- Carothers equation
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".
This equation was proposed by
Wallace Carotherswho invented nylonin 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 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 "X"n = 50, and "p" = 99% is required for "X"n = 100.
Related to the Carothers equation are the following equations::where::*"X"w is the "weight average degree of polymerization",:*"M"n is the
number average molecular weight,:*"M"w is the weight average molecular weight,:*"M"o is the molecular weight of the repeating monomerunit,:*"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%.
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
viscosityof the medium.
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