First slide

Concept of potential energy

Question

The potential energy between two atoms in a molecule is given by $U (x) = \frac{a}{x^{12}} - \frac{b}{x^{6}}$ ; where a and b are positive constants and x is the distance between the atoms. The atom is in stable equilibrium when

Difficult

A

$x = \sqrt[6]{\frac{11 a}{5 b}}$
B

$x = \sqrt[6]{\frac{a}{2 b}}$
C

$x = 0$
D

$x = \sqrt[6]{\frac{2 a}{b}}$

Solution

Condition for stable equilibrium $F = - \frac{dU}{dx} = 0$
$\Rightarrow - \frac{d}{dx} [\frac{a}{x^{12}} - \frac{b}{x^{6}}] = 0 \Rightarrow - 12 {ax}^{- 13} + 6 {bx}^{- 7} = 0$
$\Rightarrow \frac{12 a}{x^{13}} = \frac{6 b}{x^{7}} \Rightarrow \frac{2 a}{b} = x^{6} \Rightarrow x = \sqrt[6]{\frac{2 a}{b}}$

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