Question

The potential energy function for the force between two atoms in a diatomic molecule can be expressed approximately as $U\left(r\right)=\frac{a}{r^{12}}-\frac{b}{r^6}$, where $a$ and $b$ are constants and $r$ is the separation between the atoms.
(a) Determine the force function $F\left(r\right)$.
(b) Find the value of for which the molecule will be in the stable equilibrium.

Answer 1

The potential energy function between the two atoms of diatomic molecules is

$u\left(r\right)=\frac{a}{r{}^{\prime 2}}-\frac{b}{r^6}$

(a) The force function $F\left(r\right)$ is calculate as

$\begin{array}{c}u\left(r\right)=\frac{a}{r{}^{\prime 2}}-\frac{b}{r^6}\\ F\left(r\right)=\frac{du\left(r\right)}{dr}\\ =\frac{d}{dr}\left(\frac{a}{r{}^{\prime 2}}-\frac{b}{r^6}\right)\\ F\left(r\right)=-12a{r}^{-13}+6r^{-7}\end{array}$

(b) The molecule will be in stable equation $F\left(r\right)$=0

$\begin{array}{c}-12r^{-13}+6r^{-7}=0\\ -12r^{-13}=-6r^{-7}\\ 2r^{-13}-r^{-7}=0\\ r^{-7}\left(2r^{-6}-1\right)=0\\ 2r^{-6}=1\\ \frac{2}{r^6}=1\\ r^6=2\\ r={\left(2\right)}^{\frac{1}{6}}\end{array}$