According to the law of equipartition of energy, the number of vibrational modes of a polyatomic gas of…
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 ( $C_P$ where $C_v$ are the specific heat capacities of the gas at constant pressure and constant volume, respectively):
$\frac{4+3 \gamma}{\gamma-1}$
$\frac{3+4 \gamma}{\gamma-1}$
$\frac{4-3 \gamma}{\gamma-1}$
$\frac{3-4 \gamma}{\gamma-1}$
Solution
A polygamic gas has 3 translational, 3 rotational and $f$ vibration modes
$U=\frac{3}{2} k_B T+\frac{3}{2} k_B T+f k_B T$
$U=(3+f) k_B T$
$C_V=(3+f) R$
$C_p=(4+f) R$
$\frac{C_p}{C_v}=\frac{4+f}{3+f}=\gamma$
$4+f=3 \gamma+f \gamma$
$4-3 \gamma=f(\gamma-1) \Rightarrow f=\frac{4-3 \gamma}{\gamma-1}$