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# Carothers equation

In step-growth polymerization, the Carothers equation (or Carothers' equation) gives the number-average degree of polymerization, Xn, for a given fractional monomer conversion, p.

$\bar{X}_n=\frac{1}{1-p}$

This equation was proposed by Wallace Carothers who invented nylon in 1935.

Notes:

• Xn is also the average chain length (in monomer units)
• p = (N0-N)/N0, where:
N0 is the number of molecules present initially
N is the number of unreacted molecules at time t
p is also a measure of the extent of reaction, or yield

A high monomer conversion is required to achieve a high number-average degree of polymerization. For example, a monomer conversion, p, of 98% is required for Xn = 50, and p = 99% is required for Xn = 100.

## Related equations

Related to the Carothers equation are the following equations:

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

where:

The last equation shows that the maximum value of the PDI is 2, which occurs at a monomer conversion of 100%.

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.