Chemistry Reference
In-Depth Information
Given an
unreacted
end
Probability of reaction of each -cooh group =P
HO
R
COR COR COR CO
O
O
O
O
Probability that
two-COOH groups
have reacted = p
2
when extent of
reaction = p
FIGURE 7.4
Concepts involved in deriving relations between molecular weight distribution and extent
of reaction for self-condensation of a bifunctional monomer.
During such a random choice of a molecule the chances that a polymer con-
taining i structural units (an i-mer) will be selected will depend directly on how
many such molecules there are in the reaction vessel. (To illustrate: If a box
contains nine red balls and one black ball, the probability that a red ball will be
selected in a blind choice is 9/10.) In other words, if one molecule is selected,
the probability that it will be an i-mer equals the mole fraction x
i
of i-mers
in the reaction mixture. We have just calculated this probability and we see
then that
p
i
2
1
x
i
5
ð
Þ
:
1
2
p
(7-22)
Equation (7-22)
is the differential number distribution function for equilibrium
step-growth polymerizations in homogeneous systems.
We now derive an expression for the differential weight distribution. The total
number, N, of molecules remaining at an extent of reaction p,is
N
N
0
ð
i
p
Þ
(7-23)
5
2
[This is the same as
Eq. (7-18)
with f
av
5
2 in this particular case.] The mole frac-
tion of i-mers x
i
is
p
i
2
1
x
i
5
N
i
=
N
0
ð
1
p
Þ
5
ð
1
p
Þ
(7-24)
2
2
where N
i
is the number of moles of i-mers. Thus
2
p
i
2
1
N
i
5
N
0
ð
1
2
p
Þ
(7-25)
If the formula weight of a monomer that has reacted at both ends equals M
0
,
then the molecular weight of M
i
of an i-mer is iM
0
, and the weight of i-mers in
