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 possessing 3 translational, 3 rotational, and $f$ vibrational degrees of freedom:

Internal energy (U) is calculated as $\frac{3}{2}k_BT + \frac{3}{2}k_BT + fk_BT$, simplifying to $(3 + f)k_BT$.

The specific heat at constant volume ($C_v$) is $(3 + f)R$.

The specific heat at constant pressure ($C_p$) is $(4 + f)R$.

The adiabatic index ($\gamma$) is the ratio $\frac{C_p}{C_v}$, which equals $\frac{4 + f}{3 + f}$.

Rearranging this equation yields $3\gamma + f\gamma = 4 + f$.

Further simplification leads to $f(\gamma - 1) = 4 - 3\gamma$.

Thus, the number of vibrational modes $f$ is given by $\frac{4 - 3\gamma}{\gamma - 1}$.