What are the different quantities used to describe the average molecular weight?

Two common experimentally determined quantities describe the average molecular weight of a polymer sample, which often contains molecules of varying sizes: the number average molecular weight (Mn) and the weight average molecular weight (Mw).

Number Average Molecular Weight (Mn)

Mn is calculated based on the number of molecules of each size present in the sample. It is essentially the total weight of all the molecules divided by the total number of molecules. More formally, it's determined from the mole fraction distribution.

• Formula: Mn = Σ (Ni * Mi) / Σ Ni

  • Where:
    • Ni = number of molecules of molecular weight Mi
    • Mi = molecular weight of molecules of type i

• Significance: Mn is highly sensitive to the presence of small molecules, as these contribute significantly to the total number of molecules. Techniques like osmometry are commonly used to determine Mn.

Weight Average Molecular Weight (Mw)

Mw, on the other hand, is calculated based on the weight fraction of each size of molecule in the sample. It's more sensitive to the presence of high molecular weight molecules, as these contribute more to the overall weight. It's determined from the weight fraction distribution.

• Formula: Mw = Σ (Wi Mi) / Σ Wi = Σ (Ni Mi^2) / Σ (Ni * Mi)

  • Where:
    • Wi = weight fraction of molecules of molecular weight Mi
    • Mi = molecular weight of molecules of type i
    • Ni = number of molecules of molecular weight Mi

• Significance: Mw is more influenced by larger molecules. Light scattering techniques are frequently used to measure Mw.

Polydispersity Index (PDI)

While not a direct average molecular weight, the Polydispersity Index (PDI) provides valuable information about the distribution of molecular weights in a sample.

• Formula: PDI = Mw / Mn

• Significance:

  • PDI = 1: Indicates a monodisperse sample (all molecules have the same molecular weight).
  • PDI > 1: Indicates a polydisperse sample (molecules have a range of molecular weights). A higher PDI signifies a broader distribution of molecular weights.

Summary Table

Quantity	Description	Sensitivity	Formula	Determination Method(s)
Number Average Molecular Weight (Mn)	Average molecular weight based on the number of molecules	Small molecules	Σ (Ni * Mi) / Σ Ni	Osmometry
Weight Average Molecular Weight (Mw)	Average molecular weight based on the weight of molecules	Large molecules	Σ (Wi Mi) / Σ Wi = Σ (Ni Mi^2) / Σ (Ni * Mi)	Light Scattering
Polydispersity Index (PDI)	Measure of the distribution of molecular weights	Broadness of dist.	Mw / Mn	Calculated from Mw & Mn

In summary, Mn and Mw provide different perspectives on the average molecular weight, with Mn being more sensitive to smaller molecules and Mw to larger molecules. The PDI gives an indication of the breadth of the molecular weight distribution.