Gas-phase carbonyl sulfide (OCS) absorbs radiation strongly near 2100 cm?¹, making it a convenient reference when spectroscopists need to calibrate their instruments. A large number of rovibrational transitions are usually visible, but the spectrum is relatively simple because the molecule is linear. To calibrate the spectrometer at much high wavenumbers, it may be necessary to look for transitions from states that have a combination of vibrational modes excited, but those states also tend to have much lower populations. From the temperature and the energy, we can estimate the population and predict whether it is realistic that the molecule can be observed by the spectrometer.
a. Find the number density of OCS molecules in a 100.0 Pa sample at 298 K that are in the vibrational state ($v_1$, $v_2$, $v_3$) = (1,0,1) and rotational state J = 10. The constants are CS-stretch $\omega_1$ = 859 cm?¹, bend $\omega_2$ = 527 cm?¹, and CO-stretch $\omega_3$ = 2080 cm?¹, B = 0.2028 cm?¹. The bending mode is a doubly degenerate vibrational motion, so it counts as two modes. Give the final value in molecules per cubic centimeter. (HINT: The rotational energy levels of a diatomic molecule are given by: $E_{rot}$ = BJ(J + 1); B = h/8$\pi$^2$Ic)
