Question

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

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

Answer 1

The correct option is D $x=\sqrt[6]{6\frac{2a}{b}}$

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

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

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