Difference Between Number Average and Weight Average Molecular Weights in Polymer Science

Difference Between Average Molecular Weights

The difference between average molecular weights lies in how they weight the molecules in a sample, with the number average molecular weight (Mn) weighting by molecule count and the weight average molecular weight (Mw) weighting by molecular mass.

Number Average Molecular Weight (Mn)

Mn is calculated by dividing the total mass of all molecules by the total number of molecules. It treats each molecule equally, regardless of its size. For example, consider three molecules weighing 100, 1000, and 10000 units. The Mn is (100 + 1000 + 10000)/3 = 3700.

This average highlights the typical molecular size if every molecule had equal influence. It is straightforward but can underestimate the effect of large molecules.

Weight Average Molecular Weight (Mw)

Mw accounts for the mass distribution by weighting each molecular weight by the fraction of the total sample mass it represents. Using the same example, the total mass is 11100 units. Fractions are 0.009, 0.09, and 0.9 respectively. Multiplying these fractions by their masses yields 0.9, 90, and 9000, summing to 9090.9 for Mw.

This average emphasizes heavier molecules, reflecting their greater influence on the material’s physical properties.

Polydispersity Index (PDI)

The relationship between Mw and Mn is expressed through the polydispersity index (PDI), calculated as PDI = Mw / Mn. In the example, PDI = 9090.9 / 3700 = 2.457. A PDI of 1 indicates uniform molecular weights, while higher values reveal a broader size distribution.

Significance in Polymer Science

Polymers and other polydisperse substances do not have a single molecular weight but a distribution. Mn and Mw characterize this distribution differently. Mw better reflects properties influenced by larger molecules, such as mechanical strength. Mn provides insight into the average chain length.

• Mn: Weights molecules equally; simple average by number.
• Mw: Weights molecules by their mass; larger molecules have more influence.
• PDI: Ratio Mw/Mn indicating molecular weight distribution breadth.

Summary of Key Points

• Mn sums molecule masses and divides by molecule count.
• Mw uses mass fractions, emphasizing larger molecules.
• PDI measures the heterogeneity of molecular weights.
• Mw correlates better with material properties in polymers.
• Both averages are essential to understand polydisperse systems.