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\begin{equation}  c^{\rm{ad}}_{s} = \sqrt{\frac{dP}{d\rho}} = \sqrt{\gamma \frac{P}{\rho}} = \sqrt{\gamma RT}  \end{equation}  $P = \frac{\left( c^{\rm{ad}}_{s}\right)^2}{\gamma} The isothermal equation of state $P=(c^{\rm{iso}}_{s})^2  \rho$ gives us that $c^{\rm{iso}}_{s} = \sqrt{RT$  and   \begin{equation}  \int{\frac{dP}{\rho}} = \frac{(c^{\rm{ad}}_{s})^2 }{\gamma -1}   \end{equation}