An ideal gas of molar mass $M$ is contained in a very tall vertical cylindrical column in a uniform gravitational field. Assuming the gas temperature to be $T$, the height at which the centre of gravity of the gas is located is (where $R$ is the universal gas constant).

$A$ 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:

The adiabatic elasticity of hydrogen gas $(\gamma = 1.4)$ at $NTP$ is

Consider two boxes containing ideal gases $A$ and $B$ such that their temperatures, pressures, and number densities are the same. The molecular size of $A$ is half of that of $B$ and the mass of molecule $A$ is four times that of $B$. If the collision frequency in gas $B$ is $32 \times 10^{18} /s$, then the collision frequency in gas $A$ is . . . . . . $/s$.

The heat energy that must be supplied to $14 \text{ g}$ of nitrogen at room temperature to raise its temperature by $48^{\circ}C$ at constant pressure is ($R$ = gas constant, Molecular weight of nitrogen $(N_2) = 28$) (in $R$)

$A$ vessel of volume $V$ contains an ideal gas having mass density $\rho$ at temperature $T$ and pressure $p$. After a portion of the gas is let out, the pressure in the vessel is decreased by $\Delta p$. The mass of the released gas is