In a molecule, the potential energy between two atoms is given by `U(x)=(a)/(x^(12))-(b)/(x^(6))`. Where `a` and `b` are positive constants and `x` is the distance between atoms. Find the value of `x` at which force is zero and minimum P.E. at that point.

Explanation

Force is zero `rArr (dU)/(dx)=0`
i.e., `a(-12)x^(-13)-b(-6)x^(-7)=0`
`(-12a)/(x^(13))+(6b)/(x^(7))=0 rArr (12 a)/(x^(13))=(6b)/(x^(7))`
`rArr x^(6)=(2a)/(b) therefore x=[(2a)/(b)]^(1//6)`
`U_(min)=a((b)/(2a))^(12//6)-b((b)/(2a))^(6//6)`
`rArr U_(min)=(ab^(2))/(4a^(2))-(b^(2))/(2a)rArr U_(min)=(-b^(2))/(4a)`