Problem $3$

Consider a collision between a molecule and an ion that interact through the following potential energy function,

$V(r)=-\frac{c}{r^3}$ for $r>d$

$V(r)=$ for $rd$

where $C$ is a positive constant, $r$ is the interparticle separation and $d$ is the hard sphere collision diameter. Denote the well depth of the potential as ${}_d$ such that $)=(d.$

Part a$.$ Derive an equation for the centrifugal barrier, written in terms of the initial energy $,$ impact parameter $b$ and constants.

Part b$.$ Derive an equation for cross section for close contact collisions where "close contact" means that the distance of closest approach is equal to the hard sphere diameter $d.$ Write your equation in terms of only the initial energy $$ and constants.

Part c$.$ Write an equation for the maximum rotational energy at close contact for collisions with initial energy $.$ Write your equation in terms of only $$ and constants.

Part d$.$ Determine an equation for the average rotational energy at close contact for collisions with initial energy $.$ Write your equation in terms of only $$ and constants. Show your work to receive credit.

Useful Information: $c=3{10}^{10}cm{s}^{-1},h=6\mathrm{.}626{10}^{-34}Js,$

$v(s^{-1})=v(c{m}^{-1}{)}^{**}c(cm{s}^{-1})$

$R=8\mathrm{.}314\frac{J}{m}$olK,$1$mole$=6\mathrm{.}022{10}^{23}$ molecules

$1a=0\mathrm{.}001k\frac{g}{m}ol$