Step-Growth Kinetics

Introduction

Carothers' equation relates conversion to average chain length, but it does not tell us how long a reactor must run. Kinetics supplies that missing time dimension. Polyesterification often depends on the concentrations of acid and alcohol end groups, with different apparent orders depending on catalysis.

Core explanation

For a simplified externally catalysed esterification where catalyst concentration is constant, the rate of acid-group consumption may be written −d[COOH]/dt = k[COOH][OH]. This is second order overall in the two reacting functional groups. If they begin at equal concentration c₀ and remain balanced, c = c₀(1 − p), so −dc/dt = kc² and integration gives 1/c − 1/c₀ = kt. A plot of 1/c against time should be linear within this model. Uncatalysed polyesterification can be self-catalysed by carboxylic acid groups: an additional acid group helps activate another acid group, giving a common idealised form −d[COOH]/dt = k[COOH]²[OH]. With balanced concentrations this becomes −dc/dt = kc³, so 1/c² − 1/c₀² = 2kt. Mechanism, catalyst and composition decide which law is appropriate; neither exponent should be assumed from a single endpoint measurement. Late in polymerisation, viscosity rises, diffusion slows and removal of water or alcohol can shift equilibrium. These effects may cause departures from a simple constant-k plot. A rate law for functional-group disappearance is not the same as the rate at which number-average molar mass rises: the Carothers relationship makes chain length particularly sensitive to the last fraction of conversion.

Step-by-step reasoning

Identify whether a catalyst concentration is effectively constant or whether carboxyl groups act as catalyst. Write the appropriate functional-group rate law. For balanced groups replace [COOH] and [OH] by c, then use c = c₀(1 − p). Integrate or inspect the supplied linear plot and state where diffusion or equilibrium may invalidate the model.

Visual explanation

Plot 1/c versus time for a second-order balanced model and 1/c² versus time for a third-order balanced self-catalysed model. A straight line in the appropriate coordinates helps identify an apparent rate law over the measured range.

Real-world analogy

A factory packing line slows as the remaining items become scarce. If some of those items also help run the packing machinery, the slowdown is sharper. Self-catalysed polyesterification similarly depends on acid groups as both reactants and catalytic participants.

Real-world example

Industrial PET synthesis controls catalyst, temperature and by-product removal to obtain high conversion in practical time. A kinetic model helps schedule the process, but viscosity and mass transport become especially important near the final high-molar-mass stage.

Why?

The concentration of unreacted ends falls as links form, so encounters become less frequent. Self-catalysis adds another concentration dependence because acid groups help activate the reaction. These mechanistic differences produce different time plots.

Common misconception

Second order is not a universal law for all step-growth reactions. It follows from a particular simplified mechanism and constant catalyst conditions. Self-catalysed esterification can show an apparent third-order dependence under an ideal model.

Worked example

Question: Balanced acid and alcohol groups each have concentration c. A constant-catalyst rate law is −dc/dt = kc². If c₀ = 0.50 mol L⁻¹ and c = 0.25 mol L⁻¹, what is p? Reasoning: c/c₀ = 1 − p = 0.5. Answer: p = 0.50; the integrated rate plot would use 1/c − 1/c₀ = 2 L mol⁻¹.

Quick check

1. What linear plot tests an ideal balanced second-order law −dc/dt = kc²? Answer: A plot of 1/c against time should be a straight line.

Exam focus

Specify which species' concentration defines the rate and whether the catalyst concentration is constant. Do not confuse the order of functional-group consumption with the molecular-weight distribution or the Carothers conversion relation.

Advanced insight

A polymer kinetics teaching source treats self-catalysed esterification with an extra acid concentration factor. At high conversion, falling acid concentration and rising viscosity can compete, making a short-range empirical fit unreliable outside its measured interval.

Summary

An ideal externally catalysed balanced polyesterification may follow −dc/dt = kc², giving a linear 1/c plot. Self-catalysis by acid groups can give −dc/dt = kc³ and a linear 1/c² plot. Actual late-stage behaviour also depends on equilibrium, by-product removal and transport.

Practice questions

1. If [COOH] = [OH] = c, what is the overall concentration dependence of k[COOH][OH]? Answer: kc², second order in c.

2. If the rate is k[COOH]²[OH] in a balanced mixture, how does it depend on c? Answer: kc³, third order in c.

3. Why can a simple rate law fail near complete conversion? Answer: Viscosity, diffusion limits, equilibrium and by-product removal can change the apparent kinetics.

4. How is conversion p related to balanced unreacted-group concentration? Answer: c = c₀(1 − p), provided the groups start balanced and are consumed together.