Carothers Equation - Related Equations

Related Equations

Related to the Carothers equation are the following equations (for the simplest case of linear polymers formed from two monomers in equimolar quantities):

\begin{matrix} \bar{X}_w & = & \frac{1+p}{1-p} \\ \bar{M}_n & = & M_o\frac{1}{1-p} \\ \bar{M}_w & = & M_o\frac{1+p}{1-p}\\ PDI & = & \frac{\bar{M}_w}{\bar{M}_n}=1+p\\ \end{matrix}

where:

  • Xw is the weight average degree of polymerization,
  • Mn is the number average molecular weight,
  • Mw is the weight average molecular weight,
  • Mo is the molecular weight of the repeating monomer unit,
  • PDI is the polydispersity index.

The last equation shows that the maximum value of the PDI is 2, which occurs at a monomer conversion of 100% (or p = 1). This is true for step-growth polymerization of linear polymers. For chain-growth polymerization or for branched polymers, the PDI can be much higher.

In practice the average length of the polymer chain is limited by such things as the purity of the reactants, the absence of any side reactions (i.e. high yield), and the viscosity of the medium.

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