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

At equilibrium, `(dU)/(dx)=0`
`12ax^(-13)=6bx^(-7) rArr x^(6)=(2a)/(b)`
`x=((2a)/(b))^(1/6)`
`U(x)=a/(x^(12))-b/x^(6)`
`=(a)/((2a)/(b))^(2)-(b)/((2a)/(b))=(ab^(2))/(4a^(2))-b^(2)/(2a)=-b^(2)/(4a)`
`U(x=oo)=0`
So, `[U(x=oo)-U ("at equilibrium"]=b^(2)/(4a)`