Grand Canonical Ensemble

Here we consider the system of fixed volume, temperature and chemical potential. The temperature may be assumed to be constant by keeping it in touch with a infinite hear source. Similarly the chemical potential may be kept constant by an infinite reservoir containing chemical potential of a particular species. Thus the grand canonical partition function may be derived similarly as the canonical partition function and a relationship may be obtained as given below.

In the derivation of other thermodynamic properties, the average number of particles in the ensemble needs to be computed:

Similarly the average energy can be obtained via

 

In the terms of measurable quantities we can give the values of internal energy as:

Similarly the expression for entropy and pressure can be derived. The derivations are done in the same manner as for canonical partition function. The expressions for entropy and pressure are