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

Derivation

The rotational partition function, $Q_{rot}$ 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. $I=\mu {r_e}^2$ $q=\sum _j^{\mathrm{\infty }}{g}_j\exp \left(\frac{-E_j}{k_{B}T}\right)$

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:

$g=2J+1$

The rotational energy of a molecule is:

$E_J=\frac{{\hslash }^2}{2I}J(J+1)$

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

$q_{rot}=\sum _j^{\mathrm{\infty }}(2J+1)\exp (\frac{-{\hslash }^2}{2k_{B}TI}J(J+1))$

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

$q_{rot}=\underset{0}{\overset{\mathrm{\infty }}{\int }}(2J+1)\exp (\frac{-{\hslash }^2}{2k_{B}TI}J(J+1))\text{d}J$

From a table of integrals:

$\int (2x+1)\exp (-ax(x+1))\text{d}x=\frac{\exp (-ax(x+1))}{a}$

Letting x = J and $a=\frac{-{\hslash }^2}{2k_{B}TI}$ we get

$q_{rot}=-\frac{-\exp (-ax(x+1))}{a}{|}_0^{\mathrm{\infty }}$

$=0-\frac{1}{a}=-\frac{1}{a}$

$=\frac{2k_{B}TI}{{\hslash }^2}$

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

$q_{rot}=\frac{2k_{B}TI}{{\hslash }^2\sigma }$

The rotational characteristic temperature ${\theta }_{rot}$ is introduced to simplify the rotational partition function expression.

${\theta }_{rot}=\frac{{\hslash }^2}{2k_{B}I}$

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

$q_{rot}=\frac{T}{\sigma {\theta }_{rot}}$