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.

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

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

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 200,000 1/3 x 1,000 Mn = Σni·Mi = 400,333 We can say that the average person lives in a city of about 400,000.

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 0.8326 = 832,600 = 200,000 x 0.1665 = 33,300 = 1,000 x 0.0008 = 0.8 200,000 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.

Calculating Mn & Mw. Mn = Σni·Mi Mw = Σwi·Mi M1 = 100 M2 = 10 n1 = ½ = ½ · 100 + ½ · 10 = 50 + 5 = 55 Σni·Mi w1 n1·M1 Mn = = ½ · 100 / 55 = 10/11 w2 n2·M2 Mn = = ½ · 10 / 55 = 1/11 Mw = = 10/11 · 100 + 1/11 ·10 = 90.9 + 0.9 = 91.8 Σwi·Mi

Calculating Mn & Mw. Mn = Σni·Mi Mw = Σwi·Mi M1 = 100 M2 = 10 = 1/11 · 100 + 10/11 · 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 · 100 + ½ ·10 = 50 + 5 = 55 Σwi·Mi

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 · 100 + 1/11 ·10 = 91.8 Mw Nr of Chains Mn Higher Av. Mw

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

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

Weight fractions Probability for chain transfer: 0.3 0.1 0.01 0.001