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

Options

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

Answer

D

Explanation

`U=(a)/(x^(12))-(b)/(x^(6))=ax^(-12)-bx^(-6)`
`F=-(dU)/(dx)=-[a(-12)x^(-13)-b(-6)x^-7]=0`
(for equilibrium)
`x=((2a)/(b))^((1)/(6))`
At `xinfty`,`U(infty)=0`
`x=((2a)/(b))^((1)/(6))`
`U_(at eq)=(axxb^2)/(4a^2)-(bxxb)/(2a)=(b^2)/(4a)-(b^2)/(2a)=-(b^2)/(4a)`
`U_(infty)-U_(at eq)=(b^2)/(4a)`