$($a$)$ What is the $T$ needed to ensure that critical nucleus radius $(r^{**})$ is $\frac{1}{2}{R}_g$ for linear polyethylene with

a $M_w=100,000\frac{g}{m}ol.$ Assume the following: $T_m=120C,H_m=178\frac{J}{g},{}_{PE}=14\mathrm{.}2m\frac{J}{m^2},$

and ${}_{PE}=0\mathrm{.}9\frac{g}{c}{m}^3.$ Find $T$ and $R_g.$

$($b$)$ Assuming the nucleus is not spherical but a square with a thickness of $10nm(1),$ what are the square

side dimensions $($width $($w$)$ and height $($h$))?$