Page 336 - Numerical Methods for Chemical Engineering

Important probability distributions 325

Equations (7.35) and (7.36) provide two equations for the two unknowns E(W A out ) and
E(W B out ), yielding
p A [M 2 + (β 2 − 1)p B M 1 ]
out
E W = (7.37)
A
1 − p A (α 1 − 1)p B (β 2 − 1)
from which E(W out ) is computed by (7.36). The expected weight attached to a randomly
B
selected group is then
out out
E(W) = M 1 + α 1 E W A )]P(M 1 + M 2 + β 2 E W B P(M 2 ) (7.38)
Gelation occurs when E(W) →∞, which happens as 1 − p A (α 1 − 1)p B (β 2 − 1), the
denominator of E(W out ), goes to zero, yielding the gel point condition
A
p A (α 1 − 1)p B (β 2 − 1) = 1 (7.39)
Using the relation between the two conversions posed by the reaction stoichiometry (7.23),
the conversion of A at the gel point is
+

β 2 N 2 1
p A,c = (7.40)
α 1 N 1 (α 1 − 1)(β 2 − 1)
For balanced end group concentrations, with α 1 = 2,β 2 = 3,
+
6
1 1
p A,c = p B,c = p c = = = 0.7071 (7.41)
(2 − 1)(3 − 1) 2
Figure 7.6 shows the predicted DP w vs. p, up to the point where it diverges to infinity as the
gel forms (denoted by vertical dashed line).
Above, we have computed the gel point only for a mixture of two reactants. This theory is
extended to the case of multiple monomers (with a more general treatment of the conditional
probabilities) in Beers & Ray (2001). This probabilistic approach can also be used to
compute the gel and sol mass fractions following gelation.

Important probability distributions

We now consider some important, and common, forms of probability distributions for a
random variable. First, we state some basic definitions.

Definition Probability distribution of a discrete random variable
Let X be a random variable that may take one of a countable number M of discrete values X 1 ,
X 2 ,..., X M . Let us conduct some very large number T of trials in which we measure the value
of X. Let N(X j ) be the number of times that we observe the value X j , M N(X j ) = T .
j=1
Then, the probability distribution of X is defined in the frequentist manner for very large T
as
M
N(X j )
P(X j ) = P(X j ) = 1 (7.42)
T
j=1