Rotational temperature

The characteristic rotational temperature (Template:Math or Template:Math) 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

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

where $\bar{B}=B/hc$ is the rotational constant, Template:Mvar is a molecular moment of inertia, Template:Mvar is the Planck constant, Template:Mvar is the speed of light, Template:Math is the reduced Planck constant and Template:Math is the Boltzmann constant.

The physical meaning of Template:Math is as an estimate of the temperature at which thermal energy (of the order of Template:Math) is comparable to the spacing between rotational energy levels (of the order of Template:Math). 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)
H2	87.6
N2	2.88
O2	2.08
F2	1.27
HF	30.2
HCl	15.2
CO2	0.561
HBr	12.2
CO	2.78

See also

• Rotational spectroscopy
• Vibrational temperature
• Vibrational spectroscopy
• Infrared spectroscopy
• Spectroscopy

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