12 Rotational and vibrational spectra Note: The masses of nuclides are listed in Table 0.2 of the Resource section. 12A General features of molecular spectroscopy Answers to discussion questions 12A.2 Doppler broadening. This contribution to the linewidth is due to the Doppler effect which shifts the frequency of the radiation emitted or absorbed when the molecules involved are moving towards or away from the detecting device. Molecules have a wide range of speeds in all directions in a gas and the detected spectral line is the absorption or emission profile arising from all the resulting Doppler shifts. The shape of a Doppler-broadened spectral line reflects the Maxwell distribution of speeds in the sample at the temperature of the experiment; hence the line broadens as the temperature is increased because the molecules acquire a wider and higher range of speeds. Doppler broadening can be significant in gas-phase samples but it can be reduced by decreasing the sample temperature. Lifetime broadening. The Doppler broadening is significant in gas-phase samples, but lifetime broadening occurs in all states of matter. This kind of broadening is a quantum mechanical effect related to the uncertainty principle in the form δE ≈ ħ/τ (eqn 12A.19) and is due to the finite lifetimes τ of the states involved in emission transitions. When τ is finite, the energy of the states is smeared out and hence the transition frequency is broadened. The rate of spontaneous emission cannot be changed; hence it is a natural limit to the breadth of a spectral line. Pressure broadening or collisional broadening. Collisional deactivation, which arises from collisions between molecules and from collision of molecules with the walls of the container, affects the rate of transition from an upper to a lower energy state. Lowering the pressure can reduce this rate. For a gas phase collisional lifetime of τ col , the mean time between collisions, the resulting collisional linewidth is δE col ~ col / τ  . Because τ col = 1/z for gases where z is the collision frequency, the kinetic model of gases implies that z is proportional to the pressure and that linewidths are proportional to the gas pressure. Thus, gas phase linewidths can be reduced by decreasing the pressure. The collisional frequency of liquid phase molecules is more difficult to define but, since pressure has little effect upon liquid density and kinetic energy, we expect pressure to have little effect upon the linewidth of liquid samples. Estimating that a liquid-phase molecule experiences a deactivating collision in the period of a vibration, the collisional linewidth is something like δE col ~ 13 21 col / ~ / (1.0 10 s) ~ 1.1 10 J τ − − × ×   or ~53 cm −1 /(τ/ps) [12A.19] as a wavenumber. Solutions to exercises 12A.1(b) The ratio of Einstein coefficients A/B is (i) ( ) ( ) ( ) 3 34 6 1 3 32 3 3 3 8 1 8π 6.626 10 Js 500 10 s 8π [12A.9] 7.73 10 J m s 2.998 10 m s A hv B c − − − − − × × × = = = × × (ii) ( ) ( ) 34 28 3 3 3 2 8π 6.626 10 J s 8π so 3.9 10 J m s λ λ 3.0 10 m c A h v B − − − − × = = = = × × 12:1