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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.

Assume zero temperature, complete ionization, constant composition, noninteracting electrons, negligible ion thermal pressure, and Newtonian stellar gravity. Ions supply almost all the mass, while electron degeneracy pressure supplies support. By the Pauli exclusion principle, the two electron spin states fill a momentum sphere up to Fermi momentum $p_F$. Counting states giveswhere ${\mu }_e$ is mean molecular weight per electron. The momentum flux of this isotropic Fermi gas isWriting $x=p_F/(m_{e}c)$ and performing the integral gives the equation of state of a cold electron gasIn the nonrelativistic limit $p_F\ll m_{e}c$, $v(p)\simeq p/m_e$, soIn the ultrarelativistic limit $p_F\gg m_{e}c$, $v(p)\simeq c$, soThese are polytropic equations of state with indices $3/2$ and $3$, respectively. The relativistic softening underlies the Chandrasekhar mass limit; Coulomb corrections, thermal effects, rotation, and general relativity are excluded from this idealized derivation.