10.6 Partition Function for a Gas
443
= q rot,0 + 2
∞
k=1
e
±(a−c)k 2 q rot,k
(10.51)
where a ≡ βA, b ≡ βB, c = βC, and
q rot,k ≡
∞
j =k
(2j + 1)e
−bj (j +1)
= e
−bk(k+1)
∞
l=0
(2k + 2l + 1)e
−b l(2k+l+1) .
For k >> 1/
√
b, q rot,k diminishes rapidly with increasing k.
10.6.6 Spherical Molecules
Spherical molecules have equal moments of inertia around all three principle axes.
Examples include methane CH 4 and carbon tetrachloride CCl 4 . The allowed energy
values have the form
j = Aj (j + 1) .
(10.52)
However, now the quantum angular momentum operator acts on functions of
the three angles that determine the orientation of the molecule in space. These
functions 22 not only depend on the possible measured values of the square of
the total angular momentum, j (j + 1) ¯
h 2 , but also on two projections of angular
momentum, with possible values m ¯
h and m ¯
h, with both m’s ranging by integers
from −j to j . For this reason, the energy levels have degeneracy (2j + 1) 2 .
The rotational molecular partition function for a spherical molecule becomes
z rot =
1
σ
∞
j =0
(2j + 1)
2 exp (−bj (j + 1))) ,
(10.53)
where σ is the symmetry number, the number of orientations of the molecule that
have identical atomic positions.
22 The functions are often taken as the Wigner rotation functions D
∗j
m m (α, β, γ ), where α, β, γ are
the Euler angles that specify the orientation of a three-dimensional body in space.