If the potential energy between two molecules is given by U = -A r 6 + B r 12, then at equilibrium, separation between molecules, and the potential energy are

(A) \left(\frac{ B }{ A }\right)^{1 / 6}, 0

(B) \left(\frac{ B }{2 A }\right)^{1 / 6},-\frac{ A ^{2}}{2 B }

(D) \left(\frac{2 B }{ A }\right)^{1 / 6},-\frac{ A ^{2}}{2 B }

JEE Mains · Physics · STD 11 - Work,energy,power and collision

If the potential energy between two molecules is given by \(U =\frac{-A }{ r ^{6}}+\frac{ B }{ r ^{12}},\) then at equilibrium, separation between molecules, and the potential energy are

Pick an option to check your answer.

Verified Solution

Correct Answer

(C) \(\left(\frac{2 B }{ A }\right)^{1 / 6},-\frac{ A ^{2}}{4 B }\)

Detailed explanation

\(U =\frac{- A }{ r ^{6}}+\frac{ B }{ r ^{12}}\)

\(F =-\frac{ dU }{ dr }=-\left( A \left(-6 r ^{-7}\right)\right)+ B \left(-12 r ^{-13}\right)\)

\(0=\frac{6 A }{ r ^{7}}-\frac{12 B }{ r ^{13}}\)

\(\frac{6 A }{12 B }=\frac{1}{ r ^{6}} \Rightarrow r =\left(\frac{2 B }{ A }\right)^{1 / 6}\)

\(U \left( r =\left(\frac{2 B }{ A }\right)^{1 / 6}\right)=-\frac{ A }{2 B / A }+\frac{ B }{4 B ^{2} / A ^{2}}\)

\(=\frac{- A ^{2}}{2 B }+\frac{ A ^{2}}{4 B }=\frac{- A ^{2}}{4 B }\)

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