Rotational temperature

The characteristic rotational temperature (θR or θrot) 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[1]

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

where \( {\displaystyle {\overline {B}}=B/hc} \) 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 kB is the Boltzmann constant.

The physical meaning of θR is as an estimate of the temperature at which thermal energy (of the order of kBT) 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	\( {\displaystyle \theta _{\mathrm {R} }} \) (K)[2]
H₂	87.6
N₂	2.88
O₂	2.08
F₂	1.27
HF	30.2
HCl	15.2
CO₂	0.561^[3]
HBr	12.2^[4]
CO	2.78^[5]

References

P. Atkins and J. de Paula "Physical Chemistry", 9th edition (W.H. Freeman 2010), p.597
P. Atkins and J. de Paula "Physical Chemistry", 9th edition (W.H. Freeman 2010), Table 12.2, Data section in appendix
P. Atkins and J. de Paula "Physical Chemistry", 9th edition (W.H. Freeman 2010), Table 16.1, p.597
P. Atkins and J. de Paula "Physical Chemistry", 10th edition, Table 12D.1, p.987

P. Atkins and J. de Paula "Physical Chemistry", 10th edition, Table 12D.1, p.987