You perform a light scattering experiment $($at a light wavelength of $690nm)$ on cellulose nitrate in acetone at $25C$ and get the following data:

$($a$)$ Calculate the weight$-$average molecular weight $(\frac{?}{\text{b}\text{a}\text{r}}(M{)}_w)$ and radius of gyration $(R{g}_g)$ for your sample $($the slope of the $c=0$ line can be read as $68\mathrm{.}77{10}^{-7}mo\frac{l}{g}$ and the intercept as $($:$8\mathrm{.}35{10}^{-7}\frac{\text{m}\text{o}\text{l}}{g}\}.$

$($b$)$ If you assume that your polymer was synthesized by an ideal polymerization, such that its molecular weight distribution is exactly equal to the most$-$probable distribution, what is the polydispersity index $($PDI$)$ and number$-$average degree of polymerization $({\stackrel{}{x}}_n)$ for your polymer $($each repeat unit of cellulose nitrate has the elemental formula $($:$C_6{H}_7{N}_3{O}_{11}\}?$

$($c$)$ Calculate the theoretical root$-$mean$-$squared end$-$to$-$end distance in the freely jointed state $($the bond lengths can be approximated as two bonds of $1\mathrm{.}79$ and one bond of $3\mathrm{.}88$ per repeat unit$).$

$($d$)$ If $C_{}$~~$20,$ calculate the unperturbed root$-$mean$-$squared end$-$to$-$end distance. What does this value of $C_{}$ tell us about the stiffness of the polymer chains? Explain.

$($e$)$ Calculate the value of the expansion coefficient, $,$ for cellulose nitrate in acetone at $25C.$ What does this value of $$ tell us about the polymer$-$solvent interactions at these conditions? Explain.