Let γ₁ be the ratio of molar specific heat at constant pressure and molar specific heat at constant volume of a monoatomic gas and γ₂ be the similar ratio of diatomic gas. Considering the diatomic gas molecule as a rigid rotator, the ratio, γ₁/γ₂ is :

Solution

Physics sir · 2 steps

Count degrees of freedom first, and γ = 1 + 2/f gives both ratios in one line.

1. Degrees of freedom. A monoatomic molecule can only move, so it has 3 translational degrees of freedom. A rigid diatomic molecule also rotates about 2 axes, so f = 3 + 2 = 5. Being rigid means it does not vibrate, so there are no vibrational terms.

   C_V = f/2R, C_P = C_V + R, γ = C_P/C_V = 1 + 2/f

   Gas	f	C_V	C_P	γ
   Monoatomic	3	3/2R	5/2R	γ₁ = 5/3
   Diatomic (rigid)	5	5/2R	7/2R	γ₂ = 7/5

2. So. γ₁/γ₂ = 5/3 × 5/7 = 25/21, which is option (b).

The idea to keep. For any ideal gas, γ = 1 + 2/f. Monoatomic has f = 3, rigid diatomic has f = 5, and a vibrating diatomic adds 2 more to make f = 7.

Answer from the NTA final answer key. Solution by PrepWiser.