ConverterHub

Carothers Equation Calculator

Enter the monomer conversion p to find the degree of polymerization.

How it is calculated

The tool applies Xn = 1 / (1 − p). p is the extent of reaction, the fraction (not the percentage) of the functional groups that have reacted, and Xn is the number-average degree of polymerization, the average number of monomer units per chain. It also reports the conversion p × 100 and the unreacted share (1 − p) × 100 in percent. The equation assumes a step-growth reaction with exactly equal amounts of the two functional groups; an excess of one monomer lowers the real Xn and is not included. Only values strictly between 0 and 1 are accepted, so 0, 1 and anything outside return no result. Results are shown with two decimals.

Worked example

Suppose a polyester reaction has reached p = 0.98, meaning 98% of the groups have reacted. The unreacted share is 1 − 0.98 = 0.02, so Xn = 1 / 0.02 = 50.00. On average each chain holds 50 monomer units. The tool also shows a conversion of 98.00% and 2.00% remaining. Pushing p from 0.98 to 0.99 would double Xn to 100, because the unreacted share is halved.

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Common mistakes

  • Typing the conversion as a percentage (98) instead of a fraction (0.98). A value of 1 or more returns no result.
  • Expecting a high conversion like 90% to give long chains. At p = 0.9 the result is only 10, so step-growth polymers need conversions very close to 1.
  • Applying the equation to chain-growth (addition) polymerization. The relation assumes step-growth reactions, where all chain sizes react with each other.
  • Ignoring stoichiometric imbalance. With unequal amounts of the two monomers the real Xn is lower than this tool shows.
  • Reading Xn as the molecular weight. Multiply Xn by the average mass of a repeat unit to get the number-average molecular weight; the tool does not do that step.

About this calculator

In step-growth (condensation) polymerization — the process behind nylon, polyester and many other engineering plastics — monomers react two at a time, building chains gradually as the reaction proceeds. The Carothers equation, Xn = 1/(1 − p), relates the number-average degree of polymerization Xn to p, the fraction of functional groups that have reacted.

The relationship is famously unforgiving: to reach a useful chain length, conversion has to be pushed very close to 100% — at p = 0.99, Xn is only 100, while getting to Xn = 1000 requires p = 0.999. This calculator applies the equation directly, and it’s a standard reference for polymer chemists and materials scientists designing and controlling reactions to hit a target molecular weight.

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