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

To find the correct match between Column - I and Column - II, let us analyze each physical quantity step-by-step:

(A) Root mean square speed of gas molecules:
The root mean square speed ($v_{\mathrm{rms}}$) of the molecules of an ideal gas is defined as the square root of the average of the squares of the speeds of the individual molecules. From the kinetic theory of gases, it is given by the formula:
$v_{\mathrm{rms}}=\sqrt{\frac{3RT}{M}}$
where $R$ is the universal gas constant, $T$ is the absolute temperature, and $M$ is the molar mass of the gas.
Therefore, (A) matches with (Q).

(B) Pressure exerted by ideal gas:
According to the kinetic theory of gases, the pressure $P$ exerted by an ideal gas on the walls of its container is due to the collisions of gas molecules. It is given by:
$P=\frac{1}{3}nm{\overline{v}}^2$
where $n$ is the number density of the molecules (number of molecules per unit volume), $m$ is the mass of a single molecule, and ${\overline{v}}^2$ is the mean square speed of the gas molecules.
Therefore, (B) matches with (P).

(C) Average kinetic energy of a molecule:
According to the law of equipartition of energy, the average kinetic energy associated with each degree of freedom of a molecule is $\frac{1}{2}{k}_{B}T$. For a monoatomic gas molecule (or considering only the translational kinetic energy of any molecule which has 3 translational degrees of freedom), the average kinetic energy is:
$K_{\mathrm{avg}}=\frac{3}{2}{k}_{B}T$
where $k_B$ is the Boltzmann constant and $T$ is the absolute temperature.
Therefore, (C) matches with (S).

(D) Total internal energy of 1 mole of a diatomic gas:
The internal energy $U$ of $\mu$ moles of an ideal gas is given by:
$U=\frac{f}{2}\mu RT$
where $f$ is the degrees of freedom. A diatomic gas has 5 degrees of freedom at normal temperatures (3 translational and 2 rotational). For $\mu =1$ mole:
$U=\frac{5}{2}RT$
Therefore, (D) matches with (R).

Combining these matches, we get:
(A) - (Q), (B) - (P), (C) - (S), (D) - (R)