Page 335 - Numerical Methods for Chemical Engineering

324 7 Probability theory and stochastic simulation

t A
A t = in
B A
A 1 a a 1 a a
2 2
aa
a
t
A
1 a
2
in aa
B
2
a
Figure 7.5 Expected weights looking “out” or “in” from functional groups.
We first consider E(W|M 1 ). If we select a type-1 unit, the mass W of the chain must at
least equal the mass M 1 of a single type-1 unit, as well as the contributions from the masses
observed by looking “outwards” from each of the α 1 acid groups (Figure 7.5). E(W out )is
A
the expected weight of the section of chain attached to a type-1 unit through one of its acid
groups. Thus,
out
E(W|M 1 ) = M 1 + α 1 E W (7.30)
A
Similarly, the expected weight observed if we select a type-2 monomer equals the weight of
a single type-2 monomer unit plus the expected weights E(W out ) looking outwards across
B
each of the β 2 base groups:
out
E(W|M 2 ) = M 2 + β 2 E W (7.31)
B
To compute E(W out ), we note that a randomly-selected acid group must be either reacted
A
or unreacted, so that we expand in the possible outcomes:
out out out
E W = E W |L P(L|a) + E W |A P(A|a) (7.32)
A A A
If the selected acid group is unreacted, there is no chain attached to this group and
E(W out |A) = 0. Also, as P(L|a) = p A ,wehave
A
out out
E W = E W (7.33)
A A |L × p A
Next, from Figure 7.5, we note that the expected weight looking “out” from an acid group
on monomer 1 equals the expected weight observed looking “in” from a base group on a
monomer of type 2, and hence

out in out
E W |L = E W = M 2 + (β 2 − 1)E W (7.34)
A B B
Here, we have used the fact that if we come “in” across one of the base groups, there are
only (β 2 − 1) other possible ways to go “out.” Combining (7.33) and (7.34) yields
out in out
E W = E W × p A = p A M 2 + (β 2 − 1)E W (7.35)
A B B
Applying the same logic to E(W B out ),

out in out
E W = E W × p B = p B M 1 + (α 1 − 1)E W (7.36)
B A A