According to the law of equipartition of energy, the number of vibrational modes r of a polyatomic gas with constant γ = Cₚ/Cᵥ is:
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According to the law of equipartition of energy, the number of vibrational modes r of a polyatomic gas with constant γ = Cₚ/Cᵥ is:

G1 Physics · Accepted Answer

Step 1: The printed key assumes a nonlinear polyatomic molecule, with 3 translational and 3 rotational active degrees of freedom. Step 2: Each active vibrational mode contributes two quadratic terms. Thus effective degrees of freedom f = 6 + 2r. Step 3: For an ideal gas, Cᵥ = (f/2)R and Cₚ = Cᵥ + R. Step 4: γ = Cₚ/Cᵥ = 1 + 2/f; therefore f = 2/(γ − 1). Step 5: Substitute 6 + 2r = 2/(γ − 1), so r = 1/(γ − 1) − 3 = (4 − 3γ)/(γ − 1). Final answer: (C).