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 b are positive constants and x is the distance between the atoms. The atoms is in stable equilibrium when

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

Answer 1

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

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

$U\left(x\right)=\frac{a}{k^{12}}-\frac{b}{x^6}$

The atom is in stable equilibrium. When,

condition for stable equilibrium

$F=-\frac{dU}{dx}=0$

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

or, $-12{ax}^{-13}+6b{x}^{-7}=0$

or, $\frac{+12a}{x^{13}}=\frac{6b}{x^7}$

or, $\frac{2a}{b}=x^6$

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