 Isothermal flow through molecular hydrogen formation

Across a thin steady transition with negligible external impulse, mass conservation and momentum conservation give the displayed relations. If the gas stays isothermal while neutral atomic hydrogen becomes molecular hydrogen, the mean particle mass doubles and $a_2^2=a_1^2/2$ for its isothermal sound speed. Let $J=u_1+a_1^2/u_1$. The downstream subsonic branch has $u_2=[J-\sqrt{J^2-2a_1^2}]/2$, ${\rho }_2/{\rho }_1=u_1/u_2$, and $p_2-p_1={\rho }_1{u}_1(u_1-u_2)>0$.

For $u_1\ll a_1$, the leading behavior is $u_2\simeq u_1/2$, ${\rho }_2\simeq 2{\rho }_1$ and $p_2\simeq p_1$, at unchanged temperature. Radiation of the molecular binding energy prevents that released energy from necessarily heating the gas; it does not, by itself, establish an isothermal thermal closure. If temperature evolution is allowed, an energy balance and a cooling model are required to determine all four changes uniquely.