Explain the differences between number-average molecular weight (Mn) and weight-average molecular weight (Mw).

Number-average molecular weight (Mn) and weight-average molecular weight (Mw) are two different ways to describe the average molecular weight of a polymer sample, which is usually a mixture of chains of varying lengths. Mn is the total weight of all the polymer molecules in a sample divided by the total number of polymer molecules. It is calculated by summing the molecular weights of all the chains and dividing by the total number of chains. Mn is sensitive to the presence of smaller molecules and gives equal weight to each molecule, regardless of its size. Mw, on the other hand, is calculated based on the weight fraction of each molecule in the sample. It gives more weight to larger molecules, as they contribute more to the overall weight of the sample. Mw is always greater than or equal to Mn. The ratio of Mw to Mn is called the polydispersity index (PDI), which provides information about the breadth of the molecular weight distribution. A PDI of 1 indicates a monodisperse polymer, where all the chains have the same molecular weight. A PDI greater than 1 indicates a polydisperse polymer, where the chains have a range of molecular weights. For example, if a polymer sample contains 10 molecules with a molecular weight of 10,000 and 1 molecule with a molecular weight of 100,000, Mn will be closer to 10,000, while Mw will be closer to 100,000.