Quantitative Physiology I / Molecular and Cellular Systems, BMEN E4001x
Notes 01 Chemical Kinetics and Equilibria
Law of Mass Action
[
]
[
]
[
]
k
*
C
*
B
*
A
reaction
of
rate
D
C
C
B
A
for
2
=
→
+
+
+
k
Note that A, B, C, etc, are thus in units of concentration, while k is of units to yield conc/time.
Note that in the following, I’ll adopt K&S notation of lower case letters for species
concentrations.
Unimolecular reaction as a general framework – first order:
B
A
k
→
A = reagent
B = product
k
= rate constant, a mix of energetic and physical factors
The reaction rate for this elementary reaction is defined as the rate of conversion of A to B,
which is equivalent to either the rate of loss of A or rate of generation of B.
In this simple case,
ak
k
A
dt
da
dt
A
d

=

=
=
]
[
]
[
From this, we get for a starting concentration of A of [A]
0
, or a
0
,
)
kt
exp(
*
]
A
[
)
t
](
A
[
0

=
This is not a very useful chemical reaction in biology, as it is unregulated. This is found more
often in decay processes.
As a first level of complexity, add the reverse reaction
B
A
1
1
k
k
←
→

This reaction is first order in both forward and reverse. An isomerization reaction, for example
b
k
a
k
dt
db
b
k
a
k
dt
da
1
1
1
1



=
+

=
differential equations
and
0
a
b
a
=
+
“conservation” equation
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View Full DocumentSolve, looking at the expression for a, and using the third relation
0
1
1
1
0
1
1
1
0
1
1
a
)
k
k
(
k
a
'
a
a
k
a
)
k
k
(
)
a
a
(
k
a
k
dt
da





+

=
+
+

=

+

=
which leads to
0
1
1
1
1
1
1
1
1
1
0
1
0
1
1
1
1
1
a
)
k
k
(
k
)
t
)
k
k
(
exp(
C
)
t
(
a
...
or
)
t
)
k
k
(
exp(
C
)
t
(
'
a
....
or
'
a
)
k
k
(
a
k
a
)
k
k
(
k
'
a
)
k
k
(
dt
'
da









+
+
+

=
+

=
+

=
+
+
+
+

=
For this example, make the assumption boundary condition that at t=0,a=a
0
0
1
1
1
0
1
1
1
0
1
1
0
1
1
1
0
a
)
k
k
(
k
C
a
k
C
)
k
k
(
a
)
k
k
(
a
)
k
k
(
k
C
a






+
=
+
+
=
+
+
+
=
Not the most informative, but let’s try to draw out. We know that a graph of [A] vs. t
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 Fall '10
 KAM
 pH, Reaction, K1

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