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 system is in stable equilibrium when -

x = 0

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x = \frac{a}{2 b}

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x = {(\frac{2 a}{b})}^{1 / 6}

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x = (\frac{11 a}{5 b})

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Solution

The correct option is C $x = {(\frac{2 a}{b})}^{1 / 6}$
Given that, $U (x) = \frac{a}{x^{12}} - \frac{b}{x^{6}}$
We know, $F = - \frac{d u}{d x} = (- 12) a x^{- 13} - (- 6 b) x^{- 7} = 0$
or
$\frac{6 b}{x^{7}} = \frac{12 a}{x^{13}}$ or $x^{6} = 12 a / 6 b = 2 a / b$ or $x = {(\frac{2 a}{b})}^{1 / 6}$
$\frac{d^{2} U}{d x^{2}} = + v e a t x = {(\frac{2 a}{b})}^{1 / 6}$

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