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. $$I=\mu {r_e}^2$$ $$q=\sum_{j}^{\infty} g_j\exp \left( \frac{-E_j}{k_B T} \right)$$

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:

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

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

$$q_{rot}=\sum_{j}^{\infty} (2J+1) \exp \left( \frac{-\hbar^2}{2 k_B T I }J(J+1)\right)$$ Since the spacings of the rotational energy levels is small, the sum can be approximated as an integral over J,

$$q_{rot}=\int_{0}^{\infty} (2J+1)\exp \left( \frac{-\hbar^2}{2 k_B T I}J(J+1)\right)\textrm{d}J$$

From a table of integrals:

$$\int (2x+1) \exp(-ax(x+1)) \textrm{d}x = \frac{\exp(-ax(x+1))}{a}$$ Letting x = J and $$ a = \frac {-\hbar^2}{2 k_B T I }$$ we get

$$q_{rot} = {-\frac{-\exp(-ax(x+1))}{a}} \bigg|_0^\infty$$
 * $$= 0 - \frac{1}{a} = -\frac{1}{a}$$
 * $$=\frac{2 k_B T I }{\hbar^2}$$

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.

$$q_{rot}=\frac{2 k_B T I}{\hbar^2\sigma}$$

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

$$\theta_{rot}=\frac{\hbar^2}{2k_B I}$$

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

$$q_{rot}=\frac{T}{\sigma\theta_{rot}}$$