Recall the canonical partition function expression for the ideal gas:
Define the thermal wavelength 
as
which has a quantum mechanical meaning as the width of the free particle distribution function. Here it serves as a useful parameter, since the canonical partition can be expressed as
The grand canonical partition function follows directly from Q(N,V,T):
Thus, the free energy is
In order to obtain the equation of state, we first compute
the average particle number 
Thus, eliminating 
in favor of in the
equation of state gives
as expected. Similarly, the average energy is given by
where the fugacity has been eliminated in favor of the average particle number. Finally, the entropy
which is the Sackur-Tetrode equation derived in the context of the canonical and microcanonical ensembles.