If the potential energy of a gas molecule is $U = \frac{M}{r^6} - \frac{N}{r^{12}}$,where $M$ and $N$ are positive constants,then the potential energy at equilibrium must be

An insulated box containing a diatomic gas of molar mass $M$ is moving with a velocity $v$. The box is suddenly stopped. The resulting change in temperature is

For a mass $m$ of an ideal gas at constant pressure $P$,the volume versus temperature graph is represented by the straight line $B$. If the mass is changed to $2m$ and the pressure is changed to $2P$,which straight line will represent the new state?

$A$ thermally isolated cylindrical closed vessel of height $8 \ m$ is kept vertically. It is divided into two equal parts by a diathermic (perfect thermal conductor) frictionless partition of mass $8.3 \ kg$. Thus,the partition is held initially at a distance of $4 \ m$ from the top. Each of the two parts of the vessel contains $0.1 \ mol$ of an ideal gas at temperature $300 \ K$. The partition is now released and moves without any gas leaking from one part of the vessel to the other. When equilibrium is reached,the distance of the partition from the top (in $m$) will be. . . . . . (take the acceleration due to gravity $g = 10 \ m/s^2$ and the universal gas constant $R = 8.3 \ J \ mol^{-1} \ K^{-1}$).

Statement-$1$: Internal energy of a gas $U = nC_VT$ is due to the random motion of gas molecules.
Statement-$2$: $A$ container is moving with speed $v$. It is suddenly stopped by a force,and the temperature of the gas increases.

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).