1. Molecular weight and molecular weight distribution.
In a polymer sample the chains hardly ever have the same length.
They have an average length.
Exceptions are for example proteins.

2. Number of chains with length i
Number average molecular weight.
Mole fraction ni =
Number of chains with length i
Total number of chains Ni
ΣNi =
Σni = 1
ΣNi·Mi
ΣNi
Mn =
·Mi Σ Ni
( ) =
= Σni·Mi

3. Weight average molecular weight.Mn
Weight fraction wi =
Weight of chains with length i
Total weight of all chains
ni·Mi
Σni·Mi =
Σwi = 1
Σni·Mi2
Σni·Mi
Mw = =
Σwi·Mi
ΣNi·Mi
ΣNi
Compare: Mn = =
Σni·Mi

4. An example: City Population Σni·Mi Amsterdam Groningen Schiermonnikoog
1,000,000
200,000
1,000
Average population:
1/3 x 1,000,000
1/3 x ,000
1/3 x ,000
Mn =
Σni·Mi
= 400,333
We can say that the average person lives in a city of about 400,000.

5. An example:
Weighted average: This is an average that would account for the fact that a large city like Amsterdam holds a larger percentage of the total population of the three cities than Schiermonnikoog.
City
Population
Amsterdam
Groningen
Schiermonnikoog
1,000,000
200,000
1,000
Mw =
Σwi·Mi
1,000,000
1,201,000
1,000,000 x
200,000 x
1,000 x
= 1,000,000 x = 832,600
= ,000 x = 33,300
= ,000 x =
200,000
1,201,000
1,000
1,201,000 +
865,900.8
We can say that the average person lives in a city of about 866,000.

6. Calculating Mn & Mw. Mn = Σni·Mi Mw = Σwi·Mi M1 = 100 M2 = 10 n1 = ½
= ½ · ½ · 10
= = 55
Σni·Mi w1
n1·M1 Mn =
= ½ · 100 / 55 = 10/11 w2
n2·M2 Mn =
= ½ · 10 / 55 = 1/11
Mw =
= 10/11 · /11 ·10
= = 91.8
Σwi·Mi

7. Calculating Mn & Mw. Mn = Σni·Mi Mw = Σwi·Mi M1 = 100 M2 = 10
= 1/11 · /11 · 10 = 18.2
Σni·Mi w1
n1·M1 Mn =
= 1/11 · 100 / 18.2 = ½ w2
n2·M2 Mn =
= 10/11 · 10 / 18.2 = ½
Mw =
= ½ · ½ ·10 = = 55
Σwi·Mi

8. ΣNi·Mi ΣNi Mn = = Σni·Mi Σni·Mi2 Σni·Mi Mw = = Σwi·Mi Mn Σwi/Mi
n1·M1
= 1/11 · 100 / 18.2 = ½
n2·M2 w2
= 10/11 · 10 / 18.2 = ½
Σni·Mi3
Σni·Mi2
Mz = =
Σzi·Mi zi
Σwi·Mi
wi·Mi
ni·Mi Mw wi
zi/Mi
Σzi/Mi Mz
z1 =
w1·M1
= ½ · 100 / 55 = 10/11
z2 =
w2·M2
= ½ · 10 / 55 = 1/11
Σzi·Mi = 10/11 · /11 ·10 = 91.8 Mw
Nr of
Chains Mn
Higher Av. Mw

9. Consequences of the Schultz Flory Distribution:
Dispersity (D): Mw / Mw = 2
Mw = weight average = wiMi , wi = 1
Mn = number average = niMi , ni = 1

10. Schulz-Flory distribution:
Chain growth and chain transfer independent of chain length

11. Weight fractions
Probability for chain transfer:
0.3
0.1
0.01
0.001