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| In [[step-growth polymerization]], the '''Carothers equation''' (or '''Carothers' equation''') gives the [[degree of polymerization]], ''X''<sub>n</sub>, for a given fractional monomer conversion, ''p''.
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| There are several versions of this equation, proposed by [[Wallace Carothers]] who invented [[nylon]] in 1935.
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| ==Linear polymers: two monomers in equimolar quantities==
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| The simplest case refers to the formation of a strictly linear polymer by the reaction (usually by condensation) of two monomers in equimolar quantities. An example is the synthesis of [[nylon-6,6]] whose formula is [-NH-(CH<sub>2</sub>)<sub>6</sub>-NH-CO-(CH<sub>2</sub>)<sub>4</sub>-CO-]<sub>n</sub>
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| from one mole of [[hexamethylenediamine]], H<sub>2</sub>N(CH<sub>2</sub>)<sub>6</sub>NH<sub>2</sub>, and one mole of [[adipic acid]], HOOC-(CH<sub>2</sub>)<sub>4</sub>-COOH. For this case<ref>Cowie J.M.G. "Polymers: Chemistry & Physics of Modern Materials (2nd edition, Blackie 1991), p.29</ref><ref>Rudin Alfred "The Elements of Polymer Science and Engineering", Academic Press 1982, p.171</ref>
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| :<math>\bar{X}_n=\frac{1}{1-p}</math>
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| In this equation
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| :*<math>\bar{X}_n</math> is the [[Molar mass distribution#Number average molecular weight|number-average]] value of the [[degree of polymerization]], equal to the average number of monomer units in a polymer molecule. For the example of nylon-6,6 <math>\bar{X}_n = 2n</math> (n diamine units and n diacid units).
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| :*<math>p=\frac{N_0-N}{N_0}</math> is the extent of reaction (or conversion to polymer), defined by
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| :*<math>N_0</math> is the number of molecules present initially
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| :*<math>N</math> is the number of molecules present after time t
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| This equation shows that a high monomer [[Conversion (chemistry)|conversion]] is required to achieve a high degree of polymerization. For example, a monomer conversion, ''p'', of 98% is required for <math>\bar{X}_n = 50</math>, and ''p'' = 99% is required for <math>\bar{X}_n = 100</math>.
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| ==Linear polymers: one monomer in excess==
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| If one monomer is present in stoichiometric excess, then the equation becomes<ref>Allcock Harry R., Lampe Frederick W. and Mark James E. "Contemporary Polymer Chemistry" (3rd ed., Pearson 2003) p.324</ref>
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| :<math>\bar{X}_n=\frac{1+r}{1+r-2rp}</math>
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| :* ''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.
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| 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 the [[limiting reagent]] monomer, p → 1 and
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| :<math>\bar{X}_n\to\frac{1+r}{1-r}</math>
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| 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.
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| ==Branched polymers: multifunctional monomers==
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| The '''functionality''' of a monomer molecule is the number of functional groups which participate in the polymerization. Monomers with functionality greater than two will introduce [[branching (chemistry)|branching]] into a polymer, and the degree of polymerization will depend on the average functionality f<sub>av</sub> per monomer unit. For a system containing N<sub>0</sub> molecules initially and equivalent numbers of two functional groups A and B, the total number of functional groups is N<sub>0</sub>f<sub>av</sub>.
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| : <math>f_{av} = \frac{\sum N_i \sdot f_i}{\sum N_i}</math>
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| And the '''modified Carothers equation''' is<ref>{{cite journal
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| | last = Carothers
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| | first = Wallace
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| | title = Polymers and polyfunctionality
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| | journal =Transaction of the Faraday Society
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| | issue =
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| | pages = 39–49
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| | publisher =
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| | location =
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| | year = 1936
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| | url =
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| | issn =
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| | doi = 10.1039/TF9363200039
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| | id =
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| | accessdate =
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| | volume = 32 }}</ref><ref>Cowie p.40</ref><ref>Rudin p.170</ref>
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| : <math>x_{n} = \frac{2}{2-pf_{av}}</math>, where p equals to <math>\frac{2(N_0-N)}{N_0 \sdot f_{av}}</math> | |
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| == Related equations ==
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| Related to the Carothers equation are the following equations (for the simplest case of linear polymers formed from two monomers in equimolar quantities):
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| :<math>
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| \begin{matrix}
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| \bar{X}_w & = & \frac{1+p}{1-p} \\
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| \bar{M}_n & = & M_o\frac{1}{1-p} \\
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| \bar{M}_w & = & M_o\frac{1+p}{1-p}\\
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| PDI & = & \frac{\bar{M}_w}{\bar{M}_n}=1+p\\
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| \end{matrix}
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| </math>
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| where:
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| :*''X''<sub>w</sub> is the ''weight average degree of polymerization'',
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| :*''M''<sub>n</sub> is the [[number average molecular weight]],
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| :*''M''<sub>w</sub> is the [[weight average molecular weight]], | |
| :*''M''<sub>o</sub> is the molecular weight of the repeating [[monomer]] unit,
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| :*''Đ (PDI, old nomenclature)'' is the [[polydispersity index]].
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| The last equation shows that the maximum value of the ''Đ'' 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 [[branching (chemistry)|branched]] polymers, the Đ can be much higher.
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| 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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| ==References==
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| {{reflist}}
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| [[Category:Polymer chemistry]]
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| [[Category:Equations]]
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