In a molecule, the potential energy between two atoms is given by `U (x) = (1)/(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 minimim `P.E` at that point.

Explanation

Force is zero `Rightarrow (du)/(dx) = 0`
`i.e., a(-12)x^(-13) - b(-6)x^(-7) = 0`
`(-12a)/(x^(13)) + (6b)/(x^(7)) = 0 Rightarrow (12a)/(x^(13)) = (6b)/(x^(7))`
`Rightarrow x^(6) = (2a)/b therefore x = ((2a)/(b))^(1/6)`
Substituting the value of x
`Rightarrow U_(min) = a((b)/(2a))^(12/6) - b((b)/(2a))^(6/6)`
`U_(min) = ((b^(2))/(4a)) - ((b^(2))/(2a)) Rightarrow U _(min) = (-b^(2))/(4a)`