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\end{align}  Then  \begin{equation}  dQ = \frac{R}{\gamma -1}dT - RT \frac{d \rho}{\rho} \rho}{\rho}.  \end{equation}  We can rewrite  \begin{equation}  \frac{d\rho}{rho} = \frac{1}{\rho} \frac{dP}{RT} - \frac{1}{\rho}\frac{P}{RT^2} dT  \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} 
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\begin{equation}  dG = SdT + \frac{dP}{\rho}  \end{equation}  Now we assume to special cases, adiabatic and isothermal flows. First assume an adiabatic flow, $dQ =0$. Then