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# Carothers equation

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

There are several versions of this equation, proposed by Wallace Carothers who invented nylon in 1935.

## Contents

• 1 Linear polymers: two monomers in equimolar quantities
• 2 Linear polymers: one monomer in excess
• 3 Branched polymers: multifunctional monomers
• 4 Related equations
• 5 References

## Linear polymers: two monomers in equimolar quantities[ edit ]

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-(CH2)6-NH-CO-(CH2)4-CO-]n
from one mole of hexamethylenediamine , H2N(CH2)6NH2, and one mole of adipic acid , HOOC-(CH2)4-COOH. For this case [1] [2]

$\displaystyle \bar X_n=\frac 11-p$

X
¯

n

=

1

1

p

\displaystyle \bar X_n=\frac 11-p

In this equation

• $\displaystyle \bar X_n$

X
¯

n

\displaystyle \bar X_n

is the 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

$\displaystyle \bar X_n=2n$

X
¯

n

=
2
n

\displaystyle \bar X_n=2n

(n diamine units and n diacid units).

• $\displaystyle p=\frac N_0-NN_0$

p
=

N

0

N

N

0

\displaystyle p=\frac N_0-NN_0

is the extent of reaction (or conversion to polymer), defined by

• $\displaystyle N_0$

N

0

\displaystyle N_0

is the number of molecules present initially as monomer

• $\displaystyle N$

N

\displaystyle N

is the number of molecules present after time t. The total includes all degrees of polymerization: monomers, oligomers and polymers.

This equation shows that a high monomer conversion is required to achieve a high degree of polymerization. For example, a monomer conversion, p, of 98% is required for

$\displaystyle \bar X_n=50$

X
¯

n

=
50

\displaystyle \bar X_n=50

, and p = 99% is required for

$\displaystyle \bar X_n=100$

X
¯

n

=
100

\displaystyle \bar X_n=100

.

## Linear polymers: one monomer in excess[ edit ]

If one monomer is present in stoichiometric excess, then the equation becomes [3]

$\displaystyle \bar X_n=\frac 1+r1+r-2rp$

X
¯

n

=

1
+
r

1
+
r

2
r
p

\displaystyle \bar X_n=\frac 1+r1+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 the limiting reagent monomer, p → 1 and

$\displaystyle \bar X_n\to \frac 1+r1-r$

X
¯

n

1
+
r

1

r

\displaystyle \bar X_n\to \frac 1+r1-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.

## Branched polymers: multifunctional monomers[ edit ]

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 into a polymer, and the degree of polymerization will depend on the average functionality fav 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 N0fav.

$\displaystyle f_av=\frac \sum N_i\cdot f_i\sum N_i$

f

a
v

=

N

i

f

i

N

i

\displaystyle f_av=\frac \sum N_i\cdot f_i\sum N_i

And the modified Carothers equation is [4] [5] [6]

$\displaystyle x_n=\frac 22-pf_av$

x

n

=

2

2

p

f

a
v

\displaystyle x_n=\frac 22-pf_av

, where p equals to

$\displaystyle \frac 2(N_0-N)N_0\cdot f_av$

2
(

N

0

N
)

N

0

f

a
v

\displaystyle \frac 2(N_0-N)N_0\cdot f_av

## Related equations[ edit ]

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

$\displaystyle \beginmatrix\bar X_w&=&\frac 1+p1-p\\\bar M_n&=&M_o\frac 11-p\\\bar M_w&=&M_o\frac 1+p1-p\\PDI&=&\frac \bar M_w\bar M_n=1+p\\\endmatrix$

X
¯

w

=

1
+
p

1

p

M
¯

n

=

M

o

1

1

p

M
¯

w

=

M

o

1
+
p

1

p

P
D
I

=

M
¯

w

M
¯

n

=
1
+
p

\displaystyle \beginmatrix\bar X_w&=&\frac 1+p1-p\\\bar M_n&=&M_o\frac 11-p\\\bar M_w&=&M_o\frac 1+p1-p\\PDI&=&\frac \bar M_w\bar M_n=1+p\\\endmatrix

where:

• Xw is the weight average degree of polymerization,
• Mn is the number average molecular weight ,
• Mw is the weight average molecular weight ,
• Mo is the molecular weight of the repeating monomer unit,
• Đ (PDI, old nomenclature) is the polydispersity index .

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 branched 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 any side reactions (i.e. high yield), and the viscosity of the medium

## References[ edit ]

1. ^ Cowie J.M.G. “Polymers: Chemistry & Physics of Modern Materials (2nd edition, Blackie 1991), p.29
2. ^ Rudin Alfred “The Elements of Polymer Science and Engineering”, Academic Press 1982, p.171
3. ^ Allcock Harry R. , Lampe Frederick W. and Mark James E. “Contemporary Polymer Chemistry” (3rd ed., Pearson 2003) p.324
4. ^ Carothers, Wallace (1936). “Polymers and polyfunctionality”. Transactions of the Faraday Society. 32: 39–49. doi : 10.1039/TF9363200039 .

5. ^ Cowie p.40
6. ^ Rudin p.170

Categories :

• Polymer chemistry
• Equations

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