Atomic Energy Dependencies Challenge

Test your understanding of how atomic phenomena scale with atomic number

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.

List-I

(P) \( E \propto Z^2 \)

(Q) \( E \propto (Z - 1)^2 \)

(R) \( E \propto Z(Z - 1) \)

(S) \( E \) is practically independent of \( Z \)

List-II

(1) energy of characteristic x-rays

(2) electrostatic part of the nuclear binding energy for stable nuclei with mass numbers in the range 30 to 170

(3) energy of continuous x-rays

(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

(A) P→4, Q→3, R→1, S→2

(B) P→5, Q→2, R→1, S→4

(C) P→5, Q→1, R→2, S→4

(D) P→3, Q→2, R→1, S→5

Solution

The correct answer is C) P→5, Q→1, R→2, S→4.

Explanation:

P → 5: The energy of radiation from hydrogen-like atoms is given by \( E \propto Z^2 \) (from Bohr's model).

Q → 1: The energy of characteristic x-rays follows Moseley's law: \( E \propto (Z - 1)^2 \), where the -1 accounts for shielding by the remaining electron in the K-shell.

R → 2: The electrostatic part of nuclear binding energy (Coulomb energy) is proportional to \( Z(Z - 1) \) because it depends on the pairwise interactions between protons.

S → 4: The average nuclear binding energy per nucleon is nearly constant (independent of Z) for stable nuclei in the given mass range, which is why the curve of binding energy per nucleon is relatively flat in this region.

Energy Dependencies Visualization

Select a phenomenon to see how its energy scales with atomic number:

Hydrogen-like transitions (E ∝ Z²) Characteristic X-rays (E ∝ (Z-1)²) Nuclear Coulomb energy (E ∝ Z(Z-1)) Binding energy per nucleon (E independent of Z)

Note: This is a simplified representation for educational purposes.

💡 Did you know? The semi-empirical mass formula (Weizsäcker formula) describes nuclear binding energy with terms that include both the Z² dependence (Coulomb term) and terms that are nearly independent of Z (volume and surface terms). This explains why option S matches with the average binding energy per nucleon!