For a diatomic gas, if γ₁=((C p)/(C v)) for rigid molecules and γ₂=((C p)/(C v)) for another diatomic molecules, but also having vibrational modes. Then, which one of the following options is correct? (Cp and Cv are specific heats of the gas at constant pressure and volume)

Solution

Physics sir · 2 steps

Vibration adds degrees of freedom, and every extra degree of freedom pulls γ down towards 1.

1. Gamma from f. By equipartition, each degree of freedom gives 1/2R to Cᵥ, so Cᵥ = f/2R and Cₚ = Cᵥ + R. Dividing the two gives the formula below.

   γ = Cₚ/Cᵥ = 1 + 2/f

   Molecule	Degrees of freedom f	γ = 1 + 2/f
   Rigid diatomic	3 translational + 2 rotational = 5	γ₁ = 7/5 = 1.4
   Diatomic with vibration	5 + 2 vibrational (kinetic + potential) = 7	γ₂ = 9/7 ≈ 1.29

2. So. A larger f makes 2/f smaller, so γ₂ = 1.29 is less than γ₁ = 1.4. The answer is (a), γ₂ < γ₁.

The idea to keep. Remember γ = 1 + 2/f. The more ways a molecule can store energy, the larger f is and the closer γ gets to 1. One vibrational mode adds 2 to f.

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