Page 35 - Radiochemistry and nuclear chemistry

24 Radiochemistry and Nuclear Chemistry

(b) Rotational energy. Taking the simplest case, a linear diatomic molecule with atomic
masses m 1 and m 2 at a distance r apart, the rotational energy is given by

Ero t = Iro t 60 2 (2.28)

where/rot is the rotational moment of inertia and w the angular velocity of rotation (radians
s-l)" [rot is calculated from

fro t = mre a r 2 (2.29)

where the reduced mass mred=(ml - l + m2-l)-t. Equation (2.28), derived from classical
mechanics, has to be modified to take into account that only certain energy states are
permitted

Ero t = h2nr(nr + 1)/(87r2/rot) (2.30)

where n r is the rotational quantum number. For transformation of energy in eV to the wave
number ~ or wavelength ~, of the corresponding photon energy, the relation

AE (eV) = 1.23980 x 10 -4 ~ (cm -l) (2.31)

is used, where ~ = 1/~,. (For blue light of about 480 nm the following relations are
obtained: 480 nm = 4.8 • 10 -7 m = 0.48 #m = 20 833 era- 1 = 2.58 eV.) The rotational
energies are normally in the range 0.001 - 0.1 eV, i.e. the wavelength region 10 -3 - 10 -5
m. The partition function for the rotational energy is obtained by introducing (2.30) into
(2.19). More complicated expressions are obtained for polyatomic and nonlinear molecules.
(c) Vibrational energy. For a diatomic molecule the vibrational energy is given by

Evi b- hewv(nv+ ~h) (2.32)

where

Wv= V2(k' Imred) ~h /(vrc) (2.33)

is the zero point vibrational frequency (the molecule is still vibrating at absolute zero, when
no other movements occur) and n v the vibrational quantum number, k' is the force constant
for the particular molecule.

2.6.4. The isotopic ratio

It is seen from this digression that the mass of the molecular atoms enters into the
partition functions for all three modes of molecular movement. The largest energy changes
are associated with the vibrational mode, and the isotopic differences are also most
pronounced here. Neglecting quantization of translational and rotational movements, one
can show that