# 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 N

_{1}molecules have molecular weight M

_{1}each, N

_{2}molecules have molecular weight M

_{2}each and so on...

Then we have-

Total molecular weight of all the N

_{1}molecules = N

_{1}M

_{1}

Total molecular weight of all the N

_{2}molecules = N

_{2}M

_{2}and so on...

Therefor, the total molecular weight of all the molecules = N

_{1}M

_{1}+ N

_{2}M

_{2}+ ... = ΣN

_{i}M

_{i}

Total number of all the molecules = N

_{1}+ N

_{2}+ N

_{3}+ ... = ΣN

_{i}

Hence, the number average molecular weight (M

_{n}) = [N

_{1}M

_{1}+ N

_{2}M

_{2}+ ...] / [N

_{1}+ N

_{2}+ N

_{3}+ ...]

or, the number average molecular weight (M

_{n}) = ΣN

_{i}M

_{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}) = [N

_{1}M

_{1}+ N

_{2}M

_{2}+ ...] / [N

_{1}+ N

_{2}+ ...]

and Weight average molecular weight (M

_{w}) = [N

_{1}M

_{1}

^{2}+ N

_{2}M

_{2}

^{2}+ ...] / [N

_{1}M

_{1}+ N

_{2}M

_{2}+ ...]

If M

_{n}= M

_{w}

[N

_{1}M

_{1}+ N

_{2}M

_{2}+ ...] / [N

_{1}+ N

_{2}+ ...] = [N

_{1}M

_{1}

^{2}+ N

_{2}M

_{2}

^{2}+ ...] / [N

_{1}M

_{1}+ N

_{2}M

_{2}+ ...]

The LHS and RHS can be eqaul only when M

_{1}= M

_{2}= ... = 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}= [N

_{1}M

_{1}+ N

_{2}M

_{2}+ N

_{3}M

_{3}+ ...] / [N

_{1}+ N

_{2}+ N

_{3}+ ...]

Number of 1st type of molecules N

_{1}= 30

Mass of 1st type of molecules M

_{1}= 20,000

Number of 2nd type of molecules N

_{2}= 40

Mass of 2nd type of molecules M

_{2}= 30,000

Number of 3rd type of molecules N

_{3}= 30

Mass of 3rd type of molecules M

_{3}= 60,000

Substituting the given data in above equation we get-

M

_{n}= 36,000