= Isothermal flow through molecular hydrogen formation
{title2=$\rho_1u_1=\rho_2u_2,\quad p_1+\rho_1u_1^2=p_2+\rho_2u_2^2$}
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 $p_2-p_1=\rho_1u_1(u_1-u_2)>0$.
For $u_1\ll a_1$, the leading behavior is $u_2\simeq u_1/2$, $\rho_2\simeq2\rho_1$ 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.
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