 Rotational temperature

For a linear rigid rotor, rotational temperature expresses the rotational energy constant in kelvins: $E_J=k_B{\theta }_{r}J(J+1)$, where $I$ is the perpendicular moment of inertia. With all integer $J\ge 0$ allowed and degeneracies $2J+1$, its rotational canonical partition function is $Z={\sum }_J(2J+1)e^{-{\theta }_{r}J(J+1)/T}$. It approaches $1+3e^{-2{\theta }_r/T}$ at low temperature and $T/{\theta }_r$ at high temperature. The latter limit gives mean rotational energy $k_{B}T$ and heat capacity $k_B$ per rotor.

Rotational temperature is a concept used in spectroscopy and thermodynamics to describe the temperature of a rotating molecule, specifically relating to its rotational energy levels. In quantum mechanics, molecules can rotate in space, and this rotation corresponds to quantized energy levels. These energy levels are influenced by the moment of inertia of the molecule and the rotational quantum numbers.