# 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 N

_{1}, N

_{2}etc. molecules have molecular weights M

_{1},M

_{2}etc. respectively.

Total molecular weight of N

_{1}molecules = N

_{1}M

_{1}

Total molecular weight of N

_{2}molecules = N

_{2}M

_{2}and so on...

The products with their respective molecular weights will be N

_{1}M

_{1}X N

_{2}M

_{2}

Sum of the products = N

_{1}M

_{1}

^{2}+ N

_{2}M

_{2}

^{2}+... = ΣN

_{i}M

_{i}

^{2}

Total weight of all molecules = N

_{1}M

_{1}+ N

_{2}M

_{2}+... = ΣN

_{i}M

_{i}

Hence, weight average molecular weight (M

_{w}) = N

_{1}M

_{1}

^{2}+ N

_{2}M

_{2}

^{2}+... / N

_{1}M

_{1}+ N

_{2}M

_{2}+...

weight average molecular weight (M

_{w}) = ΣN

_{i}M

_{i}

^{2}/ ΣN

_{i}M

_{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}) = [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.

### Equal number of molecules with M_{1} = 1,00000 and M_{2} = 10,000 are mixed. Calculate weight average molecular weight.

## Answer

Let, N_{1}= N

_{2}= 10

Given-

M

_{1}= 100000

M

_{2}= 10000

Putting these values in the formula we get-

weight average molecular weight (M

_{w}) = 91818