One mole of an ideal gas undergoes a process $P = \frac{2V^2}{1+V^2} \, Pa$. The change in temperature of the gas when the volume changes from $V = 1 \, m^3$ to $V = 2 \, m^3$ is:

$A$ sample of oxygen gas and a sample of hydrogen gas both have the same mass,same volume,and the same pressure. The ratio of their absolute temperatures is (Molecular weights of $O_2$ and $H_2$ are $32$ and $2$ respectively).

$10 \ g$ of a gas at $STP$ occupies a volume of $2 \ L$. At what temperature will the volume be doubled,if pressure and amount of the gas remain constant?

$A$ vessel is filled with an ideal gas at a pressure of $10 \text{ atm}$ and temperature $27^{\circ} C$. Half of the mass is removed from the vessel and the temperature of the remaining gas is increased to $87^{\circ} C$. Then the pressure of the gas in the vessel will be (in $\text{ atm}$)

The tyre of a bicycle has a volume of $2 \times 10^{-3} \ m^3$. Initially,the tube is filled to $75 \%$ of its volume with air at an atmospheric pressure of $10^5 \ N \ m^{-2}$. When a rider is on the bicycle,the area of contact of the tyre with the road is $24 \times 10^{-4} \ m^2$. The mass of the rider with the bicycle is $120 \ kg$. If a pump delivers a volume of $500 \ cm^3$ of air in each stroke,then the number of strokes required to inflate the tyre is $(g = 10 \ m \ s^{-2})$.

For a perfect gas,two pressures $P_{1}$ and $P_{2}$ are shown in the figure. The graph shows: