At $0\,^{\circ}\text{C}$,the value of the density of a fixed mass of an ideal gas divided by its pressure is $x$. At $100\,^{\circ}\text{C}$,this value will be

$A$ jar $P$ is filled with a gas having pressure,volume,and temperature $P, V, T$ respectively. Another gas jar $Q$ is filled with a gas having pressure $2P$,volume $\frac{V}{4}$,and temperature $2T$. The ratio of the number of molecules in jar $P$ to those in jar $Q$ is:

$A$ vessel of volume $8 \, L$ contains an ideal gas at $300 \, K$ and $2 \, atm$ pressure. The gas is allowed to leak until the pressure becomes $125 \, kPa$. Calculate the number of moles that leaked out if the temperature remains constant.

$A$ long cylindrical pipe of radius $20 \,cm$ is closed at its upper end and has an airtight piston of negligible mass as shown. When a $50 \,kg$ mass is attached to the other end of the piston,it moves down by a distance $\Delta l$ before coming to equilibrium. Assuming air to be an ideal gas,$\Delta l / l$ (see figure) is close to ($g=10 \,m/s^2$,atmospheric pressure is $10^5 \,Pa$):

Air is pumped into an automobile tube up to a pressure of $200\, kPa$ in the morning when the air temperature is $22^{\circ}C$. During the day,the temperature rises to $42^{\circ}C$ and the tube expands by $2\%$. The pressure of the air in the tube at this temperature will be approximately ...... $kPa$.

$O_2$ gas is filled in a vessel. If the pressure is doubled and the temperature becomes four times,how many times will its density become?