Dear Prabhanjan
Consider a property which is only sensitive to the number of molecules present — a property that is not influenced by the size of any particle in the mixture. The best example of such properties are the colligative properties of solutions such as boiling point elevation, freezing point depression, and osmotic pressure. For such properties, the most relevant average molecular weight is the totalweight of polymer divided by the number of polymer molecules. This average molecular weightfollows the conventional definition for the mean value of any statistical quantity. In polymer science, it is called the number average molecular weight — MN.
To get a formula for MN , we must first realize that the molecular weight distribution is not a continuous function of M. Rather, only discrete values of M are allowed. The possible values of M are the various multiples of the monomer molecular weight — M0. By monomer molecular weight we mean the weight per monomer that appears in the polymer chain. For condensation reactions, for example, where molecules of water are typically lost from the monomers during reaction, we will
take M0 as the monomer molecular weight less any weight loss due to the polymerization reaction. The possible values of M make up a set of numbers with discrete values labeled Mi. Let Ni be the number of polymers with molecular weight Mi. Then the total weight of all polymers is
Total Weight = Sigma (NiMi) i=1 to ∞
and the total number of polymer molecules is
Total Number = Sigma (Ni) i=1 to ∞
Now, Consider of polymer property which depends not just on the number of polymer molecules but on the size or weight of each polymer molecule. A classic example is light scattering. For such a property we need a weight average molecular weight. To derive the weight average molecular weight, replace the appearance of the number of polymers of molecular weight i or Ni in the
number average molecular weight formula with the weight of polymer having molecular weight i or NiMi . The result is
MW = Sigma (NiMi2) i=1 to ∞devided with Sigma (NiMi) i=1 to ∞
By noting that NiMi/ Sigma (NiMi) i=1 to ∞
is the weight fraction of polymer with molecular weight i, wi, an
alternative form for weight average molecular weight in terms of weight fractions
MW = Sigma (wiMi) i=1 to ∞
Comparing this expression to the expression for number average molecular weight in terms of number fraction we see that MN is the average Mi weighted according to number fractions and that MW is the average Mi weighted according to weight fractions. The meanings of their names are thus apparent.
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