Specific Heats And Gamma

Evaluating this ratio uses the specific heat relations for an ideal gas found in NCERT Class 11, Chapter 13 (Kinetic Theory). A monatomic gas has three translational degrees of freedom, giving molar heat capacity at constant volume $C_V=\frac{3}{2}R$. By Mayer's relation $C_P=C_V+R$, so $C_P=\frac{5}{2}R$. The ratio of specific heats is therefore $\gamma =\frac{C_P}{C_V}=\frac{5/2R}{3/2R}=\frac{5}{3}\approx 1.67$. The option 1.40 corresponds to a diatomic gas with five degrees of freedom, not a monatomic one. The option 1.33 corresponds to a polyatomic gas with six degrees of freedom. The option 1.00 is physically impossible because $C_P$ always exceeds $C_V$ for a gas, since extra energy is needed to do expansion work at constant pressure. The ratio gamma is physically important because it governs adiabatic processes and the speed of sound in the gas, both of which depend on how energy is partitioned among molecular motions. A magnitude check confirms the answer: $\gamma$ must always be greater than one and, for the fewest degrees of freedom, takes its maximum common value of $\frac{5}{3}$, matching the behaviour of noble gases like argon.