$A$ container $A$ contains a gas at pressure $P$,volume $V$,and temperature $T$. Another container $B$ contains a gas at pressure $2P$,volume $V/4$,and temperature $2T$. The ratio of the number of molecules in container $A$ to that in container $B$ is:

$A$ 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 $\,cm$)

For one gram mole of a gas,the value of $R$ in the equation $PV = RT$ is nearly ...... $cal/K$

One mole of an ideal gas passes through a process where pressure and volume obey the relation $P = P_0 \left[ 1 - \frac{1}{2} \left( \frac{V_0}{V} \right)^2 \right]$. Here $P_0$ and $V_0$ are constants. Calculate the change in the temperature of the gas if its volume changes from $V_0$ to $2V_0$.

The gas law $\frac{PV}{T} = \text{constant}$ is true for

If the pressure of an ideal gas contained in a closed vessel is increased by $0.5\%,$ the increase in temperature is $2 \, K.$ The initial temperature of the gas is ...... $^\circ C$