An alternating copolymer is known to have a ( number-average molecular weight of \(100,000 \mathrm{~g} / \mathrm{mol}\) and a degree of polymerization of 2210 . If one of the repeat units is ethylene, which of styrene, propylene, tetrafluoroethylene, and vinyl chloride is the other repeat unit? Why?

To find the average molecular weight of the entire copolymer, we must first determine the molecular weight of the ethylene repeat unit. The ethylene repeat unit has the chemical structure of \(\mathrm{CH_{2}CH_{2}}\). The molecular weight of ethylene repeat unit: M(C) \(= 12.01 \mathrm{~g/mol}\) M(H) \(= 1.01 \mathrm{~g/mol}\) Ethylene repeat unit molecular weight (\(\mathrm{M_{ethylene}}\)) = \(2 \times (12.01 + (2 \times 1.01)) = 28.04 \mathrm{~g/mol}\).

It is necessary to determine the molecular weights of the other possible repeat units - styrene, propylene, tetrafluoroethylene, and vinyl chloride: 1. Styrene repeat unit has the chemical structure of \(\mathrm{C_{8}H_{8}}\), so its molecular weight can be calculated as follows: M(C) \(= 12.01 \mathrm{~g/mol}\) M(H) \(= 1.01 \mathrm{~g/mol}\) Styrene repeat unit molecular weight (\(\mathrm{M_{styrene}}\)) = \(8 \times 12.01 + 8 \times 1.01 = 104.15 \mathrm{~g/mol}\). 2. Propylene repeat unit is \(\mathrm{CH_{2}CHCH_{3}}\), so its molecular weight: Propylene repeat unit molecular weight (\(\mathrm{M_{propylene}}\)) = \(3 \times 12.01 + 6 \times 1.01 = 42.08 \mathrm{~g/mol}\). 3. Tetrafluoroethylene repeat unit is \(\mathrm{CF_{2}CF_{2}}\), so its molecular weight: M(C) \(= 12.01 \mathrm{~g/mol}\) M(F) \(= 19.00 \mathrm{~g/mol}\) Tetrafluoroethylene repeat unit molecular weight (\(\mathrm{M_{tetrafluoroethylene}}\)) = \(2 \times (12.01 + 2 \times 19.00) = 100.02 \mathrm{~g/mol}\). 4. Vinyl chloride repeat unit is \(\mathrm{CH_{2}CHCl}\), so its molecular weight: M(C) \(= 12.01 \mathrm{~g/mol}\) M(H) \(= 1.01 \mathrm{~g/mol}\) M(Cl) \(= 35.45 \mathrm{~g/mol}\) Vinyl chloride repeat unit molecular weight (\(\mathrm{M_{vinylchloride}}\)) = \(12.01 + 1.01 + 12.01 + 35.45 = 62.48 \mathrm{~g/mol}\).

We are given that the number-average molecular weight of the copolymer is \(100,000 \mathrm{~g/mol}.\) We are also given that the degree of polymerization is 2210. Degree of polymerization (DP) = 2210. Molecular weight of the copolymer \((\mathrm{M_{copolymer}})\) = \(\frac{\mathrm{M_{ethylene} + M_{other~repeat~unit}}}{2} \times \mathrm{DP}\) Now we will check the options one by one: 1. Styrene: \(\frac{\mathrm{28.04 + 104.15}}{2} \times 2210 = 145,417 \mathrm{~g/mol} \) 2. Propylene: \(\frac{\mathrm{28.04 + 42.08}}{2} \times 2210 = 77,184 \mathrm{~g/mol} \) 3. Tetrafluoroethylene: \(\frac{\mathrm{28.04 + 100.02}}{2} \times 2210 = 140,721 \mathrm{~g/mol} \) 4. Vinyl chloride: \(\frac{\mathrm{28.04 + 62.48}}{2} \times 2210 = 99,656 \mathrm{~g/mol} \) Comparing these calculated molecular weights to the given molecular weight of \(100,000 \mathrm{~g/mol}\), we can see that the copolymer's molecular weight is closest to the case of vinyl chloride, so the other repeat unit is vinyl chloride.