Kinetics

The interactive plot below shows the solutions to a range of rate equations commonly encountered in chemical kinetics.

The specific rate equation can be selected through the dropdown box (chemical equations in the table below), and the values of the various rate constants and initial concentrations of reagents adjusted to see the effect they have on the kinetics. The plot shows the concentrations of species $A, B, C...$ against time. Details on the models used, info on how it works, and some ideas for interesting things to explore/use in teaching are below the plot.

$$ \begin{array}{cc} \hline \textsf{Dropdown Heading} & \textsf{Chemical Equation} \\ \hline \textsf{A->B} & A \xrightarrow{k_1} B\\ \textsf{A->B->C (sequential)} & A \xrightarrow{k_1} B \xrightarrow{k_2} C \\ \textsf{A->B->C (parallel)} & A \xrightarrow{k_1} B ~\textsf{and}~ A \xrightarrow{k_2} C \\ \textsf{A+B->C} & A + B \xrightarrow{k_1} C \\ \textsf{A->B (autocatalysis)} & A\xrightarrow{k_1} B \\ \textsf{A+B<->C->D} & A+B \underset{k_2}{\stackrel{k_1}{\rightleftharpoons}}C \xrightarrow{k_3} D \\ \textsf{A+B<->C->D+A} & A+B \underset{k_2}{\stackrel{k_1}{\rightleftharpoons}} C \xrightarrow{k_3} D+A \\ \hline \end{array} $$

$k_1$ = 1 $k_2$ = 1 $k_3$ = 1 Presets:

$[A]_0$ = 1 $[B]_0$ = 0 $[C]_0$ = 0 $[D]_0$ = 0

ODE Solver Options

Step Size = 0.1 Max Time = 10

Note on how it works: I've written a simple (and frankly very satisfying) implementation of the Runge-Kutta (RK4) method in TypeScript that solves these equations numerically and reasonably quickly (this seemed easier than working out the analytical solutions for everything, and makes it trivial to add more examples later). You might experience some lag if you crank the step size down and the time range up to max, however (controls above).

Detail on the selectable models:

A->B.
- $A \xrightarrow{k_1} B$
- The simplest case. Useful to illustrate that as $A$ goes down, $B$ goes up, etc...
A->B->C sequential.
- $A \xrightarrow{k_1} B \xrightarrow{k_2} C$
- Sequential reaction. Useful illustration of the steady-state approximation (when $k_1 << k_2$).
A->B->C parallel.
- $A \xrightarrow{k_1} B$ and $A \xrightarrow{k_2} C$
- Parallel decay of A. Useful illustration of branching (test how the relative sizes of $k_1$ and $k_2$ affect the product ratio.)
A+B->C.
- $A + B \xrightarrow{k_1} C$
- Simplest case of a bimolecular reaction. Useful to illustrate how if $[B]=0$, nothing happens.
A->B (autocatalysis).
- $A\xrightarrow{k_1} B$, but where $B$ catalyses the reaction.
- Interesting example where the kinetics are described by the differential logistic equation. Similar behaviour to things like population dynamics models from biology.
A+B<->C->D.
- $A+B \rightleftarrows C \xrightarrow{k_3} D$
- Useful illustration of a pre-equilibrium. Play with $k_1$ and $k_2$ and see how the amount of $C$ changes.
A+B<->C->D+A.
- $A+B \rightleftarrows C \xrightarrow{k_3} D+A$
- As above but where $A$ is regenerated - as in enzyme kinetics (Michaelis-Menten mechanism). In this case, $A$ is the enzyme, $B$ is the substrate, $C$ is the enzyme-substrate complex, and $D$ is the product. Play with the rate constants and see when the traditional assumptions hold.

For models 4 to 7, ensure that $[B]_0 \neq 0$, or you won't get an interesting output! Note that controls that are not used in a given model (e.g. $k_2$ or $[C]_0$ for model 1) will just have no effect on the output.