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The Principle of Equal Reactivity of Functional Groups

The reactivity of a functional group is

independent of molecular size unaffected by reaction of the other functional group(s) in the

monomer

* Each of the possible reactions is equally probable

* Relative preponderances of the reactions depend only onthe relative numbers of each type of molecular species

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Carothers Theory

Number-average degree of polymerization nx : NNo

xn

=

No: number of molecules present initiallyN: number of molecules remained after a time tof polymerization

Extent of reaction p(assuming NA = NB)

pN

No

No

NNo

initiallypresentgroupsfunctionalofnumber

reactedhavethatgroupsfunctionalofnumberp

=

==

1

1

Carothers equation:p

xn

=1

1

In order to produce high molecular weight polymers with usefulphysical properties, the extent of reaction pmust be very high.

unitrepeattheinunitsmonomerofNumber

unitrepeattheofweightMolecularM

MxM

o

onn

=

=

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Reactant ratio r(for RA2 + RB2 and NANB system)

B

A

N

Nr = (r< 1)

NA: number of A functional groups present initiallyNB: number of B functional groups present initially

Since there are two functional groups per molecule

2

)1(

2

rNNo

NNNo

B

BA

+=

+=

Define pin terms of NA (minority)number of unreacted A groups = NA pNA = rNB(1 p)number of unreacted B groups = NB pNA = NB(1 rp)

2

)21(

2

)1()1( rprNrpNprNN BBB

+=

+=

General Carothers equation

rprr

NNoxn

211

+ +==

For RA2 + RB2 + RB system

RBB

A

NN

Nr

2+=

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Kinetics

General elementary reaction for step polymerization

~~-A + B-~~ + catalyst = ~~-AB-~~ + catalyst

So the rate of reaction is given by

]][][['][

catalystBAkdt

Ad=

Since [catalyst] is a constant (k= k[catalyst])

]][[][ BAkdtAd =

For a reaction with equimolar stoichiometry, [A] = [B] = c

2kcdt

dc=

Integration =c t

kdtc

dc

0 02gives

ktcc

corkt

cco

o

o

== 111

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Recognizing co/c= No/Nktc

po=

1

1

1

This equation also applies to reactions without catalyst (k= k)

For self-catalyzed step polymerization(assuming A groups catalyze the reaction)

]][][["][

ABAkdt

Ad=

Again letting [A] = [B] = c3

"ckdt

dc=

Upon integration as before gives

tkcc

cortk

cc

o "21)("211 2

0

2

2

0

2==

In terms of p

tkcp

"21)1(

1 202

=

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Ring Formation

Intramolecular reaction of terminal functional groups on the samemolecule results in the formation of rings

The ease of ring formation depends on the ring size:

5-, 6-, and 7-membered rings are stable and often form inpreference to linear polymer 3-, 4-, and 8- to 11-membered rings are unstable and usually

are not formed

12-membered and larger rings are more stable but theirprobability of formation is low

Many step polymerization (bimolecular processes) are performed inbulk to depress the unimolecular process of ring formation.

R C OH

O

HOH2O O

R

O

C