228
to be of nth order (or, more specifically, would have an obvious reaction order of n):
ath order in relation to OH and bth order in relation of NCO). Since all the systems
used in the reactions had an NCO/OH ratio of 1 it resulted in [NCO] = [OH] at any
point during polymerization process. The kinetic expression, associated to a specific
velocity k n , can therefore written as
−
[
] = − [ ] = [
] [ ] = [
]
d
dt
d
dt
k
k
n
a
b
n
n
NCO
O H
NCO OH
N CO ,
(1)
d
dt
K n
n
α
α
=
−
(
)
1
,
(2)
where
α = [
] − [
]
[
]
NCO
NCO
NCO
0
0
(3)
is the degree of conversion, K
k
NCO
n
n
n
= [
] [
]
−
NCO 0
1
0
,
being the initial concertration of NCO groups at time t = 0.
If the complete reaction between all NCO groups and all OH groups had an
enthalpy of ΔH _ 0, then α =
( )
H t
H
∆ 0
, (where H(t) is the accumulated energy liberated
at a time t during the reaction), and Eq. (2) could be rewritten as
1
1
0
0
∆
∆
H
dH
dt
K
H t
H
n
n
=
−
( )


 


 
(4)
The specific velocity k _ n can be related to the temperature T through an Arrhenius
dependence relationship.
k
A e
n
E RT
A
=
−
0
/
(5)
where R is the universal gas constant,  A 0 is an apparent frequency factor, E A an
apparent energy of activation. Substitution of (5) in (4) finally yields:
dH dt
A
H
H H t
n
E
RT
n
A
/
)
= ( )
− ( )
 
 
−
−
0
0
1
0
∆
∆
e
,
(6)
Time and temperature are related by the heating rate:
T T
d T
dr
= + ⇒
=
0
β
β ,
(7)
where T 0 is the initial temperature and β is the heating rate. Substitution of (7) in (4)
would result in
S. Dhanuskar et al.