Chemistry Labs

Polymer chemistry

Molecular weight and its distribution

Define number- and weight-average molar masses and dispersity, calculate them from chain populations, interpret polymer molecular-weight distributions, and compare major measurement methods and their limitations.

1 Lower secondary2 Upper secondary3 Undergraduate4 Advanced5 Research

IntuitionA polymer sample contains many chain sizes

A small molecule has a well-defined molar mass, but a polymer sample usually contains short and long chains together. Counting every chain gives one average; asking where most of the mass resides gives another. The whole distribution—not one number—helps determine processing and performance.

Compare idealized chain-length weight distributions at the same number-average degree of polymerization. The curves illustrate how broader distributions raise the weight average; real data depend on sample and detector response.

SchoolAverages and dispersity

Definition: Number-average molar mass, Mn

For Nᵢ molecules of molar mass Mᵢ, the number average is Mₙ = ΣNᵢMᵢ/ΣNᵢ. Each molecule has equal weight in the count, so numerous short chains strongly affect Mₙ.

Definition: Weight-average molar mass, Mw

M_w weights each chain by its mass: M_w = ΣNᵢMᵢ²/ΣNᵢMᵢ. Equivalently, if wᵢ is the mass fraction at Mᵢ, M_w = ΣwᵢMᵢ. The high-mass tail therefore matters disproportionately.

Definition: Dispersity

Dispersity Đ = M_w/Mₙ is dimensionless and, for a distribution of positive chain masses, is at least 1. Đ = 1 only for a monodisperse population (all chains have the same molar mass); larger values indicate a broader distribution but do not specify its shape.

Mn=∑iNiMi∑iNi,Mw=∑iNiMi2∑iNiMi=∑iwiMi,Đ=MwMnM_n=\frac{\sum_i N_iM_i}{\sum_i N_i},\qquad M_w=\frac{\sum_i N_iM_i^2}{\sum_i N_iM_i}=\sum_i w_iM_i,\qquad Đ=\frac{M_w}{M_n}

Example: Calculate both averages from chain counts

A sample contains one chain at 10 kg mol⁻¹, two at 20 kg mol⁻¹, and one at 30 kg mol⁻¹. Calculate Mₙ, M_w, and Đ.

Solution

There are 4 chains and total mass 10 + 2(20) + 30 = 80 in the population units, so Mₙ = 80/4 = 20 kg mol⁻¹. The squared-mass sum is 1(10²)+2(20²)+1(30²)=1,800 (kg mol⁻¹)²; M_w = 1,800/80 = 22.5 kg mol⁻¹. Therefore Đ=22.5/20=1.125.

Example: Calculate from mass fractions

A two-component polymer population has 40% of its mass at 10 kg mol⁻¹ and 60% at 30 kg mol⁻¹. Find Mₙ, M_w, and Đ using mass fractions.

Solution

M_w = ΣwᵢMᵢ = 0.40(10)+0.60(30)=22 kg mol⁻¹. Since wᵢ=NᵢMᵢ/ΣNⱼMⱼ, Mₙ = 1/Σ(wᵢ/Mᵢ)=1/[0.40/10+0.60/30]=1/0.06=16.67 kg mol⁻¹. Đ=22/16.67≈1.32.

Common molecular-mass averages and measurement sensitivities
QuantityWeighting / interpretation
MₙEqual per molecule; colligative methods respond to chain count
M_wMass-weighted; static light scattering is strongly weighted toward large chains
M_zHigher moment, emphasizes the high-mass tail

UndergraduateDistributions, polymerization statistics, and measurement

Mz=∑iNiMi3∑iNiMi2,Xn=MnM0,Đ=MwMnM_z=\frac{\sum_i N_iM_i^3}{\sum_i N_iM_i^2},\qquad X_n=\frac{M_n}{M_0},\qquad Đ=\frac{M_w}{M_n}

For an ideal step-growth polymerization of bifunctional A–A and B–B monomers at stoichiometric balance, with equal reactivity and negligible cyclization, the most-probable number distribution has Xₙ=1/(1−p) and X̄w=(1+p)/(1−p). Thus Đ=1+p; at p=0.99, Xₙ=100, X̄w=199, Đ=1.99. This is a model limit, not a universal distribution for every synthesis.

Size-exclusion chromatography (SEC, often called GPC) separates dissolved chains by hydrodynamic volume: larger coils generally elute earlier through a porous column. With concentration detection and calibration, the elution trace is converted to a molar-mass distribution. Conventional calibration reports apparent masses relative to standards of a specified polymer and solvent; branching, solvent quality, and polymer–column interactions can bias the result.

What common methods measure
MethodTypical information and caveat
SEC/GPC + concentration detectorDistribution and calibrated averages; conventional calibration is hydrodynamic-volume dependent
SEC + multi-angle light scatteringMolar mass across elution slices without column-standard calibration, if concentration and dn/dc are known; aggregation and low signal are concerns
Osmometry / end-group analysisNumber-average-sensitive; reliability falls when chains are very large or end groups are difficult to quantify
ViscometryProvides viscosity-average mass through polymer-specific Mark–Houwink parameters, not Mn or Mw directly

References

  • Molecular Size Distribution in Linear Condensation Polymers · Paul J. Flory, 1945
  • Principles of Polymer Chemistry · Paul J. Flory, 1953
  • Textbook of Polymer Science, 3rd Edition · Fred W. Billmeyer, Jr., 1984