$A$ diatomic gas of molecular weight $30 \, g/mol$ is filled in a container at $27 \, ^\circ C$. It is moving at a velocity of $100 \, m/s$. If it is suddenly stopped,the rise in temperature of the gas is:

Fill in the blanks:
$(i)$ At low $......$ and high $......$ temperature,real gases behave as an ideal gas.
$(ii)$ $......$ is a measure of the average kinetic energy of a gas.
$(iii)$ The kinetic energy of a gas with mass $m$ is $E$. Its momentum will be $......$.
$(iv)$ At $......$ temperature,$v_{rms}$ will be double that of $v_{rms}$ at $0^{\circ} C$.

$CO_2$ $(O - C - O)$ is a triatomic gas. The average kinetic energy of $1\,g$ of the gas is: (where $N$ is Avogadro's number,$k$ is Boltzmann's constant,and the molar mass of $CO_2 = 44\,g/mol$)

$A$ 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 $3$.

Two closed vessels of same volume are joined through a narrow tube and both vessels are filled with air of pressure $90 \text{ kPa}$ and temperature $400 \text{ K}$. Keeping the temperature of one vessel constant at $400 \text{ K}$, the temperature of the second vessel is raised to $500 \text{ K}$. The final pressure in the vessels is . . . . . . $\text{ kPa}$.

Three vessels of equal capacity have gases at the same temperature and pressure. The first vessel contains neon (monatomic),the second contains chlorine (diatomic),and the third contains uranium hexafluoride (polyatomic). Do the vessels contain equal number of respective molecules? Is the root mean square speed of molecules the same in the three cases? If not,in which case is $v_{rms}$ the largest?