What is an integrated rate law?

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Answer 1 · 2014-08-08T21:44:12

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.

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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