Q.3 (a) In a polymer sample 30% molecules have a molecular mass 20,000, 40% have molecular mass 30,000 and the 30% have 60,000. Calculate the weight average, number average molecular weight and polydispersity of polymer sample.
Best answer
Given (number fractions of molecules): take 100 molecules, so Ni = 30, 40, 30.
| Ni | Mi | NiMi | NiMi2 |
|---|---|---|---|
| 30 | 20,000 | 6.0 × 105 | 1.2 × 1010 |
| 40 | 30,000 | 12.0 × 105 | 3.6 × 1010 |
| 30 | 60,000 | 18.0 × 105 | 10.8 × 1010 |
| Σ = 100 | 36.0 × 105 | 15.6 × 1010 |
Number-average molecular weight:
\( \bar M_n = \dfrac{\sum N_iM_i}{\sum N_i} = \dfrac{36\times10^5}{100} = 36{,}000 \)
Weight-average molecular weight:
\( \bar M_w = \dfrac{\sum N_iM_i^2}{\sum N_iM_i} = \dfrac{15.6\times10^{10}}{36\times10^5} = 43{,}333 \)
Polydispersity index:
\( PDI = \dfrac{\bar M_w}{\bar M_n} = \dfrac{43333}{36000} = 1.20 \)
M̄w ≈ 43,333; M̄n = 36,000; PDI ≈ 1.20
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