The speed of sound in an ideal gas at a given temperature $T$ is $v$. The rms speed of gas molecules at that temperature is $v_{\text{rms}}$. The ratio of the velocities $v$ and $v_{\text{rms}}$ for helium and oxygen gases are $X$ and $X^{\prime}$,respectively. Then,$\frac{X}{X^{\prime}}$ is equal to

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 $= 2 \; \mathring{A}$)

Assertion $(A):$ 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 volume.
Reason $(R):$ The molecules of gas collide with each other and the velocities of the molecules change due to the collision.

$A$ thermally insulated vessel contains an ideal gas of molecular mass $M$ and ratio of specific heats $\gamma$. It is moving with speed $v$ and it is suddenly brought to rest. Assuming no heat is lost to the surroundings,its temperature increases by

At constant pressure, equal amounts of heat are supplied to a monatomic gas and a diatomic gas separately. The ratio of the increases in internal energies of the two gases 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.