For an ideal gas with pressure $P$,temperature $T$,and volume $V$,where the mass of each molecule is $m$,what is the density of the gas? ($k$ is the Boltzmann constant)

For $H_2, O_2$,and $He$ gases at the same temperature and pressure,the volumes are equal. Which of the following has the highest number of atoms?

$A$ gas at $27^{\circ}C$ has a volume $V$ and pressure $P$. On heating,its pressure is doubled and volume becomes three times. The resulting temperature of the gas will be ...... $^{\circ}C$.

$A$ flask containing air at $27\,^{\circ}C$ at atmospheric pressure is corked up. $A$ pressure of $2.5\,atm$ would force the cork out. The temperature at which it happens is

$A$ container holds $10 \ kg$ of gas at a pressure of $10^7 \ N/m^2$. How much gas in $kg$ must be removed so that the final pressure becomes $2.5 \times 10^6 \ N/m^2$? (Assume temperature remains constant.)

$A$ cylinder containing gas at $27\,^{\circ}C$ is divided into two parts of equal volume,each $100\,cm^3$,and at equal pressure by a piston of cross-sectional area $10.85\,cm^2$. The gas in one part is raised in temperature to $100\,^{\circ}C$ while the other is maintained at the original temperature. The piston and walls are perfect insulators. How far will the piston move during the change in temperature? (in $cm$)