A polymer sample consists of equal masses of molecules with a molecular weight of 1.0×10⁴ g/mol and molecules of 1.0×10⁵ g/mol. Which correctly gives the number-average molecular weight Mn and the weight-average molecular weight Mw?

Answer: C. Mn ≈ 1.8×10⁴, Mw ≈ 5.5×10⁴

With equal masses w of each, the number of molecules is proportional to the mass divided by the molecular weight. The number-average molecular weight is Mn = total mass/total number of molecules = 2/(1/1.0×10⁴ + 1/1.0×10⁵) ≈ 1.8×10⁴ g/mol. The weight-average molecular weight is the average weighted by mass fraction, Mw = 0.5 × 1.0×10⁴ + 0.5 × 1.0×10⁵ = 5.5×10⁴ g/mol. By number there are ten times as many small molecules, so Mn is pulled toward the small side, and Mw ≥ Mn always holds. The pair with Mn and Mw swapped has the order reversed, and the pair with both as a simple average ignores the difference between weighting by number and by mass. The dispersity Mw/Mn ≈ 3.0 is a measure of the breadth of the molecular weight distribution.