Editing Equation of state (section)

== Ideal gas law ==

=== Classical ideal gas law ===
The classical [[ideal gas law]] may be written
<math display="block">pV = nRT.</math>

In the form shown above, the equation of state is thus
<math display="block">f(p, V, T) = pV - nRT = 0.</math>

If the [[Perfect gas|calorically perfect]] gas approximation is used, then the ideal gas law may also be expressed as follows
<math display="block">p = \rho(\gamma - 1)e</math>
where <math>\rho</math> is the [[number density]] of the gas (number of atoms/molecules per unit volume), <math>\gamma = C_p/C_v</math> is the (constant) adiabatic index ([[heat capacity ratio|ratio of specific heats]]), <math>e = C_v T</math> is the internal energy per unit mass (the "specific internal energy"), <math>C_v</math> is the specific heat capacity at constant volume, and <math>C_p</math> is the specific heat capacity at constant pressure.

=== Quantum ideal gas law ===
Since for atomic and molecular gases, the classical ideal gas law is well suited in most cases, let us describe the equation of state for elementary particles with mass <math>m</math> and spin <math>s</math> that takes into account quantum effects. In the following, the upper sign will always correspond to [[Fermi–Dirac statistics]] and the lower sign to [[Bose–Einstein statistics]]. The equation of state of such gases with <math>N</math> particles occupying a volume <math>V</math> with temperature <math>T</math> and pressure <math>p</math> is given by<ref>Landau, L. D., Lifshitz, E. M. (1980). Statistical physics: Part I (Vol. 5). page 162-166.</ref>

<math display="block">p= \frac{(2s+1)\sqrt{2m^3k_\text{B}^5T^5}}{3\pi^2\hbar^3}\int_0^\infty\frac{z^{3/2}\,\mathrm{d}z}{e^{z-\mu/(k_\text{B} T)}\pm 1}</math>
where <math>k_\text{B}</math> is the [[Boltzmann constant]] and <math>\mu(T,N/V)</math> the [[chemical potential]] is given by the following implicit function
<math display="block">\frac{N}{V}=\frac{(2s+1)(m k_\text{B}T)^{3/2}}{\sqrt 2\pi^2\hbar^3}\int_0^\infty\frac{z^{1/2}\,\mathrm{d}z}{e^{z-\mu / (k_\text{B} T)}\pm 1}.</math>

In the limiting case where <math>e^{\mu / (k_\text{B} T)}\ll 1</math>, this equation of state will reduce to that of the classical ideal gas. It can be shown that the above equation of state in the limit <math>e^{\mu/(k_\text{B} T)}\ll 1</math> reduces to

<math display="block">pV = N k_\text{B} T\left[1\pm\frac{\pi^{3/2}}{2(2s+1)} \frac{N\hbar^3}{V(m k_\text{B} T)^{3/2}}+\cdots\right]</math>

With a fixed number density <math>N/V</math>, decreasing the temperature causes in [[Fermi gas]], an increase in  the value for pressure from its classical value implying an effective repulsion between particles (this is an apparent repulsion due to quantum exchange effects not because of actual interactions between particles since in ideal gas, interactional forces are neglected) and in [[Bose gas]], a decrease in pressure from its classical value implying an effective attraction. The quantum nature of this equation is in it dependence on s and '''ħ'''.