Question

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 1

Hint: 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:
$$\begin{gathered} \overline{M_n}={\displaystyle \frac{N_1{M}_1+N_2{M}_2+N_3{M}_3+.....}{N_1+N_2+N_3+.....}} \\ \text{Where,} \\ \overline{M_n}=\text{ Number - average molecular mass} \\ {N}_1,N_2,...,N_n=\text{ Number of different molecules present in the polymer} \\ {M}_1,M_2,...,M_n=\text{ Mass of different molecules present in the polymer} \end{gathered}$$

Complete step by step answer: Step by step solution:
Given data:
​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 $M_3=60,000$
Using the relation of number average molecular mass which is stated below:
$$\begin{gathered} \overline{M_n}={\displaystyle \frac{N_1{M}_1+N_2{M}_2+N_3{M}_3+.....}{N_1+N_2+N_3+.....}} \\ \text{Where,} \\ \overline{M_n}=\text{ Number - average molecular mass} \\ {N}_1,N_2,...,N_n=\text{ Number of different molecules present in the polymer} \\ {M}_1,M_2,...,M_n=\text{ Mass of different molecules present in the polymer} \end{gathered}$$
Substituting the given data in above equation we get,
$$\begin{gathered} \overline{M_n}={\displaystyle \frac{(30\times 20,000)+(40\times 30,000)+(30\times 60,000)}{30+40+30}} \\ \Rightarrow \overline{M_n}=36,000 \end{gathered}$$
Therefore, option B is correct.

Additional Information: There are 2 types for calculation of average molecular masses of polymers.
-Number average molecular mass (generally determined by osmotic pressure measurement)
-Weight average molecular mass (generally determined by light scattering, ultra centrifuge or sedimentation etc.)
-These methods are adopted because of the polymerization phenomena, this phenomena is very complicated and because of this synthetic polymers are formed with different lengths. So to resolve this average value of molecular mass is taken.
-To determine the homogeneity of a polymer PDI is used. PDI (Polydispersity Index) is the ratio of weight average molecular mass and number average molecular mass.

Note: While solving these questions it is required to pay attention to whether the question is about number average molecular mass or weight average molecular mass, because both of them give distinctive values.