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Kinetic Theory
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The mean free path of molecules in an ideal gas A is half that of another ideal gas B. The diameter of the spherical molecules of gas A is twice the diameter of the molecules of B. If number densities of the gases A and B are $n_A$ and $n_B$, respectively, the correct option is:$\lambda=\frac{1}{\sqrt{2}\pi d^2n} \Rightarrow n\propto\frac{1}{\lambda d^2}$. $\frac{n_A}{n_B}=\frac{\lambda_B}{\lambda_A}\times\left(\frac{d_B}{d_A}\right)^2=2\times\left(\frac{1}{2}\right)^2=\frac{1}{2}$, so $n_A=\frac{n_B}{2}$.
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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:Total degrees of freedom $F=\frac{2}{\gamma-1}$. Each vibrational mode contributes 2, so $6+2f=\frac{2}{\frac{8}{7}-1}=14 \Rightarrow 2f=8 \Rightarrow f=4$.