 ```CAROTHERS THEORY
Carothers developed a simple method of analysis
for predicting the molar mass of polymer prepared
by step polymerization.
 He recognized that the number average degree of
polymerization with respect to monomer unit.

No
Xo 
N
where:
No  number of molecules present initially
N  number of molecules remaining after a time (t)
of polymerization
ASSUMPTION FOR CAROTHERS THEORY

There are equal numbers of functional groups,

Xn can be related to the extent of reaction P at
time t.
Number of functional groups that have reacted
P = _________________________________________________________ 1.13
Number of functional groups present initially
He also assume that the probability that any
functional group present initially has reacted.

Therefore the total number of molecules
decreases by one for each pair-wise
reaction between functional groups.
No  N
N
N
P
, P  1
, P 1 
No
No
No
N No
1
1 P 
,

No N 1  P

Combining equation (1.12) and (1.13) gives the
Carothers equation.
1
Xn 
1 P
This equation is applicable to RA2 + RB2 , ARB and
RA2 polymerization’s in which there is an exact
stoiciometric balance in the number of mutually
reactive functional groups.
KINETICS OF STEP
POLYMERIZATION REACTION

First we assume that the functional groups have equal
reactivity.
 So we can apply a single rate constant for the step-wise
reactions.
 Rate of reaction (overall) is defined as the rate of
decrease in the concentration of one or other of the
functional groups.
i.e.
 dA   dB
Rate of reaction 

Where:
dt
dt
A &amp; B are the reactive group.
Usually in step polymerization a catalyst is used.
     A  B      cat      AB      cat
The rate of theforward reaction   [A][B][cat ]
dA 

dt

Where:
 K’ is the rate constant of the reaction.
 Since the concentration of the catalyst does not
change as the reaction proceed.
  ' [cat ]
dA 
 [A][ B]
dt
when
[A] = [B] = C at equal molar stoiciometry.
dC
 C 2
dt
where C  Co at t o

Integrate
c

t
 dc
c
2 
co

dt
to
Co
No


C
N
1
1
1

 t
C Co
N
 C  Co
No
1
 t
Co
1 No
(
 1)  t
Co N
No
 1  Co t
N
N

1 P
No
N
Co
No

1

 1  C o t
1 P

This equation also applies to reaction which
proceed in the absence of catalyst.
 Certain step polymerization are self-catalyzed i.e.
one of the types of functional group acts as a cat.
(E.g.
carboxylic
acid
groups
in
a
polyesterization).
dA 

 [A][ B][ A]
dt
if [A] = [B] = C
dC
3

 C
dt
if
we integrate
c
t

 dc
co
or in
c
3

dt
1
C
to
2

1
Co
2
 2t
term of the extent of reaction
1
1  P2
 1  2Co2 t
These two equation derived assuming the
depolymerization is negligible. This assumption
is approach by in polycondensation reaction by
the elimination of the product as it form.

Normally the date are presented in a plot in the
following form

(a)
(b)

1
1
_____
________
1
(1-P)2

-P
t
t
Step polymerization of
Polymerization of
decamethylene glycol
ethylene glycol with