Number Average Molecular Weight

Number Average and Weight Average Molecular Mass MCQs

Number Average Molar Weight (M_n)

Number average molecular weight is the mole fraction of molecules in a polymer sample. It is a way of determining the molecular mass of a polymer. It gives the average of the molecular masses of the individual macromolecules. In other words, when total molecular weight of all the molecules of a sample is divided by the total number of molecules, the result obtained is called the number average molecular weight.
Let N₁ molecules have molecular weight M₁ each, N₂ molecules have molecular weight M₂ each and so on...
Then we have-

Total molecular weight of all the N₁ molecules = N₁M₁
Total molecular weight of all the N₂ molecules = N₂M₂ and so on...
Therefor, the total molecular weight of all the molecules = N₁M₁ + N₂M₂ + ... = ΣN_iM_i
Total number of all the molecules = N₁ + N₂ + N₃ + ... = ΣN_i
Hence, the number average molecular weight (M_n) = [N₁M₁ + N₂M₂ + ...] / [N₁ + N₂ + N₃ + ...]
or, the number average molecular weight (M_n) = ΣN_iM_i / ΣN_i

Weight 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.

A sample of polymer contains 30% molecules with molecular mass 20,000, 40% with molecular mass 30,000 and 30% with molecular mass 60,000. What is the number average molecular mass of the polymer ?
A. 27,000
B. 36,000
C. 43,000
D. 64,000

Answer

Number average molecular mass is the ratio of total mass of all the molecules of a given sample to the total number of molecules. The number average molecular mass is calculated by given relation-
M_n = [N₁M₁ + N₂M₂ + N₃M₃ + ...] / [N₁ + N₂ + N₃ + ...]
Number of 1st type of molecules N₁ = 30
Mass of 1st type of molecules M₁ = 20,000
Number of 2nd type of molecules N₂ = 40
Mass of 2nd type of molecules M₂ = 30,000
Number of 3rd type of molecules N₃ = 30
Mass of 3rd type of molecules M₃ = 60,000
Substituting the given data in above equation we get-
M_n = 36,000