Question
An ideal gas is made of polyatomic molecules. Each of the molecules has three translational, three rotational and $f$ number of vibrational modes. If the ratio of heat capacities $C_{P}/C_{V}$ of the gas is 8/7, then the value of $f$ is:
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Solution
Sol. [Key Concept: Degrees of freedom of a polyatomic gas → Cv, γ]
The total internal energy for one mole of gas, $U$
$$\begin{matrix} & \ = \left( \frac{3}{2}k_{B}T + \frac{3}{2}k_{B}T + fk_{B}T \right)N_{A} \\ & \ = (3 + f)k_{B}TN_{A} = (3 + f)RT \end{matrix}$$
$$\begin{matrix} \therefore & C_{V} = \frac{dU}{dT} = (3 + f)R \\ C_{P} = & C_{V} + R = (4 + f)R \\ \Rightarrow & \frac{C_{P}}{C_{V}} = \frac{4 + f}{3 + f} = \frac{8}{7}\text{~}\text{(given)}\text{~} \\ \Rightarrow & f = 4 \end{matrix}$$
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