Page 332 - Instant notes

Physical Chemistry 318

to the energy levels. Since rotational motion does not change the electric dipole of a
molecule, the physical selection rule for rotational energy transitions is:

the molecule must possess a permanent electric dipole (i.e. the molecule must be polar)

Symmetric molecules with a center of inversion, for example homonuclear diatomics
(H 2, O 2) and symmetric polyatomic molecules (CO 2, CH 4 and SF 6) do not have
permanent electric dipoles and therefore do not give rise to rotational spectra.
Heteronuclear diatomics (e.g. HCl) and polyatomic molecules with no center of
inversion (e.g. NH 3) do have rotational spectra.
The specific selection rules summarizing allowed transitions between rotational
energy levels are:
∆J=±1 ∆K=0

For transitions between energy levels with quantum numbers J and J+1 the energy
change is:
∆E=E J+1−EJ=B(J+1)(J+2)−BJ(J+1)=2B(J+1)

i.e. the energies of allowed rotational transitions are 2B, 4B, 6B,…. Therefore the
rotational spectra of a polar linear molecule and a polar symmetric top molecule consist
of a series of lines at frequencies separated by energy 2B (Fig. 1). Rotational
spectroscopy is of often called microwave spectroscopy because values of B are such
that the energies of rotational transitions correspond to the microwave region of the
−1
spectrum. For example, B=1.921 cm for carbon monoxide, so the 2B transition in the
CO rotational spectrum occurs at a wavelength of 2.6 mm.
Microwave spectroscopy is useful for determining bond lengths from the moments of
inertia derived from the separation of the lines in a rotational spectrum.

Rotational intensities

The intensity of a rotational transition is proportional to the population of the initial
rotational energy level of the transition (see Topic I2) which depends on the Boltzmann
partition law for that energy, , and, the degeneracy of the level. The angular
momentum of rotation is quantized, and each rotational level has a degeneracy of 2J+1
corresponding to the allowed orientations of the rotational angular momentum vector
with respect to an external axis. Therefore the total relative population of a rotational
level of a linear rigid rotor is

The degeneracy factor causes the intensity of rotational transitions to increase linearly
with J, whereas the Boltzmann factor causes the intensity to decrease exponentially with
J. The net effect is a rotational spectrum with an intensity distribution that passes through
a maximum (Fig. 2). The value of J max with the maximum population (and therefore