Page 334 - Numerical Methods for Chemical Engineering

The theory of probability 323

If the number of trials N exp that we perform is very large, we expect
N exp
1
E(W) = W ≈ W v (7.19)
N exp
v=1
To predict the gel point, we compute E(W) at the specified conversions
[A] [B]
p A = 1 − p B = 1 − (7.20)
[A] 0 [B] 0
Unlike our previous discussion, we do not assume that p A = p B . We start with N 1 monomers
of type 1 per unit volume, each with α 1 acid groups, and N 2 monomers per unit volume of
type 2, each with β 2 base groups. The initial acid and base end group concentrations are

[A] 0 = α 1 N 1 [B] 0 = β 2 N 2 (7.21)
As the numbers of acid and base groups consumed by reaction are equal,
[A] 0 − [A] = [B] 0 − [B] ⇒ [A] 0 p A = [B] 0 p B (7.22)

Hence, the conversions of the acid and base groups are related,

[A] 0 α 1 N 1
p B = p A = p A (7.23)
[B] 0 β 2 N 2
The conditional probability that if we select an acid group, it is unreacted, is
(7.24)
P(A|a) = 1 − p A
and the conditional probability that it has reacted to form a linkage is

P(L|a) = p A (7.25)
As any randomly-selected acid group must be either reacted or unreacted, these two condi-
tional probabilities must sum to 1:

P(L|a) + P(A|a) = 1 (7.26)
Similarly for the base group, we have the conditional probabilities

P(B|b) = 1 − p B P(L|b) = p B (7.27)
From these conditional probabilities, we compute E(W). First, we note that if we randomly
select a monomer unit at random, the probabilities that it is of type 1 or 2 are
N 1 N 2
P(M 1 ) = P(M 2 ) = (7.28)
N 1 + N 2 N 1 + N 2
Here M 1 and M 2 denote the events that the randomly-selected monomer unit is type 1 or
type 2 respectively.
If E(W|M 1 )isthe conditional expectation of W when we have selected a type 1 monomer
unit, and E(W|M 2 ) is the corresponding value for a type-2 monomer, we expand E(W)in
terms of the mutually exclusive events M 1 and M 2 :

E(W) = E(W|M 1 )P(M 1 ) + E(W|M 2 )P(M 2 ) (7.29)