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 Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, liquid, gas ...

molecule A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bio ...

in a low pressure gas, neglecting the interactions of molecules, the

canonical ensemble In statistical mechanics, a canonical ensemble is the statistical ensemble that represents the possible states of a mechanical system in thermal equilibrium with a heat bath at a fixed temperature. The system can exchange energy with the hea ...

q_T

can be approximated by:Donald A. McQuarrie, ''Statistical Mechanics'', Harper \& Row, 1973 :

q_T = \frac\,

where

\Lambda = \frac

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 In physics, the thermal de Broglie wavelength (\lambda_, sometimes also denoted by \Lambda) is roughly the average de Broglie wavelength of particles in an ideal gas at the specified temperature. We can take the average interparticle spacing in t ...

, ''h'' is the

Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivalen ...

, ''m'' is the mass of a molecule, ''k''_B is the

Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas consta ...

and ''T'' is the

absolute temperature Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics. Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic ...

. 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) = \frac

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.

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Partition functions {{statisticalmechanics-stub

See also

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