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).

$A$ fixed mass of gas at constant pressure occupies a volume $V$. The gas undergoes a rise in temperature so that the r.m.s. velocity of the molecules is doubled. The new volume will be

$A$ closed vessel contains a mixture of two diatomic gases $A$ and $B$. Molar mass of $A$ is $16$ times that of $B$ $(M_A = 16 M_B)$ and mass of gas $A$ contained in the vessel is $2$ times that of $B$ $(m_A = 2 m_B)$. Which of the following statements are true?
$(i)$ Average kinetic energy per molecule of $A$ is equal to that of $B$.
$(ii)$ Root mean square value of translational velocity of $B$ is four times that of $A$.
$(iii)$ Pressure exerted by $B$ is eight times that exerted by $A$.
$(iv)$ Number of molecules of $B$ in the cylinder is eight times that of $A$.

$Assertion :$ Air pressure in a car tyre increases during driving.
$Reason :$ Absolute zero temperature is not zero energy temperature.

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

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: