This procedure can be applied to any number of
reactions. The challenge is preparing the initial conditions andmeasuring the initial change in concentration precisely versus
time.
[link] provides an
overview of the rate laws for several reactions. A variety ofreaction orders are observed, and they cannot be easily correlated
with the stoichiometry of the reaction.
Rate laws for various reactions
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Concentrations as a function of time and the reaction
half-life
Once we know the rate law for a reaction, we
should be able to predict how fast a reaction will proceed. Fromthis, we should also be able to predict how much reactant remains
or how much product has been produced at any given time in thereaction. We will focus on the reactions with a single reactant to
illustrate these ideas.
Consider a first order reaction like
,
for which the rate law must be
From Calculus, it is possible to use
[link] to find the function
which tells us the concentration
as a function of time. The result is
or equivalently
[link] reveals that, if a reaction is first order, we can plot
versus time and get a straight line with slope equal to
.
Moreover, if we know the rate constant and the initialconcentration, we can predict the concentration at any time during
the reaction.
An interesting point in the reaction is the
time at which exactly half of the original concentration of
has been
consumed. We call this time the
half life of the reaction and denote it as
.
At that time,
.
From
[link] and using the
properties of logarithms, we find that, for a first orderreaction
This equation tells us that the half-life of a
first order reaction does not depend on how much material we startwith. It takes exactly the same amount of time for the reaction to
proceed from all of the starting material to half of the startingmaterial as it does to proceed from half of the starting material
to one-fourth of the starting material. In each case, we halve theremaining material in a time equal to the constant half-life in
[link] .
These conclusions are only valid for first
order reactions. Consider then a second order reaction, such as thebutadiene dimerization discussed
above . The general
second order reaction
has the rate law
Again, we can use Calculus to find the
function
from
[link] . The result is most
easily written as
Note that, as
increases,
increases, so
decreases.
[link] reveals that,
for a reaction which is second order in the reactant
, we can
plot
as a function of time to get a straight line with slope equal to
. Again, if
we know the rate constant and the initial concentration, we canfind the concentration
at any time of interest during the reaction.
