What is the difference between Mn and Mw?

Mn weights each chain equally and is sensitive to short chains; it comes from end-group analysis and colligative methods such as osmometry. Mw weights each chain by its mass, is sensitive to long chains and comes from light scattering. Always Mn ≤ Mv ≤ Mw ≤ Mz, with equality only for a monodisperse sample. Their ratio Đ = Mw/Mn is the dispersity (formerly the polydispersity index, PDI): living anionic polymerization can approach 1.01–1.1, free-radical polymers are usually 1.5–2 or higher, and ideal step growth tends to 2 at high conversion.

How do I use the Carothers equation?

For a stoichiometric AA + BB or AB step-growth polymerization, Xn = 1/(1 − p), where p is the fraction of functional groups that have reacted: p = 0.99 gives Xn = 100 and p = 0.995 gives 200. With an imbalance r = NA/NB ≤ 1 it becomes Xn = (1 + r)/(1 + r − 2rp), so p = 0.99 and r = 0.98 give only Xn = 50. A monofunctional reagent gives r = NA/(NB + 2NB′). Even at p = 1, Xn cannot exceed (1 + r)/(1 − r), which is why step growth needs an exact 1:1 ratio. Xn here counts structural units, and one nylon-6,6 repeat unit contains two.

How do I get the viscosity-average molecular weight?

Use the Mark–Houwink equation [η] = K·Mᵃ, so Mv = ([η]/K)^(1/a). K and a depend on the polymer, solvent and temperature; a is 0.5 in a theta solvent and typically 0.6–0.8 in good solvents. The presets are typical textbook example values and published constants vary, so for real work take K and a from the Polymer Handbook or measure [η] for a few standards of known M and use the built-in log[η]–log M fit.

How big is a polymer coil?

For a freely jointed chain of n bonds of length l, the mean-square end-to-end distance is ⟨R²⟩ = n·l². Fixing the bond angle θ gives ⟨R²⟩ = n·l²(1 − cos θ)/(1 + cos θ), about 2n·l² for tetrahedral carbon. A real chain adds the characteristic ratio C∞, about 6.7 for polyethylene. The radius of gyration of an ideal chain is Rg = √(⟨R²⟩/6). Polyethylene of Mn = 28,000 has about 1,996 backbone C–C bonds of 0.154 nm, giving an RMS end-to-end distance near 17.8 nm and Rg near 7.3 nm.

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➰Polymer Molecular Weight Calculator - Mn, Mw, Dispersity, DP and Carothers Equation

A polymer sample is a mixture of chains of different lengths, so its molar mass is described by averages: the number average Mn = ΣNiMi/ΣNi, the weight average Mw = ΣNiMi²/ΣNiMi, the z-average Mz = ΣNiMi³/ΣNiMi² and the dispersity Đ = Mw/Mn. For one chain each of 10,000, 20,000 and 30,000 g/mol, Mn = 20,000, Mw ≈ 23,333 and Đ ≈ 1.167; mixing equal masses of 50,000 and 200,000 standards gives Mn = 80,000 and Mw = 125,000. Paste a table of (Ni, Mi) or (wi, Mi) pairs and this calculator returns Mn, Mw, Mz, the viscosity average Mv for a chosen Mark–Houwink exponent a, number and weight fractions and a bar chart, with every sum shown. The degree-of-polymerization tab knows the repeat units of 22 common polymers, including PE, PP, PS, PVC, PMMA, PTFE, PET, nylon-6, nylon-6,6 and polycarbonate. It computes M₀ from the formula and Xn = (Mn − end groups)/M₀; polyethylene with Mn = 28,000 has Xn ≈ 998. For step growth the Carothers equation Xn = (1 + r)/(1 + r − 2rp) gives Xn = 100 at p = 0.99 and r = 1, but only 50 when r = 0.98. You can also work out r from functional-group counts, including monofunctional reagents, and plot the Flory most-probable number and weight distributions (Đ = 1 + p). Further tabs cover the Mark–Houwink equation [η] = K·Mᵃ in both directions with a log–log fit of K and a, and chain dimensions: freely jointed, freely rotating and C∞ end-to-end distances, radius of gyration and Kuhn length.

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How to use Polymer Molecular Weight Calculator

1Pick a tab: distribution, degree of polymerization, Carothers, Mark–Houwink, chain size or the polymer table
2Paste distribution data or enter Mn, extent of reaction p, ratio r, K and a, or click an example
3Read the averages, degree of polymerization, distribution plots and step-by-step working, and copy any result

FAQ

What is the difference between Mn and Mw?: Mn weights each chain equally and is sensitive to short chains; it comes from end-group analysis and colligative methods such as osmometry. Mw weights each chain by its mass, is sensitive to long chains and comes from light scattering. Always Mn ≤ Mv ≤ Mw ≤ Mz, with equality only for a monodisperse sample. Their ratio Đ = Mw/Mn is the dispersity (formerly the polydispersity index, PDI): living anionic polymerization can approach 1.01–1.1, free-radical polymers are usually 1.5–2 or higher, and ideal step growth tends to 2 at high conversion.
How do I use the Carothers equation?: For a stoichiometric AA + BB or AB step-growth polymerization, Xn = 1/(1 − p), where p is the fraction of functional groups that have reacted: p = 0.99 gives Xn = 100 and p = 0.995 gives 200. With an imbalance r = NA/NB ≤ 1 it becomes Xn = (1 + r)/(1 + r − 2rp), so p = 0.99 and r = 0.98 give only Xn = 50. A monofunctional reagent gives r = NA/(NB + 2NB′). Even at p = 1, Xn cannot exceed (1 + r)/(1 − r), which is why step growth needs an exact 1:1 ratio. Xn here counts structural units, and one nylon-6,6 repeat unit contains two.
How do I get the viscosity-average molecular weight?: Use the Mark–Houwink equation [η] = K·Mᵃ, so Mv = ([η]/K)^(1/a). K and a depend on the polymer, solvent and temperature; a is 0.5 in a theta solvent and typically 0.6–0.8 in good solvents. The presets are typical textbook example values and published constants vary, so for real work take K and a from the Polymer Handbook or measure [η] for a few standards of known M and use the built-in log[η]–log M fit.
How big is a polymer coil?: For a freely jointed chain of n bonds of length l, the mean-square end-to-end distance is ⟨R²⟩ = n·l². Fixing the bond angle θ gives ⟨R²⟩ = n·l²(1 − cos θ)/(1 + cos θ), about 2n·l² for tetrahedral carbon. A real chain adds the characteristic ratio C∞, about 6.7 for polyethylene. The radius of gyration of an ideal chain is Rg = √(⟨R²⟩/6). Polyethylene of Mn = 28,000 has about 1,996 backbone C–C bonds of 0.154 nm, giving an RMS end-to-end distance near 17.8 nm and Rg near 7.3 nm.

Features

Mn, Mw, Mz, Mv and dispersity Đ from Ni–Mi or wi–Mi data
Number- and weight-fraction table with a bar chart
22 common polymer repeat units with M₀ from the formula, plus optional end groups
Number- and weight-average degree of polymerization, or Mn from a known DP
Carothers equation: Xn from p and r, required p for a target Xn, monofunctional reagents
Flory most-probable distribution plots with Xw and Đ = 1 + p
Mark–Houwink intrinsic viscosity ↔ molar mass and a log–log fit of K and a
Freely jointed, freely rotating and real-chain (C∞) size, radius of gyration and Kuhn length

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