The potential energy of two atoms in a molecule can sometimes be approximated by the Morse function, U (r) = ARe (R-01.5 - 1) 2 where r is the distance between the two atoms and A, R, and S are positive constants with S << R. Sketch this function for 0 < r < cc. Find the equilibrium separation rip, at which U (r) is minimum. Now write r = rox so that x is the displacement from equilibrium, and show that, for small displacements, U has the approximate form U = const kx 2. That is, Hooke's law applies. What is the force constant k?

Question

Hayley Daniel

Physics

18 October, 17:57

The potential energy of two atoms in a molecule can sometimes be approximated by the Morse function, U (r) = ARe (R-01.5 - 1) 2 where r is the distance between the two atoms and A, R, and S are positive constants with S << R. Sketch this function for 0 < r < cc. Find the equilibrium separation rip, at which U (r) is minimum. Now write r = rox so that x is the displacement from equilibrium, and show that, for small displacements, U has the approximate form U = const kx 2. That is, Hooke's law applies. What is the force constant k?

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Answers (1)

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Jayda Randall · Answer 1

The constant Hooke law is

k = 2 * A / S

Explanation:

To the force constant k Hooke law can use the Morse function as a:

U (r) = A * [ e ⁽ ᵇ ⁻ ⁿ / ˣ ⁾ - 1 ] ² - 1 ]

b = R, n = r₀, x = s

U (r) = A * [ e ⁽ ᵇ ⁻ ⁿ / ˣ ⁾ - 1 ] e ⁽ ᵇ ⁻ ⁿ / ˣ ⁾ * ( - ¹ / ₓ) = 0

e ⁽ ᵇ ⁻ ⁿ / ˣ ⁾ = 1

r₀ = R

r = r₀ + x, r = R + x

U (x) = - A + A / 2 * 2 / s * x²

U (x) = const + ¹/₂ * k * x²

Solve k'

k = 2 * A / S