Biomedical Engineering Reference
In-Depth Information
where
r
is the rate of reaction. From stoichiometry, the concentration of A, B, and C are related
through
C
A
¼ C
A
0
1
2
C
C
¼ C
B
(7.32)
Applying the simplest form (power-law rate expression) of kinetics to this homogeneous
reaction leads to
C
¼ k
f
C
A
0
1
n
d
C
C
d
t
¼ k
f
ðC
A
C
B
Þ
n
2
k
b
C
n
k
b
C
C
C
C
(7.33)
2
Here,
k
f
,
k
b
, and
n
are termed kinetic constants. We also know that the chemical equilibrium is
established at infinite time or the rate of change of
C
C
is zero at infinite time, and
k
f
,
k
b
, and
n
are related by
C
A
0
0:5
C
C
C
C
n
13
29
n
k
b
¼
k
f
¼
k
f
(7.34)
t
/N
Thus, two parameters can be varied and the experimental data can be used to estimate these
parameters. To this point, we have the regression model:
d
t
¼ k
f
C
A
0
1
n
k
f
13
29
n
d
C
C
C
C
C
C
(7.35)
2
which is a nonlinear ordinary differential regression model.
Determining the parameters:
k
f
and
n
of
Eqn (7.35)
from the data in
Table 7.3
is not an easy
task. This problem is tackled traditionally through linear regressions. Because straight lines
are visually pleasing and easy to spot the scattering of experimental data around the line, the
regression is normally done by transforming the model to a linear model. There are two
general methods used in the literature: the integral methods and the differential methods,
all of which attempts to reduce the differential model, such as
Eqn (7.35)
, to a linear (alge-
braic) model. The linearization of the kinetic model in terms of regression needs is not
unique in some cases. For example,
the well-known two parameters (
r
max
and
K
m
)
Michaelis
e
Menten equation
r
max
S
K
m
þ S
r ¼
(7.36)
can be linearized in multiple ways. They include
(1)
the Lineweaver
e
Burk plot (the double reciprocal plot 1/
r
~1/
S
)
r
¼
1
r
max
þ
K
m
r
max
S
(7.37)
In terms of two new parameters (
a
0
¼
1/
r
max
and
a
1
¼
K
m
/
r
max
), the regression model is
linear:
y
¼
a
0
þ
a
1
x
. Here,
y
¼
1/
r
and
x
¼
1/
S
.
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