Statistical Thermodynamics and Rate Theories/Translational partition function

Derivation of Translational Partition FunctionEdit

A Molecular Energy State or is the sum of available translational, vibrational, rotational and electronic states available. The Translational Partition Function gives a "sum over" the available microstates.


The derivation begins with the fundamental partition function for a canonical ensemble that is classical and discrete which is defined as:


where j is the index,   and  is the total energy of the system in the microstate

For a particle in a 3D box with length  , mass   and quantum numbers   the energy levels are given by:


Substituting the energy level equation   for   in the partition function



Using rules of summations we can split the above formula into a product of three summation formulas


Defining the dimensions of the box (Particle In A Box Model) in each direction to be equivalent  


Because the spacings between translational energy levels are very small they can be treated as continuous and therefore approximate the sum over energy levels as an integral over n


Using the substitutions   and   The integral simplifies to


From The list of definite integrals the simplified integral has a known solution:




Re-substituting   and  



Since   is length and