In a polymer sample, 20% of the molecules have molecular mass 10,000, 50% have molecular mass 25,000 and 30% have molecular mass 40,000. Calculate the number average and weight average molecular weights and the polydispersity index of the sample.

Answer: \(\bar M_n=26{,}500\); \(\bar M_w=30{,}660\); PDI \(=1.157\).

Take 100 molecules: \(n_1=20,\ M_1=10000\); \(n_2=50,\ M_2=25000\); \(n_3=30,\ M_3=40000\).

Number average molecular weight:

\[\bar M_n=\frac{\sum n_iM_i}{\sum n_i}=\frac{20(10000)+50(25000)+30(40000)}{100}=\frac{200000+1250000+1200000}{100}=\frac{2650000}{100}=26500\]

Weight average molecular weight:

\(\sum n_iM_i^2=20(10^8)+50(6.25\times10^8)+30(1.6\times10^9)=2\times10^9+3.125\times10^{10}+4.8\times10^{10}=8.125\times10^{10}\)

\[\bar M_w=\frac{\sum n_iM_i^2}{\sum n_iM_i}=\frac{8.125\times10^{10}}{2.65\times10^6}=30660\]

Polydispersity index:

\[\text{PDI}=\frac{\bar M_w}{\bar M_n}=\frac{30660}{26500}=1.157\]

Since PDI > 1, the sample is polydisperse (chains of different lengths); \(\bar M_w\) is larger than \(\bar M_n\) because heavier molecules count more in the weight average.