2.2.5 Ideal gas with constant specific heats (polytropic gas)

In this case, we obtain much simpler relations:

$$e = c_v T =\frac{p}{\rho} \frac{1}{\gamma-1}$$ (70)

$$h = c_p T =\frac{p}{\rho} \frac{\gamma}{\gamma-1}$$ (71)

$$s = c_v \left\{\ln p - \gamma \ln \rho \right\} + const.$$ (72)

$$\gamma \equiv \frac{c_p}{c_v} = \Gamma_1 = \Gamma_2 = \Gamma_3 = const.$$ (73)

$$c_p - c_v = \frac{p}{\rho T} \equiv \mathcal{R}$$ (74)

$$c_v = \mathcal{R} \frac{1}{\gamma-1} = c_v^\prime$$ (75)

$$c_p = \mathcal{R} \frac{\gamma}{\gamma-1} = c_p^\prime$$ (76)

$$\chi_{T} = \chi_{\rho} = \delta = 1$$ (77)

$$c_s \equiv \sqrt{\left(\frac{\partial p}{\partial \rho}\right)_{s}} = \sqrt{\gamma \frac{p}{\rho}}$$ (78)

$$c_T \equiv \sqrt{\left(\frac{\partial p}{\partial \rho}\right)_{T}} = \sqrt{\frac{p}{\rho}}$$ (79)