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\frac{d\rho}{\rho} = \frac{1}{\rho} \frac{dP}{RT} - \frac{1}{\rho}\frac{P}{RT^2} dT  \end{equation}  so that  \begin{equation} \begin{align}  dQ = &=  R\left( 1 + \frac{1}{\gamma-1}\right)dT - \frac{d \rho}{\rho}.  \end{equation} \rho}{\rho} \\  &= \frac{\gamma R}{\gamma-1}\right)dT - \frac{d \rho}{\rho}  \end{align}  Let's also introduce the enthalpy $h$ in the limit that the number of particles in the system is constant,  \begin{equation}