Molecular weight data for some polymer are tabulated here. Compute (a) the number average molecular weight, and (b) the weight-average molecular weight. (c) If it is known that this material's degree of polymerization is \(477,\) which one of the polymers listed in Table 14.3 is this polymer? Why? $$\begin{array}{ccc} \hline \text {Molecular Weight} & & \\ \text {Range }(\mathrm{g} / \mathrm{mol}) & \boldsymbol{x}_{\boldsymbol{i}} & \boldsymbol{w}_{\boldsymbol{i}} \\ \hline 8,000-20,000 & 0.05 & 0.02 \\ 20,000-32,000 & 0.15 & 0.08 \\ 32,000-44,000 & 0.21 & 0.17 \\ 44,000-56,000 & 0.28 & 0.29 \\ 56,000-68,000 & 0.18 & 0.23 \\ 68,000-80,000 & 0.10 & 0.16 \\ 80,000-92,000 & 0.03 & 0.05 \\ \hline \end{array}$$

Calculate the number average molecular weight from the given mole fractions and molecular weight ranges: $$\overline{M}_{n}=(0.05 \times 14,000)+(0.15 \times 26,000)+(0.21 \times 38,000)+(0.28 \times 50,000)+(0.18 \times 62,000)+(0.10 \times 74,000)+(0.03 \times 86,000)$$ $$\overline{M}_{n}=47,520$$ Calculate the weight average molecular weight using the given weight fractions: $$\overline{M}_{w}=\frac{(0.02 \times 14,000)+(0.08 \times 26,000)+(0.17 \times 38,000)+(0.29 \times 50,000)+(0.23 \times 62,000)+(0.16 \times 74,000)+(0.05 \times 86,000)}{(0.02+0.08+0.17+0.29+0.23+0.16+0.05)}$$ $$\overline{M}_{w}=54,230$$ Compute the molecular weight of the monomer using the degree of polymerization value of 477: $$\text{DP} = \frac{\overline{\mathrm{M}}_{\mathrm{n}}}{M_{\text{monomer}}}$$ $$M_{\text{monomer}} = \frac{\overline{\mathrm{M}}_{\mathrm{n}}}{\text{DP}}$$ $$M_{\text{monomer}} = \frac{47,520}{477}$$ $$M_{\text{monomer}} \approx 99.6$$ Compare the molecular weight of the monomer (99.6) to the polymers listed in Table 14.3 to identify the polymer. The polymer with a monomer molecular weight closest to 99.6 is poly(ethylene oxide).