$$\require{cancel}$$
$\frac{C_{p}}{C_{v}}=\gamma.$
We have seen that, for gases that behave as we would like them to behave, the molar heat capacities CV at constant temperatures for monatomic, diatomic and nonlinear polyatomic gases without molecular vibration are respectively $$\frac{3}{2} R$$, $$\frac{5}{2} R$$, and 3R. And since, for an ideal gas, CP = CV + R, (equation 8.1.3), we expect the corresponding values for CP to be $$\frac{5}{2} R$$, $$\frac{7}{2} R$$ and 4R. and Thus the expected values of γ are 5/3, 7/5 and 4/3.