Find the approximate number of molecules contained in a vessel of volume $7 \text{ litres}$ at $0^{\circ}C$ and a pressure of $1.3 \times 10^{5} \text{ Pa}$.

The pressure $(P)$ and temperature $(T)$ relationship of an ideal gas obeys the equation $PT^2 = \text{constant}$. The volume expansion coefficient of the gas will be:

$A$ cylinder containing gas at $27\,^{\circ}C$ is divided into two parts of equal volume,each $100\,cm^3$,and at equal pressure by a piston of cross-sectional area $10.85\,cm^2$. The gas in one part is raised in temperature to $100\,^{\circ}C$ while the other is maintained at the original temperature. The piston and walls are perfect insulators. How far will the piston move during the change in temperature? (in $cm$)

Two different isotherms representing the relationship between pressure $p$ and volume $V$ at a given temperature for the same ideal gas are shown for masses $m_1$ and $m_2$ of the gas respectively in the figure given. Then:

At constant pressure,the ratio of the increase in volume of an ideal gas per degree rise in Kelvin temperature to its original volume is $(T =$ absolute temperature of the gas$)$

$A$ vessel of volume $8\, L$ contains an ideal gas at $300\, K$ and $2\, atm$ pressure. The gas is allowed to leak until the pressure becomes $125\, kPa$. Calculate the number of moles that leaked out if the temperature remains constant.