The potential energy between two atoms in a molecule is given by U(x) = a x^ 12 - 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

The potential energy between two atoms in a molecule is given by …

MCQ

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

A
$x = \sqrt[6]{{\frac{{11a}}{{5b}}}}$
B
$x = \sqrt[6]{{\frac{a}{{2b}}}}$
C
$x = 0$
✓
$x = \sqrt[6]{{\frac{{2a}}{b}}}$

✓

Answer

Correct option: D.

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

d
(d)Condition for stable equilibrium $F = - \frac{{dU}}{{dx}} = 0$

$⇒$ $ - \frac{d}{{dx}}\left[ {\frac{a}{{{x^{12}}}} - \frac{b}{{{x^6}}}} \right] = 0$

$⇒$ $ - 12a{x^{ - 13}} + 6b{x^{ - 7}} = 0$

$⇒$ $\frac{{12a}}{{{x^{13}}}} = \frac{{6b}}{{{x^7}}}$

$⇒$ $\frac{{2a}}{b} = {x^6}$

$⇒$ $x = \sqrt[6]{{\frac{{2a}}{b}}}$

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