6.2 Diatomic Molecules
271
of 0.19R. The anharmonic contribution to the thermodynamic properties of CO is
thus quite small even for temperatures of the order of 1000 K.
In hindsight, however, this should not be unexpected, as a temperature of
1000 K is still slightly less than one-third the value of the characteristic vibrational
temperature, so that many excited vibrational states are still thermally inaccessible
to CO molecules even at that temperature.
Example 6.2 Iodine Bromide, IBr.
The spectroscopic constants for IBr are given by Huber and Herzberg [3]
as ω e (IBr) = 258.640 cm −1 and ω e x e (IBr) = 0.8140 cm −1 , so that x e (IBr) =
0.00303 and vib (IBr) = 381.7 K. Here, too, we shall examine the contributions to
U vib (T )−N 0 0 and S vib (T ) arising from the leading anharmonicity correction to the
vibrational energy of a diatomic molecule for the two representative temperatures
300 K and 1000 K.
For IBr at temperature 300 K, the anharmonicity contribution to the molar
vibrational internal energy U vib (T ) − N 0 0 is obtained from Eq. (6.2.40) as 0.89R,
which is 0.60% of the value 148.57R contributed by the SHO model expression.
For temperature T = 1000 K, the anharmonic contribution of 0.15R is 1.83% of
the SHO model contribution of 821.27R. The SHO model contribution to the molar
entropy of IBr for T = 300 K is found to be 0.824R, while the anharmonicity
contribution is obtained as 0.004R, and represents 0.5% of the contribution at the
level of the SHO model. The corresponding values for the entropy contributions
for T = 1000 K are 1.969R from the SHO model expression and 0.026R from the
anharmonicty correction, which is 1.3% of the SHO model contribution.
As for the CO example previously considered, the final outcome for IBr should
not be too surprising, as T = 300 K is already very close to the characteristic
vibrational temperature for IBr, and T = 1000 K is nearly three times as large, so
that many IBr vibrational levels will be thermally accessible at both temperatures.
6.2.3 The Diatomic Rotational Partition Function
The Rigid-Rotor Approximation
The concept of angular momentum is quite important in dealing with electronic
motion in atoms and with both electronic and nuclear motions in molecules. Its
description is simplest for a particle of mass m moving in circular motion about
a fixed point. With a small additional step we may also consider the case of the
rotational motion of two atoms in a diatomic molecule about their common centreof-mass. To see how this comes about, let us consider a rigid rotor (RR) consisting
of two atoms, with masses m 1 and m 2 separated by a fixed distance r = r 1 + r 2 and
rotating about their centre-of-mass, as shown in Fig. 6.4.
271
of 0.19R. The anharmonic contribution to the thermodynamic properties of CO is
thus quite small even for temperatures of the order of 1000 K.
In hindsight, however, this should not be unexpected, as a temperature of
1000 K is still slightly less than one-third the value of the characteristic vibrational
temperature, so that many excited vibrational states are still thermally inaccessible
to CO molecules even at that temperature.
Example 6.2 Iodine Bromide, IBr.
The spectroscopic constants for IBr are given by Huber and Herzberg [3]
as ω e (IBr) = 258.640 cm −1 and ω e x e (IBr) = 0.8140 cm −1 , so that x e (IBr) =
0.00303 and vib (IBr) = 381.7 K. Here, too, we shall examine the contributions to
U vib (T )−N 0 0 and S vib (T ) arising from the leading anharmonicity correction to the
vibrational energy of a diatomic molecule for the two representative temperatures
300 K and 1000 K.
For IBr at temperature 300 K, the anharmonicity contribution to the molar
vibrational internal energy U vib (T ) − N 0 0 is obtained from Eq. (6.2.40) as 0.89R,
which is 0.60% of the value 148.57R contributed by the SHO model expression.
For temperature T = 1000 K, the anharmonic contribution of 0.15R is 1.83% of
the SHO model contribution of 821.27R. The SHO model contribution to the molar
entropy of IBr for T = 300 K is found to be 0.824R, while the anharmonicity
contribution is obtained as 0.004R, and represents 0.5% of the contribution at the
level of the SHO model. The corresponding values for the entropy contributions
for T = 1000 K are 1.969R from the SHO model expression and 0.026R from the
anharmonicty correction, which is 1.3% of the SHO model contribution.
As for the CO example previously considered, the final outcome for IBr should
not be too surprising, as T = 300 K is already very close to the characteristic
vibrational temperature for IBr, and T = 1000 K is nearly three times as large, so
that many IBr vibrational levels will be thermally accessible at both temperatures.
6.2.3 The Diatomic Rotational Partition Function
The Rigid-Rotor Approximation
The concept of angular momentum is quite important in dealing with electronic
motion in atoms and with both electronic and nuclear motions in molecules. Its
description is simplest for a particle of mass m moving in circular motion about
a fixed point. With a small additional step we may also consider the case of the
rotational motion of two atoms in a diatomic molecule about their common centreof-mass. To see how this comes about, let us consider a rigid rotor (RR) consisting
of two atoms, with masses m 1 and m 2 separated by a fixed distance r = r 1 + r 2 and
rotating about their centre-of-mass, as shown in Fig. 6.4.