2.3.5 Ideal gas with constant specific heats (polytropic gas)

In this case, we obtain much simpler relations:

$$e = c_v T =\frac{p}{\rho} \frac{1}{\gamma-1}$$ (83)

$$h = c_p T =\frac{p}{\rho} \frac{\gamma}{\gamma-1}$$ (84)

$$s = c_v \left\{\ln p - \gamma \ln \rho \right\} + \mathrm{const.}$$ (85)

$$\gamma \equiv \frac{c_p}{c_v} = \Gamma_1 = \Gamma_2 = \Gamma_3 = \mathrm{const.}$$ (86)

$$c_p - c_v = \frac{p}{\rho T} \equiv \mathcal{R}$$ (87)

$$c_v = \mathcal{R} \frac{1}{\gamma-1} = c_v^\prime$$ (88)

$$c_p = \mathcal{R} \frac{\gamma}{\gamma-1} = c_p^\prime$$ (89)

$$\chi_{T} = \chi_{\rho} = \delta = 1$$ (90)

$$c_s \equiv \sqrt{\left(\frac{\partial p}{\partial \rho}\right)_{s}} = \sqrt{\gamma \frac{p}{\rho}}$$ (91)

$$c_T \equiv \sqrt{\left(\frac{\partial p}{\partial \rho}\right)_{T}} = \sqrt{\frac{p}{\rho}}$$ (92)