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))=0rArr(12a)/(x^(13))=(6b)/(x^(7))`
`rArrx^(6)=(2a)/(b)thereforex=[(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)rArrU_(min)=(-b^(2))/(4a)`