Rotational temperature
# Rotational temperature

The characteristic rotational temperature $(θ_R$ or $θ_{rot})$ is commonly used in statistical thermodynamics to simplify the expression of the rotational partition function and the rotational contribution to molecular thermodynamic properties. It has units of temperature and is defined as

    $\theta _{\mathrm {R} }={\frac {hc{\overline {B}}}{k_{\mathrm {B} }}}={\frac {\hbar ^{2}}{2k_{\mathrm {B} }I}},$

where ${\overline {B}}=B/hc$ is the rotational constant, I is a molecular moment of inertia, h is the Planck constant, c is the speed of light, ħ = h/2π is the reduced Planck constant and $k_B$ is the Boltzmann constant.

The physical meaning of $θ_R$ is as an estimate of the temperature at which thermal energy (of the order of $k_BT)$ is comparable to the spacing between rotational energy levels (of the order of hcB). At about this temperature the population of excited rotational levels becomes important. Some typical values are given in the table. In each case the value refers to the most common isotopic species.

| Molecule | $\theta _{\mathrm {R} }$ (K) |
| - | - |
| $H_2$ | 87.6 |
| $N_2$ | 2.88 |
| $O_2$ | 2.08 |
| $F_2$ | 1.27 |
| HF | 30.2 |
| HCl | 15.2 |
| $CO_2$ | 0.561 |
| HBr | 12.2 |
| CO | 2.78 |
