Final answer:
The integrated rate equation for a first-order reaction is derived by integrating the differential rate equation and can be expressed using logarithms or as an exponential decay function, which relates reactant concentration to time elapsed.
Step-by-step explanation:
To derive the integrated rate equation for the rate constant of a first-order reaction, we start with the differential rate law for a first-order reaction, which is given as:
rate = -d[A]/dt = k[A]
Integrating this equation with respect to time gives:
ln[A] - ln[A]o = -kt
Where [A] is the concentration of the reactant at any time t, [A]o is the initial concentration of the reactant, k is the rate constant, and t is time. By rearranging the terms, we get the integrated rate law:
ln[A]o - ln[A] = kt
Alternately, the integrated rate law can be represented in exponential form:
[A] = [A]oe-kt
This equation allows calculation of the concentration of reactant at any time t when the initial concentration and the rate constant are given.