Calculate the number average molecular mass Mn¯ and weight average molecular mass Mw¯ of a polymer in which 30% molecules have a molecular mass 20,000; 40% have 30,000 and the rest 30% have 60,000.

# Calculate the number average molecular mass $\overline{{M}_{n}}$ and weight average molecular mass $\left(\overline{{M}_{w}}\right)$ of a polymer in which 30% molecules have a molecular mass 20,000; 40% have 30,000 and the rest 30% have 60,000.

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### Solution:

The polymer contains ${\mathrm{n}}_{1}=30,{\mathrm{M}}_{1}=20,000;$ ${\mathrm{n}}_{2}=40,{\mathrm{M}}_{2}=30,000$ and ${\mathrm{n}}_{3}=30,{\mathrm{M}}_{3}=60,000$

So, $\overline{{\mathrm{M}}_{\mathrm{n}}}=\frac{{\mathrm{n}}_{1}{\mathrm{M}}_{1}+{\mathrm{n}}_{2}{\mathrm{M}}_{2}+{\mathrm{n}}_{3}{\mathrm{M}}_{3}}{{\mathrm{n}}_{1}+{\mathrm{n}}_{2}+{\mathrm{n}}_{3}}$

$\begin{array}{l}=\frac{30×20,000+40×30,000+30×60,000}{30+40+30}\\ =\frac{10,000\left(60+120+180\right)}{100}=36,000\end{array}$

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