$A$ spherical balloon of radius $3 \ cm$ containing helium gas has a pressure of $48 \times 10^{-3} \ bar$. At the same temperature,the pressure of a spherical balloon of radius $12 \ cm$ containing the same amount of gas will be........ $\times 10^{-6} \ bar$.

$A$ gas at $5 \, bar$ pressure is heated from $0 \, ^\circ C$ to $546 \, ^\circ C$ and its volume is reduced to one-third of its initial volume. Calculate the final pressure. (in $, bar$)

If $20 \ cm^3$ of gas at $1 \ atm$ is expanded to $50 \ cm^3$ at constant $T$,what is the final pressure?

$100 \, g$ of an ideal gas is kept in a cylinder of $416 \, L$ volume at $27^{\circ} C$ under $1.5 \, bar$ pressure. The molar mass of the gas is $.... \, g \, mol^{-1}$. (Nearest integer) (Given : $R = 0.083 \, L \, bar \, K^{-1} \, mol^{-1}$)

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 $m^3$)

Derive the relation between density $(d)$ and molar mass $(M)$ for an ideal gas.