$A$ balloon contains $500 \ m^3$ of helium at $27^{\circ}C$ and $1 \ atm$ pressure. The volume of the helium at $-3^{\circ}C$ temperature and $0.5 \ atm$ pressure will be ..... $m^3$.

One mole of a perfect gas in a cylinder fitted with a piston has a pressure $P,$ volume $V,$ and temperature $T.$ If the temperature is increased by $1 \, K$ keeping pressure constant,the increase in volume is

Absolute zero $(0\, K)$ is that temperature at which

The equation of an ideal gas in terms of pressure $(P)$,absolute temperature $(T)$,and density $(d)$ is:

Two containers $C_{1}$ and $C_{2}$ of volumes $V$ and $4V$ respectively,hold the same ideal gas and are connected by a thin horizontal tube of negligible volume with a valve which is initially closed. The initial pressures of the gas in $C_{1}$ and $C_{2}$ are $p$ and $5p$,respectively. Heat baths are employed to maintain the temperatures in the containers at $300 \, K$ and $400 \, K$,respectively. The valve is now opened. Select the correct statement.

Which of the following shows the correct relationship between the pressure $P$ and density $\rho$ of an ideal gas at constant temperature?