Which of the following graphs represents the variation of $\beta = -(dV/dP)$ with pressure $P$ for an ideal gas kept at a constant temperature?

$A$ flask contains argon and chlorine in the ratio of $2: 1$ by mass. The temperature of the mixture is $27\,^{\circ}C$. Obtain the ratio of
$(i)$ average kinetic energy per molecule,and
$(ii)$ root mean square speed $v_{rms}$ of the molecules of the two gases.
Atomic mass of argon $= 39.9\,u$; Molecular mass of chlorine $= 70.9\,u$.

Initially,a gas of diatomic molecules is contained in a cylinder of volume $V_{1}$ at a pressure $P_{1}$ and temperature $250\, K$. Assuming that $25\%$ of the molecules get dissociated,causing a change in the number of moles. The pressure of the resulting gas at temperature $2000\, K$,when contained in a volume $2V_{1}$,is given by $P_{2}$. The ratio $\frac{P_{2}}{P_{1}}$ is:

The oxygen molecule has a mass of $5.30 \times 10^{-26} \; kg$ and a moment of inertia of $1.94 \times 10^{-46} \; kg \cdot m^{2}$ about an axis through its centre perpendicular to the line joining the two atoms. Suppose the mean speed of such a molecule in a gas is $500 \; m/s$ and that its kinetic energy of rotation is two-thirds of its kinetic energy of translation. Find the average angular velocity of the molecule.

According to the kinetic theory of gases,which of the following statements are correct?
$A$. The motion of the gas molecules freezes at $0^{\circ} C$.
$B$. The mean free path of gas molecules decreases if the density of molecules is increased.
$C$. The mean free path of gas molecules increases if temperature is increased keeping pressure constant.
$D$. Average kinetic energy per molecule per degree of freedom is $\frac{3}{2} k_{B} T$ (for monoatomic gases).
Choose the most appropriate answer from the options given below:

$STATEMENT-1$: The total translational kinetic energy of all the molecules of a given mass of an ideal gas is $1.5$ times the product of its pressure and its volume. Because
$STATEMENT-2$: The molecules of a gas collide with each other and the velocities of the molecules change due to the collision.