The number average molecular weight (Mn) and weight average molecular weight (Mw) are two different ways of expressing the average molecular weight of a polymer sample, each emphasizing different aspects of the molecular weight distribution.
Number Average Molecular Weight (Mn)
The number average molecular weight (Mn) is a statistical average of the molecular weights of all the polymer chains in a sample. It's determined by dividing the total weight of the polymer sample by the total number of polymer molecules in that sample. In essence, Mn gives equal weight to each molecule, regardless of its size.
Mathematically, Mn is defined as:
Mn = Σ(Ni * Mi) / ΣNi
Where:
- Ni = Number of molecules of molecular weight Mi
- Mi = Molecular weight of the i-th molecule
Mn is sensitive to the presence of smaller molecules in the sample. A small amount of low molecular weight polymer can significantly decrease the Mn value.
Weight Average Molecular Weight (Mw)
The weight average molecular weight (Mw) is also a statistical average, but it's based on the weight fraction of each molecule rather than the number of molecules. Larger molecules contribute more to Mw than smaller molecules.
Mathematically, Mw is defined as:
Mw = Σ(Wi Mi) / ΣWi = Σ(Ni Mi^2) / Σ(Ni * Mi)
Where:
- Wi = Weight fraction of molecules of molecular weight Mi
- Mi = Molecular weight of the i-th molecule
- Ni = Number of molecules of molecular weight Mi
Mw is more sensitive to the presence of larger molecules in the sample. A small amount of high molecular weight polymer can significantly increase the Mw value.
Polydispersity Index (PDI)
The ratio of Mw to Mn, known as the Polydispersity Index (PDI), provides information about the breadth of the molecular weight distribution.
PDI = Mw / Mn
- A PDI of 1 indicates a perfectly uniform polymer (all chains have the same molecular weight). This is often difficult to achieve in practice.
- A PDI greater than 1 indicates that the polymer sample contains chains with varying molecular weights. The larger the PDI, the broader the molecular weight distribution.
Significance of Mn and Mw
Both Mn and Mw are important parameters for characterizing polymers, as they influence the polymer's physical and mechanical properties.
- Mn: Affects properties that depend on the number of chain ends, such as tensile strength, elongation, and certain solution properties like osmotic pressure.
- Mw: Affects properties that are more sensitive to the size of the polymer chains, such as viscosity, melt flow, and impact strength.
Understanding both Mn and Mw, as well as the PDI, is crucial for selecting polymers for specific applications and for controlling the properties of polymer products.
