50. The interatction energy E between two spherical particles is plotted as a function of the distance (r)
between them. When r
repulsive. When r≥a, they attract via van der Waals attraction. Which one of the following plots
correctly represents the interaction energy between the above two particles?

Introduction

The interaction energy between two spherical particles changes with their separation distance. When particles move too close to each other (r < a), they experience strong repulsion due to electron cloud overlap. At larger distances (r ≥ a), van der Waals forces dominate and cause attraction. Graphing this behavior helps us visualize when forces push particles apart or pull them together. Let’s analyze the interaction energy trend and identify the correct plot.

Understanding the Concept

📌 Given:

• For r < a → Net force is repulsive, so interaction energy is positive.

• For r ≥ a → They attract via van der Waals forces, so interaction energy becomes negative and gradually approaches zero at long distances.

⚠ Positive energy ⇒ repulsion
✔ Negative energy ⇒ attraction

This shape resembles the well-known Lennard-Jones potential.

Correct Option: (D)

Option (D) shows:

• Steep positive energy at very small r (strong repulsion)

• Energy minimum below zero at r ≈ a (attractive region)

• Energy slowly approaching zero as r → ∞ (van der Waals tail)

This is exactly what the statement describes.

Why the Other Options Are Incorrect

❌ Option (A)

• Shows a minimum at r < a, suggesting attraction before repulsion.

• But problem states that for all r < a, the interaction is repulsive.

❌ Option (B)

• Only shows positive energy, gradually decreasing but never becoming negative.

• No negative region ⇒ no attraction, contradicts van der Waals behavior.

❌ Option (C)

• Shows negative energy initially, then rising.

• Implies attraction even when particles are extremely close.

• Opposite of stated strong repulsion at r < a.

Final Answer

✔ The correct plot representing the interaction energy as a function of distance is Option (D).

⭐ Key Takeaway

• Short distances → strong repulsion due to overlapping electron clouds

• Intermediate distances (~a) → attraction due to van der Waals forces

• Large distances → particles no longer feel each other → energy → 0