Translational partition function

In statistical mechanics, the translational partition function, q T {\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 q T {\displaystyle q_{T}} can be approximated by: q T = V Λ 3 {\displaystyle q_{T}={\frac {V}{\Lambda ^{3}}}\,} where Λ = h 2 π m k B T {\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, kB 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 QT ≫ 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 ) = q T ( T ) N N ! {\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.

Physical function in thermodynamics

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

q T = V Λ 3 {\displaystyle q_{T}={\frac {V}{\Lambda ^{3}}}\,} where Λ = h 2 π m k B T {\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

Q T ( T , N ) = q T ( T ) N N ! {\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

[edit]

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

References

[edit]

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

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