Biomedical Engineering Reference
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and the decomposition rate of the n-oligomer (with n
1 intermonomer bonds) is
O
r
n; n/
c
1in1ð
6
:
74
Þ
¼ð
n
1Þ
k
C
n
½
H
(6.82)
ð2Þ
2
where C
n
is the mole concentration of n-oligomer, [H
2
O*] is the concentration of “free” acti-
vated water molecules, and is k
(2)
the reaction rate constant.
For reaction
(6.76)
, we apply the same assumption that all the n
1 intermonomer bonds
are equally reactive,
H
þ
r
n; n/
c
1in1ð
6
:
76
Þ
¼ð
n
1Þ
k
ð4Þ
½
HX
n
OH
$
(6.83)
Combing the fast-equilibrium step, reaction
(6.75)
,
Eqn (6.83)
is reduced to
H
þ
r
n; n/
c
1in1ð
6
:
76
Þ
¼ð
n
1Þ
k
K
C
n
½
(6.84)
ð4Þ
ð3Þ
where K
(3)
is the equilibrium constant of reaction
(6.75)
. Similarly, reactions
(6.77)
e
(6.79)
give
H
þ
½
O
þ
¼ð
O
r
n; n/
c
1in1ð
6
:
76
Þ
¼ð
n
1Þ
k
k
C
n
½
H
n
1Þ
k
K
K
C
n
½
H
(6.85)
ð7Þ
ð6Þ
ð7Þ
ð6Þ
ð5Þ
3
2
Combing the reaction rates from reactions
(6.74), (6.76), and (6.79)
, one can obtain the
following rate law based on
Eqns (6.81), (6.82), and (6.85)
,
r
m;n/m
¼ 2
k
C
n
(6.86)
H
r
n;n/
c
1in1
¼ð
n
1Þ
k
C
n
(6.87)
H
where k
H
is the rate constant of breaking one single intermonomeric unit bond. The overall
rate constant for the intermonomeric unit bond breaking is given by
ð3Þ
½H
þ
þ
ð6Þ
½H
3
O
þ
þ
k
H
¼
k
K
k
K
k
ð2Þ
½H
2
O
(6.88)
ð4Þ
ð7Þ
Thus, the net formation rate of m-oligomer is given by
X
N
X
N
r
m
¼
r
m;i/m
þ
r
m;m/
c
1im1
¼
2
k
C
i
ð
m
1Þ
k
C
m
(6.89)
H
H
i
¼
m
þ1
i
¼
m
þ1
which is valid for all oligomers (excluding monosaccharides). For monosaccharides, one
additional reaction
(6.80)
occurs. Combining
Eqn (6.89)
and the rate from reaction
(6.80)
,
the rate of formation of monomeric unit is
X
N
r
1
¼ 2
k
C
i
k
C
(6.90)
H
D
1
i
¼ 2
where k
D
is the rate constant for the degradation of monomeric sugar. For different sugars,
this constant can have different values.
For the formation of monomeric sugars,
Eqn (6.90)
can be reduced to
r
1
¼ 2
k
C
S0
ð2
k
H
þ
k
D
Þ
C
(6.91)
1
H
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