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
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
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
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
k
k
C n ½
H
n
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
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
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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