The temperature of a metallic sphere of radius $R$ is increased by a small amount $\Delta T$. If the linear coefficient of thermal expansion of the metal is $\alpha$, the approximate increase in the volume of the sphere is:

In an adiabatic expansion, the temperature of one mole of an ideal monatomic gas ( $\gamma=5 / 3$ ) decreases from 60 K to 50 K . The work done by the gas in the process is:

(Take the universal gas constant as $R=8.3 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$ )

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:

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: