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:

$q_{T}={\frac {V}{\Lambda ^{3}}}=V{\frac {(2\pi mkT)^{\frac {3}{2}}}{h^{3}}}$, where $\Lambda =h{\sqrt {\frac {\beta }{2\pi m}}}$ and $\beta ={\frac {1}{k_{B}T}}$.

Here, $V$ is the volume of the container holding the molecule, $\Lambda$ 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 $\Lambda$ is much less than any dimension of the volume the atom or molecule is in. Since typical values of $\Lambda$ 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 $\rho \Lambda <<1$ where $\rho$ 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)