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

## Derivation

The rotational partition function, ${\displaystyle Q_{rot}}$  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. ${\displaystyle I=\mu {r_{e}}^{2}}$  ${\displaystyle q=\sum _{j}^{\infty }g_{j}\exp \left({\frac {-E_{j}}{k_{B}T}}\right)}$

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:

${\displaystyle g=2J+1}$

The rotational energy of a molecule is:

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

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

${\displaystyle q_{rot}=\sum _{j}^{\infty }(2J+1)\exp \left({\frac {-\hbar ^{2}}{2k_{B}TI}}J(J+1)\right)}$

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

${\displaystyle q_{rot}=\int _{0}^{\infty }(2J+1)\exp \left({\frac {-\hbar ^{2}}{2k_{B}TI}}J(J+1)\right){\textrm {d}}J}$

From a table of integrals:

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

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

${\displaystyle q_{rot}={-{\frac {-\exp(-ax(x+1))}{a}}}{\bigg |}_{0}^{\infty }}$

${\displaystyle =0-{\frac {1}{a}}=-{\frac {1}{a}}}$
${\displaystyle ={\frac {2k_{B}TI}{\hbar ^{2}}}}$

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

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

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

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

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

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