For an ideal gas,the number of moles per litre in terms of its pressure $P$,gas constant $R$,and temperature $T$ is:

$2 \times 10^6$ molecules of $N_2$ gas enter into a vessel having a volume of $400 \, mL$ at $400 \, K$ temperature. Find the pressure in $atm$ and $bar$. $[R = 0.082 \, L \, atm \, mol^{-1} \, K^{-1}]$,$[1 \, atm = 1.013 \, bar]$

Under what conditions will a pure sample of an ideal gas not only exhibit a pressure of $1 \, atm$ but also a concentration of $1 \, mol \, L^{-1}$ $(R = 0.082 \, L \, atm \, mol^{-1} \, K^{-1})$?

Define the gas constant $R$ in terms of work.

An ideal gas weighing $9.0 \ g$ occupies a volume of $8.2 \ L$ at $300 \ K$ and $1 \ atm$ pressure. The molar mass of the gas is .....

An $LPG$ (Liquefied Petroleum Gas) cylinder weighs $15.0 \ kg$ when empty. When full, it weighs $30.0 \ kg$ and shows a pressure of $3.0 \ atm$. In the course of usage at $27^{\circ}C$, the mass of the full cylinder is reduced to $24.2 \ kg$. The volume of the used gas in cubic metre at the normal usage condition ($1 \ atm$ and $27^{\circ}C$) is (assume $LPG$ to be normal butane and it behaves ideally). (in $m^3$)