If one mole of the polyatomic gas is having two vibrational modes and β is the ratio of molar specific heats for polyatomic gas ( β = C_P/C_V ) then the value of β is :

Solution

Physics sir · 3 steps

Count the degrees of freedom, and give each vibrational mode two of them, not one.

1. Translation and rotation. A non-linear polyatomic molecule moves in 3 directions and rotates about 3 axes. That gives 3 + 3 = 6 degrees of freedom.
2. Vibration counts double. Each vibrational mode stores both kinetic and potential energy, so it adds 2 degrees of freedom. Two modes add 2 × 2 = 4, so f = 6 + 4 = 10.

   C_V = f/2R = 5R, C_P = C_V + R = 6R

3. So. β = C_P/C_V = 6R/5R = 1.2, which is option (c).

The idea to keep. Use γ = 1 + 2/f. For a polyatomic gas start from f = 6, then add 2 for every vibrational mode, because vibration carries kinetic and potential energy.

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