JEE Challenger
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PhyMedAdv 2025 • Paper 1

Functional Dependency of Energy on Atomic Number in Atomic and Nuclear Phenomena

List-I shows various functional dependencies of energy (EE) on the atomic number (ZZ). 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.

List-IList-II(P) E∝Z2(1) energy of characteristic x-rays(Q) E∝(Z−1)2(2) electrostatic part of the nuclear binding energy for stable nuclei with mass numbers in the range 30 to 170(R) E∝Z(Z−1)(3) energy of continuous x-rays(S) E is practically independent of Z(4) average nuclear binding energy per nucleon for stable nuclei with mass number in the range 30 to 170(5) energy of radiation due to electronic transitions from hydrogen-like atoms\begin{array}{ll} \textbf{List-I} & \textbf{List-II} \\ \text{(P) } E \propto Z^2 & \text{(1) energy of characteristic x-rays} \\ \text{(Q) } E \propto (Z - 1)^2 & \text{(2) electrostatic part of the nuclear binding energy for stable nuclei with mass numbers in the range 30 to 170} \\ \text{(R) } E \propto Z(Z - 1) & \text{(3) energy of continuous x-rays} \\ \text{(S) } E \text{ is practically independent of } Z & \text{(4) average nuclear binding energy per nucleon for stable nuclei with mass number in the range 30 to 170} \\ & \text{(5) energy of radiation due to electronic transitions from hydrogen-like atoms} \end{array}

Options

A

P →\rightarrow 4, Q →\rightarrow 3, R →\rightarrow 1, S →\rightarrow 2

B

P →\rightarrow 5, Q →\rightarrow 2, R →\rightarrow 1, S →\rightarrow 4

C

P →\rightarrow 5, Q →\rightarrow 1, R →\rightarrow 2, S →\rightarrow 4

Correct
D

P →\rightarrow 3, Q →\rightarrow 2, R →\rightarrow 1, S →\rightarrow 5

Topics & Concepts

AtomsBohr ModelX-Rays

Step-by-Step Solution

To find the correct match between List-I and List-II, we analyze the ZZ-dependence for each phenomenon:

  1. Energy of radiation due to electronic transitions from hydrogen-like atoms: According to the Bohr model, the energy levels of a hydrogen-like atom with atomic number ZZ are given by: En=−13.6 Z2n2 eVE_n = -\frac{13.6 \, Z^2}{n^2} \text{ eV} The transition energy between levels n1n_1 and n2n_2 is: ΔE=13.6 Z2(1n12−1n22)∝Z2\Delta E = 13.6 \, Z^2 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \propto Z^2 Therefore, (P) →\rightarrow (5).

  2. Energy of characteristic X-rays: According to Moseley's law, the frequency ν\nu (and thus the photon energy E=hνE = h\nu) of characteristic X-rays (like KαK_\alpha radiation) is given by: E∝ν∝(Z−b)2E \propto \nu \propto (Z - b)^2 For the KK-series, the screening constant is b=1b = 1, giving: E∝(Z−1)2E \propto (Z - 1)^2 Therefore, (Q) →\rightarrow (1).

  3. Electrostatic part of the nuclear binding energy: In the semi-empirical mass formula (liquid drop model), the Coulomb repulsion energy between ZZ protons in a nucleus with ZZ protons is proportional to the number of proton pairs, which is Z(Z−1)2\frac{Z(Z-1)}{2}: ECoulomb=−acZ(Z−1)A1/3∝Z(Z−1)E_{\text{Coulomb}} = -a_c \frac{Z(Z - 1)}{A^{1/3}} \propto Z(Z - 1) Therefore, (R) →\rightarrow (2).

  4. Average nuclear binding energy per nucleon: For stable nuclei in the mass number range 30≤A≤17030 \le A \le 170, the binding energy per nucleon (BEA)\left(\frac{\text{BE}}{A}\right) is nearly constant (around 8.5 MeV/nucleon8.5\text{ MeV/nucleon}) due to the saturation property of nuclear forces. Hence, it is practically independent of ZZ. Therefore, (S) →\rightarrow (4).

Matching the entries:

  • P→5\text{P} \rightarrow 5
  • Q→1\text{Q} \rightarrow 1
  • R→2\text{R} \rightarrow 2
  • S→4\text{S} \rightarrow 4

This corresponds to Option (C).

Functional Dependency of Energy on Atomic Number in Atomic and Nuclear Phenomena | Physics PYQ Solution - JEE Challenger