Translational partition function

In statistical mechanics, the translational partition function, ${\displaystyle q_{T}}$ 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 ${\displaystyle q_{T}}$ can be approximated by:[1]

${\displaystyle q_{T}={\frac {V}{\Lambda ^{3}}}\,}$ where ${\displaystyle \Lambda ={\frac {h}{\sqrt {2\pi mk_{B}T}}}}$

Here, V is the volume of the container holding the molecule (volume per single molecule so, e.g., for 1 mole of gas the container volume should be divided by the Avogadro number), 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

${\displaystyle Q_{T}(T,N)={\frac {q_{T}(T)^{N}}{N!}}}$

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 this approximation breaks down significantly.