Degree of Polymerization Calculator

Calculate number-average and weight-average degree of polymerization from molecular weight data.

Number-average molecular weight Mn (g/mol) (g/mol)

Weight-average molecular weight Mw (g/mol) (g/mol)

Monomer molecular weight M0 (g/mol) (g/mol)

Styrene=104, ethylene=28, vinyl chloride=62.5

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Why Polymers Need Two Molecular Weights

A polymer sample is not one molecule repeated — it is a distribution of chain lengths. Two averages are therefore needed, and their ratio characterises how broad the distribution is.

Average	Weights by	Sensitive to	Measured by
M_n (number)	Number of chains	Short chains	Osmometry, end-group analysis
M_w (weight)	Mass of chains	Long chains	Light scattering
M_z (z-average)	Higher mass moment	Very long chains	Ultracentrifugation

X_n = M_n/M₀ PDI = M_w/M_n

Because M_w weights long chains more heavily, it is always at least as large as M_n. PDI is therefore always ≥ 1, and equals exactly 1 only for a perfectly uniform sample — which in practice occurs only for proteins and DNA, where synthesis is template-directed.

PDI	Distribution	Typical origin
1.0	Monodisperse	Proteins, DNA
1.02–1.1	Very narrow	Living anionic, RAFT, ATRP
1.5–2.0	Moderate	Step-growth polymerisation
2–5	Broad	Free radical polymerisation
> 10	Very broad	Branched, e.g. LDPE

Distribution breadth matters practically. Short chains act as plasticisers, softening the material and lowering its melting range; very long chains raise melt viscosity and complicate processing. Two samples with identical M_n but different PDI behave quite differently.

Worked Examples

Example 1: Polystyrene: Mn=50000, Mw=120000, M0=104

Xn=481, Xw=1154

Result: PDI=2.4 — broad distribution (free radical)

Typical FRP polystyrene

Example 2: Living polymerization: Mn=50000, Mw=52000

PDI=1.04 — very narrow

Result: RAFT or anionic polymerization

Controlled radical polymerization target: PDI<1.1

Example 3: Step-growth polymerisation

Polyester at 99% conversion, Carothers equation X_n = 1/(1−p)

Result: X_n = 100

Step-growth needs very high conversion for useful chain length. At 95% conversion X_n is only 20 — which is why these polymerisations are driven so hard.

Example 4: Comparing at equal M_n

Two samples, M_n = 50,000, PDI 1.05 versus 2.5

Result: Same X_n, very different properties

The broad sample contains both short chains that soften it and long chains that raise melt viscosity. Identical average, different material.

Common Mistakes

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Assuming PDI can be below 1

M_w is always at least M_n because it weights heavier chains more. A calculated PDI under 1 means the input values are wrong.

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Using M_w to find the degree of polymerisation

X_n is defined from M_n, the number average. Using M_w overstates chain length substantially for broad distributions.

⚠️

Forgetting to subtract end groups

For precise work, M_n includes the initiator and terminator fragments. For long chains this is negligible, but for oligomers it matters.

⚠️

Treating PDI as a quality measure alone

Narrow distribution is not always better. Some applications need a broad distribution for processability, where short chains aid flow.

Frequently Asked Questions

PDI (dispersity) physical meaning?

Ideal monodisperse: PDI=1.0. Living anionic: PDI=1.01-1.1. Controlled radical (RAFT, ATRP): PDI=1.1-1.5. Free radical: PDI=1.5-2.0. Step-growth: PDI→2.0 at high conversion. PDI affects material properties: processing, strength.

Mn vs Mw vs Mz?

Mn: counts chains equally (affects colligative properties). Mw: weights by mass (affects mechanical properties). Mz: higher average (affects melt flow). For narrow distributions (PDI~1): Mn≈Mw≈Mz. GPC measures all three simultaneously.

Why is PDI always at least 1?

Because M_w weights chains by mass, giving longer chains more influence than M_n does. The two are equal only when every chain is identical.

What PDI does free radical polymerisation give?

Typically 2 to 5, because chain initiation, propagation and termination occur throughout the reaction, producing chains of widely varying length.

How do living polymerisations achieve narrow PDI?

All chains initiate at nearly the same moment and grow at the same rate without termination, so they end up close to the same length — PDI near 1.05.

Which average should I use for degree of polymerisation?

M_n, the number average, since X_n counts monomer units per chain on a per-molecule basis.

Formula Explorer connections

Interpretation: This formula connects molecular properties, material structure or environmental transport to a macroscopic behavior or exposure estimate. Assumption: Use parameters measured for the same material, solvent, temperature and environment. Empirical correlations may not transfer outside their calibration range.

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