The rate law for the reaction *A* + *B* → *P* is:

Unlike the 2nd order reaction of the type *A* → *P* with a rate law of , eq11 cannot be integrated unless we express [*B*] in terms of [*A*], or [*A*] and [*B*] in terms of a third variable *x*. Since A and B react in a ratio of 1:1, the remaining concentrations of *A* and *B* at any time of the reaction are:

where [*A _{0}*] and [

*B*] are the initial concentrations of A and B respectively.

_{0}Differentiating both sides of [*A*] = [*A _{0}*] –

*x*with respect to time, we have

Therefore, we can rewrite eq11 as

where *a* = [*A _{0}*] and

*b*= [

*B*]

_{0}Integrating eq12 using the partial fraction expression of ,

After some algebra, we have,

which is equivalent to