= Isobaric thermal stability of a heated ideal gas
{title2=$s=(1-\alpha)H/(c_pT_0)$}
Let the net cooling per unit mass of an <ideal gas> be $\mathcal L=A\rho T^\alpha-H$, with positive constants $A,H$. At fixed <pressure> and composition, $\rho T$ is constant, so $\mathcal L=CT^{\alpha-1}-H$. At a thermal equilibrium $T_0$, $CT_0^{\alpha-1}=H$. The <specific heat capacity at constant pressure> gives $c_p\,d(\delta T)/dt=-(\partial\mathcal L/\partial T)_p\delta T$. Thus $\delta T\propto e^{st}$ with the displayed growth rate. The equilibrium is stable for $\alpha>1$, unstable for $\alpha<1$, and marginal at linear order for $\alpha=1$. In particular, optically thin <Bremsstrahlung> with $\alpha=1/2$ is unstable under this heating law and the isobaric constraint. Fixed-density perturbations have a different criterion.
Back to article page
