Question

What is an integrated rate law?

Answer 1

An integrated rate law is an equation that expresses the concentrations of reactants or products as a function of time.

An integrated rate law comes from an ordinary rate law.

See What is the rate law?.

Consider the first order reaction

A → Products

The rate law is: rate = `r = k["A"]`

But `r = -(Δ["A"])/(Δt)`, so

`-(Δ["A"])/(Δt) = k["A"]`

If you don't know calculus, don't worry. Just skip ahead 8 lines to the final result.

If you know calculus, you know that, as the Δ increments become small, the equation becomes

`-(d["A"])/(dt) = k["A"]` or

`(d["A"])/(["A"]) = -kdt`

If we integrate this differential rate law, we get

`ln["A"]_t = -kt` + constant

At `t` = 0, `["A"]_t = ["A"]_0`, and `ln["A"]_0` = constant

So the integrated rate law for a first order reaction is

`ln ["A"] = ln["A"]_0 - kt`

This equation is often written as

`ln((["A"])/["A"]_0) = -kt` or

`(["A"])/(["A"]_0) = e^(-kt)` or

In the same way, we can derive the integrated rate law for any other order of reaction.