JEE Main 2022 · 24 June · evening · Single correct · +4 / −1
Potential energy as a function of r is given by U = A/r¹⁰ - B/r⁵, where r is the interatomic distance, A and B are positive constants. The equilibrium distance between the two atoms will be :
- (a) ( A/B )^(1/5)
- (b) ( B/A )^(1/5)
- (c) ( 2A/B )^(1/5)
- (d) ( B/2A )^(1/5)
Answer and solution
Answer (c) ( 2A/B )^(1/5)
Solution
Physics sir · 4 steps
Equilibrium means zero force, and force is -dU/dr, so set the slope of U to zero and solve for r.
- Condition for equilibrium. The force between the atoms is F = -dU/dr. At equilibrium the net force is zero, so dU/dr = 0.
- Differentiate. Use d/dr(r⁻ⁿ) = -n r⁻ⁿ⁻¹ on each term. The powers 10 and 5 come down as factors.
dU/dr = -10A/r¹¹ + 5B/r⁶ = 0
- Solve for r. This gives 10A/r¹¹ = 5B/r⁶. Multiply both sides by r¹¹: 10A = 5B r⁵, so r⁵ = 2A/B.
- So. The equilibrium distance is r = (2A/B)^(1/5), which is option (c).
The idea to keep. Equilibrium is where the force vanishes, that is, where dU/dr = 0. It is not where U = 0. Always differentiate, and keep the powers that come down as coefficients.
Answer from the NTA final answer key. Solution by PrepWiser.