Entropy and canonical partition functions

We are going to express the average entropy of a system using the partition function of the canonical ensemble. 

Determining the internal energy using relation [5.31], we obtain a second form in which to express the entropy  

just like the internal energy, entropy is expressed as a function of the only canonical partition function of the system. 

In the case of a mixture with several constituents, we obtain  

Having obtained the internal energy and the entropy, we can conclude that the canonical partition function of the system can completely define the system in the thermodynamic plane. We can therefore ejqtress any thermodynamic function using the canonical partition function, expressed as variables amount of matter N), volume (V) and temperature (7), i.e. the canonical variables associated with the Helmholtz function 7.  

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