Ideal gas

An ideal gas obeys $pV=N{k}_{B}T$ and neglects intermolecular interactions except during elastic collisions.

The mean free path is the average distance travelled between collisions. For dilute targets with number density $n$ and effective collision cross-section ${\sigma }_c$, $\ell \sim 1/(n{\sigma }_c)$; identical-particle conventions introduce factors such as $\sqrt{2}$.

For an unclustered population of fully covering circular disc absorbers, the inverse mean free path is proper number density times mean projected area. A comoving number density must first be multiplied by $(1+z{)}^3$. The incidence then constrains the proper radius; covering factor and multiplicity of detectable lines can change the inference.

The mean molecular weight is the average mass per freely moving gas particle in units of the atomic mass unit. A fully ionized hydrogen-helium gas with mass fractions $X,Y$ has ${\mu }^{-1}=2X+3Y/4$.

The mean molecular weight per electron is the mass carried per free electron in units of atomic mass $m_u$. Fully ionized hydrogen has ${\mu }_e=1$, while helium, carbon, and oxygen have approximately ${\mu }_e=2$. It converts electron number density to bulk mass density in the equation of state of a cold electron gas.

The specific-heat ratio of an ideal gas is $\gamma =C_p/C_V>1$.

A gas is nondegenerate when its occupation numbers are low enough that quantum statistics reduce to the Maxwell-Boltzmann distribution. A common criterion is $n{\lambda }_{\mathrm{th}}^3\ll 1$, where ${\lambda }_{\mathrm{th}}$ is the thermal de Broglie wavelength.

For a fixed amount of ideal gas, internal energy depends only on temperature. If $C_V$ is constant, $dE=C_{V}dT$ and hence $E=C_{V}T$ after a choice of energy zero.

For an ideal gas of $N$ particles,Thus, with $\gamma =C_p/C_V$, one has $N{k}_B/C_V=\gamma -1$.