Intermolecular Pair Potentials

A generalized intermolecular pair potential that describes the interactions between particles at the atomic level may be written as:

$U(r)=\frac{n}{n-m}(\frac{n}{m}{)}^{\frac{m}{n-m}}[(\frac{}{r}{)}^n-(\frac{}{r}{)}^m]$

Where $$ is the depth of the well $($indicative of the strength of the interaction$),$ is the distance at which the intermolecular potential between the two particles is zero, and $r$ is the distance between particles.

Arguably the most common intermolecular pair potential is the Lennard$-$Jones potential, which is also called the $"12-6"$ potential $($where $n=12$ and $m=6)$ which is used for computational convenience but may be less accurate than other potentials in certain conditions, like the $"9-3"$ potential, which is used to model interactions between particles of a fluid and solid particles of a flat wall.

a$)(15$ points$)$ Plot the $12-6$ and $9-3$ Lennard$-$Jones potentials for two argon atoms $(=3\mathrm{.}4($:$,=1\mathrm{.}7{10}^{-21}(J)\}.$ On your plot, clearly indicate the values of $$ and $$ and provide the distance at which the potential reaches its minimum, $r_{\text{m}\text{i}\text{n}}.$