Page 99 - Carrahers_Polymer_Chemistry,_Eighth_Edition

62 Carraher’s Polymer Chemistry

FIGURE 3.5 Jar with capsules, each capsule containing a single polymer chain where the capsule size is the
same and independent of the chain size, illustrating the number-average dependence on molecular weight.

N i

M i

FIGURE 3.6 Molecular weight distribution for a polydisperse polymer sample constructed from “capsule-
derived” data for the number-average situation such as given in Figure 3.5.

on statistical approaches that can be described mathematically and which correspond to physical
measurements of specifi c values.
The number average value, corresponding to a measure of chain length average, is called
the number-average molecular weight, M . Physically, the number-average molecular weight can
n
be measured using any technique that “counts” the molecules; that is, it is directly dependent on the
number of chains. These techniques include vapor phase and membrane osmometry, freezing point
lowering, boiling point elevation, and end-group analysis.
We can describe the number average using a jar filled with plastic capsules such as those used to

contain tiny prizes (Figure 3.5). Here, each capsule contains one polymer chain. All the capsules are
of the same size, regardless of the size of the polymer chain. Capsules are then withdrawn, opened,
and the individual chain length measured and recorded. A graph such as that shown in Figure 3.6
can be constructed from the data with the maximum value being the number-average molecular
weight. The probability of drawing a particular capsule is dependent on the number of each capsule
and not on the size of the chain within the capsule.
The weight-average molecular weight is similarly described, except that the capsule size corre-
sponds to the size of polymer chain contained within it (Figure 3.7). In this approach, the probability

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