A. Gordon Briggs'
Loughborough University Leicestershire, England
Vibrational Frequencies of Sulfur Dioxide Determination and application
Quite apart from their own considerable intrinsic interest, experiments in spectroscopy, if properly selected and designed, have the great merit of bringing together different areas of chemistry in a quite natural manner, although of course they are not unique in this respect. In the experiment described below, the recording of part of the unresolved vibration-rotation spectrum of gaseous sulfur dioxide (1,2,17, 19) enables the student to obtain the three fundamental frequencies of the molecule, heat capacity data, bond angle, and stretching and bending force constants. The stretching force constant can be compared with those of the sulfur trioxide molecule (5) and the sulfur monoxide radical (4), calculated from data in the references cited or from supplied abstracts of them. Most students are impressed by the wealth of results derivable from the relatively small amount of practical work which is involved in this experiment. They should be made aware, though, that the reasonable concordance of the spectroscopic and calorimetric heat capacities, of the vibration-rotation and electron diffraction bond angle values, and the similarity of the calculated SO, SO2, and SO8 force constants is a convincing verification of the general soundness of the different elaborate theories, and especially of any assumptions necessary to the development of these theories.
tively, either the equations given below may be used to calculate the bond angle, or, for a more advanced student, it may be found from the calorimetric value for the standard entropy, determined by Giauque (15), in conjunction with the statistical entropy equation (6). Recent work (21) indicates a potential curve with a double minimum but this does not affect the room tenip(mturc nic~:~slm.rnrut+. If \.ibrltior~:~l ml~nrl~~onirity is urglected, a n :l>sumption which introduces no great errors where unexcited vibrational levels are involved, then it can be shown, following Herzberg (2) that
where mo or ms is the mass in grams of a single atom of oxygen or sulfur, respectively; 2a is the angle between the S-0 bonds; kl is the stretching motion force constant and k6/lz is the bending motion force constant (1 is the equilibrium S-0 distance and 6 the change in bond angle during a vibration). The above equations lead to the further result that wL
+ 2mo/ms)(n2 + + v s 2 ) v 8 w / v l W + 2(1 + mo/ms)(l + 2mo/ms)(v~'/vtzvrz) V?
0 (4) in which w = 1 (2mo/ms) sin2u. I t should be noted that the vlt in eqns. (1)-(4) represent absolute frequencies (in Hz) which are obtained from the v, (in cm-I) of the infrared spectral measurements by multiplying the latter by the velocity of light, c = 3 X 10'0 cm sec-'.
Force Constants and Bond Angle
Sulfur dioxide is a symmetrical, non-linear molecule and therefore has 3N - 6, i.e., threefundamentalvibrational modes (I,,??, 6). These are symmetrical stretching (",), bending (v?), and antisymmetrical stretching (v8), and give rise to fairly strong infrared absorption hands. Certain overtone ( 2 4 and combination (vi vj) bands can also be observed. Bjerrum's valence force model (2) can be applied to the sulfur dioxide molecule and assumes that (a) along the direction of each bond a restoring force acts during a vibration and opposes displacement of the atoms from their equilibrium internuclear separation and (b) that a restoring force opposes any change in the angle between the two S - 0 bonds during a bending vibration. The use of Bjerrum's model means that t,here are more normal vibrational modes (three) than there are force constants (two) so that the latter are over-determined. This permits a check on the validity of the model using a value for the bond angle from an independent source such as an electron diffraction study (14). Alterna-
where for sulfur dioxide Qvib = [l - exp(-hv,/kT)I-'[I
Present Address: Department of Chemistry, University Toronto, Toronto 181, Ontario (until Aug. 1, 1'JiO).
[l - exp(-hv&T)l-' (6) From the well-known relationship of thermodynamics, C . = dE/dT, and eqn. (5) we have
Vibrational Contribution to Heat Capacity
The vibrational component of heat capacity, C,,,isi, varies with temperature and is calculable from the vibrational partition function, QYib, if the frequencies (vI,v2,v3)of the normal modes of vibration have been measured (5). From the standard equations of statistical mechanics (6), assuming that we have harmonic oscillations, we find that the vibrational contribution to the internal energy is given by
Volume 47, Number 5,
Muy 1970 / 391
xhich can be xritt,en for present purposes in the form
in vhich ui = livJkT and the summation is carried out over the three normal modes of vibration. Apparatus
Although more sophisticated instruments can be employed, a simple double-beam infrared recording spectrophotometer fitted xith a diffraction grating or sodium chloride prism and covering the range 650 to 4000 or 5000 cm-I is quite adequate for the present purpose. Such an instrument does not record the vp bending fundamental near 518 em-' and the student is expected to deduce a value for v2. A 10-cm pathlength gas cell fitted ~vithsodium chloride windom d l give spectral peaks of suitable intensity. It can be filled with sulfur dioxide to any required pressure by means of a simple lowvacuum line in a fume hood. A sulfur dioxide cylinder and the gas cell are connected most conveniently to the vacuum line with thicl