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Unlike small molecules with definite molecular weights, polymers are a mixture of individual polymer chains of varying lengths, each with a unique molecular weight. So, the molecular weight of a polymer is expressed as an average value based on the average size of the polymer chains. The two most common forms of averages used for polymers are the number average molecular weight and weight average molecular weight.

The number average molecular weight (Mn) is the summation of the number fractions (x) of the unique polymer chains multiplied by their molecular weights (M).

Figure1

Here, each number fraction (xj) is the ratio of the number of polymer chains with a particular molecular weight (nj) to the total number of polymer chains.

Figure2

The number average molecular weight is a good measure of the chain length of the polymer.

Table 1: Calculation of the number fraction and number average molecular weight (NAMW) of a polymer sample with polymer chains of different weights

Chain No. of chains MW of chain Number fraction NAMW
1 3 10 3/21 30.71
2 5 20 5/21
3 9 35 9/21
4 4 50 4/21
Total 21

Alternatively, the weight average molecular weight (Mw) is the sum of the weight fractions (w) of the polymer chains multiplied by their molecular weights.

Figure3

Here, each weight fraction (wj) is the ratio of the total weight of all polymer chains with a specific molecular weight (njMj) to the total weight of the polymer sample.

Figure4

Weight average molecular weight gives an idea about the statistical size of the polymer.

Table 2: Calculation of the weight fraction and weight average molecular weight (WAMW) of a polymer sample with polymer chains of different weights.

Chain No. of chains MW of chain Total weight Weight fraction WAMW
1 3 10 30 30/645 82.66
2 5 20 100 100/645
3 9 35 315 315/645
4 4 50 200 200/645
Total 21 645

Typically, the weight average molecular weight is higher than the number average molecular weight because the larger chains in a sample weigh more than the smaller chains.

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