10.6 Partition Function for a Gas
441
10.6.4 Linear Molecule
If the molecule is linear, it has no effective moment of inertia around its axial
symmetry axis. (The electrons are too light.) Examples include oxygen O=O,
nitrogen N≡N, carbon monoxide C≡O, carbon dioxide O=C=O, nitrogen oxide
N=O, hydrogen cyanide HC≡N, and acetylene HC≡CH.
Moments of inertia for an axial symmetric molecule about the two perpendicular
axes are equal. The energies become
rot
l,m = Bl(l + 1)
(10.47)
for l = 0, 1, 2, · · · and m an integer satisfying −l ≤ m ≤ l. The constant B =
h 2 /(8π 2 I B ). As there are 2l + 1 values of m all giving the same energy, the energy
degeneracy is 2l + 1. If the partition function sum is carried over energy levels
(instead of quantum states), then
z rot =
1
σ
l
(2l + 1) exp (−bl(l + 1))) ,
(10.48)
in which we have used b ≡ βB. The constant σ is called the ‘symmetry constant’,
described below.
If the molecule is reflection symmetric, then there are symmetry restrictions
imposed on the angular sums. A molecule after a rotation of 180 ◦ that carries the
atoms into their counterparts will change the overall wave function for the molecule
by a factor (−1) J , where J is the overall spin of the molecule. For example, oxygen16 nuclei have spin zero, while the electronic wave function for the molecule in its
ground state has spin one, and so is odd under the rotation. Since oxygen-16 atoms
are bosons, oxygen molecules are bosons, and the total wave function must be even
under the rotation that exchanges the atomic position. This restricts l to be odd.
For temperatures significantly higher than θ rot,B = ¯
h 2 /(2k B I B ) (as shown
in Table 10.3) we can account (approximately) for the symmetry restriction by
including a symmetry number σ in the partition function, as shown in Eq. (10.48).
For methane, CH 4 , there are four symmetry axes, and three rotations about each
which returns the molecule into an indistinguishable state, so σ CH 4 = 12. Room
and body temperature is much bigger than θ rot,B . The approximation using σ in the
rotational partition function sum gets better with higher temperatures.
10.6.5 Oblate and Prolate Molecule
Oblate molecules have I A = I B < I C . A pancake is an oblate object. Examples
include ammonia NH 3 , cyclobutadiene C 4 H 4 and benzene C 6 H 6 .
441
10.6.4 Linear Molecule
If the molecule is linear, it has no effective moment of inertia around its axial
symmetry axis. (The electrons are too light.) Examples include oxygen O=O,
nitrogen N≡N, carbon monoxide C≡O, carbon dioxide O=C=O, nitrogen oxide
N=O, hydrogen cyanide HC≡N, and acetylene HC≡CH.
Moments of inertia for an axial symmetric molecule about the two perpendicular
axes are equal. The energies become
rot
l,m = Bl(l + 1)
(10.47)
for l = 0, 1, 2, · · · and m an integer satisfying −l ≤ m ≤ l. The constant B =
h 2 /(8π 2 I B ). As there are 2l + 1 values of m all giving the same energy, the energy
degeneracy is 2l + 1. If the partition function sum is carried over energy levels
(instead of quantum states), then
z rot =
1
σ
l
(2l + 1) exp (−bl(l + 1))) ,
(10.48)
in which we have used b ≡ βB. The constant σ is called the ‘symmetry constant’,
described below.
If the molecule is reflection symmetric, then there are symmetry restrictions
imposed on the angular sums. A molecule after a rotation of 180 ◦ that carries the
atoms into their counterparts will change the overall wave function for the molecule
by a factor (−1) J , where J is the overall spin of the molecule. For example, oxygen16 nuclei have spin zero, while the electronic wave function for the molecule in its
ground state has spin one, and so is odd under the rotation. Since oxygen-16 atoms
are bosons, oxygen molecules are bosons, and the total wave function must be even
under the rotation that exchanges the atomic position. This restricts l to be odd.
For temperatures significantly higher than θ rot,B = ¯
h 2 /(2k B I B ) (as shown
in Table 10.3) we can account (approximately) for the symmetry restriction by
including a symmetry number σ in the partition function, as shown in Eq. (10.48).
For methane, CH 4 , there are four symmetry axes, and three rotations about each
which returns the molecule into an indistinguishable state, so σ CH 4 = 12. Room
and body temperature is much bigger than θ rot,B . The approximation using σ in the
rotational partition function sum gets better with higher temperatures.
10.6.5 Oblate and Prolate Molecule
Oblate molecules have I A = I B < I C . A pancake is an oblate object. Examples
include ammonia NH 3 , cyclobutadiene C 4 H 4 and benzene C 6 H 6 .