mole-1 K-1 at 298 K and 37.8268 J mole-1 K-1 at 500 K. Comparison of these results to the given values1 of 30.5 J mole-1 K-1 and 37.7 J mole-1 K-1 yield errors of 2.4% and 0.3% respectively.


The assignment of the overtone and combination bands to specific groupings of fundamentals was hampered by a failure to record the wavenumbers at all of the peaks on the spectrum. As a result, some values were estimated from the graph, an obvious possible source of error. Nearly all of the peaks show an indication of rotational fine structure.

During the experiment, the NaBr windows were inspected and both were found to have fingerprints on them. This may explain the transmittance values that were obtained at greater than 100% for some spectra. It is possible that differences in the way the cell was installed between calibration and the actual readings changed the amount of light absorbed by the apparatus itself.

In addition, the apparatus lacked an adequate ventilation mechanism to expel the SO2 before the CO2 was used as SO2 could not be safely drawn through the vacuum pump. In hindsight, it would have been preferable to measure CO2 first so that the apparatus could be simply evacuated before admission of the SO2. As it was, the SO2 had to be passively purged into the hood and it is likely that some SO2 was drawn through the vacuum pump as it was evacuated before the CO2 runs.

It seems strange that the calculated values for molar heat capacity at constant volume agree quite closely while equation 3 fails by more than a factor of two. As the data is the same for both, and the calculations have been scrutinized, it must be assumed that the valence force is not very accurate for the conditions in this experiment.

The spectra obtained clearly show the expected 3 peaks for SO2 and two peaks for CO2. The overlap of peaks due to symmetric and antisymmetric stretches in CO2 is somewhat apparent from the asymmetry of the fine structure as the peaks interfered. It must, however, be noted that this effect is certainly not conclusive of overlapping.


1. D.P. Shoemaker, C.W. Garland, J.W. Nibler, Experiments in Physical Chemistry, 6th ed., exp. 35, The McGraw-Hill Companies (1996).