Specific Heat Capacities Of Gases

Question

For a diatomic ideal gas the molar specific heat at constant volume is 5R/2, so what is the value of its ratio of specific heats gamma?

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From NCERT Class 11, Chapter 12 (Thermodynamics) and its link to kinetic theory, the two molar specific heats of an ideal gas are connected by Mayer's relation, $C_P - C_V = R$, where $R$ is the universal gas constant; the difference arises because a gas heated at constant pressure also performs expansion work. For a diatomic gas the molar specific heat at constant volume is $C_V = \dfrac{5R}{2}$, accounting for three translational and two rotational degrees of freedom under the equipartition principle. Then $C_P = C_V + R = \dfrac{5R}{2} + R = \dfrac{7R}{2}$. The ratio of specific heats follows as $\gamma = \dfrac{C_P}{C_V} = \dfrac{7R/2}{5R/2} = \dfrac{7}{5} = 1.4$. The option 5/3 is wrong because that is the value for a monatomic gas with $C_V = 3R/2$. The option 9/7 is wrong because it corresponds to a polyatomic gas with additional vibrational modes, not a simple diatomic gas. The option 3/2 is wrong because it is a value of $C_V/R$, not a ratio of the two specific heats at all. A plausibility check confirms the answer: $\gamma$ for any gas must exceed 1 since $C_P > C_V$, and the diatomic value 1.4 matches the well-known experimental result for air, which is largely diatomic nitrogen and oxygen.

Specific Heat Capacities Of Gases

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