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For an ideal gas  \begin{align}  d\epsilon &= \frac{R}{\gamma -1}dT \\  PdV &= -\left(\gamma -1\right)\epsilon \frac{d \rho}{\rho} = RT \frac{d \rho}{\rho}  \end{align}  Then  \begin{equation}  dQ = \frac{R}{\gamma -1}dT - RT \frac{d \rho}{\rho}  \end{equation}  Let's also introduce the enthalpy $h$ in the limit that the number of particles in the system is constant,  \begin{equation}  dh = TdS + VdP = TdS + \frac{dP}{\rho}  \end{equation}  and also the Gibbs free energy $G$ in the same paticle conserving limit  \begin{equation}  dG = SdT + \frac{dP}{\rho}  \end{equation}  In the adiabatic limit $TdS = dQ =0$ and $dh = \frac{dP}{\rho}$. So for an adiabatic ideal gas  \begin{equation}  \int{\frac{dP}{\rho}} = h   \end{equation}