How big is a polymer?

Usually, "short" polymers are called oligomers. When we start getting to tens, hundreds, thousands of repeat units or more, we have polymers. They can be huge. Remember back to general chemistry, where you learned how to calculate the molar mass of a molecule? Similarly, we can describe the "size" of a polymer using molar mass (or molecular weight) which is typically defined in units of g/mol and abbreviated as "M". Another way of describing the size of a polymer is by its degree of polymerization. Degree of polymerization (described by "x") is the number of repeat units in a polymer. If we know the molar mass of a polymer, how can we figure out the degree of polymerization? Similarly, if we know degree of polymerization, how can we figure out molar mass? All we need to know is the molar mass of the repeat unit!

Let us set up this relationship, where the new variable M₀ is the molar mass of the repeat unit:

$$M=x{\overline{M}}_0$$

Where:

• $M$ = molar mass
• $x$ = degree of polymerization
• $M_0$ = molar mass of the repeat unit

We see that the total molar mass of the polymer is just a function of the degree of polymerization and the molar mass of the repeat unit.

PROBLEM 2

A polymer with molar mass of 35,200 g/mol. has a degree of polymerization of 800. Which polymer, of the ones shown below, could it be?

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ANSWER 2

D. poly(vinyl alcohol)

PROBLEM 3

What is the molar mass of polyethylene (shown below) which has a degree of polymerization of 100?

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ANSWER 3

$$M=x{\overline{M}}_0= 100*\frac{28g}{mol}=\frac{2,800g}{mol}$$

PROBLEM 4

The following monomer is polymerized (condensation polymerization with loss of water). If the degree of polymerization is 100, what is the molar mass of the polymer?

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ANSWER 4

$M$ = 7,200 g/mol

1. First, draw the polymer in terms of the repeat unit:
   
2. Determine the molecular formula of the repeat unit:
   3 Carbon, 2 Oxygen, 4 Hydrogen
3. Calculate the molar mass of the repeat unit, ${\overline{M}}_0$:
   3 (12 g/mol) + 2 (16 g/mol) + 4 (1 g/mol) = 72 g/mol
4. Calculate polymer molar mass, $M=x{\overline{M}}_0$:
   $M=100\ast \frac{72\text{g}}{\text{mol}}=\text{7,200 g/mol}$

What if our polymer has more than one monomer, or more than one type of repeat unit? In that case, we define the mean ${\overline{M}}_0$ of the copolymer as just a weighted average of the repeat unit, where the weights are the mole fraction (X) of each type of repeat unit.

$${\overline{M}}_0={{\displaystyle \sum }}^{\text{}}{X}_j{M}_0^j$$

Where:

• $M$ = molar mass
• $x$ = degree of polymerization
• $M_0$ = molar mass of the repeat unit

Example:

What is the mean molar mass of the repeat unit for a copolymer comprised of 20 mol% styrene, 30 mol% methyl methacrylate, and 50% vinyl chloride? The chemical structures of the monomers and the resulting copolymer are shown below.

$${\overline{M}}_0=0.2*\frac{104g}{mol}+0.3*\frac{100g}{mol}+0.5*\frac{62.5g}{mol}=\frac{82.05g}{mol}$$