Suppose we have a chemical reaction where a linear chain can form (reversibly) from building blocks (e.g. atoms or molecules):

$$\ce{X_$i$ + X_$k$ <=>[$k_\mathrm{fwd}$][$k_\mathrm{rev}$] X_$i$X_$k$}$$

where $k_\mathrm{fwd}$ and $k_\mathrm{rev}$ are rate constants.

What is the distribution of polymer lengths (at equilibrium)? What are some canonical references that derive these distributions under variations of the above polymerization process (e.g. step versus chain growth)?

  • $\begingroup$ I allowed myself to edit your question a bit by merging both forward and reverse reactions into one (since you are speaking of equilibrium) and I also changed notations for the rate constants to the standardized ones $k_\mathrm{fwd}$ and $k_\mathrm{rev}$ to avoid ambiguity. $\endgroup$
    – andselisk
    May 28, 2019 at 4:37
  • $\begingroup$ When this exercise come from? $\endgroup$
    – Alchimista
    May 28, 2019 at 9:44
  • $\begingroup$ As the expression is written at present you have numerous pairs of species in equilibrium. There is no way for them to grow. You should look for references to kinetics of radical and of condensation polymerisation. Many chemical kinetics texts cover these topics. $\endgroup$
    – porphyrin
    May 28, 2019 at 10:21

1 Answer 1


Concentrations are a geometric series

First, let's focus on the reactions where one of the reactants is a monomer $\ce{X_1}$:

$$\ce{X_1 + X_n <=> X_{n+1}}$$

and call the equilibrium constant for this reaction $K_{\mathrm{extend}}$.

From this, we can figure out the equilibrium constant for making a polymer from monomers:

$$\ce{n X_1 <=> X_n}$$

$$K_n = (K_{\mathrm{extend}})^{n-1} = \frac{[X_n]}{[X_1]^n}$$

So we can express the concentration of a polymer $X_n$ dependent on the monomer concentration:

$$[X_n] = (K_{\mathrm{extend}})^{n-1} [X_1]^n = \frac{1}{K_{\mathrm{extend}}}(K_{\mathrm{extend}} \cdot [X_1])^n$$

This is a geometric series (if you take the concentration of polymers as a series, the ratio of neighbors in the series is constant). Because the amount of matter is finite, $K_{\mathrm{extend}}\cdot [X_1] < 1$ for an equilibrium scenario (as opposed to ending up with a single chain containing all the monomers). So the concentration of $[X_1]$ is the highest, and concentrations drop with increase of $n$.

Expression for $[X_1]$

Let $[X]_\mathrm{tot}$ be the total concentration of monomer, no matter whether in a polymer or not.

$$ [X]_\mathrm{tot} = \sum_{j=1}^{\infty} j[X_j]$$ $$ = \frac{1}{K_{\mathrm{extend}}} \sum_{j=1}^{\infty} j (K_{\mathrm{extend}}\cdot [X_1])^j $$

Using $$\sum_{j=1}^{\infty} j \beta^j = \frac{\beta}{(\beta - 1)^2}$$

we get

$$ [X]_\mathrm{tot} = \frac{[X_1]}{(K_{\mathrm{extend}}\cdot [X_1] - 1)^2}$$

which you can solve for the monomer concentration.

Relationship to real polymerization reactions

The polymerization reactions in chemistry and biochemistry that I am aware of are kinetically controlled. There are fibers (such as actin) in biochemistry that are dynamic (i.e. there is polymerization and depolymerization), but the processes are affected by nucleotide binding, complicating matters. I would be interested to hear whether the OP has an example that behaves like this.

  • $\begingroup$ here there is polymerization/depolymerization too because the end of the polymer can come off. can you say what properties of actin fibers change the dynamics you outlined? $\endgroup$
    – user9277
    May 30, 2019 at 13:27
  • $\begingroup$ Can you point to a book or article which contains this derivation? $\endgroup$
    – Tony Ruth
    Jul 6, 2022 at 15:45
  • 1
    $\begingroup$ I did the derivation from scratch, but here is a more comprehensive treatment that has the result as the simplest 1D case: hal.archives-ouvertes.fr/hal-01565995/document. @TonyRuth $\endgroup$
    – Karsten
    Jul 6, 2022 at 20:39
  • 1
    $\begingroup$ @TonyRuth Here is another reference that considers real systems: pubs.acs.org/doi/10.1021/jp981592z $\endgroup$
    – Karsten
    Jul 6, 2022 at 20:45

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