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

The correct option is (A) - (Q), (B) - (P), (C) - (S), (D) - (R).

Let us analyze each entry given in Column - I and match it with the correct kinetic theory expression in Column - II based on the image provided:

1. Item (A): Root mean square speed of gas molecules
The root mean square speed ($v_{\text{rms}}$) of an ideal gas molecule is given by the kinetic theory formula:

$v_{\text{rms}}=\sqrt{\frac{3RT}{M}}$

Where:
- $R$ is the universal gas constant.
- $T$ is the absolute temperature.
- $M$ is the molar mass of the gas.
Therefore, (A) matches with (Q).

2. Item (B): Pressure exerted by ideal gas
According to the kinetic theory of gases, the pressure ($P$) exerted by a gas is given by:

$P=\frac{1}{3}nm{\overline{v}}^2$

Where:
- $n$ is the number of molecules per unit volume.
- $m$ is the mass of each molecule.
- ${\overline{v}}^2$ is the mean square speed of the molecules.
Therefore, (B) matches with (P).

3. Item (C): Average kinetic energy of a molecule
The average translational kinetic energy ($E$) of a single gas molecule at temperature $T$ is given by:

$E=\frac{3}{2}{k}_{\text{B}}T$

Where $k_{\text{B}}$ is Boltzmann's constant.
Therefore, (C) matches with (S).

4. Item (D): Total internal energy of 1 mole of a diatomic gas
A diatomic gas molecule has $f=5$ degrees of freedom at room temperature (3 translational + 2 rotational).
The total internal energy ($U$) for $n=1$ mole of gas is given by:

$U=\frac{f}{2}RT=\frac{5}{2}RT$

Therefore, (D) matches with (R).

Combining all the individual matches:
(A) → (Q)
(B) → (P)
(C) → (S)
(D) → (R)