Molecules in the atmosphere are attracted by the gravitational force of the Earth. Explain why all of them do not fall onto the Earth,just like an apple falling from a tree.

An ideal gas of molar mass $M$ is contained in a very tall vertical cylindrical column in a uniform gravitational field. Assuming the gas temperature to be $T$, the height at which the centre of gravity of the gas is located is (where $R$ is the universal gas constant).

$A$ closed cylindrical vessel contains $N$ moles of an ideal diatomic gas at a temperature $T$. On supplying heat,the temperature remains the same,but $n$ moles dissociate into atoms. The heat supplied is .........

$A$ rigid diatomic gas having molar mass $M$ is contained in an insulated container. The container is moving with velocity $V$. If it is stopped suddenly, the change in temperature is ($R$ - gas constant).

The figure shows a plot of $PV/T$ versus $P$ for $1.00 \times 10^{-3} \; kg$ of oxygen gas at two different temperatures.
$(a)$ What does the dotted plot signify?
$(b)$ Which is true: $T_{1} > T_{2}$ or $T_{1} < T_{2}$?
$(c)$ What is the value of $PV/T$ where the curves meet on the $y$-axis?
$(d)$ If we obtained similar plots for $1.00 \times 10^{-3} \; kg$ of hydrogen,would we get the same value of $PV/T$ at the point where the curves meet on the $y$-axis? If not,what mass of hydrogen yields the same value of $PV/T$ (for the low-pressure,high-temperature region of the plot)?
(Molecular mass of $H_{2} = 2.02 \; u$,of $O_{2} = 32.0 \; u$,$R = 8.31 \; J \; mol^{-1} K^{-1}$.)

Match Column $- I$ and Column $- II$ and choose the correct match from the given choices.

Column $- I$	Column $- II$
$(A)$ Root mean square speed of gas molecules	$(P)$ $\frac{1}{3} n m \bar{v}^{2}$
$(B)$ Pressure exerted by ideal gas	$(Q)$ $\sqrt{\frac{3 RT}{M}}$
$(C)$ Average kinetic energy of a molecule	$(R)$ $\frac{5}{2} RT$
$(D)$ Total internal energy of $1$ mole of a diatomic gas	$(S)$ $\frac{3}{2} k_{B} T$