An ideal gas equation can be written as $P = \frac{\rho R T}{M_{0}}$,where $\rho$ and $M_{0}$ are respectively,

An ideal gas is expanding such that $PT^2 = \text{constant}$. The coefficient of volume expansion of the gas is

The following graph represents the $T-V$ curves of an ideal gas (where $T$ is the temperature and $V$ is the volume) at three pressures $P_1, P_2$ and $P_3$ compared with those of Charles's law represented as dotted lines. Then the correct relation is:

An ideal gas at pressure $P$ and temperature $T$ is enclosed in a vessel of volume $V$. Some gas leaks through a hole from the vessel and the pressure of the enclosed gas falls to $P'$. Assuming that the temperature of the gas remains constant during the leakage, the number of moles of the gas that have leaked is

An insulated cylindrical vessel is fitted with an insulated piston of negligible weight and negligible thickness at the midpoint of the vessel. The cylinder contains a gas at $0^{\circ} C$. When the gas is heated to $100^{\circ} C$, the piston moves through a length of $5 \, cm$. The length of the cylindrical vessel in $cm$ is:

One mole of a perfect gas in a cylinder fitted with a piston has a pressure $P,$ volume $V,$ and temperature $T.$ If the temperature is increased by $1 \, K$ keeping pressure constant,the increase in volume is