JEE Main 2020 · 6 September · morning · Single correct · +4 / −1

If the potential energy between two molecules is given by
U = - A/r⁶ + B/r¹²,
then at equilibrium, separation between molecules, and the potential energy are :

  1. (a) ( 2B/A )^(1/6), - A²/4B
  2. (b) ( 2B/A )^(1/6), - A²/2B
  3. (c) ( B/A )^(1/6),0
  4. (d) ( B/2A )^(1/6), - A²/2B
Answer and solution

Answer (a) ( 2B/A )^(1/6), - A²/4B

Solution

Physics sir · 4 steps

Equilibrium means zero force, so set dU/dr = 0, then put that r back into U.

  1. Force from energy. The force between the molecules is F = -dU/dr. At equilibrium the net force is zero, so dU/dr = 0.

    dU/dr = 6A/r⁷ - 12B/r¹³ = 0 ⇒ r⁶ = 2B/A

  2. Equilibrium separation. So r₀ = (2B/A)^(1/6). Keep r₀⁶ = 2B/A and r₀¹² = 4B²/A² ready for the next step.
  3. Energy at equilibrium. U = -A/(2B/A) + B/(4B²/A²) = -A²/2B + A²/4B = -A²/4B.
  4. So. The separation is (2B/A)^(1/6) and the potential energy is -A²/4B, which is option (a).

The idea to keep. For any U(r), equilibrium is where dU/dr = 0. Find r₀ from that, then substitute back and keep every term of U, not just the attractive one.

Answer from the NTA final answer key. Solution by PrepWiser.

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Asked about 90 times in JEE Main since 2020. This question is from 2020.