Problem $3(15$ points$)$: You are requested to make a branched poly$($styrene$)$ with a lower glass transition

temperature than the linear homopolymer at infinite molecular weight $(T_g=100C).$ Your supervisor

requests that you make a branched poly$($styrene$)$ with a $T_g$ of $70C$ and a total degree of polymerization $$

$1000$ repeat units. For the linear polymer of the same degree of polymerization, the $T_g$ was found to be $98$

$C.$ You are given that the thermal expansion coefficient for the free volume is $8\mathrm{.}0{10}^{-5}{K}^{-1}$ and the

density is $1\mathrm{.}0gc{m}^{-3}$ for the linear polymer and $0\mathrm{.}98gc{m}^{-3}$ for the branched poly$($styrene$).$

a$)$ Write down reaction steps on how you could possible make such a copolymer with the following

chemicals supplied to you.

b$)$ Determine the fraction of grafts in the branched poly$($styrene$).$

c$)$ Give an experimental method besides glass transition measurement to determine the

degree of branching.

d$)$ You perform GPC on the branched poly$($styrene$)$ to confirm its molecular weight relative

to linear poly$($styrene$)$ standards. How would you interpret the results for the molecular

weight analysis?