Question 4 of 7: Step-growth kinetics — Nylon-12 (A–B monomer) vs. Nylon-6,6 (A–A + B–B)

Question 4: Step-growth kinetics — Nylon-12 (A–B monomer) vs. Nylon-6,6 (A–A + B–B) (Part B — 30 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

This is a derivation-and-comparison question: essay-style development of the rate law first, then the explicit \(p(t)\) and \(\overline{X}_n(t)\) expressions, then the structural comparison.

Polymerization equations

Nylon-12 forms by self-condensation of the single A–B monomer 12-aminododecanoic acid, each molecule of which carries one amine (A) and one carboxyl (B):

\(n\ \text{H}_2\text{N(CH}_2)_{11}\text{COOH} \;\xrightarrow{\text{cat.}}\; \text{H}\!-\![\text{HN(CH}_2)_{11}\text{CO]}_n\!-\!\text{OH} + (n-1)\,\text{H}_2\text{O}\)

Nylon-6,6 forms by condensation of two bifunctional monomers — hexamethylenediamine (A–A) and adipic acid (B–B) — which react only A-with-B (amine + acid → amide):

\(n\ \text{H}_2\text{N(CH}_2)_6\text{NH}_2 + n\ \text{HOOC(CH}_2)_4\text{COOH} \;\xrightarrow{}\; \text{[HN(CH}_2)_6\text{NHCO(CH}_2)_4\text{CO]}_n + (2n-1)\,\text{H}_2\text{O}\)

Kinetic derivation (catalyzed)

Assumptions. (i) equal and independent functional-group reactivity (Flory’s principle — rate constant independent of chain length); (ii) an external acid catalyst present at constant concentration, so the esterification/amidation is first-order in each of the two reacting groups and the catalyst term folds into a constant \(k\); (iii) exact stoichiometry; (iv) the condensation water is continuously removed so the reaction is irreversible; (v) constant volume.

Let \(c\) be the concentration of un-reacted amine groups. For the A–B monomer the amine and carboxyl concentrations are equal at all times, \([\text{NH}_2]=[\text{COOH}]=c\), so

\(\displaystyle -\frac{dc}{dt}=k[\text{COOH}][\text{NH}_2]=k\,c^2.\)

Integrating from \(c_0\) at \(t=0\):

\(\displaystyle \frac{1}{c}-\frac{1}{c_0}=kt \quad\Longrightarrow\quad c=\frac{c_0}{1+c_0 k t}.\)

Define the conversion (fraction of functional groups reacted) \(p=(c_0-c)/c_0=1-c/c_0\). Then

\(\displaystyle \boxed{\,p(t)=\frac{c_0 k t}{1+c_0 k t}\,} \qquad\text{and}\qquad 1-p=\frac{1}{1+c_0 k t}.\)

By the Carothers relation the number-average degree of polymerization is \(\overline{X}_n=1/(1-p)\), hence its explicit time dependence:

\(\displaystyle \boxed{\,\overline{X}_n=\frac{1}{1-p}=1+c_0 k t\,} \qquad \overline{M}_n=\overline{X}_n\,M_0 .\)

So for the catalyzed reaction \(\overline{X}_n\) grows linearly with time (a self-catalyzed melt polycondensation, by contrast, is third-order and gives \(\overline{X}_n^2\propto t\)).

Comparison with Nylon-6,6

When the diamine and diacid are charged in exactly equimolar amounts, the amine and carboxyl concentrations are again equal, \([\text{NH}_2]=[\text{COOH}]=c\), and the identical rate law applies: the \(p(t)\) and \(\overline{X}_n(t)=1+c_0kt\) expressions are mathematically the same. The essential difference is stoichiometric robustness:

• Nylon-12 (A–B monomer): every molecule carries one A and one B, so stoichiometry is automatically exact and cannot drift. High \(\overline{X}_n\) is limited only by conversion \(p\).
• Nylon-6,6 (A–A + B–B): stoichiometry must be balanced by careful weighing (or by forming the 1:1 “nylon salt” first). Any mole imbalance, expressed as the ratio \(r=N_A/N_B\le1\), caps the attainable degree of polymerization at \(\overline{X}_n=(1+r)/(1+r-2rp)\), and even at complete conversion (\(p=1\)) \(\overline{X}_{n,\max}=(1+r)/(1-r)\). A 1 % excess of one monomer (\(r=0.99\)) limits \(\overline{X}_n\) to about 199 no matter how long one reacts.

Final polymer structures

Nylon-12: a single repeat unit with one amide per 12 carbons, \(-[\text{NH}-(\text{CH}_2)_{11}-\text{CO}]-_n\) (repeat \(\text{C}_{12}\text{H}_{23}\text{NO}\)).

Nylon-6,6: a two-unit (diamine–diacid) repeat, \(-[\text{NH}-(\text{CH}_2)_6-\text{NH}-\text{CO}-(\text{CH}_2)_4-\text{CO}]-_n\), i.e. alternating hexamethylenediamine and adipic-acid residues.