Question

A polymeric sample in which 30% molecules have a molecular mass 20,000, 40% have 30,000 and the rest 30% have 60,000. The $({{\bar{M}}_{n}})$ and $({{\bar{M}}_{w}})$ of this sample are,
(A) 36,000, 43,333
(B) 43,000, 36,000
(C) 72,000, 86,000
(D) 86,000, 72,000

Answer 1

Hint: $({{\bar{M}}_{n}})$ and $({{\bar{M}}_{w}})$are the Number average molecular weight and Weight average molecular weight of the polymer. Just substitute the given values in their formulae and calculate the answer.

Complete answer:
Polymers are the macromolecular substances formed entirely due to bonding between small repeating subunits called Monomers. Many Monomers come together and form long chains of repeating units to give rise to Polymers.
The Molecular weights of a polymer can be found out in two forms:
1) Number Average Molecular Weight $({{\bar{M}}_{n}})$ - Number Average Molecular Weight $({{\bar{M}}_{n}})$ is the average weight based on number of polymeric chains present in the sample. It is given by,
\[{{\bar{M}}_{n}}~=\dfrac{\sum{{{N}_{i}}{{M}_{i}}}}{\sum{{{N}_{i}}}}\]
where Ni is the number of polymeric chains and Mi is the molecular mass of each polymeric chain.
2) Weight Average Molecular Weight $({{\bar{M}}_{w}})$- Weight Average Molecular Weight $({{\bar{M}}_{w}})$ is the average weight based on mass of polymeric chains present in the sample. It is given by,
     \[{{\bar{M}}_{w}}~=\dfrac{\sum{{{N}_{i}}{{M}_{i}}^{2}}}{\sum{{{N}_{i}}{{M}_{i}}}}\]
In this problem, there are three types of molecules in the given polymeric sample having three different molecular masses. So, let’s first calculate ${{\bar{M}}_{n}}$
     \[{{\bar{M}}_{n}}~=\dfrac{\sum{{{N}_{i}}{{M}_{i}}}}{\sum{{{N}_{i}}}}=\dfrac{(30\times 20000)+(40\times 30000)+(30\times 60000)}{30+40+30}\]
\[{{\bar{M}}_{n}}~=36000\]
Similarly, we can calculate ${{\bar{M}}_{w}}$using the formula and we get,
     \[{{\bar{M}}_{w}}~=\dfrac{\sum{{{N}_{i}}{{M}_{i}}^{2}}}{\sum{{{N}_{i}}{{M}_{i}}}}=\dfrac{(30\times {{20000}^{2}})+(40\times {{30000}^{2}})+(30\times {{60000}^{2}})}{30+40+30}\]
\[{{\bar{M}}_{w}}~=43,333\]
Therefore, \[{{\bar{M}}_{n}}~=36000\]and \[{{\bar{M}}_{w}}~=43,333\].

So, option (A) is the correct answer.

Note:
Don’t get mistaken by the terms 30%, 40%, etc. It just indicates that in total 100 molecules are present in the polymeric sample and out of that 30 molecules have a certain weight and similarly, 40 molecules have another weight and so on. Just substitute these values in place of \[{{N}_{i}}\].