Question

If \[{{N}_{1}},{{N}_{2}},{{N}_{3}},...\].are the number of molecules with molecular masses \[{{M}_{1}},{{M}_{2}},{{M}_{3}},....\] respectively, then mass average molar mass is expressed as:-
(a)- \[\dfrac{\sum{NiM{{i}^{2}}}}{\sum{NiMi}}\]

(b)- \[\dfrac{\sum{NiMi}}{\sum{Ni}}\]

(c)- \[\dfrac{\sum{M{{i}^{2}}}}{\sum{Ni}}\]

(d)- \[\dfrac{\sum{NiMi}}{\sum{Mi}}\]

Answer 1

Hint:
The mass average molar mass is defined as the total mass of the polymer divided by the total number of molecules. The mass average molar mass can be calculated by multiplying the mass fraction of fraction 1 with a molecular mass contribution by the fraction and then calculating for n number of molecules.

Complete step by step answer:
In computing the molecular mass of a polymer we can also use the mass fraction of the molecules to get the mass average molecular mass.
In the question it is given \[{{N}_{1}},{{N}_{2}},{{N}_{3}},...\].are the number of molecules, so
Total no. of molecules is given by:

\[{{N}_{1}}+{{N}_{2}}+{{N}_{3}}+.....+{{N}_{l}}=\sum{N}i\]

The molar mass contribution by fraction 1 = \[\dfrac{{{N}_{1}}{{M}_{1}}}{\sum{{{N}_{i}}}}\]

Similarly, molecular mass contribution by other fraction is given by: \[\dfrac{{{N}_{1}}{{M}_{1}}}{\sum{{{N}_{i}}}},\dfrac{{{N}_{2}}{{M}_{2}}}{\sum{{{N}_{i}}}},.......,\dfrac{{{N}_{l}}{{M}_{l}}}{\sum{{{N}_{i}}}}\]

So, total mass of the polymer = W = \[{{\sum{{{N}_{i}}M}}_{i}}\]
Mass of fraction 1 = \[{{N}_{1}}{{M}_{1}}\]
Mass fraction of fraction 1 = \[\dfrac{{{N}_{1}}{{M}_{1}}}{W}=\dfrac{{{N}_{1}}{{M}_{1}}}{\sum{{{N}_{i}}{{M}_{i}}}}\]

Molar mass contribution by fraction 1 is given by: \[\dfrac{{{N}_{1}}{{M}_{1}}{{M}_{1}}}{\sum{{{N}_{i}}{{M}_{i}}}}=\dfrac{{{N}_{1}}{{M}_{1}}^{2}}{\sum{{{N}_{i}}{{M}_{i}}}}\]

So, for n number of molecules = \[\dfrac{\sum{{{N}_{i}}{{M}_{i}}^{2}}}{\sum{{{N}_{i}}{{M}_{i}}}}\]

For all synthetic polymers, the total mass is always greater than the total number of molecules. If they were to be equal, the polymer sample may be considered as perfectly homogenous (each molecule has the same molecular mass) but this doesn't happen.

So, the correct option is (a).

Note:
There are a number average molecular mass concepts in this topic, which is similar. So don't get confused between them. And also mass average molar mass is different from average molar mass value.