The ratio of the speed of sound in a monatomic gas at $27^{\circ} C$ and the root-mean-square (rms) speed of the molecules of the same gas at a temperature of $127^{\circ} C$ is

Consider the following statements for air molecules in an air-tight container:
$(I)$ The average speed of molecules is larger than root mean square speed.
$(II)$ Mean free path of molecules is larger than the mean distance between molecules.
$(III)$ Mean free path of molecules increases with temperature.
$(IV)$ The rms speed of nitrogen is smaller than oxygen molecule.
Which of the above statements are correct?

Fill in the blanks:
$(i)$ At absolute zero,the volume of an ideal gas is ...... .
$(ii)$ With an increase in temperature,the pressure of a gas ...... .
$(iii)$ All molecular motion will stop at ...... .
$(iv)$ Due to ...... at higher altitudes from the surface of the Earth,the air becomes cooler.

$A$ light container having a diatomic gas enclosed within is moving with velocity $v$. The mass of the gas is $M$ and the number of moles is $n$. The kinetic energy of the gas with respect to the ground is:

In the kinetic theory of gases,which of these statements is/are true?
$(i)$ The pressure of a gas is proportional to the mean speed of the molecules.
$(ii)$ The root mean square speed of the molecules is proportional to the pressure.
$(iii)$ The rate of diffusion is proportional to the mean speed of the molecules.
$(iv)$ The mean translational kinetic energy of a gas is proportional to its kelvin temperature.

An ideal gas has $N$ molecules in a closed box at temperature $T_1$ and pressure $P_1$. If the number of molecules in the box is doubled and the total kinetic energy is kept the same,what will be the new pressure $P_2$ and temperature $T_2$?