The temperature $(T)$ of one mole of an ideal gas varies with its volume $(V)$ as $T = -\alpha V^3 + \beta V^2$,where $\alpha$ and $\beta$ are positive constants. The maximum pressure of the gas during this process is ............

$A$ flat plate is moving normal to its plane through a gas under the action of a constant force $F$. The gas is kept at a very low pressure. The speed of the plate $v$ is much less than the average speed $u$ of the gas molecules. Which of the following options is/are true?

An ideal gas at pressure $P$ and temperature $T$ is expanding such that $PT^3 = \text{constant}$. The coefficient of volume expansion of the gas is . . . . . . .

$A$ flask contains argon and chlorine in the ratio of $2: 1$ by mass. The temperature of the mixture is $27\,^{\circ}C$. Obtain the ratio of
$(i)$ average kinetic energy per molecule,and
$(ii)$ root mean square speed $v_{rms}$ of the molecules of the two gases.
Atomic mass of argon $= 39.9\,u$; Molecular mass of chlorine $= 70.9\,u$.

An ideal gas filled in a cylinder occupies volume $V$. The gas is compressed isothermally to the volume $V/3$. Now,the cylinder valve is opened and the gas is allowed to leak keeping temperature same. What percentage of the number of molecules should escape to bring the pressure in the cylinder back to its original value (in $\%$)?

$A$ 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 $Hg$):