The internal energy of $10 \, g$ of nitrogen at $S.T.P.$ is about ......... $J$.

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.

The average translational kinetic energy and the $r.m.s.$ speed of molecules in a sample of oxygen gas at $300 \, K$ are $6.21 \times 10^{-21} \, J$ and $484 \, m/s$ respectively. The corresponding values at $600 \, K$ are nearly (assuming ideal gas behaviour):

Select the $incorrect$ statement$(s)$.

$A$ flask contains argon and chlorine in the ratio of $2: 1$ by mass. The temperature of the mixture is $27\,^{\circ}C$. Obtain the ratio of
$(i)$ average kinetic energy per molecule,and
$(ii)$ root mean square speed $v_{rms}$ of the molecules of the two gases.
Atomic mass of argon $= 39.9\,u$; Molecular mass of chlorine $= 70.9\,u$.

Statement $(I)$: Gas thermometers are less sensitive than liquid thermometers.
Statement $(II)$: The ratio of the universal gas constant and Avogadro's number is called Boltzmann's constant.
Statement $(III)$: The density of a given mass of a gas at constant pressure is inversely proportional to its absolute temperature.
The correct option among the following is: