# What is an integrated rate law?

Aug 8, 2014

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 = $r = k \left[\text{A}\right]$

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

$\left(d \left[\text{A"])/(["A}\right]\right) = - k \mathrm{dt}$

If we integrate this differential rate law, we get

$\ln {\left[\text{A}\right]}_{t} = - k t$ + constant

At $t$ = 0, ${\left[\text{A"]_t = ["A}\right]}_{0}$, and $\ln {\left[\text{A}\right]}_{0}$ = constant

So the integrated rate law for a first order reaction is

$\ln {\left[\text{A"] = ln["A}\right]}_{0} - k t$

This equation is often written as

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

$\left({\left[\text{A"])/(["A}\right]}_{0}\right) = {e}^{- k t}$ or

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