Statistical Thermodynamics and Rate Theories/Translational partition function

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Derivation of Translational Partition Function[edit]

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:

Therefore,

Re-substituting and


Since is length and