Molar Mass Distributions

When you first learned about the molar mass of molecules, you learned that the molar mass is linked to the identity of the compound; for example, H₂O always had a molar mass of 18 g/mol. If the molar mass wasn't 18g/mol, it couldn't be water! The situation is very different for polymers. Take polypropylene, for example:

Polypropylene

Source: Lauren Zarzar

The molar mass of this polymer could be 420 g/mol (if degree of polymerization is 10) or 21,000 g/mol (if the degree of polymerization is 500). Although vastly different in molar mass, both of these molecules are polypropylene. For polymers, there is almost always a molar mass distribution. An example distribution is given in Figure 1.10. Although this curve looks continuous, we know that in fact it cannot be - the mass of the polymer does change in discreet units, depending on the size of the repeat unit. However, we do typically draw the distributions as continuous function.

Figure 1.10: A typical weight-fraction molar mass distribution curve for a polymer with the most probable distribution of molar mass and a repeat unit molar mass 100 g mol^-1.

Source: Based on figure 1.4 from Young, Robert J., and Peter A. Lovell.
Introduction to Polymers, Third Edition, CRC Press, 2011.

Because there is a distribution in molar mass, we have a choice as to how to actually define a characteristic molar mass for a sample. There are three general approaches for calculating the molar mass from a distribution, giving us three different values: ${\overline{M}}_n$(number average molar mass), ${\overline{M}}_w$, (weight average molar mass) and ${\overline{M}}_z$(z-average molar mass). You'll notice from Figure 1.10 that $M_n<M_w<M_z$. Each value of molar mass is defined differently.

$M_n$ is defined as the sum of the products of the molar mass of each size of polymer multiplied by its mole fraction (X). Recall from general chemistry that a mole fraction is equal to the ratio of number of moles (or molecules) of a type of polymer (N) divided by the total number of moles (or molecules). Basically, this is the same as your "average" arithmetic mean!

$${\overline{M}}_n={{\displaystyle \sum }}^{\text{}}{X}_i{M}_i=\frac{{{\displaystyle \sum }}^{\text{}}{N}_i{M}_i}{{{\displaystyle \sum }}^{\text{}}{N}_i}$$

Sometimes it's easier for us to work in weight fractions ($w_i$) rather than mole fractions, since mass is often easier to measure. The weight fraction $w_i$ is defined as the mass of molecules of molar mass $M_i$ divided by the total mass of all the molecules present:

$$w_i=\frac{N_i{M}_i}{{{\displaystyle \sum }}^{\text{}}{N}_i{M}_i}$$

Thus, we can define $M_w$:

$${\overline{M}}_w={{\displaystyle \sum }}^{\text{}}{w}_i{M}_i=\frac{{{\displaystyle \sum }}^{\text{}}{N}_i{M}_i^2}{{{\displaystyle \sum }}^{\text{}}{N}_i{M}_i}$$

Compare ${\overline{M}}_w$ to ${\overline{M}}_n$ — do you notice how ${\overline{M}}_w$ is a function if $M_i$ squared? Therefore, bigger polymers have a greater influence on ${\overline{M}}_w$ than they do on ${\overline{M}}_n$, skewing the value of ${\overline{M}}_w$ to be larger than ${\overline{M}}_n$.