Weight-average molecular weight

 

Example 1

It has been determined that polyvinyl acetate (Table 1) has the molecular size distribution shown in Figure. What are the "weight-average" molecular weight and the average degree of polymerization?

 

Table 1. Molecular weight distribution 

Range of molecular weights, gm/mol wt

(MW)i, gm/mol wt

Wi, fraction

(Wi)(MW)i, gm/mol wt

5,000 - 10,000

7,500

0.12

900

10,000 - 15,000

12,500

0.18

2,250

15,000 - 20,000

17,500

0.26

4,550

20,000 - 25,000

22,500

0.21

4,725

25,000 - 30,000

27,500

0.14

3,850

30,000 - 35,000

32,500

0.09

2,925

 

 

Sum = 1.00

Sum = 19,200

 

 

Figure. Polymer size distribution

 

Solution: 

On the basis of 100 g polyvinyl acetate,

Mw = {Sum [(Wi)(MW)i]}/{Sum Wi} = 19,200 g/(ave mol wt)

Mer weight of vinyl acetate:

C4H6O2 = 48 + 6 + 32 = 86 g/mer wt

DP = 19200/86 = 224

Units: (g/ave mol wt mers )/(g/mer wt ave mol wt ) *

 

 

Number-average molecular weight

 

Example 2

Determine the "number-average" molecular weight for the polymer of Example 1.

 

Solution:

On the basis of 100 g polyvinyl acetate.

 

Table 2 

Range of molecular weights, gm/mol wt

Molecules

(Xi) (MW)i

5,000 - 10,000

(12)(AN)*/7,500

(12) (AN)

10,000 - 15,000

(18)(AN)/12,500   

(18) (AN)

15,000 - 20,000

(26)(AN)/17,500

(26) (AN)

20,000 - 25,000

(21)(AN)/22,500

(2 1) (AN)

25,000 - 30,000'

(14)(AN)/27,500

(14)(AN)

30,000 - 35,000

(9)(AN)/32,500

(9) (AN)

 

(0.00624) (AN)

(100) (AN)

 

100(AN)/ (0.00624)(AN)= 16,010 gm/mole

 

  * AN = Avogadro's number.

As seen from the above example, the number-average molecular weight is 16,010, while the weight-average molecular, determined before, was 19,200. Whenever there is a distribution in sizes, the "number-average" molecular weight will be less than the "weight-average" molecular weight, because of the large numbers of small molecules in the smaller weight fractions. If all the molecules were the same size, the two averages would be identical.

 

 


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E-mail: ksoderholm@dental.ufl.edu

© 1999, Karl-Johan M. Söderholm