Editing Equipartition theorem (section)

===Diatomic gases===
{{See also|Two-body problem|Rigid rotor|Harmonic oscillator}}

A diatomic gas can be modelled as two masses, {{math|''m''<sub>1</sub>}} and {{math|''m''<sub>2</sub>}}, joined by a [[spring (device)|spring]] of [[Hooke's law|stiffness]] {{math|''a''}}, which is called the ''rigid rotor-harmonic oscillator approximation''.<ref name="mcquarrie_2000c">{{cite book | last = McQuarrie | first = DA | year = 2000 | title = Statistical Mechanics | edition = revised 2nd | publisher = University Science Books | isbn = 978-1-891389-15-3 | pages = [https://archive.org/details/statisticalmecha00mcqu_0/page/91 91–128] | url = https://archive.org/details/statisticalmecha00mcqu_0/page/91 }}</ref> The classical energy of this system is
<math display="block">H = 
\frac{\left| \mathbf{p}_1 \right|^2}{2m_1} + 
\frac{\left| \mathbf{p}_2 \right|^2}{2m_2} + 
\frac{1}{2} a q^2,</math>
where {{math|'''p'''<sub>1</sub>}} and {{math|'''p'''<sub>2</sub>}} are the momenta of the two atoms, and {{mvar|q}} is the deviation of the inter-atomic separation from its equilibrium value. Every degree of freedom in the energy is quadratic and, thus, should contribute {{math|{{frac|1|2}}''k''<sub>B</sub>''T''}} to the total average energy, and {{math|{{frac|1|2}}''k''<sub>B</sub>}} to the heat capacity. Therefore, the heat capacity of a gas of ''N'' diatomic molecules is predicted to be {{math|7''N''·{{frac|1|2}}''k''<sub>B</sub>}}: the momenta {{math|'''p'''<sub>1</sub>}} and {{math|'''p'''<sub>2</sub>}} contribute three degrees of freedom each, and the extension {{mvar|q}} contributes the seventh. It follows that the heat capacity of a mole of diatomic molecules with no other degrees of freedom should be {{math|1={{sfrac|7|2}}''N''<sub>A</sub>''k''<sub>B</sub> = {{sfrac|7|2}}''R''}} and, thus, the predicted molar heat capacity should be roughly 7&nbsp;cal/(mol·K). However, the experimental values for molar heat capacities of diatomic gases are typically about 5&nbsp;cal/(mol·K)<ref name="Wueller_1896" /> and fall to 3&nbsp;cal/(mol·K) at very low temperatures.<ref name="Eucken_1912" /> This disagreement between the equipartition prediction and the experimental value of the molar heat capacity cannot be explained by using a more complex model of the molecule, since adding more degrees of freedom can only ''increase'' the predicted specific heat, not decrease it.<ref name="maxwell_1875" /> This discrepancy was a key piece of evidence showing the need for a [[Quantum mechanics|quantum theory]] of matter.

[[Image:Chandra-crab.jpg|thumb|left|upright=1.25|Figure 6. A combined X-ray and optical image of the [[Crab Nebula]]. At the heart of this nebula there is a rapidly rotating [[neutron star]] which has about one and a half times the mass of the [[Sun]] but is only 25 km across. The equipartition theorem is useful in predicting the properties of such neutron stars.]]