Question

The potential energy function for the force between two atoms in a diatomic molecule is approximately given by $U\left(x\right)=\frac{a}{x^{12}}-\frac{b}{x^6}$, where a and b are constants and x is the distance between the atoms. If the dissociation energy of the molecule is $d=\left[U\left(x=\infty \right)-U_{atequilibrium}\right],D$ is

• A. $\frac{b^2}{2a}$
• B. $\frac{b^2}{12a}$
• C. $\frac{b^2}{4a}$
• D. $\frac{b^2}{6a}$

Answer 1

Answer: (C)

$U\left(x\right)=\frac{a}{x^{12}}-\frac{b}{x^6}$
$U\left(x=\infty \right)=0$
as, $F=-\frac{dU}{dx}=-\left[\frac{12a}{x^{13}}+\frac{6b}{x^7}\right]$
at equilibrium, $F=0$
$\therefore x^6=\frac{2a}{b}$
$\therefore U_{atequilibrium}=\frac{a}{{\left(\frac{2a}{b}\right)}^2}-\frac{b}{\left(\frac{2a}{b}\right)}=\frac{-b^2}{4a}$
$\therefore D=\left[U\left(x=\infty \right)-U_{atequilibrium}\right]=\frac{b^2}{4a}$