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):

(1) \(\frac{4 + 3\gamma}{\gamma - 1}\)

(2) \(\frac{3 + 4\gamma}{\gamma - 1}\)

(3) \(\frac{4 - 3\gamma}{\gamma - 1}\)

(4) \(\frac{3 - 4\gamma}{\gamma - 1}\)

Answer 1

Correct option is (3) \( \frac{4 - 3\gamma}{\gamma - 1}\)

A polygamic gas has 3 translational, 3 rotational and f vibration modes

\(U = \frac 32 k_BT + \frac 32 k_BT + fk_B T\)

\(U = (3 + f)k_BT\)

\(C_V = (3 + f)R\)

\(C_P = (4 +f)R\)

\(\frac{C_P}{C_V} = \frac{4+f}{3 + f}=\gamma\)

\(4 +f = 3\gamma + f\gamma\)

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

\(\Rightarrow f = \frac{4 - 3\gamma}{\gamma - 1}\)