10.6 Partition Function for a Gas
439
In terms of the thermal de Broglie wavelength,
z trans =
V
3
T
.
(10.45)
10.6.2 The Internal Parts
Because we have separated the energies of rotation, vibration and electron orbitals,
and nuclear states, we can focus on the partition function for each type of energy
carried by the molecules.
Consider the contributions to the partition function in a dilute gas of molecules. 20
The zero of energy is usually taken so that when the participating particles are far
separated, they have zero energy. ‘Far separated’ is an idealized concept, since even
separated particles in the laboratory usually encounter containment boundaries, in
which case they have a finite lowest translational energy. This is the Heisenberg
uncertainty principle at play. But if the boundaries are macroscopically separated
by a distance L, the lowest translational energy will be o = h 2 /(8mL 2 ). Even
the energy for a hydrogen atom in a cubic millimeter container will be about 2 ×
10 −19 eV, a negligible amount in comparison to other energies involved.
10.6.3 The Rotational Part
Because electrons are so light compared to nuclei, single isolated atoms do not show
atomic rotational excitations at ordinary temperatures. But thermal excitation of
rotational states of molecules (with two or more atoms) in a gas at room temperature
is likely, as the energy required is typically of the order of a thousandth of an electron
volt, while at room temperature, the average kinetic energies are about 0.025 eV.
If we neglect centrifugal distortion, 21 the energy of rotation will be that of a
‘rigid’ molecule, for which the energy in Hamiltonian form is
20 If the gas becomes too dense, or changes into a liquid or a solid, the intermolecular interactions
inhibit molecular rotations, and cause the vibrations of molecules to couple together into
vibrational states for clusters of molecules, or even coherent vibrations of a whole crystal of
molecules.
21 If we needed to included centrifugal distortion, then to first order there would be an additive
term in the energy proportional to L 4 , and, if there is vibration along with rotation, there would
be another ‘vibro-rotation’ energy term proportional to L 2 (n + 1/2), where n is the vibrational
excitation quantum number.
439
In terms of the thermal de Broglie wavelength,
z trans =
V
3
T
.
(10.45)
10.6.2 The Internal Parts
Because we have separated the energies of rotation, vibration and electron orbitals,
and nuclear states, we can focus on the partition function for each type of energy
carried by the molecules.
Consider the contributions to the partition function in a dilute gas of molecules. 20
The zero of energy is usually taken so that when the participating particles are far
separated, they have zero energy. ‘Far separated’ is an idealized concept, since even
separated particles in the laboratory usually encounter containment boundaries, in
which case they have a finite lowest translational energy. This is the Heisenberg
uncertainty principle at play. But if the boundaries are macroscopically separated
by a distance L, the lowest translational energy will be o = h 2 /(8mL 2 ). Even
the energy for a hydrogen atom in a cubic millimeter container will be about 2 ×
10 −19 eV, a negligible amount in comparison to other energies involved.
10.6.3 The Rotational Part
Because electrons are so light compared to nuclei, single isolated atoms do not show
atomic rotational excitations at ordinary temperatures. But thermal excitation of
rotational states of molecules (with two or more atoms) in a gas at room temperature
is likely, as the energy required is typically of the order of a thousandth of an electron
volt, while at room temperature, the average kinetic energies are about 0.025 eV.
If we neglect centrifugal distortion, 21 the energy of rotation will be that of a
‘rigid’ molecule, for which the energy in Hamiltonian form is
20 If the gas becomes too dense, or changes into a liquid or a solid, the intermolecular interactions
inhibit molecular rotations, and cause the vibrations of molecules to couple together into
vibrational states for clusters of molecules, or even coherent vibrations of a whole crystal of
molecules.
21 If we needed to included centrifugal distortion, then to first order there would be an additive
term in the energy proportional to L 4 , and, if there is vibration along with rotation, there would
be another ‘vibro-rotation’ energy term proportional to L 2 (n + 1/2), where n is the vibrational
excitation quantum number.