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# What is an integrated rate law?

An integrated 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 .

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.

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