The Carothers Equation

Introduction

Step-growth polymers become very long only near complete functional-group conversion. The Carothers equation quantifies this fact for an ideal balanced bifunctional system: number-average degree of polymerisation Xₙ = 1/(1 − p), where p is the fraction of relevant groups reacted.

Core explanation

Imagine N₀ bifunctional starting molecules with balanced complementary groups and no branching or ring formation. Each intermolecular coupling joins two molecules into one and reduces the number of separate molecules by one. If a fraction p of possible link-forming groups has reacted, the number of intermolecular couplings is pN₀ and the remaining chain count is N = N₀(1 − p). The number-average degree of polymerisation is starting units divided by chains: Xₙ = N₀/N = 1/(1 − p). This simplified derivation relies on balanced bifunctional groups and on each counted reaction joining separate molecules. At p = 0.90, Xₙ = 10; at p = 0.99, Xₙ = 100; at p = 0.999, Xₙ = 1000. The last percentage point of conversion therefore matters enormously. If monomer types are stoichiometrically imbalanced or monofunctional chain stoppers are added, a modified relation must be used. Cyclisation, side reactions and loss of functional groups can also lower attainable chain length. In an A–A plus B–B system the degree may be counted in individual monomer molecules or paired repeat units, so specify the convention before translating Xₙ into molar mass. The equation predicts a number average, not the length of every single chain; real products have a distribution.

Step-by-step reasoning

Express p as a decimal rather than a percentage. Compute 1 − p and take its reciprocal. Check that a large desired Xₙ requires p close to one. If solving backward, rearrange p = 1 − 1/Xₙ. State the ideal balanced bifunctional assumptions before interpreting the numerical result.

Visual explanation

Plot Xₙ on a vertical axis against p from zero to one. The curve rises slowly at first, reaches 10 at 0.90 and 100 at 0.99, then climbs steeply toward the right edge without a finite value at p = 1 in the ideal model.

Real-world analogy

To make a line of a hundred linked paper clips, almost every available connection must be made. If one clip in ten remains unconnected at random, the line fragments into much shorter pieces on average.

Real-world example

A polyester plant may reach 99% functional-group conversion yet obtain an average chain length near 100 starting units under ideal conditions. Pushing conversion another fraction of a percent can strongly affect fibre-forming molar mass.

Why?

Each bond-forming step reduces the molecule count. The denominator 1 − p is therefore the fraction of initial molecules remaining as distinct chains in the ideal counting model. As that count approaches zero relative to N₀, average chain length rises.

Common misconception

The equation does not say that every chain has length 100 when p = 0.99. It gives a number average across a broad distribution. Nor can p = 1 literally produce an infinite chain in a finite real vessel; ideal assumptions break down.

Worked example

Question: An ideal balanced bifunctional step-growth system reaches p = 0.98. Find Xₙ. Reasoning: The unreacted fraction is 1 − 0.98 = 0.02. Divide one by this fraction. Answer: Xₙ = 50 starting units per chain on a number-average basis.

Quick check

1. What conversion is needed for ideal Xₙ = 200? Answer: p = 1 − 1/200 = 0.995, or 99.5%.

Exam focus

Write the assumptions and the count basis. Convert a stated percent into decimal p, and distinguish number-average chain length from a monodisperse product. Check whether stoichiometric imbalance or chain stopping invalidates the simple form.

Advanced insight

Even with p extremely high, small amounts of monofunctional impurities can cap chains. The sensitivity of Xₙ to both conversion and stoichiometry explains why industrial step-growth polymerisation requires careful reagent purity and end-group control.

Summary

For ideal balanced bifunctional step growth, Xₙ = 1/(1 − p). It follows from counting how each intermolecular coupling reduces the number of molecules. High degree of polymerisation needs p very close to one; imbalance, cyclisation and side reactions limit real systems.

Practice questions

1. Find Xₙ when p = 0.90 under ideal assumptions. Answer: Xₙ = 1/(1 − 0.90) = 10.

2. Find p required for Xₙ = 1000. Answer: p = 1 − 1/1000 = 0.999, or 99.9%.

3. Is Xₙ the length of every chain in the sample? Answer: No. It is the number average over a distribution of chain lengths.

4. Give one reason a real experiment could fall below this prediction. Answer: Stoichiometric imbalance, a monofunctional impurity, cyclisation or side reactions can stop ideal intermolecular growth.