An ideal polyatomic gas has molecules with three translational, three rotational, and f independent vibrational modes. If the heat-capacity ratio Cₚ/Cᵥ = 8/7, find f.
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Question

An ideal polyatomic gas has molecules with three translational, three rotational, and f independent vibrational modes. If the heat-capacity ratio Cₚ/Cᵥ = 8/7, find f.

G1 Physics · Accepted Answer

Step 1: Three translation and three rotation modes contribute six quadratic degrees of freedom. Step 2: Each vibrational normal mode contributes two quadratic terms (kinetic and potential), giving total degrees of freedom F = 6 + 2f. Step 3: Equipartition gives Cᵥ = (F/2)R and Cₚ = Cᵥ + R. Step 4: Hence γ = Cₚ/Cᵥ = 1 + 2/F = 8/7. Therefore 2/F = 1/7, so F = 14. Step 5: 6 + 2f = 14 ⇒ 2f = 8 ⇒ f = 4. Final answer: 4 vibrational modes (Option A).