Chemistry Reference
In-Depth Information
in the design of practical polymerization mixtures. The major reasons for lack of
agreement between calculated and observed gel points probably include intra-
molecular reactions, loss of functional groups through side reactions, unequal
reactivities of functional groups of the same ostensible type, and the possibility
that estimations correspond to initial formation of three-dimensional polymer
whereas most experimental observations reflect
the presence of massive
amounts of gel.
Because of these factors, molecular weight calculations are used mainly for
systematic modifications of formulations which have unsatisfactory property bal-
ances rather than for accurate predictions of gel points under practical operating
co
nditions. The design of branched condensation polymers still relies heavily on
X
n
estimates, since these are less complicated than the theoretically more accurate
X
w
method described in the next section of this chapter.
7.4.3
Molecular Weight Distribution in Equilibrium Step-Growth
Polymerizations
The calculations described in this section yield estimates of the molecular weight
distribution of the reaction mixture during a step-growth polymerization that is
proceeding according to the assumptions outlined in
Section 7.4.1
.
It will be helpful first to review some very simple principles of probability
before calculating the relation between the degree of conversion and molecular
weight distribution.
If an event can happen in a ways and fail to happen in b ways then the proba-
bility of success (or happening) in a single trail is u
b). Thus the proba-
bility of heads in a single coin toss is 1/2. Similarly, the probability of failure in a
single trial is v
a/(a
5
1
b/(a
b).Ifu
0 the event is impossible, and if u
1 the event
5
1
5
5
is a certainty.
If u
1
is the probability of event E
1
happening and u
2
is the probability of event
E
2
, the likelihood that E
2
occurs after E
1
happened is u
1
u
2
if E
2
depends on E
1
(i.e., if the happening of E
1
affects the probability of the occurrence of E
2
).
Some events are mutually exclusive. (For example, a coin toss may produce a
head or a tail. The occurrence of one excludes the happening of the other event.)
In that case the probability that either event happens is u
1
1
u
2
.
The coin toss example given here involves a priori probabilities. These are
probabilities whose magnitudes can be calculated ahead of the event from the
number of different ways in which a given event may occur or fail to happen. We
are concerned here with actuarial probabilities, which are similar to estimates of
chances used by insurance companies to calculate life insurance premiums. It is
obviously impossible, for example, to estimate the life expectancy of a person
who is now 21 years old by calculating the number of ways in which he or she
may live or die. The life expectancy of an individual is equated to that experi-
enced in the past by a large number of similar people.
