278
6 Molecular Systems
To understand the origin of the small maximum in C V ,rot (T ), we may utilize an
alternative, but equivalent, form for C V ,rot (T ) given in Eq. (6.1.11a), namely,
C V ,rot (T )
Nk B
=
1
2
j,j
p j p j
jj
k B T
2
,
(6.2.62a)
in terms of the rotational angular momentum quantum numbers j, j . The rotational
fractional population expression is given by
p j (T ) = (2j + 1)e
−j (j+1)) rot /T /z rot ,
(6.2.62b)
with z rot (T ) given by Eq. (6.2.59). The quantity jj /k B is given in terms of the
characteristic temperature rot and the difference jj ≡ |j − j | between the
rotational quantum numbers as
jj /k B = (2j + 1 + j )) rot .
(6.2.62c)
According to Eqs. (6.2.62), the rotational heat capacity obeys a law of corresponding states, as C V ,rot (T ) is solely a function of the ratio T // rot , and hence all
diatomic ideal gases will behave in the same fashion. The behaviours of expression
(2.5.20a) and its components as functions of T // rot for T // rot ≤ 5 are illustrated
in Fig. 6.6.
The corresponding states behaviour of the normalized rotational heat capacity
C V ,rot /(N k B ) as a function of T // rot indeed shows a small maximum for T // rot
0.75 and decays to the classical high-temperature value unity by T // rot 2. For
HD, for example, the maximum should occur at about T ≈ 50 K, and C V ,rot (T )
should assume the classical value 1 by T = 130 K. For comparison, for N 2 the
maximum in C V rot /(N k B ) would occur for T 2 K, and C V ,rot /(N k B ) would
attain its classical value by T = 6 K. It is also clear from this figure that the
maximum in the temperature dependence of C V ,rot /(N k B ) is almost entirely due
to the change in the occupation of the first excited rotational state as the temperature
is increased.
Corrections to the Classical Expression
A better approximation to the exact value for z rot (T ) for a diatomic molecule may
be obtained from the Euler–Maclaurin summation formula obtained in Section 5
of Appendix B.1. The Euler–Maclaurin summation formula for a sum of the type
n
j =0
f (j), in which f (j) is a function defined on the integers and continuous for all
noninteger numbers, is
6 Molecular Systems
To understand the origin of the small maximum in C V ,rot (T ), we may utilize an
alternative, but equivalent, form for C V ,rot (T ) given in Eq. (6.1.11a), namely,
C V ,rot (T )
Nk B
=
1
2
j,j
p j p j
jj
k B T
2
,
(6.2.62a)
in terms of the rotational angular momentum quantum numbers j, j . The rotational
fractional population expression is given by
p j (T ) = (2j + 1)e
−j (j+1)) rot /T /z rot ,
(6.2.62b)
with z rot (T ) given by Eq. (6.2.59). The quantity jj /k B is given in terms of the
characteristic temperature rot and the difference jj ≡ |j − j | between the
rotational quantum numbers as
jj /k B = (2j + 1 + j )) rot .
(6.2.62c)
According to Eqs. (6.2.62), the rotational heat capacity obeys a law of corresponding states, as C V ,rot (T ) is solely a function of the ratio T // rot , and hence all
diatomic ideal gases will behave in the same fashion. The behaviours of expression
(2.5.20a) and its components as functions of T // rot for T // rot ≤ 5 are illustrated
in Fig. 6.6.
The corresponding states behaviour of the normalized rotational heat capacity
C V ,rot /(N k B ) as a function of T // rot indeed shows a small maximum for T // rot
0.75 and decays to the classical high-temperature value unity by T // rot 2. For
HD, for example, the maximum should occur at about T ≈ 50 K, and C V ,rot (T )
should assume the classical value 1 by T = 130 K. For comparison, for N 2 the
maximum in C V rot /(N k B ) would occur for T 2 K, and C V ,rot /(N k B ) would
attain its classical value by T = 6 K. It is also clear from this figure that the
maximum in the temperature dependence of C V ,rot /(N k B ) is almost entirely due
to the change in the occupation of the first excited rotational state as the temperature
is increased.
Corrections to the Classical Expression
A better approximation to the exact value for z rot (T ) for a diatomic molecule may
be obtained from the Euler–Maclaurin summation formula obtained in Section 5
of Appendix B.1. The Euler–Maclaurin summation formula for a sum of the type
n
j =0
f (j), in which f (j) is a function defined on the integers and continuous for all
noninteger numbers, is