Expression of internal energy

In expression [5.30], the partition function Zc may depend on variables other than temperature, such as volume, for example the derivative here is therefore a partial derivative with regard to temperature, thus keeping the other variables constant, giving the expression of the internal energy  

We find that only the temperature and the canonical partition function must be known to calculate the internal energy of the system. 

In the case of a mixture with several constituents A, B, C, etc., we easily obtain  

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Let us take our set of variables Sq defined by the relation [1.4] and the expression of internal energy according to relation [1.5], We define the characteristic function Fp, with canonical variables in set by the following transformation, which is called a Legendre transformation ... 

With the expressions of internal energy and entropy, we can calculate all other thermodynamic functions defined by them. This is the same for the primary and secondary partial derivatives of these thermodynamic functions, i.e. the conjugate variables of the problem variables including pressure (or its opposite), entropy, the chemical potential and the thermodynamic coefficients which are the secondary derivatives of thermodynamic functions. Below are the results, easily obtained for any of the variables. 

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