Adapted from problems $2\mathrm{.}18$ and $2\mathrm{.}19(40$ marks$)$ The attractive $(E_A)$ and repulsive $(E_R)$

potential energies are described as such, where $A,B,$ and $n$ are constants and $r$ is the inter$-$atomic radius:

$E_A=-\frac{A}{r},E_R=\frac{B}{r^n}$

c$.$ For a $K^{+}-C{l}^{-}$ion bond pair, we have $A=1\mathrm{.}436e\frac{V}{n}m,B=5\mathrm{.}86{10}^{-6}e\frac{V}{n}{m}^9,$ and $n=9.$ Plot

and identify $E_A,E_R,$ and $E_N$ for this system for $0\mathrm{.}2nmr0\mathrm{.}8nm$

d$.$ Based on the plot, estimate $r_0$ and $E_0$

e$.$ The equilibrium bond length $(r_0)$ and equilibrium bond energy $(E_0)$ occur at the bond length which

$minimizes{E}_N.$ Based on this knowledge, find an expression for $r_0$ and $E_0$ in terms of $A,B,$ and $n.$

Hint: remember optimization problems; the optimum occurs when the derivative is $0.$ Start by finding

$\frac{d{E}_N}{dr}$

f$.$ Based on $($e$),$ calculate $r_0$ and $E_0,$ and compare to the graphical estimations from part $($d$).$