The density of water is $1000 \; kg \; m^{-3}$. The density of water vapour at $100 \; ^{\circ}C$ and $1 \; atm$ pressure is $0.6 \; kg \; m^{-3}$. The volume of a molecule multiplied by the total number gives what is called molecular volume. What is the average distance (in $\mathring{A}$) between atoms (interatomic distance) in water? (Radius of a water molecule $= 2 \; \mathring{A}$)

One mole of a monoatomic ideal gas undergoes a quasistatic process, which is depicted by a straight line joining points $(V_{0}, T_{0})$ and $(2 V_{0}, 3 T_{0})$ in a $V-T$ diagram. What is the value of the heat capacity of the gas at the point $(V_{0}, T_{0})$?

$A$ container holds $10^{26} \text{ molecules/m}^3$ each of mass $3 \times 10^{-27} \text{ kg}$. Assume that $1/6$ of the molecules move with velocity $2000 \text{ m/s}$ directly towards one wall of the container while the remaining $5/6$ of the molecules move either away from the wall or in perpendicular direction,and all collisions of the molecules with the wall are elastic.

At a temperature of $300 \ K$,the average translational kinetic energy and $rms$ speed of a sample of oxygen gas are $6.21 \times 10^{-21} \ J$ and $484 \ m/s$ respectively. At $600 \ K$,these values will be respectively: (Assume ideal gas behavior)

$A$ horizontal uniform glass tube of $100 \, cm$ length, sealed at both ends, contains a $10 \, cm$ mercury column in the middle. The temperature and pressure of the air on either side of the mercury column are $81^{\circ} C$ and $76 \, cm$ of mercury, respectively. If the air column at one end is kept at $0^{\circ} C$ and the other end at $273^{\circ} C$, the pressure of the air which is at $0^{\circ} C$ is (in $cm$ of $Hg$):

$A$ vessel contains $14\,g$ of nitrogen gas at a temperature of $27^{\circ}\,C$. The amount of heat to be transferred to the gas to double the r.m.s. speed of its molecules will be $......J$ (Take $R = 8.32\,J\,mol^{-1}K^{-1}$)