For an ideal gas,the density of the gas is $\rho_0$ when the temperature and pressure of the gas are $T_0$ and $P_0$ respectively. When the temperature of the gas is $2 T_0$,its pressure becomes $3 P_0$. The new density will be:

An ideal monoatomic gas is confined in a cylinder by a spring-loaded piston of cross-section $8.0 \times 10^{-3} \, m^2$. Initially,the gas is at $300 \, K$ and occupies a volume of $2.4 \times 10^{-3} \, m^3$,and the spring is in its relaxed state as shown in the figure. The gas is heated by a small heater until the piston moves out slowly by $0.1 \, m$. The force constant of the spring is $8000 \, N/m$ and the atmospheric pressure is $1.0 \times 10^5 \, N/m^2$. The cylinder and the piston are thermally insulated. The piston and the spring are massless,and there is no friction between the piston and the cylinder. The final temperature of the gas will be: (Neglect the heat loss through the lead wires of the heater. The heat capacity of the heater coil is also negligible.) (in $, K$)

The equation of state for $n$ moles of an ideal gas is $PV = nRT$,where $R$ is a constant. The $SI$ unit for $R$ is

$4$ moles of an ideal gas is at $0^\circ C$. At constant pressure,it is heated to double its volume. Its final temperature will be ...... $^\circ C$.

The $PV$ versus $T$ graph for equal masses of $H_2$,$He$,and $O_2$ is shown in the figure. Choose the correct alternative.

$A$ cylinder contains $10 \, kg$ of gas at a pressure of $10^7 \, N/m^2$. What amount of gas in $kg$ must be removed so that the final pressure becomes $2.5 \times 10^6 \, N/m^2$?