The potential energy (U) of a diatomic molecule is a function dependent on r (interatomic distance) as
U = α/r¹⁰ - β/r⁵ - 3
where, α and β are positive constants. The equilibrium distance between two atoms will be ( 2α/β )^(a/b), where a = ___________.

Solution

Physics sir · 4 steps

At equilibrium the force is zero, so set dU/dr = 0 and solve for r.

1. Equilibrium condition. The force between the atoms is F = -dU/dr. At the equilibrium distance this force is zero, so dU/dr = 0. The constant -3 disappears when you differentiate.

   dU/dr = -10α/r¹¹ + 5β/r⁶ = 0

2. Solve for r. This gives 5β/r⁶ = 10α/r¹¹. Multiply both sides by r¹¹ to get 5β r⁵ = 10α, so r⁵ = 2α/β.
3. Match the form. Taking the fifth root, r = (2α/β)^(1/5). Comparing with (2α/β)^(a/b) gives a = 1 and b = 5.
4. So. a = 1.

The common slip: 5. Stopping at r⁵ = 2α/β and reading the power 5 as a gives 5. You still have to take the fifth root, so the exponent is 1/5 and a = 1.

The idea to keep. Equilibrium in any potential energy curve means zero force, that is dU/dr = 0. Constants in U never affect where equilibrium is.

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