View source for Rotational temperature

The '''characteristic rotational temperature''' ({{math|''θ''<sub>R</sub>}} or {{math|''θ''<sub>rot</sub>}}) 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<ref name=Atkins>P. Atkins and J. de Paula "Physical Chemistry", 9th edition (W.H. Freeman 2010), p.597</ref>

:<math>\theta_{\mathrm{R}} = \frac{hc \overline{B}}{k_{\mathrm{B}}} = \frac{\hbar^2}{2k_{\mathrm{B}}I},</math>

where  <math>\overline{B} = B/hc</math> is the [[Rigid rotor#Quantum mechanical linear rigid rotor|rotational constant]], {{mvar|I}} is a molecular [[moment of inertia]], {{mvar|h}} is the [[Planck constant]], {{mvar|c}} is the [[speed of light]], {{math|1=''ħ'' = ''h''/2''π''}} is the [[reduced Planck constant]] and {{math|''k''<sub>B</sub>}} is the [[Boltzmann constant]].

The physical meaning of {{math|''θ''<sub>R</sub>}} is as an estimate of the temperature at which [[thermal energy]] (of the order of {{math|''k''<sub>B</sub>''T''}}) is comparable to the spacing between rotational energy levels (of the order of {{math|''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.
{| class="wikitable sortable"
|-
! Molecule !! <math>\theta_{\mathrm{R}}</math> (K)<ref name="AtkinsTable">P. Atkins and J. de Paula "Physical Chemistry", 9th edition (W.H. Freeman 2010), Table 13.2, Data section in appendix</ref>
|-
| H<sub>2</sub>  || 87.6
|-
| N<sub>2</sub>  || 2.88
|-
| O<sub>2</sub>  || 2.08
|-
| F<sub>2</sub> || 1.27
|-
| HF || 30.2
|-
| HCl || 15.2
|-
| CO<sub>2</sub>  || 0.561<ref>P. Atkins and J. de Paula "Physical Chemistry", 9th edition (W.H. Freeman 2010), Table 16.1, p.597</ref>
|-
|HBr
|12.2<ref>P. Atkins and J. de Paula "Physical Chemistry", 10th edition, Table 12D.1, p.987</ref>
|-
|CO
|2.78<ref>P. Atkins and J. de Paula "Physical Chemistry", 10th edition, Table 12D.1, p.987</ref>
|}
==References==
{{reflist}}

==See also==
*[[Rotational spectroscopy]]
*[[Vibrational temperature]]
*[[Vibrational spectroscopy]]
*[[Infrared spectroscopy]]
*[[Spectroscopy]]

{{DEFAULTSORT:Rotational Temperature}}
[[Category:Atomic physics]]
[[Category:Molecular physics]]


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