Editing Fermi gas (section)

===Thermodynamic quantities===
====Degeneracy pressure====
[[File:Quantum ideal gas pressure 3d.svg|thumb|Pressure vs temperature curves of classical and quantum ideal gases (Fermi gas, [[Bose gas]]) in three dimensions. Pauli repulsion in fermions (such as electrons) gives them an additional pressure over an equivalent classical gas, most significantly at low temperature.]]
By using the [[first law of thermodynamics]], this internal energy can be expressed as a pressure, that is
<math display="block">P = -\frac{\partial E_{\rm T}}{\partial V} = \frac{2}{5}\frac{N}{V}E_{\mathrm{F}}= \frac{(3\pi^2)^{2/3}\hbar^2}{5m}\left(\frac{N}{V}\right)^{5/3},</math>
where this expression remains valid for temperatures much smaller than the Fermi temperature. This pressure is known as the '''degeneracy pressure'''. In this sense, systems composed of fermions are also referred as [[degenerate matter]].

Standard [[star]]s avoid collapse by balancing thermal pressure ([[plasma (physics)|plasma]] and radiation) against gravitational forces. At the end of the star lifetime, when thermal processes are weaker, some stars may become white dwarfs, which are only sustained against gravity by [[electron degeneracy pressure]]. Using the Fermi gas as a model, it is possible to calculate the [[Chandrasekhar limit]], i.e. the maximum mass any star may acquire (without significant thermally generated pressure) before collapsing into a black hole or a neutron star. The latter, is a star mainly composed of neutrons, where the collapse is also avoided by neutron degeneracy pressure.

For the case of metals, the electron degeneracy pressure contributes to the compressibility or [[bulk modulus]] of the material.

====Chemical potential====
{{See also|Fermi level}}
Assuming that the concentration of fermions does not change with temperature, then the total chemical potential ''μ'' (Fermi level) of the three-dimensional ideal Fermi gas is related to the zero temperature Fermi energy ''E''<sub>F</sub> by a [[Sommerfeld expansion]] (assuming <math>k_{\rm B}T \ll E_{\mathrm{F}}</math>):
<math display="block">\mu(T) = E_0 + E_{\mathrm{F}} \left[ 1- \frac{\pi ^2}{12} \left(\frac{k_{\rm B}T}{E_{\mathrm{F}}}\right) ^2 - \frac{\pi^4}{80} \left(\frac{k_{\rm B}T}{E_{\mathrm{F}}}\right)^4 + \cdots \right], </math>
where ''T'' is the [[temperature]].<ref>{{cite web|title=Statistical Mechanics of Ideal Fermi Systems |url=http://www.uam.es:80/personal_pdi/ciencias/jgr/pdfs/fermi.pdf| last=Kelly|first=James J.|date=1996| website=Universidad Autónoma de Madrid|url-status=dead |archive-url=https://web.archive.org/web/20180412225816/http://www.uam.es/personal_pdi/ciencias/jgr/pdfs/fermi.pdf| archive-date=2018-04-12| access-date=2018-03-15}}</ref><ref>{{cite web |title=Degenerate Ideal Fermi Gases |url=http://www.physics.usyd.edu.au/ugrad/sphys_old/sphys_webct/PHYS3905_SM/TSM12.pdf |url-status=dead|archive-url=https://web.archive.org/web/20080919073627/http://www.physics.usyd.edu.au/ugrad/sphys_old/sphys_webct/PHYS3905_SM/TSM12.pdf| archive-date=2008-09-19 |access-date=2014-04-13}}</ref>

Hence, the [[internal chemical potential]], ''μ''-''E''<sub>0</sub>, is approximately equal to the Fermi energy at temperatures that are much lower than the characteristic Fermi temperature ''T''<sub>F</sub>. This characteristic temperature is on the order of 10<sup>5</sup> [[kelvin|K]] for a metal, hence at room temperature (300 K), the Fermi energy and internal chemical potential are essentially equivalent.