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 constant and 'x' is the distance between atoms. Find the value of 'x' at which force is zero and minimum P.E at that point

In a molecule, the potential energy between two atoms is given by U(x)

1.	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 constant and 'x' is the distance between atoms. Find the value of 'x' at which force is zero and minimum P.E at that point
Answer» Solution :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 (12a)/(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))/(2a)`