$A$ closed cylindrical vessel contains $N$ moles of an ideal diatomic gas at a temperature $T$. On supplying heat,the temperature remains the same,but $n$ moles dissociate into atoms. The heat supplied is .........

$A$ cylindrical tube $AB$ of length $l$, closed at both ends, contains an ideal gas of $1 \text{ mol}$ having molecular weight $M$. The tube is rotated in a horizontal plane with constant angular velocity $\omega$ about an axis perpendicular to $AB$ and passing through the edge at end $A$. If $P_{A}$ and $P_{B}$ are the pressures at $A$ and $B$ respectively, then (Consider the temperature is same at all points in the tube):

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 $0^{\circ} C$.

Nitrogen gas is filled in an insulated container. If $\alpha$ fraction of moles dissociates without exchange of any energy,then the fractional change in its temperature is ..............

$A$ 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 the heat required to increase the rms speed of $4$ moles of a diatomic gas from $v$ to $\sqrt{3} v$ is $83.1 \ kJ$, then the initial temperature of the gas is (Universal gas constant $R = 8.31 \ J \ mol^{-1} \ K^{-1}$) (in $^{\circ} C$)