Translational partition function and quantum mechanics

Let us consider a gas molecule with mass m free to displace in a container with the volume A/v/available for the displacement of the molecule. LetXi, X2 andXs be the length, width and height of the container  

Quantum mechanics is used to calculate the energy levels of a molecule with mass /w in a recipient with lengthX], i.e.  

With the translational energies being very narrow in a typical laboratory container, we can replace the equation with an integral and write the contribution according to Ox] in the form  

By disregarding 1 with regard to n, through which we can change the variable u =n Pe, we obtain  

The complete translational partition function for the three directions will therefore be  

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