Question

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.

• 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
• D. P \( \rightarrow \) 3, Q \( \rightarrow \) 2, R \( \rightarrow \) 1, S \( \rightarrow \) 5

Hint:

Use known empirical relationships: - \( Z^2 \) for hydrogen-like atoms, - \( (Z - 1)^2 \) for characteristic x-rays, - \( Z(Z - 1) \) for electrostatic nuclear binding, - and flat nuclear binding energy per nucleon in mid-mass nuclei.

Answer 1

The Correct Option is C

(P) \( E \propto Z^2 \) 
This is the energy dependence for hydrogen-like atoms (Bohr model). The energy of electronic transitions in such atoms varies as: \[ E_n = - \frac{Z^2}{n^2} \cdot \text{constant} \] \[ \Rightarrow \text{P} \rightarrow 5 \quad \text{(Energy of electronic transitions in hydrogen-like atoms)} \] (Q) \( E \propto (Z - 1)^2 \) 
This is the empirical formula for characteristic x-rays (Moseley’s law), accounting for screening by inner electrons: \[ E = a (Z - 1)^2 \Rightarrow \text{Q} \rightarrow 1 \] (R) \( E \propto Z(Z - 1) \) 
This is the electrostatic (Coulomb) part of nuclear binding energy between protons, modeled as: \[ E_{\text{Coulomb}} \propto \frac{Z(Z - 1)}{A^{1/3}} \Rightarrow \text{R} \rightarrow 2 \] (S) \( E \) practically independent of \( Z \) 
The average nuclear binding energy per nucleon for stable nuclei (mass number 30 to 170) is nearly constant, i.e., independent of \( Z \): \[ \Rightarrow \text{S} \rightarrow 4 \] 

• P \( \rightarrow \) 5
• Q \( \rightarrow \) 1
• R \( \rightarrow \) 2
• S \( \rightarrow \) 4