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468 13 STATISTICAL THERMODYNAMICS
0 10 20 30 40
0.00
0.05
0.10
0.15
J
P(
J)
1000 K
100 K
Figure 13.14
distribution peaks before tailing o�: at the lower temperature the peak
comes at a lower value of J and the distribution is narrower. Even at
the lowest temperature, many states have signi�cant probabilities of being
occupied.
�e probability distribution over the vibrational states is given by
P(υ) = nυ
N
= 1
qV
e−hc ν̃υ/kT qV = (1 − e−hc ν̃/kT)−1
Figure 13.15 shows plots of P(υ) for 100 K and 1000 K. Because the vi-
brational temperature is so high (θV = 3122 K) at 100 K only the ground
state is occupied, and even at 1000 K there is only a small probability of
the �rst excited state being occupied.
0 1 2
0.0
0.2
0.4
0.6
0.8
1.0
υ
P(
υ)
1000 K
100 K
Figure 13.15
(b) �e rotational partition function of a heteronuclear diatomic is given by
[13B.11–542], qR = ∑J(2J+1)e−βhcB̃ J(J+1).�is expression can be written