Summation

Summation is used in the different definitions of molecular weight. The definition for Number average molecular weight makes sense, but the other three definitions are abstract concepts. An example molecular weight calculation is thus shown below, and it is recommended that you work through this problem on your own prior to the test.

Assume a polymer with monomer of molecular weight 100.
Ignore that the end groups have molecular weight 101 because of an addition hydrogen, and assume a chain with a degree of polymerization of 50 has a molecular weight of 5000.
There are
- 2 chains with D.P.=60

If asked to calculate the average, you would calculate

5(10) + 3(30) + 2(60) = 50 + 90 + 120 = 260, and then,
260 / (5 + 3 + 2) = 26
and conclude that the average D.P is 26, and the average chain has a molecular weight of 2600.
so you can figure out the answer for number average molecular weight using math you already are familiar with--now look at how this same math is produced using the formalism for summation.

Now, look at the definition of number average molecular weight, specifically, the part between the two equality signs:

Let i, the counter, run from 1 to 3.

M(1) is 1000 M represents the molecular weight of an individual polymer molecule
M(2) is 3000
M(3) is 6000

and

N(1) is 5 N(i) represents the number of polymer molecules with molecular weight M(i)
N(2) is 3
N(3) is 2

becomes 5(1000) + 3(3000) + 2(6000) and

becomes 5 + 3 + 2, and you saw all this above when asked to calculate the average.

More on Definite and Indefinite Summation available.

Here is the formula Weight Average Molecular Weight:

using the numbers from above, the calculation becomes:



5(1000)(1000) + 3(3000)(3000) + 2(6000)(6000)   5,000,000 + 27,000,000 + 72,000,000
--------------------------------------------- = ----------------------------------- =
        5(1000) + 3(3000) + 2(6000)                     5000 + 9000 + 12000



104,000,000
----------- = 4000
  26,000

The weight average molecular weight is 4,000. Note, on a test, check to see that your answer to falls between the lowest molecular weight (1000 in the above problem) and the highest molecular weight (6000 in the above problem) as a test of the reasonability of your answer.

From here you should go to Linear Combination

Last Update- January 8, 1995- wld