Carothers Theory: Stoichiometric Balance of Reactive Groups

We will now consider in greater depth the theoretical treatment of linear step-growth polymerization. Specifically, we will examine the Carothers equation, which relates degree of polymerization to extent of reaction. We learned about degree of polymerization in Lesson 2, but what is extent of reaction?

If we assume equal amounts of mutually reactive groups (e.g., $A$ and $B$ groups), then we can define the extent of reaction $(p)$ as

$p = \frac{# f u n c t i o n a l g r o u p s r e a c t e d}{# f u n c t i o n a l g r o u p s i n i t i a l l y}$

Extent of reaction is really just the probability that any single functional group that was present at the start has reacted. Consider the mixture of bifunctional monomers in Figure 3.1.

Figure 3.1: Stoichiometric balance of A and B

Source: Lauren Zarzar

What is the extent of reaction shown in Figure 3.1? $A$ and $B$ each represent a functional group, so there were 16 functional groups (2 on each of 8 molecules) initially. Looking at the products, we find that 6 of those functional groups are now part of new bonds, so they reacted. We plug these values our equation for extent of reaction:

$p = \frac{# f u n c t i o n a l g r o u p s r e a c t e d}{# f u n c t i o n a l g r o u p s i n i t i a l l y} = \frac{6}{16} = 0.375$

But notice, there is another way to represent extent of reaction in terms of number of molecules, where $N$ is the number of molecules left after the reaction and $N_{O}$ is the number of molecules initially. Again, using Figure 3.1, where our initial number of molecules is 8 and our final number of molecules is 5:

$p = \frac{N_{O} - N}{N_{O}} = \frac{8 - 5}{8} = 0.375$

Let’s rearrange this new expression for extent of reaction:

$\frac{N_{O}}{N} = \frac{1}{1 - p}$

This value, $N_{O} / N$ , should be familiar, because it’s the same as the number average degree of polymerization!

${\bar{x}}_{n} = \frac{N_{O}}{N}$

We combine the equations for extent of reaction with number average degree of polymerization to yield the Carothers equation for stoichiometric balance of reactive groups:

${\bar{x}}_{n} = \frac{1}{1 - p}$

So what does this equation tell us?

If you want polymers of any significant length (i.e., high degree of polymerization, ${\bar{x}}_{n}$ ) you need very high extent of reaction $(p)$ !