21.3 : Polymers: Defining Molecular Weight

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 (M_n) is the summation of the number fractions (x) of the unique polymer chains multiplied by their molecular weights (M).

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

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 (M_w) is the sum of the weight fractions (w) of the polymer chains multiplied by their molecular weights.

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

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.