$A$ closed hollow insulated cylinder is filled with gas at $0^{\circ} C$ and also contains an insulated piston of negligible weight and negligible thickness at the middle point. The gas in one side of the piston is heated to $100^{\circ} C$. If the piston moves $5 \,cm$, the total length of the hollow cylinder is: (in $\,cm$)

$A$ thin tube sealed at both ends is $100 \ cm$ long. It lies horizontally, the middle $20 \ cm$ containing mercury and two equal ends containing air at standard atmospheric pressure. If the tube is now turned to a vertical position, by what amount will the mercury be displaced? (Given: cross-section of the tube can be assumed to be uniform) ........ $cm$

By using the ideal gas equation $PV = \mu RT$,write the value,unit,and dimension of $R$.

$A$ container contains $32 \,g$ of $O_2$ at a temperature $T$. The pressure of the gas is $P$. An identical container containing $4 \,g$ of $H_2$ at a temperature $2T$ has a pressure of ............

When the temperature of a gas at constant volume is increased by $5^{\circ}C$,its pressure increases by $0.5\%$ of its initial value. The initial temperature of the gas is..... $K$.

Two vessels $A$ and $B$ contain oxygen. The volume of $B$ is twice that of $A$,the pressure of $B$ is thrice that of $A$ and the temperature of $B$ is half of that of $A$. Then,find the ratio of number of molecules of oxygen in vessels $A$ and $B$.