Editing Equipartition theorem (section)

===Non-ideal gases===
{{See also|Virial expansion|Virial coefficient}}

In an ideal gas the particles are assumed to interact only through collisions. The equipartition theorem may also be used to derive the energy and pressure of "non-ideal gases" in which the particles also interact with one another through [[conservative force]]s whose potential {{math|''U''(''r'')}} depends only on the distance {{mvar|r}} between the particles.<ref name="pathria_1972" /> This situation can be described by first restricting attention to a single gas particle, and approximating the rest of the gas by a [[spherical symmetry|spherically symmetric]] distribution. It is then customary to introduce a [[radial distribution function]] {{math|''g''(''r'')}} such that the [[probability density function|probability density]] of finding another particle at a distance {{mvar|r}} from the given particle is equal to {{math|4''πr''<sup>2</sup>''ρg''(''r'')}}, where {{math|1=''ρ'' = ''N''/''V''}} is the mean [[density]] of the gas.<ref name="mcquarrie_2000b">{{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/254 254–264] | url = https://archive.org/details/statisticalmecha00mcqu_0/page/254 }}</ref> It follows that the mean potential energy associated to the interaction of the given particle with the rest of the gas is
<math display="block">\langle h_{\mathrm{pot}} \rangle = \int_0^\infty 4\pi r^2 \rho U(r) g(r)\, dr.</math>
The total mean potential energy of the gas is therefore <math> \langle H_\text{pot} \rangle = \tfrac12 N \langle h_{\mathrm{pot}} \rangle </math>, where {{mvar|N}} is the number of particles in the gas, and the factor {{frac|1|2}} is needed because summation over all the particles counts each interaction twice.
Adding kinetic and potential energies, then applying equipartition, yields the ''energy equation''
<math display="block">H = \langle H_{\mathrm{kin}} \rangle + \langle H_{\mathrm{pot}} \rangle = 
\frac{3}{2} Nk_\text{B}T + 2\pi N \rho \int_0^\infty r^2 U(r) g(r) \, dr.</math>
A similar argument,<ref name="pathria_1972" /> can be used to derive the ''pressure equation''
<math display="block">3Nk_\text{B}T = 3PV + 2\pi N \rho \int_0^\infty r^3 U'(r) g(r)\, dr.</math>