A polyatomic ideal gas has 24 vibrational modes. What is the value of γ?

Solution

Physics sir · 4 steps

Each vibrational mode stores energy twice, as kinetic and potential, so 24 modes add a full 24R to C_V.

1. Translation and rotation. A polyatomic molecule has 3 translational and, being non-linear, 3 rotational degrees of freedom. Each gives 1/2R per mole, so together they give 3R to C_V.
2. Vibration. Each vibrational mode has a kinetic and a potential energy term, so it gives 2 × 1/2R = R. The 24 modes add 24R, and the total is f = 3 + 3 + 2 × 24 = 54.

   C_V = 3R + 24R = 27R, C_P = C_V + R = 28R, γ = C_P/C_V = 28/27

3. Cross-check. The formula γ = 1 + 2/f = 1 + 2/54 gives the same value, 28/27 = 1.037.
4. So. γ ≈ 1.037, just above 1. The only option close to it is (c), 1.03.

The idea to keep. Count every vibrational mode twice, once for kinetic and once for potential energy. Many degrees of freedom push γ = 1 + 2/f down toward 1.

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