# Ideal gas partition function

The canonical ensemble partition function, *Q*,
for a system of *N* identical particles each of mass *m* is given by

where *h* is Planck's constant, *T* is the temperature and is the Boltzmann constant. When the particles are distinguishable then the factor *N!* disappears. is the Hamiltonian
corresponding to the total energy of the system.
*H* is a function of the *3N* positions and *3N* momenta of the particles in the system.
The Hamiltonian can be written as the sum of the kinetic and the potential energies of the system as follows

Thus we have

This separation is only possible if is independent of velocity (as is generally the case). The momentum integral can be solved analytically:

Thus we have

The integral over positions is known as the
configuration integral,
(from the German *Zustandssumme* meaning "sum over states")

In an ideal gas there are no interactions between particles so .
Thus for every gas particle.
The integral of 1 over the coordinates of each atom is equal to the volume so for *N* particles
the *configuration integral* is given by where *V* is the volume.
Thus we have

If we define the de Broglie thermal wavelength as where

one arrives at (Eq. 4-12 in ^{[1]})

where

is the single particle translational partition function.

Thus one can now write the partition function for a real system can be built up from
the contribution of the ideal system (the momenta) and a contribution due to
particle interactions, *i.e.*

## References[edit]

- ↑ Terrell L. Hill "An Introduction to Statistical Thermodynamics" (1960) ISBN 0486652424