Helmholtz energy and development of the virial

We know that the Helmholtz energy (relation [5.43]) is expressed using the following canonical partition function  

By combining this relation and relation [7.89], the Helmholtz energy of the perfect gas is written  

According to the definition of the Helmholtz energy function, we can easily deduce that the pressure is given by the opposite of the derivative of this Helmholtz energy with regard to volume  

The termz of equation [7.91], which represents the partition function of the non-interacting gas molecnle, contains a volume resulting from the contribution of three degrees of translation to the partition function we therefore write the Helmholtz energy in the form  

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