1. Root Mean Square Speed ($v_{rms}$): The formula for the RMS speed of an ideal gas is $v_{rms} = \sqrt{\frac{3RT}{M}}$. Thus, (P) matches with (2).
2. Pressure ($P$): According to the kinetic theory, pressure is given by $P = \frac{1}{3}nm\bar{v}^2$, where $n$ is number density and $m$ is mass of a molecule. Thus, (Q) matches with (1).
3. Average Kinetic Energy per Molecule ($E$): The average translational kinetic energy per molecule depends only on temperature and is given by $E = \frac{3}{2}k_BT$. Thus, (R) matches with (4).
4. Internal Energy of a Diatomic Gas ($U$): A diatomic gas has 5 degrees of freedom ($f=5$) at moderate temperatures (3 translational + 2 rotational). The internal energy per mole is $U = \frac{f}{2}RT = \frac{5}{2}RT$. Thus, (S) matches with (3).

Conclusion: P-2, Q-1, R-4, S-3.