Weight Average Molecular Weight

Number Average and Weight Average Molecular Mass MCQs

Weight Average Molar Weight (M_w)

Weight average molecular weight is the weight fraction of molecules in a polymer sample. It is another way of determining the molecular mass of a polymer. It gives the average of the molecular masses of the individual macromolecules in the polymer sample.
When the total molecular weight of group of molecules having particular molecular weights are multiplied with their respective molecular weight, the product are added and the sum is dived by the total weight of all molecules, the result obtained is called weight average molecular weight. Let N₁, N₂ etc. molecules have molecular weights M₁,M₂ etc. respectively.
Total molecular weight of N₁ molecules = N₁M₁
Total molecular weight of N₂ molecules = N₂M₂ and so on...
The products with their respective molecular weights will be N₁M₁ X N₂M₂

Sum of the products = N₁M₁² + N₂M₂² +... = ΣN_iM_i²
Total weight of all molecules = N₁M₁ + N₂M₂ +... = ΣN_iM_i
Hence, weight average molecular weight (M_w) = N₁M₁² + N₂M₂² +... / N₁M₁ + N₂M₂ +...
weight average molecular weight (M_w) = ΣN_iM_i² / ΣN_iM_i

Number Average Molecular Weight (M_w)

Under what condition Number Average Molecular Weight and Weight Average Molecular Weight are same ?

We know that-
Number average molecular weight (M_n) = [N₁M₁ + N₂M₂ + ...] / [N₁ + N₂ + ...]
and Weight average molecular weight (M_w) = [N₁M₁² + N₂M₂² + ...] / [N₁M₁ + N₂M₂ + ...]
If M_n = M_w
[N₁M₁ + N₂M₂ + ...] / [N₁ + N₂ + ...] = [N₁M₁² + N₂M₂² + ...] / [N₁M₁ + N₂M₂ + ...]
The LHS and RHS can be eqaul only when M₁ = M₂ = ... = M_i
That means all molecules should have same molecular weight. In that case average has no significance.

Equal number of molecules with M₁ = 1,00000 and M₂ = 10,000 are mixed. Calculate weight average molecular weight.

Answer

Let, N₁ = N₂ = 10
Given-
M₁ = 100000
M₂ = 10000
Putting these values in the formula we get-
weight average molecular weight (M_w) = 91818