Translational partition function

In statistical mechanics, the translational partition function, $q_T$ is that part of the partition function resulting from the movement (translation) of the center of mass. For a single atom or molecule in a low pressure gas, neglecting the interactions of molecules, the Canonical Ensemble $q_T$ can be approximated by:

$q_T = \frac{V}{\Lambda^3}\,$ where $\Lambda = \frac{h}{\sqrt{2\pi m k_B T }}$

Here, V is the volume of the container holding the molecule, Λ is the Thermal de Broglie wavelength, h is the Planck constant, m is the mass of a molecule, k_B is the Boltzmann constant and T is the absolute temperature. This approximation is valid as long as Λ is much less than any dimension of the volume the atom or molecule is in. Since typical values of Λ are on the order of 10-100 pm, this is almost always an excellent approximation.

When considering a set of N non-interacting but identical atoms or molecules, when Q_T ≫ N , or equivalently when ρ Λ ≪ 1 where ρ is the density of particles, the total translational partition function can be written

$Q_T(T,N) = \frac{ q_T(T)^N }{N!}$

The factor of N! arises from the restriction of allowed N particle states due to Quantum exchange symmetry . Most substances form liquids or solids at temperatures much higher than when when this approximation breaks down significantly.

See also

• Rotational partition function
• Vibrational partition function
• Partition function (mathematics)