According to the law of equipartition of energy, the number of vibrational… | Degrees of freedom and specific heats of gases

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

A $\dfrac{4+3\gamma}{\gamma-1}$

B $\dfrac{3+4\gamma}{\gamma-1}$

C $\dfrac{4-3\gamma}{\gamma-1}$

D $\dfrac{3-4\gamma}{\gamma-1}$

The answer is C.

Solution

A polyatomic gas has 3 translational, 3 rotational, and f vibrational modes. $U=\left(3+f\right)k_BT$ (each vibrational mode contributing $k_BT$). So $C_V=(3+f)R$, $C_P=(4+f)R$, and $\gamma=\dfrac{4+f}{3+f}$. Solving, $4+f=3\gamma+f\gamma\Rightarrow4-3\gamma=f(\gamma-1)\Rightarrow f=\dfrac{4-3\gamma}{\gamma-1}$.

Concept Degrees of freedom and specific heats of gases

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

Solution

Kinetic Theory in past papers

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