The rate constant can be calculated from any one of the experiments, for instance the first one:
If we square the second equation and rearrange, we get
If we now eliminate in the first equation using
this one, we obtain
or
Therefore
Note that this phenomenological equilibrium constant, unlike the thermodynamic equilibrium constant, has units.
We therefore need to compare plots of and
vs t. If one plot is clearly linear while the
others isn't, that will tell us the order of reaction.
First however, we need to calculate a from b.
Since the production of each B requires 2 A's and the reaction
is irreversible, the long-time value of b is half the initial
amount of A, i.e.
.
Moreover,
. The data can therefore be transformed
to
The graphs are as follows:
The second plot ( vs t) is clearly linear while
the other isn't. The order of the reaction with
respect to A is therefore one. The rate constant (the
negative of the slope of the plot) is
or
.
The time derivative of c(t) is
The right-hand side of the rate equation is
so the differential equation is satisfied.