According to the law of equipartition of energy, the number of vibrational modes of a polyatomic gas

Question

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

Professor PC Thomas Classes · Answer

(c) : Polyatomic molecule has 3 translational, 3 rotational and vibrational modes. According to law of equipartition of energy, $ \begin{aligned} & U=\left(\frac{3 R T}{2}+\frac{3 R T}{2}+f R Tight) \ & U=3 R T+f R T \ & C_V=\frac{d u}{d t}=3 R+f R \Rightarrow f=\frac{C_V-3 R}{R}=\frac{C_V}{R}-3 \ & 	ext { As, } C_V=\frac{R}{\gamma-1} \ & 	herefore \quad f=\frac{R}{(\gamma-1) R}-3 \Rightarrow \frac{1}{\gamma-1}-3=\frac{4-3 \gamma}{\gamma-1} \end{aligned} $ \ So, the correct option is (c).