Biomedical Engineering Reference
In-Depth Information
where C
0
is the total molar concentration of monomers and oligomers.
S
X
N
C
S0
¼
C
i
(6.92)
i
¼ 1
Equation
(6.90)
describes the rate for individual oligomer units. However, this is not
convenient in most cases as the number of molecular species is not easy to determine.
Lumped kinetics can be derived based on E
qn (6.90)
. Summing up all the oligomers, we
obtain
n
"
X
#
X
N
X
N
N
r
S1
¼
nr
n
¼
2
k
C
i
ð
n
1Þ
k
C
n
k
C
1
¼
k
C
(6.93)
H
H
D
D
1
n
¼ 1
n
¼1
i
¼
n
þ1
S0
¼
P
r
n
¼
P
2
P
C
i
P
P
N
N
N
N
N
r
k
ð
n
1Þ
k
C
n
k
C
1
¼
k
ð
n
1Þ
C
n
k
C
H
H
D
H
D
1
n
¼1
n
¼ 1
n
¼ 2
n
¼ 1
i
¼
n
þ1
¼
k
H
ð
C
S1
C
S0
Þ
k
C
D
1
(6.94)
S
where the subscript
1 denotes the mixture measured based on total number of monomeric
S
units, and subscript
0 denotes the mixture measured based on total number of molecules.
X
N
C
S1
¼
iC
i
(6.95)
i
¼ 1
Equations
(6.93) and (6.94)
indicate that the reaction mixture in the hydrolyzate solution
can be characterized based on the total number of moles and the total number of monomeric
units. Effectively, all the oligomers are lumped into two “pseudo-oligomer species:
S
0 and
S
1.”
Since H
3
O
þ
and H
þ
are of the same effect, we have
H
þ
¼10
pH
O
þ
þ½
½
H
(6.96)
3
where pH is the pH value of the aqueous solution. Reactions
(6.73)
and
(6.77)
lead to
½H
2
O
0
½H
2
O
¼
(6.97)
10
pH
1 þ
K
where [H
2
O*] is the concentration of “free” activated water and [H
2
O*]
0
is the total concen-
tration of activated water. Thus, the rate constant for the hydrolysis is given by
k
2
1
10
pH
þ
k
H
¼
k
(6.98)
3
10
pH
1 þ
k
This section shows that simple kinetic expressions can be obtained for otherwise a compli-
cated problem when proper assumptions can be made. For acid hydrolysis of polymers,
the reaction kinetics may be approximated by three “first-order reaction rate” relationships
of lumped component concentrations: 1) moles of monomers in the reaction mixture, 2) total
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