Partition Function of an Ideal Monoatomic Gas

The energy per molecule e and the entropy per molecule s in an ideal gas are calculated from the partition function. 

To compute the partition function of a freely moving atom, the summation over / in (4.34) has to be replaced by an integration over the variables defining the energy of a free atom. The translational energy of an atom (or center of mass motion of a molecule) is 

The movement of the atom is defined by its momenta and spatial coordinates. These define what is known as the phase space. The minimum volume per representative point in phase space equals as follows from Heisenberg s uncertainty principle dx dpx h. The normalized space density element becomes 

In the definition of the partition function (4.34), the index i sums all elements in space. In the present case it is replaced by an integral over momenta and coordinates of the atom 

This is the value of the translational partition function in three dimensions. V is the volume available per atom. Thus the translational partition function increases with increasing mass and temperature and decreasing pressure (because the volume per molecule enlarges). Once the partition function is known, the energy and entropy per atom of the ideal gas can be computed. Substitution of (4.40) into (4.36) gives 

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