# A two-step reaction

A turns into B, with rate constant k_{1}, which
turns into C, with rate constant k_{2}. Now,
B can turn back into A, with rate constant k_{-1}.
The previous example showed the
result of *k*_{-1} = 0.
Analytical solutions to the equations for these reactions are
not available, but numerical solutions may readily be found.
The *Steady State Approximation* and the
*Pre-Equilibrium Approximation* are often used to analyse
systems of this type.

Type:

`demo3 `*k*_{1} k_{-1} k_{2}

to plot the changes in concentrations for systems of this type,
calculated numerically.

`demo3ssa `*k*_{1} k_{-1} k_{2}

`demo3pea `*k*_{1} k_{-1} k_{2}

`demo3comp `*k*_{1} k_{-1} k_{2}

give the results for the *Steady State Approximation* and the
*Pre-Equilibrium Approximation*, respectively. `demo3comp`
plots the results for the numerical solutions to the differential
equations and the formation of C according to both approximations,
on the same graph.

## Examples

`demo3comp 0.2 1 1.1`

The concentration of B is always low, so the *Steady State Approximation*
(magenta line) works well. The *Pre-Equilibrium Approximation*
(cyan line) does not. The red, green and blue lines show the
concentrations of A, B and C, respectively, as the reaction
proceeds, found by numerically solving the differential
equations for the reaction. The changes in A and B using the two
approximations can be seen by typing: `demo3ssa 0.2 1 1.1`
and `demo3pea 0.2 1 1.1`.

`demo3comp 1.1 1 0.1`

In this example, A and B rapidly reach equilibrium, so the
*Pre-Equilibrium Approximation*
(cyan line) works much better than the
*Steady State Approximation*
(magenta line). The difference between the numerical solution
and the *Pre-Equilibrium Approximation* occurs at the beginning,
whilst A and B attain equilibrium.

`demo3comp 1 1.1 1.2`

In this example, the concentration of B is not constant, and A and B
do not come close to reaching an equilibrium. Neither approximation
works very well.

These graphs were created using
Gnuplot
Next Example

© 1997 J M Goodman