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Using occupation number representation for identical particles, the grand canonical partition function is expressed in terms of single particle partition function. This is then used to discuss cases of identical fermions and identical bosons. Mean occupation number for fermions and bosons is obtained.

In this article various thermodynamic functions are expressed in terms of the grand canonical partition function.
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We derive the Boltzmann relation for entropy using the canonical partition function. This derivation uses the fact that the number of micro states \(\Omega(E)\) as a function energy has a sharp peak around the mean energy.

The ideal gas equation \(pV=NkT\) is good approximation for low densities. In this section a scheme of obtaining corrections to the ideal gas equation is discussed.

For a system in equilibrium with a heat bath at temperature \(T\) the energy is not constant. We start with the Boltzmann relation \(S=k\log \Omega\). It will be shown that if of a microstate has energy \(E_r\), its probability \(p_r\) is proportional to \(e^{-\beta E_r}\) and is given by
\[p_r = \frac{\exp(-\beta E_r)}{Z} \]
where \(Z\) is function of \(T,V,N\), called the canonical partition function and is given by
\[Z = \sum_{\text{MS }r} \exp(-\beta E_r).\]
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