# Ideal gas partition function

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

\[Q_{NVT}=\frac{1}{N!}\frac{1}{h^{3N}}\int\int d{\mathbf p}^N d{\mathbf r}^N \exp \left[ - \frac{H({\mathbf p}^N,{\mathbf r}^N)}{k_B T}\right]\]

where *h* is Planck's constant, *T* is the temperature and \(k_B\) is the Boltzmann constant. When the particles are distinguishable then the factor *N!* disappears. \(H(p^N, r^N)\) 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

\[H({\mathbf p}^N, {\mathbf r}^N)= \sum_{i=1}^N \frac{|{\mathbf p}_i |^2}{2m} + {\mathcal V}({\mathbf r}^N)\]

Thus we have

\[Q_{NVT}=\frac{1}{N!}\frac{1}{h^{3N}}\int d{\mathbf p}^N \exp \left[ - \frac{|{\mathbf p}_i |^2}{2mk_B T}\right] \int d{\mathbf r}^N \exp \left[ - \frac{{\mathcal V}({\mathbf r}^N)} {k_B T}\right]\]

This separation is only possible if \({\mathcal V}({\mathbf r}^N)\) is independent of velocity (as is generally the case). The momentum integral can be solved analytically:

\[\int d{\mathbf p}^N \exp \left[ - \frac{|{\mathbf p} |^2}{2mk_B T}\right]=(2 \pi m k_B T)^{3N/2}\]

Thus we have

\[Q_{NVT}=\frac{1}{N!} \frac{1}{h^{3N}} \left( 2 \pi m k_B T\right)^{3N/2} \int d{\mathbf r}^N \exp \left[ - \frac{{\mathcal V}({\mathbf r}^N)} {k_B T}\right]\]

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

\[Z_{NVT}= \int d{\mathbf r}^N \exp \left[ - \frac{{\mathcal V}({\mathbf r}^N)} {k_B T}\right]\]

In an ideal gas there are no interactions between particles so \({\mathcal V}({\mathbf r}^N)=0\). Thus \(\exp(-{\mathcal V}({\mathbf r}^N)/k_B T)=1\) 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 \(V^N\) where *V* is the volume. Thus we have

\[Q_{NVT}=\frac{V^N}{N!}\left( \frac{2 \pi m k_B T}{h^2}\right)^{3N/2}\]

If we define the de Broglie thermal wavelength as \(\Lambda\) where

\[\Lambda = \sqrt{h^2 / 2 \pi m k_B T}\]

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

\[Q_{NVT}=\frac{1}{N!} \left( \frac{V}{\Lambda^{3}}\right)^N = \frac{q^N}{N!}\] where \[q= \frac{V}{\Lambda^{3}}\] 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.*

\[Q_{NVT}=Q_{NVT}^{\rm ideal} ~Q_{NVT}^{\rm excess}\]

## References

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

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