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 calculate the molecular weight of a single repeat unit, we will use the number-average molecular weight (\(\bar{M}_\mathrm{n}\)) and the degree of polymerization (DP) as follows: $$ \bar{M}_\mathrm{n} = DP \times M_\mathrm{repeat} $$ Rearranging the equation to get the molecular weight of a single repeat unit (\(M_\mathrm{repeat}\)): $$ M_\mathrm{repeat} = \frac{\bar{M}_\mathrm{n}}{DP} $$ Substitute the given values: $$ M_\mathrm{repeat} = \frac{100,000 \mathrm{g/mol}}{2210} $$

Calculate the molecular weight of the known repeat unit, ethylene (C\(_2\)H\(_4\)) as follows: Ethylene molecular weight: (2 × 12.01) + (4 × 1.01) = 28.06 g/mol The molecular weight of a single repeat unit from the alternating copolymer consists of both ethylene and the unknown repeat unit, so we can determine the molecular weight of the unknown repeat unit using: $$ M_\mathrm{unknown} = M_\mathrm{repeat} - M_\mathrm{ethylene} $$

Calculate the molecular weights of the given options (styrene, propylene, tetrafluoroethylene, and vinyl chloride) and compare them to \(M_\mathrm{unknown}\). The option with a molecular weight closest to \(M_\mathrm{unknown}\) is the other repeat unit in the alternating copolymer. Molecular weights of the given options: - Styrene: (C\(_8\)H\(_8\)) = (8 × 12.01) + (8 × 1.01) = 104.16 g/mol - Propylene: (C\(_3\)H\(_6\)) = (3 × 12.01) + (6 × 1.01) = 42.09 g/mol - Tetrafluoroethylene: (C\(_2\)F\(_4\)) = (2 × 12.01) + (4 × 19.00) = 100.04 g/mol - Vinyl chloride: (C\(_2\)H\(_3\)Cl) = (2 × 12.01) + (3 × 1.01) + (1 × 35.45) = 62.50 g/mol Based on molecular weights, one of the given options should be the other repeat unit in the copolymer.