Reaction Kinetics, Examples of Important Rate Laws

Consider a reaction of Butyl chloride in water to form butyl alchohol:

C4H9Cl + H2O -> C4H9OH + HCl

Careful measurement of the concentration of the reacting butyl chloride shows that as the reactant concentration drops, so does the overall reaction rate. This is because the reaction must involve a butyl chloride molecule in a rate limiting step (which may an elementary reaction step, as well), so the rate is proportional to the chloride concentration, i.e. the reaction is first order in butyl chloride. A plot of the butyl chloride concentration as a function of time shows this behavior graphically:

Note that the rate law is a relationship that holds for an instantaneous rate, and that an average rate of reaction over a finite interval of time is not the same thing. The instantaneous rate is measured by determining the average rate of reaction over shorter and shorter time intervals, until the instantaneous rate is determined as a limit. A particular instantaneous rate, the rate of reaction just after the reactants are mixed is called the Initial Rate, and will be very useful in our kinetic studies in the near future.
Consider the ISOMERIZATION of Methyl Isonitrile to Acetonitrile

Panel (a) is the time dependent concentration (pressure) of the reactant.

Panel (b) is the natural log of the pressure, which appears to be a straight line. This is indicative of a first order reaction, expected for a unimolecular reaction like an isomerization. The integrated rate law for a first order reaction is

Another way to look at the same data is to determine the HALF LIFE of the reaction, i.e. the time it takes to deplete half of the reactants.

the half life, or t1/2 can be derived from the integrated rate law as:

t1/2 = (ln 2) / k

The half life is a constant (independent of concentration) for a first-order reaction only!

An example of a second-order reaction is that of the gas-phase decomposition of NO2
2 NO2 -> 2 NO + O2

Panel (a) shows a plot of the log of the concentration of reactant versus time. It is NOT a straight line and the reaction is NOT first order in reactant.

Panel (b) shows the inverse of the concentration versus time which is a straight line for a second order reaction. This plot is a straight line because the integrated rate law for a second order reaction is

For a summary of important relations for simple (zeroth, first, and second order) reactions, see my table.

Kinetics of Simple Rate Laws: A Summary    (some example problems)
Consider the reaction of A -> Products.
The rate of this reaction is defined as the negative time rate of change of the reactant concentration or

If the reaction had a more complicated stoichiometry, it will still have a rate that can be defined in terms of any one of the reagents or products.
For instance, if a reaction 2A + 3B -> C + 4D is observed, the rate of that reaction is defined as:

The kinetics of any reaction depend on the reaction mechanism, or rate law, and the initial conditions. If we assume for the reaction A -> products there is there is an initial concentration of reactant of [A]0 at time t=0, and the rate law is an integral order in A, then we can summarize the kinetics of the reaction as follows:
Zeroth Order First Order Second Order
Rate Law
Integrated Rate Law
Units of Rate Constant   {k}
Linear Plot [A]  vs.  t ln([A])  vs.  t 1/[A]  vs.  t