7/31/2019 Physical Chemistry 05
1/2
Colby College
Integrating Rate Laws Using the Finite Difference
Approximation
The finite difference method can be used to integrate kinetic
rate laws. The finite differenceapproach is used byMatLab, Stella,
and the Kinetics Mechanism Simulation applet. For
example, the first order rate law for A products is:
d[A]
dt= k [A] 1
Approximating the derivative with finite differences gives:
[A]
t= k [A] 2
Multiplying by
t gives:[A] = k [A] t 3
where
[A] = [A](t+t) [A](t) 4
with [A](t) the concentration of A at time t and [A](t+t) the
concentration of A in the next timeinterval. Substituting Eq. 4
into Eq. 3 with some rearrangement gives:
[A](t+t) = [A](t) k [A] t 5
or simplifying the notation gives:
[A]2 = [A]1 k [A]1t 6
where [A]1 is the current value of the concentration and [A]2 is
the next concentration at t + t.
For another example, an A + B products mechanism gives the
corresponding finite differenceapproximation:
[A]2 = [A]1 k [A]1[B]1t 7
The trick in using the finite difference approach is to use a t
that is small enough. Inpractice, you should run your calculation
with two different t values and compare. If the runsdiffer
significantly, choose an even smaller t and run again. There are
two ways to decrease tin the Kinetics Mechanism Simulation Applet.
Choosing a smaller Maximum time or a larger
number of steps both decrease the time step size, since t = (max
time)/(number of steps).The Kinetics Mechanism Simulation
applet,MatLab,MathCad, and Stella also use some
more sophisticated techniques to decrease the errors inherent in
a finite difference integration,
but the form of Eq. 5 is still the basis. [You can view the
finite difference equations in Stella by
clicking on the triangular shaped button that is just above the
2 symbol on the left-hand border
of the main window.]
Table I is an example spreadsheet for a first-order rate law,
with stoichiometry A products,to show how well the finite
difference approximation works, based on Eq. 6. The exact column
isfrom the exact integrated time dependence:
