The canonical partition function for the ideal gas is much easier to evaluate than the microcanonical partition function. Recall the expression for the canonical partition function Q(N,V,T):
Note that this can be expressed as
since the Hamiltonian is completely separable. Evaluating the Gaussian integral gives us the final result immediately:
The expressions for the energy
gives rise to the results E=3NkT/2 and PV=NkT just as for the microcanonical ensemble. Note also that the entropy S(N,V,T) given by
which reduces to the microcanonical expression exactly if we use the fact that :
Thus, the canonical and microcanonical ensembles gives rise to exactly the same thermodynamics!! Let us now look more carefully at the expression for the entropy.