$A$ thin tube sealed at both ends is $100 \ cm$ long. It lies horizontally, the middle $20 \ cm$ containing mercury and two equal ends containing air at standard atmospheric pressure. If the tube is now turned to a vertical position, by what amount will the mercury be displaced? (Given: cross-section of the tube can be assumed to be uniform) ........ $cm$

The equation of state for $5 \, g$ of oxygen at a pressure $P$ and temperature $T$,when occupying a volume $V$,will be (where $R$ is the gas constant).

At $NTP$,the mass of one litre of air is $1.293 \; g$. The value of the specific gas constant will be ...... $J/(K \cdot g)$.

$A$ perfect gas of a given mass is heated first in a small vessel and then in a large vessel,such that their volumes remain unchanged. The $P-T$ curves are

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

An engineer is given a fixed volume $V_m$ of metal with which to construct a spherical pressure vessel. Assuming the vessel has thin walls and is pressurized to its bursting point,the amount of gas the vessel can contain,$n$ (in moles),depends only on $V_m$ $(m^3)$,the temperature $T$ $(K)$,the ideal gas constant $R$ $(J/(K \cdot mol))$,and the tensile strength of the metal $\sigma$ $(N/m^2)$. Which of the following gives $n$ in terms of these parameters?