276
6 Molecular Systems
partition function in terms of the sum over a set of appropriate energy states, we
obtain
z rot (T ) =
states
e
−ββ states =
j,m
e
−ββ jm =
levels
ω levels e
−ββ levels
=
∞
j =0
(2j + 1)e
−ββ j ,
(6.2.58)
for the partition function associated with rotational states of a heteronuclear
diatomic molecule.
We shall prefer to express z rot in the form
z rot (T ) =
∞
j =0
(2j + 1)e
−j (j+1)) rot /T ,
(6.2.59)
with rot , called ‘the characteristic rotational temperature’, defined as
rot ≡
¯
h 2
2I k B
.
(6.2.60)
At temperatures T such that T rot , the first few terms of the summation will
suffice, and we may approximate z rot by
z rot (T ) 1 + 3e
−2 rot /T
+ 5e
−6 rot /T
+ · · · .
For high temperatures (i.e., for T rot ), the quantum mechanical expression
(6.2.59) can be approximated by its classical limit when typical rotational energy
level separations are small in comparison with k B T , so that many rotational levels
are thermally accessible. The classical limit is obtained by replacing the quantum
mechanical sum over the rotational quantum number j by an integral over the
classical rotational angular momentum j , namely,
z
cl
rot (T )
∞
0
(2j + 1)e
−j (j+1)) rot /T dj
=
∞
0
e
−j (j+1)) rot /T d[j (j + 1)] .
From this last expression we can evaluate z cl
rot (T ) readily as
z
cl
rot (T )
e −j (j+1)) rot /T
− rot /T
∞
0
=
T
rot
,
(6.2.61a)