Problem $2$

For a $K^{+}-C{l}^{-}$ion pair, attractive and repulsive energies $E_A$ and $E_R,$ respectively, depend on the distance between the ions $r,$ according to

$E_A=\frac{-1\mathrm{.}436}{r}$

$E_R=\frac{5\mathrm{.}86{10}^{-6}}{r^9}$

For these expressions, energies are expressed in electron volts per $K^{+}-C{l}^{-}$pair, and $r$ is the distance in nanometers. The net energy $E_N$ is just the sum of the preceding two expressions.

$($a$)$ Superimpose on a single plot $E_A,E_R$ and $E_N$ versus $r$ up to $1\mathrm{.}0$ nm $.$

$($b$)$ Based on this plot, determine $($i$)$ the equilibrium spacing $r_0$ between $K^{+}-C{l}^{-}$ions, and $($ii$)$ the magnitude of the bonding energy $E_0$ between the ions.

$($c$)$ Also calculate the $r_0$ and $E_0$ mathematically and compare your results with the graphical results from part $($b$).$