Chemistry Reference
In-Depth Information
In arithmetic, this is equivalent to noting that if S successes have already been
experienced in n trials then the relative frequency of success is S/n and the
lim
n
-N
ð
is taken to be the probability of success in a single trial. This is
the kind of probability we shall use in the following paragraphs. The rules for
combining such probabilities are as given earlier.
The concepts we use to develop relations between the degree of conversion
and molecular weight distribution of the reaction mixture in equilibrium step-
growth polymerizations are most clearly illustrated with reference to the self-
polymerization of a monomer that contains two coreactive groups. An example
would be a hydroxy acid that can undergo self-polymerization according to
S
=
n
Þ
n HO
R
C
OH
H
O
R
C
OH +
n -1
H
2
O
n
(7-21)
O
O
This is a so-called AB-type monomer. The final equations are very similar to
those which can be developed for the reaction AA- and BB-type monomers, as in
Fig. 7.3
, for example. The derivation in the latter case, however, requires some
steps that are beyond the scope of this text. The reader is referred to the original
source
[2]
for a more complete description of the ideas involved.
We are assuming, as before, that the probability of reaction and the reaction
rate of either group in
Eq. (7-21)
is independent of the size of the molecule to
which this group is attached.
Imagine that we can reach into the reaction mixture at time t when the extent
of reaction is p and select an unreacted functional group of either kind. The
probability that the other end of the structural unit we have picked has reacted
is p, since a fraction p of all functional groups of that type will have reacted. If
this other end has reacted, then the unreacted end we picked is attached to at
least one other monomer residue. That means that the probability that an
unreacted end is part of a molecule that contains at least two monomer units is
p. The other group on the second monomer from the end we picked could also
have reacted, however, and the probability that this single event may have hap-
pened is still equal to p. The probability that the two similar groups we have
mentioned will have reacted equals p
2
, and this equals the probability that our
initial end is attached to a molecule that contains at least three monomer resi-
dues. By extension, the probability that the molecule in question contains at
least i monomer residues will be p
i
2
1
. The concepts involved here are summa-
rizedalsoin
Fig. 7.4
.
We now wish to know the probability that our initial unreacted end is attached
to a molecule that contains exactly i monomer residues. This requires that there
will have been i
1 reactions, with a net probability of p
i
2
1
, followed by a non-
reaction. The probability that the last functional group has not reacted is 1
2
p
(since the probability that it has either reacted or has not reacted must equal 1).
The combined probability that the initial random selection of a molecule found a
macromolecule with i monomer residues is evidently p
i
2
1
(1
2
2
p).
