To derive the integrated rate law for a first-order reaction, let’s consider a generic reaction:

A → Products

The rate of this reaction can be expressed as:

Rate = -d[A]/dt

Where [A] represents the concentration of A and t is time. The negative sign indicates that the concentration of A decreases over time.

We can rearrange the equation as follows:

-d[A] = k dt

Integrating both sides of the equation:

∫(-d[A]) = ∫k dt

The integration of the left side gives:

-[A] = kt + C

Where C is the constant of integration. At time t=0, the initial concentration of A is [A]0. Substituting these values into the equation:

-[A]₀ = k(0) + C

C = -[A]₀

Therefore, the integrated rate law for a first-order reaction is:

-ln([A]/[A]₀) = kt

This can also be written as:

ln([A]₀/[A]) = kt

Where [A]₀ is the initial concentration of A, [A] is the concentration of A at time t, k is the rate constant of the reaction, and t is the reaction time.

The integrated rate law for a first-order reaction shows that the natural logarithm of the ratio of initial concentration to the concentration at a given time is directly proportional to the rate constant multiplied by time.