Polymer Molecular Weight Calculator (Mn, Mw, PDI)

Understanding the polymer molecular weight calculator — concept and principle

A polymer sample never contains chains of one length; it is a distribution. To describe it, polymer scientists use averages: the number-average molar mass Mn, the weight-average molar mass Mw, and their ratio, the dispersity or polydispersity index (PDI). This calculator computes all three from a list of chain counts (or mole fractions) and their molar masses, for example from a GPC result divided into slices.

How it is calculated

• Mn = ΣNᵢMᵢ ÷ ΣNᵢ
• Mw = ΣNᵢMᵢ² ÷ ΣNᵢMᵢ
• PDI = Mw ÷ Mn

Nᵢ is the number of chains (or moles, or mole fraction) with molar mass Mᵢ in g/mol. The two lists must have the same length and order.

The principle

Mn is an ordinary average: total mass divided by number of chains, so every molecule counts equally. Mw weights each chain by its mass, so it answers the question "if I pick a random gram of polymer, what is the molar mass of the chain it belongs to?". Heavy chains therefore pull Mw up more than Mn. The two are equal only when all chains are identical; otherwise Mw is larger and PDI exceeds 1. Different properties follow different averages: colligative properties such as osmotic pressure depend on Mn, while light scattering measures Mw, and melt viscosity is very sensitive to the high-mass end.

Worked example

1. Defaults: N = 10, 30, 40, 20 chains with M = 10,000, 20,000, 50,000, 100,000 g/mol.
2. ΣN = 100 and ΣNM = 100,000 + 600,000 + 2,000,000 + 2,000,000 = 4.7 × 10⁶.
3. Mn = 4.7 × 10⁶ ÷ 100 = 47,000 g/mol.
4. ΣNM² = 1 × 10⁹ + 1.2 × 10¹⁰ + 1 × 10¹¹ + 2 × 10¹¹ = 3.13 × 10¹¹.
5. Mw = 3.13 × 10¹¹ ÷ 4.7 × 10⁶ = 66,595.7 g/mol.
6. PDI = 66,595.7 ÷ 47,000 = 1.417.

How to read the result

A PDI of 1 means a uniform (monodisperse) sample. Well-controlled living polymerisations can approach about 1.1 or below, ideal step-growth polymers tend toward 2, and many industrial free-radical polymers are broader. A higher PDI at the same Mn means more very long and very short chains, which affects processing, toughness and brittleness.

Common mistakes

• Entering weight fractions in place of chain numbers. With weight fractions wᵢ, Mw = ΣwᵢMᵢ and Mn = 1 ÷ Σ(wᵢ/Mᵢ), so the same formula gives the wrong answer.
• Comparing GPC results calibrated with different standards; molar masses relative to polystyrene are not absolute.
• Reporting PDI without Mn or Mw; the ratio alone says nothing about chain length.

Frequently asked questions

Is PDI the same as dispersity Đ?

Yes. IUPAC now recommends the term dispersity and the symbol Đ for Mw/Mn.

What is Mz?

The next higher average, ΣNᵢMᵢ³ ÷ ΣNᵢMᵢ², which is even more sensitive to the longest chains.

Why is molar mass written in g/mol or Da?

Both express the mass of one mole of chains; 1 g/mol corresponds to 1 dalton per molecule.

Sources: IUPAC (2009). Dispersity in polymer science. Pure and Applied Chemistry, 81(2), 351–353. · Odian, G. Principles of Polymerization. Wiley. · Young, R. J. & Lovell, P. A. Introduction to Polymers. CRC Press.