Misplaced Pages

This article relies largely or entirely on a single source. Relevant discussion may be found on the talk page. Please help improve this article by introducing citations to additional sources. Find sources: "Translational partition function" – news · newspapers · books · scholar · JSTOR (March 2024)

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:

${\displaystyle q_{T}={\frac {V}{\Lambda ^{3}}}\,}$ 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.

See also

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

References

1. Donald A. McQuarrie, Statistical Mechanics, Harper \& Row, 1973

v t e Statistical mechanics
Theory	Principle of maximum entropy ergodic theory
Statistical thermodynamics	Ensembles partition functions equations of state thermodynamic potential: U H F G Maxwell relations
Models	Ferromagnetism models Ising Potts Heisenberg percolation Particles with force field depletion force Lennard-Jones potential
Mathematical approaches	Boltzmann equation H-theorem Vlasov equation BBGKY hierarchy stochastic process mean-field theory and conformal field theory
Critical phenomena	Phase transition Critical exponents correlation length size scaling
Entropy	Boltzmann Shannon Tsallis Rényi von Neumann
Applications	Statistical field theory elementary particle superfluidity Condensed matter physics Complex system chaos information theory Boltzmann machine

This quantum mechanics-related article is a stub. You can help Misplaced Pages by expanding it.

• v
• t
• e

This article about statistical mechanics is a stub. You can help Misplaced Pages by expanding it.

• v
• t
• e

• Partition functions
• Quantum physics stubs
• Statistical mechanics stubs