Instep-growth polymerization, theCarothers equation (orCarothers' equation) gives thedegree of polymerization,X_n, for a given fractionalmonomer conversion,p.

There are several versions of this equation, proposed byWallace Carothers, who inventednylon in 1935.

The simplest case refers to the formation of a strictly linear polymer by the reaction (usually by condensation) of twomonomers in equimolar quantities. An example is the synthesis ofnylon-6,6 whose formula is[−NH−(CH₂)₆−NH−CO−(CH₂)₄−CO−]_nfrom one mole ofhexamethylenediamine,H₂N(CH₂)₆NH₂, and one mole ofadipic acid,HOOC−(CH₂)₄−COOH. For this case^[1][2]

${\displaystyle {\overline{X}}_n=\frac{1}{1-p}}$

In this equation

• ⁠${\displaystyle {\overline{X}}_n}$⁠ is thenumber-average value of thedegree of polymerization, equal to the average number of monomer units in a polymer molecule. For the example of nylon-6,6${\displaystyle {\overline{X}}_n=2n}$ (n diamine units andn diacid units).
• ${\displaystyle p=\frac{N_0-N}{N_0}}$ is the extent of reaction (or conversion to polymer), defined by
  • ⁠${\displaystyle N_0}$⁠ is the number of molecules present initially as monomer
  • ⁠${\displaystyle N}$⁠ is the number of molecules present after timet. The total includes all degrees of polymerization: monomers,oligomers and polymers.

This equation shows that a high monomerconversion is required to achieve a high degree of polymerization. For example, a monomer conversion,p, of 98% is required for⁠${\displaystyle {\overline{X}}_n}$⁠ = 50, andp = 99% is required for⁠${\displaystyle {\overline{X}}_n}$⁠ = 100.

If one monomer is present instoichiometric excess, then the equation becomes^[3]

${\displaystyle {\overline{X}}_n=\frac{1+r}{1+r-2rp}}$

• r is the stoichiometric ratio of reactants, the excess reactant is conventionally the denominator so that r < 1. If neither monomer is in excess, then r = 1 and the equation reduces to the equimolar case above.

The effect of the excess reactant is to reduce the degree of polymerization for a given value of p. In the limit of complete conversion of thelimiting reagent monomer, p → 1 and

${\displaystyle {\overline{X}}_n\to \frac{1+r}{1-r}}$

Thus for a 1% excess of one monomer, r = 0.99 and the limiting degree of polymerization is 199, compared to infinity for the equimolar case. An excess of one reactant can be used to control the degree of polymerization.

Thefunctionality of a monomer molecule is the number of functional groups which participate in the polymerization. Monomers with functionality greater than two will introducebranching into a polymer, and the degree of polymerization will depend on the average functionality f_av per monomer unit. For a system containing N₀ molecules initially and equivalent numbers of two functional groups A and B, the total number of functional groups is N₀f_av.

${\displaystyle f_{av}=\frac{\sum N_i\cdot f_i}{\sum N_i}}$

And themodified Carothers equation is^[4][5][6]

${\displaystyle x_n=\frac{2}{2-p{f}_{av}}}$, where p equals to${\displaystyle \frac{2(N_0-N)}{N_0\cdot f_{av}}}$

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

where:

• X_w is theweight 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,
• Đ is thedispersity index. (formerly known as polydispersity index, symbol PDI)

The last equation shows that the maximum value of theĐ is 2, which occurs at a monomer conversion of 100% (or p = 1). This is true for step-growth polymerization of linear polymers. Forchain-growth polymerization or forbranched polymers, the Đ 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 anyside reactions (i.e. high yield), and theviscosity of the medium.

• Polymer chemistry
• Equations

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