Rate laws – integral form

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:

H_2O_2(aq)\rightleftharpoons H_2O(l)+\frac{1}{2}O_2(g)

The rate law is experimentally determined to be first order: rate = k[H2O2]. 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:

-\frac{d[H_2O_2]}{dt}=k[H_2O_2]\; \; \; \; \; \; \; \; 12

Rearranging and integrating eq12,


where [H2O2]0 is the concentration of the peroxide at t = 0, i.e. its initial concentration. We then get:

ln\frac{[H_2O_2]}{[H_2O_2]_0}=-kt\; \; \; or\; \; \; ln[H_2O_2]=-kt+ln[H_2O_2]_0\; \; \; \; \; \; \; \; 13

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 vAB is:

ln\frac{[A]}{[A]_0}=-vkt\; \; \; or\; \; \; ln[A]=-vkt+ln[A]_0\; \; \; \; \; \; \; \; 14

For a zero order reaction, e.g. the decomposition of excess N2O on hot platinum, N_2O\rightarrow N_2+\frac{1}{2}O_2, the rate equation is -\frac{d[N_2O]}{dt}=k, and by integrating both sides of the differential rate equation, its integral form is:

[N_2O]=-kt+[N_2O]_0\; \; \; \; \; \; \; \; 15

In general, a species, A, that participates in a zero order reaction of vAB has the equation:

[A]=-vkt+[A]_0\; \; \; \; \; \; \; \; 16

For a second order reaction, e.g.  NO2 + CONO + CO2 , the rate equation is:


and its integral form is:

\frac{1}{[NO_2]}=kt+\frac{1}{[NO_2]_0}\; \; \; \; \; \; \; \; 17

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

\frac{1}{[A]}=vkt+\frac{1}{[A]_0}\; \; \; \; \; \; \; \; 18

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


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