$A$ very tall vertical cylinder is filled with a gas of molar mass $M$ under isothermal conditions at temperature $T$. The density and pressure of the gas at the base of the container are $\rho_0$ and $p_0$,respectively. Choose the correct statement$(s)$ if gravity is assumed to be constant throughout the container.

The oxygen molecule has a mass of $5.30 \times 10^{-26} \; kg$ and a moment of inertia of $1.94 \times 10^{-46} \; kg \cdot m^{2}$ about an axis through its centre perpendicular to the line joining the two atoms. Suppose the mean speed of such a molecule in a gas is $500 \; m/s$ and that its kinetic energy of rotation is two-thirds of its kinetic energy of translation. Find the average angular velocity of the molecule.

Two identical bulbs are filled with gas at $N.T.P.$ One bulb is placed in ice and the other in a hot bath, such that the pressure becomes $1.5$ times the initial pressure. What is the temperature of the hot bath in $^\circ C$?

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.

$A$ fixed thermally conducting cylinder has a radius $R$ and height $L_0$. The cylinder is open at its bottom and has a small hole at its top. $A$ piston of mass $M$ is held at a distance $L$ from the top surface,as shown in the figure. The atmospheric pressure is $P_0$.
$1.$ The piston is now pulled out slowly and held at a distance $2L$ from the top. The pressure in the cylinder between its top and the piston will then be
$(A) P_0$ $(B) \frac{P_0}{2}$ $(C) \frac{P_0}{2} + \frac{Mg}{\pi R^2}$ $(D) \frac{P_0}{2} - \frac{Mg}{\pi R^2}$
$2.$ While the piston is at a distance $2L$ from the top,the hole at the top is sealed. The piston is then released,to a position where it can stay in equilibrium. In this condition,the distance of the piston from the top is
$(A) \left(\frac{2P_0 \pi R^2}{\pi R^2 P_0 + Mg}\right)(2L)$ $(B) \left(\frac{P_0 \pi R^2 - Mg}{\pi R^2 P_0}\right)(2L)$ $(C) \left(\frac{P_0 \pi R^2 + Mg}{\pi R^2 P_0}\right)(2L)$ $(D) \left(\frac{P_0 \pi R^2}{\pi R^2 P_0 - Mg}\right)(2L)$
$3.$ The piston is taken completely out of the cylinder. The hole at the top is sealed. $A$ water tank is brought below the cylinder and put in a position so that the water surface in the tank is at the same level as the top of the cylinder as shown in the figure. The density of the water is $\rho$. In equilibrium,the height $H$ of the water column in the cylinder satisfies
$(A) \rho g(L_0 - H)^2 + P_0(L_0 - H) + L_0 P_0 = 0$
$(B) \rho g(L_0 - H)^2 - P_0(L_0 - H) - L_0 P_0 = 0$
$(C) \rho g(L_0 - H)^2 + P_0(L_0 - H) - L_0 P_0 = 0$
$(D) \rho g(L_0 - H)^2 - P_0(L_0 - H) + L_0 P_0 = 0$
Give the answer for questions $1, 2$ and $3$.

$A$ horizontal uniform glass tube of $100 \ cm$ length, sealed at both ends, contains a $10 \ cm$ mercury column in the middle. The temperature and pressure of air on either side of the mercury column are $31^{\circ} C$ and $76 \ cm$ of mercury, respectively. If the air column at one end is kept at $0^{\circ} C$ and the other end at $273^{\circ} C$, then the pressure of air which is at $0^{\circ} C$ is (in $cm$ of $Hg$):