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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.