According to the law of equipartition of energy, the number of vibrational modes of a polyatomic gas of constant γ = (Cp)/(Cv) is ( CP and CV are the specific heat capacities of the gas at constant pressure and constant volume, respectively): (a) (4+3γ)/(γ-1) (b) (3+4γ)/(γ-1) (c) (4-3γ)/(γ-1) (d) (3-4γ)/(γ-1)

Answer: (c). Polyatomic with f vibrational modes: CV = (3 + f)R. γ = 1 + R/CV = 1 + 1/3+f ⇒ 3 + f = (1)/(γ - 1) f = (1)/(γ-1) - 3 = (4 - 3γ)/(γ - 1) So the answer is (c).

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Specific Heats of Gases - Cp, Cv, γ & Mayer's Relation

According to the law of equipartition of energy, the number of vibrational modes of a polyatomic gas of constant $\gamma = \frac{C_p}{C_v}$ is ( $C_P$ and $C_V$ are the specific heat capacities of the gas at constant pressure and constant volume, respectively):

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Answer: (c)

Polyatomic with $f$ vibrational modes: $C_V = (3 + f)R$ .
$\gamma = 1 + \dfrac{R}{C_V} = 1 + \dfrac{1}{3+f} \Rightarrow 3 + f = \dfrac{1}{\gamma - 1}$
$f = \dfrac{1}{\gamma-1} - 3 = \dfrac{4 - 3\gamma}{\gamma - 1}$
So the answer is (c).

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