View source for Translational partition function

{{Short description|Physical function in thermodynamics}}
{{one source |date=March 2024}}
{{Use American English|date = February 2019}}
In [[statistical mechanics]], the '''translational partition function''', <math>q_T</math> is that part of the [[partition function (statistical mechanics)|partition function]] resulting from the movement (translation) of the [[center of mass]]. For a single [[atom]] or [[molecule]] in a low pressure gas, [[Ideal gas | neglecting the interactions of molecules]], the [[canonical ensemble]] <math>q_T</math> can be approximated by:<ref>Donald A. McQuarrie, ''Statistical Mechanics'', Harper \& Row, 1973 </ref>

:<math>q_T = \frac{V}{\Lambda^3}\,</math> where <math> \Lambda = \frac{h}{\sqrt{2\pi m k_B T }}</math> 

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''<sub>B</sub> is the [[Boltzmann constant]] and ''T'' is the [[Thermodynamic temperature|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 [[Thermal de Broglie wavelength | 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<sub>T</sub> ≫ N '', or equivalently when ρ Λ ≪ 1 where ρ is the density of particles, the total translational partition function can be written

:<math>
Q_T(T,N) = \frac{ q_T(T)^N }{N!}
</math>

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

==See also==
* [[Rotational partition function]]
* [[Vibrational partition function]]
* [[Partition function (mathematics)]]

==References==
<references />

{{Statistical mechanics topics}}

[[Category:Partition functions]]


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