Statistical Thermodynamics and Rate Theories/Rotational partition function of a linear molecule

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The rotational partition function, is a sum over state calculation of all rotational energy levels in a system, used to calculate the probability of a system occupying a particular energy level. The open form of the partition function is an infinite sum, as shown below. By making a few substitutions and replacing the sum with an integral, an algebraic expression for the rotational partition function can be derived.

The degeneracy, g, of a rotational energy level, j, is the number of different measurable states that have the same energy. For rotational energy levels, this is given by:

The rotational energy of a molecule is:

Substituting these values into the open form of the partition function, we get

Since the spacings of the rotational energy levels is small, the sum can be approximated as an integral over J,

From a table of integrals:

Letting x = J and we get

A symmetry factor is introduced to account for the nuclear spin states of homonuclear diatomic molecules. has a value of 2 for homonuclear diatomics and 1 for other linear molecules.

The rotational characteristic temperature is introduced to simplify the rotational partition function expression.

The physical meaning of the characteristic rotational temperature is an estimate of which thermal energy is comparable to energy level spacing. Substituting this into the partition function gives us