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

Answer

B

Explanation

`U_((x)) =(a)/(x^(12)) - (b)/(x^(6))`
`F=-(U(x))/(dx)=+(12 a)/(x^(13)) - (6b)/(x^(7))`
At normal distance between the molecules, F=0
`:. +(12a)/(x^(13)) - (6b)/(x^(7))=0` or `x((2a)/(b))^(1//6)`
`U_((oo))=(a)/(oo^(12)) - (b)/(oo^(6))=0`
Dissociation energy of molecule,
`D=U_((x=oo)) - U_("at equilibrium")`
`=0-((a)/(x^(12))-(b)/(x^(6)))` where, `x=((2a)/(b))^(1//6)`
`D=(-a)/(x^(12))+(b)/(x^(6))=-(a)/((2a//b)^(12//6))+(b)/((2a//b)^(6//6))`
`=-(b^(2))/(4a)+(b^(2))/(2a) = (b^(2))/(4a)`