Differential Equations for Chemical Kinetics
Example of Differential Equations for Chemical Kinetics
Chemical reactions of a wide variety can be modeled with coupled
(often nonlinear) differential equations. These describe the time
evolution of the concentrations of the various chemical species:
reactants, intermediaries, catalysts, and products. Such problems
are quite simple to set up and solve with Mathematica. The
function NDSolve
can numerically integrate the differential equations that arise.
The resulting concentrations can be plotted as a function of time
and also be used to accurately compute the expected concentration
of the molecular species.
Reaction with an intermediate
Here is the reaction,
where the mechanism of the reaction proceeds through an intermediate
species .
The rate equations corresponding to this reaction are
where ,
,
and
represent the concentrations of the species ,
,
and
at the time .
The parameters ,
,
and
(the "rate constants") are considered to be constant here.
This might be the case if the reaction proceeds in contact with
a heat bath. If the temperature of the mixture of reactants is allowed
to be variable, then there needs to be an additional set of equations
supplied which model the temperature variation of ,
,
and
with time.
Although these equations do not have an explicit analytic solution,
they can be numerically integrated using the built in numerical
differential equation solver, NDSolve.
Here is the solution for a particular choice of the rate constants
,
,
and .
In this example the initial concentrations of
and
are equal and that of the intermediary
is zero.
The result is expressed in terms of a set of replacement rules
that give the functions as InterpolatingFunction objects. The notation
,
for example, is a shorthand for the numerical information needed
by the interpolation algorithm to reproduce the solution.
Here is plot of the result (we multiply the concentration of
by a factor of 50 so that it can be seen in the graph on the same
scale as
and ).
The colors of the curves for ,
,
and ,
are black, red, and green respectively.
For this example, as the reaction proceeds, the intermediary is
created and then settles down to an approximately steady state value.
Reaction with Catalysts
The rate equations corresponding to this reaction are
Here is a particular case of the integration of these differential
equations. The initial concentrations of
and
are equal and the rate constants
and chosen
as shown.
Here is plot of the result. The colors of the curves for ,
,
,
and
are black, red, green, and blue respectively.
It can be seen that, as the reaction proceeds towards completion,
the concentrations of the catalysts
and
eventually return to their initial values.
