CE 561 Lecture Notes
Fall 2009
p. 1 of 14
Day 4: Laplace transform methods for solving rate equations; Stochastic (kinetic Monte
Carlo) methods for modeling reacting systems
Preliminary comments:
Matrix methods discussed last week work only for
linear
differential equations – which means
that they work only for 0
th
or 1
st
order reactions (any other order leads to nonlinear terms in the
rate equations), and of course only work for systems at constant temperature (where the rate
“constants” are constant – the Arrhenius temperature dependence introduces even stronger
nonlinearities in nonisothermal systems).
The same is true for Laplace transform methods of
solving rate equations, which are described below.
We can occasionally, through luck or
cleverness, find an analytical solution to nonlinear ODE’s, but there is not a general procedure
for doing so like there is for linear equations.
A common practice in experimental chemical kinetics is to perform experiments under
pseudo
firstorder
conditions.
This means that all reactants except one are supplied in large excess, so
that all but one concentration effectively remain constant, i.e. for
A + B
→
C + D with
r
=
kC
A
C
B
,
one might do experiments with a large excess of B so that
C
B
is approximately constant.
Then
the reaction is
pseudofirstorder
with
r=k
eff
C
A
, where the constant concentration of B is lumped
into the effective rate constant (
k
eff
=kC
B
)
.
If this cannot be done, and there is no “clever” solution to the nonlinear equations, then the
equations must be solved numerically.
Laplace transform methods for solution of rate equations
An alternative method for integrating the rate equations (which are a system of 1
st
order ODE’s)
is to use the Laplace transform.
We may remember from process control class that the Laplace
transform is an integral transform that converts differentiation into multiplication.
The definition
of the transform is:
F s
f t
e
f t dt
st
( )
( )
( )
=
=
−
∞
z
L
0
The inverse transform is
f t
F s
i
e
F s ds

st
i
i
( )
( )
( )
=
=
− ∞
+ ∞
z
L
1
1
2
π
γ
This is often called the
complex inversion formula
.
We generally look up both the transform and
inverse transform in tables (or use Mathematica or Maple).
It is relatively easy to do the
transform using the formula above, but the inverse transform is more difficult.
The inverse
transform requires taking an integral over a contour in the complex plane.
This requires some
knowledge of complex analysis, and we won’t cover it further.
If we are using the transform to
solve ODE’s, and we use tables to find the transform and inverse transform, then the Laplace
transform method simply becomes a convenient way of tabulating solutions of ordinary
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Fall 2009
p. 2 of 14
differential equations.
Applied in that way, this method is simply a systematic means of looking
up the known answer to the problem.
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 Fall '09
 Rate Equations

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