Integrated Rate Equations

Why we need an integrated rate law

The differential rate law tells us how the rate of a reaction depends on the concentration of the reactant(s) at any instant. The trouble is that this rate is an instantaneous rate — experimentally, it is obtained as the slope of the tangent drawn to the concentration-vs-time curve at a chosen instant $t$. Reading a tangent's slope off a curve is fiddly and imprecise, and it makes it hard to pin down the rate law — and hence the order — of a reaction with confidence.

We can sidestep this by integrating the differential rate equation. Integration converts a statement about the rate of change of concentration into an explicit equation connecting the concentration itself to time. This is exactly the kind of relationship we can test directly against experimental data, because concentration at a given time is something we can actually measure, whereas an instantaneous rate has to be inferred.

What integration buys us

Once the differential rate equation for a reaction is integrated, we get a formula of the form "concentration as a function of time," built around the rate constant $k$. Such an integrated rate equation lets us:

• calculate the rate constant $k$ directly from a table of (concentration, time) readings, without ever having to draw a tangent, and
• test whether a proposed order fits the data, since the integrated equation predicts a very specific mathematical relationship (often a straight line on a suitably chosen plot) between concentration and time. …