Question

Match the LIST-I with LIST-II 

LIST-I (Parameters)	LIST-II (Expressions)
A. Potential energy of a system of two atoms	I. \( \alpha = \sum_{j} \pm \frac{1}{p_{ij}} \)
B. Madelung constant	II. \( U_{ij} = \pm \frac{q^2}{r} \)
C. Coulomb electrostatic energy	III. \( U = -\frac{A}{r^2} + \frac{B}{r^{10}} \)
D. Cohesive energy	IV. \( U_{ij} = \lambda e^{-r_{ij}/\rho} \pm \frac{q^2}{r_{ij}} \)

Choose the correct answer from the options given below:

• A. A - II, B - I, C - III, D - IV
• B. A - III, B - II, C - I, D - IV
• C. A - I, B - II, C - IV, D - III
• D. A - III, B - I, C - II, D - IV

Hint:

Start with the most fundamental and easily recognizable pairs. The Coulomb energy (C-II) and the general form of interatomic potential (A-III) are often the easiest to identify, which can help eliminate incorrect options quickly.

Answer 1

The Correct Option is D

Step 1: Concept Overview:
This question tests your understanding of solid-state physics by requiring you to match key concepts with their corresponding mathematical expressions.
Step 2: Detailed Explanation:
A. Potential energy of two atoms: The potential energy \(U\) between two atoms includes both attractive and repulsive forces. Expression III, \( U = -\frac{A}{r^2} + \frac{B}{r^{10}} \), represents this balance (characteristic of Mie or Lennard-Jones potentials). Therefore, A corresponds to III.
B. Madelung constant: The Madelung constant (\(\alpha\)) quantifies the electrostatic interaction energy in a crystal lattice. It's calculated as a sum over lattice points. Expression I, \( \alpha = \sum_{j} \pm \frac{1}{p_{ij}} \), where \(p_{ij}\) is the normalized distance, defines the Madelung constant. Therefore, B corresponds to I.
C. Coulomb electrostatic energy: Coulomb's law describes the electrostatic potential energy between point charges. Expression II, \( U_{ij} = \pm \frac{q^2}{r} \), represents this energy between two ions with charge \(q\) at a distance \(r\). Therefore, C corresponds to II.
D. Cohesive energy: The cohesive energy of an ionic crystal is the total potential energy per ion pair at equilibrium. This includes Coulomb attraction and short-range repulsion. Expression IV, \( U_{ij} = \lambda e^{-r_{ij}/\rho} \pm \frac{q^2}{r_{ij}} \), models this total potential energy and is used to determine the cohesive energy. Therefore, D corresponds to IV.
Step 3: Solution:
The correct matches are A-III, B-I, C-II, D-IV.