The potential energy function for the force between tow atoms in a diatomic molecule is approximately given by `U(x)=(a)/(x^(12))-(b)/(x^(6))`, where a and b are constants and x is the distance between the atoms. If the dissociation energy of the molecule is `D=[U(x=oo)-U_("at equilibrium")],D` is

Options

A. `(b^2)/(2a)`
B. `(b^2)/(12a)`
C. `(b^2)/(4a)`
D. `(b^2)/(6a)`

Answer

C

Explanation

`U = a/(x^12) - b/(x^6)`
Force , `F = -(dU)/(dx) = -d/(dx) ((a)/(x^12) - (b)/(x^6)) = -[(-12a)/(x^(13)) + (6b)/(x^7)] = [(12a)/(x^13) - (6b)/(x^7)]`
At equilibrium , F = 0, `" " :. " " (12a)/(x^13) - (6b)/(x^7) = 0 or x^6 = (2a)/b`
`U_("atequlibrium") = a/(((2a)/(b))^(2)) - b/(((2a)/(b))) = (ab^2)/(4a^2) - (b^2)/(2a) = (b^2)/(4a) - (b^2)/(2a) = -(b^2)/(4a)`
`U_(x = oo) = 0, " "D = U_(x = oo) - U_("atequilibrium") = [0 - (-(b^2)/(4a))] = (b^2)/(4a)`.