Using Lennard$-$Jones model for the potential energy $(V)$ as a function of the distance $(r)$ between particles, derive, at which separation:

a$)$ potential energy is zero

b$)$ force is zero

c$)$ potential energy minimum $($Hint: $d\frac{V}{d}r=0)$

d$)$ force is minimum $($Hint: $d\frac{F}{d}r=0)$

For Argon, at the values of scaled distance $\frac{r}{r_0}$ between $0\mathrm{.}95$ and $2\mathrm{.}05,$ with the step size $0\mathrm{.}05$ calculate:

e$)$ potential energies $V,$ attractive forces $F_a,$ repulsive forces $F_r$ and total forces $F$ and list in a Table with the types of net interactions. $($Please attach Excel worksheet$)$

f$)$ Plot $V$ versus $r.$

g$)$ Plot three forces $,$ and $($:$F\}$ versus $r$ on the same chart.

h$)$ Mark the points on the charts corresponding to the distances in $"$a$),$ b$),$ c$)$ and d$)".$

a$)$ Write momentum, heat and mass transfer equations by molecular mechanism. Explain all terms and their units.

$\backslash$table$[[\backslash$table$[[$Transferred$],[$quantity$]],$Equation,Explanation of terms and their units$],[$Momentum$,,],[$Heat$,,],[$Mass$,,]]$