For the interaction between two atoms in a diatomic molecule, the potential energy is approximately U(x) = a/x¹² - b/x⁶, where a and b are constants and x is the separation between the atoms. If the dissociation energy is defined as D = U(x = ∞) - U at the equilibrium separation, then D eq

Question

For the interaction between two atoms in a diatomic molecule, the potential energy is approximately U(x) = a/x¹² - b/x⁶, where a and b are constants and x is the separation between the atoms. If the dissociation energy is defined as D = U(x = ∞) - U at the equilibrium separation, then D equals:

Accepted Answer

b²/4a. Set dU/dx=0: -12a/x^13 + 6b/x^7 = 0 -> x^6 = 2a/b. Then U_eq = a/x^12 - b/x^6 = a(b/2a)^2 - b(b/2a) = b^2/4a - b^2/2a = -b^2/4a. Since U(infinity)=0, D = 0 - (-b^2/4a) = b^2/4a.

Solution

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