Biomedical Engineering Reference
In-Depth Information
where r
j
is the rate of reaction for species j; r
i
is the reaction rate for reaction i as
written; N
R
is the total number of reactions; and
n
ij
is the stoichiometric coefficient of
species j in reaction i. For example, elementary reaction system described by
(6.30)
and
(6.31)
yield
r
1
¼
k
C
C
B
k
C
AB
(6.34)
1
1
A
r
2
¼
k
C
AB
(6.35)
2
r
A
¼
r
(6.36)
1
r
B
¼
r
(6.37)
1
r
C
¼
r
(6.38)
2
r
AB
¼
r
1
r
(6.39)
2
There are no assumptions or approximations involved in these equations.
While
Eqns (6.34)
(6.39)
are convenient to use due to the simplistic nature of the reac-
tion system, in other cases, one wish to simplify the reaction as there could be hundreds or
even thousands of reaction steps in some practical cases. Usually, we simplify the reac-
tions by considering the relative magnitude of the reaction rates as not all the reactions
areprecededatequationrates.Somereaction species in the mixture are of low level,
even difficult to detect with conventional techniques. All these information would allow
us to simplify the reaction system. This is the subject of discussion in the following
subsections.
e
6.3.1. Fast-Equilibrium Step Approximation
Some reactions or steps are faster than others in a reaction system. In the
simplistic example, we have two reactions. Suppose one reaction is much slower
than the other one, then the slower reaction would be the one controlling the overall
reaction rate. For example, if reaction
(6.31)
is the rate-limiting step or reaction
(6.30)
proceeds much faster than reaction
(6.30)
, we would have reaction
(6.30)
to be in
equilibrium while deriving a suitable rate expression. Since reaction
(6.31)
is slow,
reaction
(6.30)
would be completed while
(6.31)
is still proceeding slowly. That is
to say
0 ¼
r
1
¼
k
C
C
B
k
C
AB
(6.40)
1
A
1
Solving
Eqn (6.40)
, we obtain
k
1
C
AB
¼
C
C
B
¼
K
C
C
(6.41)
A
1
A
B
k
1
Thus, the concentration of AB* can be expressed as a function of C
A
and C
B
, elim-
inating the need for us to deal with the species AB*. Furthermore, if we assume that
K
1
is small, i.e. the concentration C
AB*
is negligible when compared with C
A
and C
B
,
then we do not need to look at the difference, for example, between C
A
and the total
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