Prepare a polymer blend from a poly$($ethylene$-$alt$-$propylene$)($h$-$PEP$)$ of weight$-$average

molecular weight $M_w=200000gmo{l}^{-1}$ and dispersity $=1\mathrm{.}01$ with perdeuterated

poly$($ethylene$-$alt$-$propylene$)($d$-$PEP$)$ of weight$-$average molecular weight $M_w=$

$225000gmo{l}^{-1}$ and dispersity $=1\mathrm{.}01.$ The Flory parameter for the $d-PE\frac{P}{h}-$PEP pair is

given by:

$=\frac{0\mathrm{.}571}{T}-6\mathrm{.}6{10}^{-4}$

where $T$ is the absolute temperature.

a$.$ Calculate and plot the normalized free energy of mixing $\frac{G_{\text{m}\text{i}\text{x}}}{N}{k}_{B}T$ versus ${}_{\text{d}\text{P}\text{E}\text{P}}$

at $T=300K,325K,350K,385K,$ and $450K$

b$.$ Find the binodal and spinodal concentrations as a function of temperature and

then use these to plot a phase diagram.

c$.$ What would be the degree of polymerization $($where both d$-$PEP and h$-$PEP have

the same degree of polymerization$)$ for the critical point to be at $T=380K?$