Application to translational degrees of freedom

The application of the formulas is simplest if the molecules possess energy in only one form. Therefore we consider a system such as a monatomic gas, which has only translational energy. For the moment we ignore any contribution of the internal electronic energy of the atom to the properties of the system. The energy of translation e, is made up of the energies in each component of the motion, so we write 

because these energies are additive, the translational partition function q, factors into a product  

The energy levels for translation are the energy levels for a particle in a box (Section 21.3). If the width of the box in the x direction is a, then the permitted values of the kinetic energy from the Schrddinger equation are 

If n is very large, then In + k 2n = (4a/h)(2mey, where the last form is obtained by solving Eq. (29.36) for 2n (the subscript x on e has been dropped). The value of the spacing 

The number of levels in the range de is g and is obtained by dividing the range by the spacing between levels  

---