- Carothers equation
In

step-growth polymerization , the**Carothers equation (or Carothers' equation)**gives the**number-average degree of polymerization**, "X"_{n}, for a given**fractional monomer conversion**, "p".:$ar\{X\}\_n=frac\{1\}\{1-p\}$

This equation was proposed by

Wallace Carothers who inventednylon in 1935.Notes::* "X"

_{n}is also the average chain length (in monomer units):* "p" = ("N"_{0}-"N")/"N"_{0}, where:::"N"_{0}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 yieldA 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 "X"_{n}= 50, and "p" = 99% is required for "X"_{n}= 100.**Related equations**Related to the Carothers equation are the following equations::$egin\{matrix\}ar\{X\}\_w\; =\; frac\{1+p\}\{1-p\}\; \backslash ar\{M\}\_n\; =\; M\_ofrac\{1\}\{1-p\}\; \backslash ar\{M\}\_w\; =\; M\_ofrac\{1+p\}\{1-p\}\backslash PDI\; =\; frac\{ar\{M\}\_w\}\{ar\{M\}\_n\}=1+p\backslash end\{matrix\}$where::*"X"

_{w}is the "weight average degree of polymerization",:*"M"_{n}is thenumber average molecular weight ,:*"M"_{w}is theweight average molecular weight ,:*"M"_{o}is the molecular weight of the repeatingmonomer unit,:*"PDI" is thepolydispersity index 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.

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