Editing Equipartition theorem (section)

===Translational energy and ideal gases===
{{See also|Ideal gas}}

The (Newtonian) kinetic energy of a particle of mass {{mvar|m}}, velocity {{math|'''v'''}} is given by
<math display="block">H_{\text{kin}} = \tfrac 1 2 m |\mathbf{v}|^2 = \tfrac{1}{2} m\left( v_x^2 + v_y^2 + v_z^2 \right),</math>
where {{math|''v<sub>x</sub>''}}, {{math|''v<sub>y</sub>''}} and {{math|''v<sub>z</sub>''}} are the Cartesian components of the velocity {{math|'''v'''}}. Here, {{mvar|H}} is short for [[Hamiltonian (quantum mechanics)|Hamiltonian]], and used henceforth as a symbol for energy because the [[Hamiltonian mechanics|Hamiltonian formalism]] plays a central role in the most [[#General formulation of the equipartition theorem|general form]] of the equipartition theorem.

Since the kinetic energy is quadratic in the components of the velocity, by equipartition these three components each contribute {{math|{{frac|1|2}}''k''<sub>B</sub>''T''}} to the average kinetic energy in thermal equilibrium. Thus the average kinetic energy of the particle is {{math|{{sfrac|3|2}}''k''<sub>B</sub>''T''}}, as in the example of noble gases above.

More generally, in a monatomic ideal gas the total energy consists purely of (translational) kinetic energy: by assumption, the particles have no internal degrees of freedom and move independently of one another. Equipartition therefore predicts that the total energy of an ideal gas of {{mvar|N}} particles is {{math|{{sfrac|3|2}} ''N''&thinsp;''k''<sub>B</sub>&thinsp;''T''}}.

It follows that the [[heat capacity]] of the gas is {{math|{{sfrac|3|2}} ''N''&thinsp;''k''<sub>B</sub>}} and hence, in particular, the heat capacity of a [[mole (unit)|mole]] of such gas particles is {{math|1={{sfrac|3|2}}''N''<sub>A</sub>''k''<sub>B</sub> = {{sfrac|3|2}}''R''}}, where ''N''<sub>A</sub> is the [[Avogadro constant]] and ''R'' is the [[gas constant]]. Since ''R'' ≈ 2 [[calorie|cal]]/([[mole (unit)|mol]]·[[Kelvin|K]]), equipartition predicts that the [[molar heat capacity]] of an ideal gas is roughly 3&nbsp;cal/(mol·K). This prediction is confirmed by experiment when compared to monatomic gases.<ref name="kundt_1876" />

The mean kinetic energy also allows the [[root mean square speed]] {{math|''v''<sub>rms</sub>}} of the gas particles to be calculated:
<math display="block">v_{\text{rms}} = \sqrt{\left\langle v^2 \right\rangle} = \sqrt{\frac{3 k_\text{B} T}{m}} = \sqrt{\frac{3 R T}{M}},</math>
where {{math|1=''M'' = ''N''<sub>A</sub>''m''}} is the mass of a mole of gas particles. This result is useful for many applications such as [[Graham's law]] of [[effusion]], which provides a method for [[enriched uranium|enriching]] [[uranium]].<ref>[https://www.nrc.gov/reading-rm/doc-collections/fact-sheets/enrichment.html Fact Sheet on Uranium Enrichment] U.S. Nuclear Regulatory Commission. Accessed 30 April 2007</ref>