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.

Tue Feb 1 14:50:00 EST 2000