Editing Equipartition theorem (section)

===Rotational energy and molecular tumbling in solution===
{{See also|Angular velocity|Rotational diffusion}}

A similar example is provided by a rotating molecule with [[principal moments of inertia]] {{math|''I''<sub>1</sub>}}, {{math|''I''<sub>2</sub>}} and {{math|''I''<sub>3</sub>}}. According to classical mechanics, the [[rotational energy]] of such a molecule is given by
<math display="block">H_{\mathrm{rot}} = \tfrac{1}{2} ( I_1 \omega_1^2 + I_2 \omega_2^2 + I_3 \omega_3^2 ),</math>
where {{math|''ω''<sub>1</sub>}}, {{math|''ω''<sub>2</sub>}}, and {{math|''ω''<sub>3</sub>}} are the principal components of the [[angular velocity]]. By exactly the same reasoning as in the translational case, equipartition implies that in thermal equilibrium the average rotational energy of each particle is {{math|{{sfrac|3|2}}''k''<sub>B</sub>''T''}}. Similarly, the equipartition theorem allows the average (more precisely, the root mean square) angular speed of the molecules to be calculated.<ref name="pathria_1972" />

The tumbling of rigid molecules—that is, the random rotations of molecules in solution—plays a key role in the [[relaxation (NMR)|relaxation]]s observed by [[nuclear magnetic resonance]], particularly [[protein nuclear magnetic resonance spectroscopy|protein NMR]] and [[residual dipolar coupling]]s.<ref>{{cite book |vauthors=Cavanagh J, Fairbrother WJ, Palmer AG 3rd, Skelton NJ, Rance M | year = 2006 | title = Protein NMR Spectroscopy: Principles and Practice | edition = 2nd | publisher = Academic Press | isbn = 978-0-12-164491-8}}</ref> Rotational diffusion can also be observed by other biophysical probes such as [[fluorescence anisotropy]], [[flow birefringence]] and [[dielectric spectroscopy]].<ref>{{cite book | last = Cantor | first = CR |author2=Schimmel PR | year = 1980 | title = Biophysical Chemistry. Part II. Techniques for the study of biological structure and function | publisher = W. H. Freeman | isbn = 978-0-7167-1189-6}}</ref>