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

Correct Answer: (A) - (Q), (B) - (P), (C) - (S), (D) - (R)

Let us analyze the entries in Column - I and find their matching expressions in Column - II from the given image:

1. (A) Root mean square speed of gas molecules:
The root mean square (rms) speed of gas molecules of molar mass $M$ at absolute temperature $T$ is given by the formula:
$v_{\text{rms}}=\sqrt{\frac{3RT}{M}}$
Comparing this with Column - II, it matches option (Q).
Therefore, (A) - (Q).

2. (B) Pressure exerted by an ideal gas:
According to the kinetic theory of gases, the pressure $P$ exerted by an ideal gas is given by:
$P=\frac{1}{3}nm{\overline{v}}^2$
Here, $n$ is the number density of the gas molecules (number of molecules per unit volume), $m$ is the mass of a single molecule, and ${\overline{v}}^2$ is the mean square speed.
Comparing this with Column - II, it matches option (P).
Therefore, (B) - (P).

3. (C) Average kinetic energy of a molecule:
The average translational kinetic energy of a single gas molecule at absolute temperature $T$ depends only on temperature and is given by:
$E=\frac{3}{2}{k}_{B}T$
Here, $k_B$ is the Boltzmann constant.
Comparing this with Column - II, it matches option (S).
Therefore, (C) - (S).

4. (D) Total internal energy of 1 mole of a diatomic gas:
A diatomic gas molecule has $5$ degrees of freedom (3 translational and 2 rotational) at moderate temperatures.
According to the law of equipartition of energy, the energy associated with each degree of freedom per mole of gas is $\frac{1}{2}RT$.
Thus, the total internal energy $U$ for $1$ mole of a diatomic gas is:
$U=5\times \left(\frac{1}{2}RT\right)=\frac{5}{2}RT$
Comparing this with Column - II, it matches option (R).
Therefore, (D) - (R).

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