Example Problem

You would like to make a polyester with $M_{n} = 5000 g / m o l$ by reacting 1 mol butane-1,4-diol and 1 mol of adipic acid. At what value of $p$ should you stop your reaction to obtain this size?

Molecular structure of Butane-1,4-diol + adipic acid

Figure 3.2: Butane-1,4-diol + adipic acid

Source: Lauren Zarzar

Let’s break this down into parts. First, let’s calculate ${\bar{M}}_{O}$ . What is ${\bar{M}}_{O}$ ? To figure that out, we need to draw the polymer that is formed:

Molecular structure of butanediol adipic acid polymer

Figure 3.3: Butane-1,4-diol adipic acid polymer

Source: Lauren Zarzar

What is the repeat unit molar mass of this polymer? We simply add up the molar masses of the atoms:

$10 C + 4 O + 16 H = 200g/mol$

However, realize that there are 2 monomers in this repeat unit! Thus,

${\bar{M}}_{O} = \frac{m o l a r m a s s r e p e a t u n i t}{# m o n i m e r u n i t s i n r e p e a t} = \frac{200 g / m o l}{2} = 100 g / m o l$

Now we know that ${\bar{M}}_{O} = 100 g / m o l$ , what is ${\bar{x}}_{n}$ ? We set up the following relationship:

$5000 g / m o l = \frac{100 g}{m o l} {\bar{x}}_{n}$

${\bar{x}}_{n} = 50$

Now we know that ${\bar{x}}_{n} = 50$ . What is the extent of reaction you need to obtain this size of polymer?

$50 = \frac{1}{1 - p}$

$p = 0.98$

So now we know Carothers theory for when we have a stoichiometric balance of $A$ and $B$ reactive groups. But what do we do if we don’t have the same number of reactive groups?