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 positive constants and $x$ is the distance between the atoms. The atom is in stable equilibrium when

• A. $x={\left(\frac{2a}{b}\right)}^{\frac{1}{6}}$
• B. $x={\left(\frac{11a}{5b}\right)}^{\frac{1}{6}}$
• C. $x=0$
• D. $x={\left(\frac{a}{2b}\right)}^{\frac{1}{6}}$

Answer 1

The correct option is A $x={\left(\frac{2a}{b}\right)}^{\frac{1}{6}}$

Using Lennard jones potential graph,

Atoms will be in stable equilibrium when potential energy between them will be minimum.
Finding the minima for stable points
$\frac{dU}{dx}=0$ (for minima)
$\frac{dU}{dx}=\frac{(\frac{a}{x^{12}}-\frac{b}{x^6})}{dx}=\frac{-12a}{x^{13}}+\frac{6b}{x^7}$
On solving further,
$\frac{2a}{b}=x^6$
$x=(\frac{2a}{b}{)}^{1/6}$