Question

Suppose 30 $$ molecules have M = 20000; 40$$ molecules have M=30000, rest of them have M=60000, then PDI is:
    (A) 1.45
    (B) 1.20
    (C) 0.83
    (D) 0.98

Answer 1

Hint: To find out the polydispersity index, first we need to work out the number average molecular mass. Then weight average molecular mass should be found. On dividing the weight average molecular mass by number average molecular mass, we get PDI.

Complete answer:
-Polydispersity index (PDI) is used as a measure of the distribution of molecular mass in a given polymer sample. Larger the PDI, the broader will be the molecular weight.
-As we know macromolecules(polymers) have a molar mass distribution due to the simplest homopolymers made of homologue chains with a different number of repeating units. Thus, they exhibit the average values of two types
    (i)Number average molecular mass ($\overline{M_n}$)
    (ii)Weight average molecular mass ( $\overline{M_w}$)
    The ratio of these average values gives us the PDI
             PDI = ${\displaystyle \frac{\overline{M_w}}{\overline{M_n}}}$
We are going to find out the $\overline{M_n}$first. The equation to find $\overline{M_n}$ is given below,
$\overline{M_n}$= ${\displaystyle \frac{\sum _{}^{}{N}_i{M}_i}{\sum _{}^{}{N}_i}}$
Consider that there are 100 total numbers of molecules. Then the percentage value in the question gives us the number of molecules (N) for each value of M. For example, N= 30 for M= 20000. Let's substitute the given values in the above equation,
$\overline{M_n}$ = ${\displaystyle \frac{(30\times 20000)+(40\times 30000)+(30\times 60000)}{(30+40+30)}}$
          =36000
Thus, we got the Number average molecular mass. Next, we are going to find out the Weight average molecular mass ( $\overline{M_w}$). The equation to find out $\overline{M_w}$is,
  $\overline{M_w}$ =${\displaystyle \frac{\sum N_i{M}_i^2}{\sum N_i{M}_i}}$
On substituting the given values, the equation becomes,
$\overline{M_w}$= ${\displaystyle \frac{30\times {(20000)}^2+40\times {(30000)}^2+30\times {(60000)}^2}{(30\times 20000)+(40\times 30000)+(30\times 60000)}}$
         =43333
Since we got both the values of $\overline{M_w}$and $\overline{M_n}$, substituting this in the equation of PDI we get,
PDI = ${\displaystyle \frac{43333}{36000}}$
       = 1.20

Therefore, the answer is option (B) 1.20.

Note:
Polydispersity index can also be calculated by the degree of polymerization method and in this method, the weight-average degree of polymerization is divided by the number average degree of polymerization. It should be noted that the value of $\overline{M_w}$ will always be greater than the number of average molecular mass $\overline{M_n}$. If PDI is 1 then the polymer chain is monodisperse and as PDI value increases the heterogeneity in cross linking, network formation etc. will be more with random arrangement.