Ideal gas partition function

Canonical Ensemble Partition function, $Q$, for a system of $N$ identical particles each of mass $m$ \begin{equation} Q_{NVT}=\frac{1}{N!}\frac{1}{h^{3N}}\int\int d{\bf p}^N d{\bf r}^N \exp \left[ - \frac{{\sf h}({\bf p}^N, {\bf r}^N)}{k_B T}\right] \end{equation} When the particles are distinguishable then the factor $N!$ disappears. ${\sf h}({\bf p}^N, {\bf r}^N)$ is the Hamiltonian (Sir William Rowan Hamilton 1805-1865 Ireland) corresponding to the total energy of the system. ${\sf h}$ 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 \begin{equation} {\sf h}({\bf p}^N, {\bf r}^N)= \sum_{i=1}^N \frac{|{\bf p}_i |^2}{2m} + {\cal V}({\bf r}^N) \end{equation} Thus we have \begin{equation} Q_{NVT}=\frac{1}{N!}\frac{1}{h^{3N}}\int d{\bf p}^N \exp \left[ - \frac{|{\bf p}_i |^2}{2mk_B T}\right] \int d{\bf r}^N \exp \left[ - \frac{{\cal V}({\bf r}^N)} {k_B T}\right] \end{equation} This separation is only possible if ${\cal V}({\bf r}^N)$ is independent of velocity (as is generally the case).

The momentum integral can be solved analytically: \begin{equation} \int d{\bf p}^N \exp \left[ - \frac{|{\bf p} |^2}{2mk_B T}\right]=(2 \pi m k_b T)^{3N/2} \end{equation} Thus we have \begin{equation} Q_{NVT}=\frac{1}{N!} \frac{1}{h^{3N}} \left( 2 \pi m k_B T\right)^{3N/2} \int d{\bf r}^N \exp \left[ - \frac{{\cal V}({\bf r}^N)} {k_B T}\right] \end{equation}

The integral over positions is known as the {\it configuration integral}, $Z_{NVT}$ \begin{equation} Z_{NVT}= \int d{\bf r}^N \exp \left[ - \frac{{\cal V}({\bf r}^N)} {k_B T}\right] \end{equation} In an ideal gas there are no interactions between particles so ${\cal V}({\bf r}^N)=0$ Thus $\exp(-{\cal V}({\bf r}^N)/k_B T)=1$ for every gas particle. The integral of 1 over the coordinates of each atom is equal to the volume so for $N$ particles the {\it configuration integral} is given by $V^N$ where $V$ is the volume. Thus we have \begin{equation} Q_{NVT}=\frac{V^N}{N!}\left( \frac{2 \pi m k_B T}{h^2}\right)^{3N/2} \end{equation} If we define the {\it de Broglie thermal wavelength} as $\Lambda$ where \begin{equation} \Lambda = \sqrt{h^2 / 2 \pi m k_B T} \end{equation} we arrive at\\ \fbox{\parbox{\columnwidth}{ \begin{equation} \label{eqpartitionideal} Q_{NVT}=\frac{1}{N!} \left( \frac{V}{\Lambda^{3}}\right)^N \end{equation} }}

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. \begin{equation} Q_{NVT}=Q_{NVT}^{\rm ideal} Q_{NVT}^{\rm excess} \end{equation}