What are number average and weight average molecular weights of a polymer and what is polydispersity?

In one line

Because a polymer sample contains chains of many different lengths, its molecular weight is expressed as an average: the number average (M_n) counts every molecule equally, while the weight average (M_w) gives more importance to heavier molecules.

Explained simply

Imagine a class where most students weigh 50 kg but a few weigh 90 kg. The simple average per student is one number; but if you ask "on average, how heavy is the person a random kilogram belongs to?", the heavy students count more. That second average is like M_w. Polymerisation is random, so some chains grow long and some stay short; one single molecular weight does not exist.

How it works

1. Group the molecules: N_i molecules of molecular weight M_i.
2. M_n = total mass / total number of molecules. Measured by colligative properties (osmotic pressure, end-group analysis).
3. M_w weights each molecule by its mass fraction. Measured by light scattering or ultracentrifugation.
4. Since heavier molecules count more, M_w ≥ M_n always.
5. Polydispersity index (PDI) = M_w/M_n. PDI = 1 means all chains have the same length (monodisperse, e.g., many natural proteins). Synthetic polymers usually have PDI greater than 1.

Key formula

\( \bar{M}_n = \dfrac{\sum N_i M_i}{\sum N_i} \)

\( \bar{M}_w = \dfrac{\sum N_i M_i^2}{\sum N_i M_i} = \dfrac{\sum w_i M_i}{\sum w_i} \)

\( \text{PDI} = \dfrac{\bar{M}_w}{\bar{M}_n} \)

• N_i = number (or mole fraction / percentage) of molecules with molecular weight M_i
• w_i = mass of molecules with molecular weight M_i (w_i = N_iM_i)
• M in g/mol; PDI has no unit

Quick example

Two chains of 10,000 and two chains of 30,000 g/mol: M_n = (2×10000 + 2×30000)/4 = 20,000. M_w = (2×10⁸ + 2×9×10⁸)/(80,000) = 2.0×10⁹/80,000 = 25,000. PDI = 1.25.

Significance: M_n relates to properties that depend on the number of molecules (colligative properties, end groups); M_w relates better to strength, toughness and melt viscosity. A narrow distribution (PDI near 1) gives more uniform properties.

Common mistakes

• Using N_iM_i² in the denominator of M_w; the denominator is ΣN_iM_i.
• Getting PDI less than 1; that signals a calculation error.
• Mixing data given as mass fractions with the number-based formula. If masses are given, use w_i directly.

How MU asks it

"Define number average and weight average molecular weight. Give their significance" (3–5 marks), usually followed by a numerical. For full marks: both definitions, both formulas with symbols, measurement methods, the M_w ≥ M_n statement and PDI.