State whether the following statements are True or False:
$(i)$ Due to collision,the density of gas molecules changes.
$(ii)$ The average kinetic energy of $1 \ g$ mass of each gas is equal at the same temperature.
$(iii)$ At the same temperature,the $v_{rms}$ of two different gases is the same.
$(iv)$ At constant temperature,if the pressure of a gas is increased,its mean free path decreases.

(D) $(i)$ False. Collisions between gas molecules do not change the total number of molecules or the volume of the container,so the density remains constant.
$(ii)$ False. The average kinetic energy per molecule depends only on temperature $(KE_{avg} = \frac{3}{2} k_B T)$. However,the average kinetic energy of a fixed mass $(1 \ g)$ depends on the molar mass of the gas $(KE_{total} = \frac{n}{2} RT = \frac{m}{M} \frac{3}{2} RT)$,which varies for different gases.
$(iii)$ False. The root mean square velocity is given by $v_{rms} = \sqrt{\frac{3RT}{M}}$. Since $v_{rms}$ depends on the molar mass $M$,it is different for different gases at the same temperature.
$(iv)$ True. The mean free path is given by $\lambda = \frac{k_B T}{\sqrt{2} \pi d^2 P}$. Since $\lambda \propto \frac{1}{P}$,increasing the pressure at a constant temperature decreases the mean free path.