List-I shows various functional dependencies of energy (E) on the atomic number (Z). Energies associated with certain phenomena are given in List-II.

Choose the option that describes the correct match between the entries in List-I to those in List-II.

To find the correct match between List-I and List-II, let us analyze each of the physical processes and their dependencies on the atomic number $Z$:

1. Entry (P): $E\propto Z^2$
According to Bohr's model of hydrogen-like atoms, the energy of an electron in the $n$-th orbit is given by:

$E_n=-13.6\times \frac{Z^2}{n^2}\text{eV}$

Therefore, the energy of radiation emitted during an electronic transition between two states is proportional to $Z^2$.
This matches entry (5) in List-II: energy of radiation due to electronic transitions from hydrogen-like atoms.
Hence, P → 5.

2. Entry (Q): $E\propto {(Z-1)}^2$
According to Moseley's law for characteristic X-rays (specifically ${\text{K}}_{\alpha }$ X-rays), the frequency $\nu$ of the emitted characteristic X-ray is given by:

$\nu \propto {(Z-b)}^2$

where $b$ is the screening constant (with $b\approx 1$ for the K-series). Since the photon energy is $E=h\nu$, we have:

$E\propto {(Z-1)}^2$

This matches entry (1) in List-II: energy of characteristic x-rays.
Hence, Q → 1.

3. Entry (R): $E\propto Z(Z-1)$
The electrostatic (Coulomb) potential energy of a nucleus is due to the mutual electrostatic repulsion between protons. If there are $Z$ protons in the nucleus, the number of proton pairs experiencing electrostatic repulsion is:

${}^Z\text{C}_2=\frac{Z(Z-1)}{2}$

Since the electrostatic energy is proportional to the number of interacting proton pairs, the electrostatic part of the nuclear binding energy is proportional to $Z(Z-1)$.
This matches entry (2) in List-II: electrostatic part of the nuclear binding energy for stable nuclei with mass numbers in the range 30 to 170.
Hence, R → 2.

4. Entry (S): $E$ is practically independent of $Z$
For stable nuclei in the mass number range $30\le A\le 170$, the average binding energy per nucleon is nearly constant (approximately $8\text{MeV/nucleon}$). Because the curve of binding energy per nucleon is highly flat in this region, the average nuclear binding energy per nucleon is practically independent of the atomic number $Z$.
This matches entry (4) in List-II: average nuclear binding energy per nucleon for stable nuclei with mass number in the range 30 to 170.
Hence, S → 4.

Combining all the matches:
P → 5, Q → 1, R → 2, S → 4