Question
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
✓
Answer
$U =\frac{- A }{ r ^{6}}+\frac{ B }{ r ^{12}}$
$F =-\frac{ dU }{ dr }=-\left( A \left(-6 r ^{-7}\right)\right)+ B \left(-12 r ^{-13}\right)$
$0=\frac{6 A }{ r ^{7}}-\frac{12 B }{ r ^{13}}$
$\frac{6 A }{12 B }=\frac{1}{ r ^{6}} \Rightarrow r =\left(\frac{2 B }{ A }\right)^{1 / 6}$
$U \left( r =\left(\frac{2 B }{ A }\right)^{1 / 6}\right)=-\frac{ A }{2 B / A }+\frac{ B }{4 B ^{2} / A ^{2}}$
$=\frac{- A ^{2}}{2 B }+\frac{ A ^{2}}{4 B }=\frac{- A ^{2}}{4 B }$
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