Which of the following graphs correctly represents the variation of $\beta = - \left( \frac{dV}{dP} \right)/V$ with $P$ for an ideal gas at constant temperature?

The expansion of an ideal gas of mass $m$ at a constant pressure $P$ is given by the straight line $D$. Then the expansion of the same ideal gas of mass $2m$ at a pressure $P/2$ is given by the straight line

The $P-T$ diagram of an ideal gas having three different densities $\rho_1, \rho_2, \rho_3$ (in three different cases) is shown in the figure. Which of the following is correct?

What does the universal gas constant $R$ represent? Write its value and unit.

The figure shows two flasks connected to each other. The volume of flask $1$ is twice that of flask $2$. The system is filled with an ideal gas at temperatures $100\, K$ and $200\, K$ respectively. If the mass of the gas in flask $1$ is $m$,what is the mass of the gas in flask $2$?

Calculate the volume of vapour of $1\,g$ of water at $1\,atm$ pressure.