An** integral rate law** mathematically expresses the rate of a reaction in terms of the initial concentration and the measured concentration of one or more reactants over a particular time.

Such a rate law can be derived from its differential form via simple calculus. Consider the decomposition of hydrogen peroxide to oxygen:

The rate law is experimentally determined to be first order: *rate* = *k*[*H _{2}O_{2}*]. If we are monitoring the progress of the reaction by measuring the change in concentration of the peroxide, the differential form of the rate law is:

Rearranging and integrating eq12,

where [*H _{2}O_{2}*]

*is the concentration of the peroxide at*

_{0}*t*= 0, i.e. its initial concentration. We then get:

Eq13 is the integral form of the first order rate law for the decomposition of hydrogen peroxide. The general equation for a species, *A*, that participates in a first order reaction of *vA* → *B* is:

For a zero order reaction, e.g. the decomposition of excess *N _{2}O* on hot platinum, , the rate equation is , and by integrating both sides of the differential rate equation, its integral form is:

In general, a species, *A*, that participates in a zero order reaction of *vA* → *B* has the equation:

For a second order reaction, e.g. *NO _{2} *+

*CO*→

*NO*+

*CO*, the rate equation is:

_{2}and its integral form is:

Once again, the generic second order rate equation that involves only one species, *A*, in a reaction, *vA* → *B*, is:

Finally, the diagram below shows the combined concentration-time plot of eq14, eq16 and eq18 for a chemical species *A*.