Question

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

• A. $\frac{4 + 3\gamma}{\gamma - 1}$
• B. $\frac{3 + 4\gamma}{\gamma -1}$
• C. $\frac{4 - 3\gamma}{\gamma -1}$
• D. $\frac{3 - 4\gamma}{\gamma - 1}$

Answer 1

The Correct Option is C

For a polyatomic gas with 3 translational, 3 rotational, and $f$ vibrational modes:

Internal energy (U) $= \frac{3}{2}k_BT + \frac{3}{2}k_BT + fk_BT = (3 + f)k_BT$

$C_v = (3 + f)R$

$C_p = (4 + f)R$

$\gamma = \frac{C_p}{C_v} = \frac{4 + f}{3 + f}$

$3\gamma + f\gamma = 4 + f$

$f(\gamma - 1) = 4 - 3\gamma$

$f = \frac{4 - 3\gamma}{\gamma - 1}$