Polymer Basics - Fundamentals of Industrial Chemistry: Pharmaceuticals, Polymers, and Business - page 101

to be 28 amu or 28 g/mol. or 28 daltons. Strictly speaking, these terms are
not synonyms, just as weight and mass are not synonyms. However, for our
purposes, the distinctions are not critical. The units g/mol are commonly used
in the industry and that is what we will use.
If we consider the molecular weight of polyethylene, we should realize
that polyethylene consists of many polymer chains of varying length. The
molecular weight of a single chain is meaningless; rather we need to consider
the average of the large number of chains. One way to think about the
average molecular weight is to add up the molecular weight of each polymer
chain and then divide by the number of chains. This is the number average
molecular weight, symbolized as M n . All polymer chains do not have an
equal importance and larger molecular weight chains can have a greater
influence than smaller molecular weight chains. Therefore, a weighted
average is often considered.
An analogy might be useful. If a 50 foot Beneteau sailboat weighs
14,000 kg and after sitting in the water, 999 barnacles, each weighing 1 g (or
0.001 kg), grow on the hull, then what is the average weight? The number
average is 14 kg, but this isn't particularly descriptive of the system. The
weighted average (for a polymer, this would be the weight average molecular
weight or M w ) is 13,999 kg, which is likely to be more descriptive of the
barnacle-ridden Beneteau. Similarly the weight average molecular weight is
more relevant for many polymer applications.
To illustrate the arithmetic, let's pick an example of a small toy boat weigh-
ing 100.00 g and three barnacles, each weighing 1.00 g. The number average
is the sum of the masses divided by the number of items. In this case, it is
a total weight of 103.0 g divided by 4 which gives us 25.75 g as the number
average weight. To calculate the weighted average, multiply each weight by
itself divided by the total weight and add the products. This gives the weighted
average of 97.12g.

100.00
103.00 =

100.00 × 97.09

1. 0 0
103.00 =

1. 0 0 × 0.01

1. 0 0
103.00 =

1. 0 0 × 0.01

1. 0 0
103.00 =

1. 0 0 × 0.01

97.12

As another example, let's consider two polymer chains, one with a
molecular weight of 40,000 g/mol and the other with a molecular weight of