Which graph represents the molar heat capacity at constant volume $(C_V)$ for a monoatomic gas?

For the equation $\beta = - \frac{1}{V} \left( \frac{dV}{dP} \right)$, identify the correct graph of $\beta$ versus $P$ for an ideal gas at constant temperature.

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).

$2.8 \ g$ of nitrogen gas $(N_2)$ is in a vessel at a temperature of $127^{\circ} C$. The amount of heat energy required to increase the rms speed of the nitrogen molecules by $41.4 \%$ is $(R = 8.31 \ J \ mol^{-1} \ K^{-1})$. (in $J$)

Select the correct statement.

In the kinetic theory of gases,which of these statements is/are true?
$(i)$ The pressure of a gas is proportional to the mean speed of the molecules.
$(ii)$ The root mean square speed of the molecules is proportional to the pressure.
$(iii)$ The rate of diffusion is proportional to the mean speed of the molecules.
$(iv)$ The mean translational kinetic energy of a gas is proportional to its kelvin temperature.