Dioxane $\left(\mathrm{C}_4 \mathrm{H}_8 \mathrm{O}_2, \rho=1.03 \mathrm{~g} / \mathrm{cm}^3, \delta=10.0\left(\mathrm{cal} / \mathrm{cm}^3\right)^{1 / 2}\right)$ and methyl propyl ketone $\left(\mathrm{C}_5 \mathrm{H}_{10} \mathrm{O}\right.$, $\left.\rho=0.80 \mathrm{~g} / \mathrm{cm}^3, \delta=9.7 \mathrm{cal} / \mathrm{cm}^3\right)^{1 / 2}$ ) are both reported to be theta solvents for 1,4-polyisoprene ( $\left.\delta=9.7 \mathrm{cal} / \mathrm{cm}^3\right)^{1 / 2}$ ) at $33^{\circ} \mathrm{C}$.
a. Show by calculation that one of these is exactly as expected, based on Flory-Huggins theory, and that the other is not.
b. The plot below shows the Flory-Huggins spinodal curves for a single molecular weight of polyisoprene in these two solvents. Based on the critical temperature of 287 K , estimate the molecular weight of the polymer.
c. The two curves show a slight but significant difference in the critical composition, $\phi_{\mathrm{c}}$ (one is 0.0275 , the other is 0.0308 ). Show by calculation that this difference is what Flory-Huggins theory predicts, and thereby identify which solvent corresponds to curve (a), and which to (b). (Hint: why was the molecular weight in question (b) only an estimate, given that $T_\theta$ and $T_{\mathrm{c}}$ are precise?)
d. In fact, based on the results in question (a), it should be clear that one of these two curves cannot correspond to experimental reality. Identify which one is "wrong," and explain why.
(Graph cant copy)