To calculate the order of the reaction for bromine, notice that experiments
1 and 2 hold the
concentration of acetone constant while doubling the concentration of
bromine. The initial rate of the
reaction is unaffected by the increase in bromine concentration, so the
reaction is zero order in
bromine. We can prove this mathematically by taking the ratio of the rates
from experiments 1 and
two:

As you can see in the above equations, by holding the concentrations of all
but one species constant
between two experiments, you can calculate the order of the reaction in a
single reactant at a time. By
similar reasoning, we can conclude that because the rate of reaction
doubled when the concentration
of acetone was doubled (cf. experiments 1 and 3) the reaction must
be first order in acetone.
However, had the rate quadrupled or octupled with a doubling of the acetone
concentration, the
reaction would have been second or third order in acetone, respectively.
In practice, you will likely
never see a reaction with an order higher than 3. If you calculate an
order higher than 3 for a
reaction, double check your math because that is highly unusual. If you
compute a fractional power
for a reactant's order, do not be discouraged; they are quite common
(especially half-order reactions).

To calculate the value of k, the rate constant, you simply plug into
the rate law the values of the
concentrations, the orders, and the rate of the reaction from any one of
the three experiments. All
three experiments should give a value of 1.64 x 10^{-4}
s^{-1}. You should
prove this to yourself.

Using the Integrated Rate Law to Determine the Rate Law

One major disadvantage to using the method of initial rates is the need to
perform multiple
experiments. Another disadvantage is that it works only for relatively
slow reactions. If your
reaction proceeds too fast, the rate that you measure will have a large range of
uncertainty. To combat these
problems, chemists have developed a method that uses data on
concentration versus time during a
reaction to infer the order of a reaction. The strategy works by comparing
concentration versus time
information to the mathematical predictions made by integrated rate
laws. This is usually
accomplished by assuming that a reaction has a certain order and making a
plot out of the data that
should be linear if the assumption about the rate law is correct. If we
guess correctly, the graph is
linear. If we are wrong, then the graph is curved and we need to choose
another order against
which to plot our data.

To practice this method, we will first need to know the forms of the
integrated rate laws for some
common reaction orders. Luckily, only orders zero through two are common
so we need to only
consider reactions with these three orders. Due to the mathematical
complexities of the problem, we
will only consider rate laws of the form rate = [A]^{n}. However,
with some experimental
tricks I will explain below, that treatment does allow us to use the method
of integrated rates for a
reaction with any arbitrary rate law.

By integrating the differential rate law (the simple rate law from Heading
), we
derive the integrated rate law for a reaction. Without proof, we present
below the forms of the integrated
rate laws for reaction orders zero through two: