Page 330 - Numerical Methods for Chemical Engineering

The theory of probability 319

Because [A] = [B] and the reaction consumes equal numbers of acid and base end
0 0
groups, p A = p B = p.At 0 ≤ p ≤ 1, let [P n ] be the concentration of chains that com-
prise n monomer units, i.e., that have a chain length or degree of polymerization equal to n.
Each of the monomers has a chain length of 1; thus, at p = 0, [P 1 ] = 2[M 1 ] = [A] and
0
0
all other [P n =1 ] = 0.
We wish to compute as functions of 0 ≤ p ≤ 1 the chain length distribution [P n ], the
number (DP n ) and weight (DP w ) averaged chain lengths, and their ratio (the polydispersity
P disp ), which is a measure of the breadth of the distribution. If P disp = 1, all chains are of
the same length, and if P disp 1 there is considerable variation in chain length.
∞ n[P n ] ∞ n [P n ]
2

n=1 n=1 DP w
∞ [P n ] ∞ n[P n ]
DP n ≡ DP w ≡ P disp ≡ (7.2)
n=1 n=1 DP n
We compute these quantities using statistical techniques developed by Paul Flory (Flory,
1953, Odian, 1991, Dotson et al., 1996).
First, we make the assumption (the equal reactivity hypothesis) that the reactivity of an
end group does not depend upon the length of the chain to which it is attached. As defined
above, p A = p B = p is the fraction of all acid or base groups that have reacted, and the
fraction that remain unreacted is (1 − p). If all groups are equally likely to have reacted or
not, and we randomly select a group, the probability that it will have reacted is p and the
probability that it will not have reacted is (1 − p).

Defining the probability of an event
Above, we have invoked an argument of frequentist statistics to define a probability p from
the conversion p. That is, the probability of observing an event E is determined by the
expected number of observations of E, N E , in a very large number N of independent random
trials,
N E
p(E) ≈ (7.3)
N
It is not necessary to invoke such a large-population argument to define a probability. We
can say that the probability of observing an event E is p(E) if we have no preference between
making the following two bets:
In a random trial, event E is observed.
or
A perfectly uniform random number generator in [0, 1] returns a value u
less than or equal to p(E).

The latter definition of a probability is less restrictive, but it also appears to be open to
subjective interpretation. That is, we are making a personal value (belief) judgement about
whichbettoaccept. Thissubjectwillberevisitedinourdiscussionofstatisticsandparameter
estimation. For now, we accept the latter definition as being more general, but note that the
frequentist approach provides a convenient choice of belief when the necessary frequency
information is available. It is always incumbent upon us to assign probabilities in such a