Semi-Grand Canonical Ensemble

Here the mixture in the fixed container is made in such a manner that only species of a particular type are allowed to pass through semi-permeable material. However all the components are allowed to contact with an infinite source of temperature. However only a particular component (suppose component 1) is allowed to interact with a reservoir of infinite capacity in chemical potential of species 1.Thus the chemical potential of component 1 remains fixed. The partition function is given by:

The Helmholtz work function can be derived as below

Let us now start the derivation of the other thermodynamic functions From definition of partition function we have

Writing the partition function in term of total differential  i.e  we get