Integrated Rate Equation
Definition and meaning of Integrated Rate Equation in chemistry.
An integrated rate equation is a mathematical expression, obtained by integrating a differential rate law, that gives the concentration of a reactant as an explicit function of time.
In more detail
While a differential rate law states how the instantaneous rate depends on concentration (rate = k[A]^n), the integrated form is more useful experimentally: plotting concentration-time data in the appropriate way (for example [A] vs. t, ln[A] vs. t, or 1/[A] vs. t) reveals which plot is linear, thereby identifying the reaction order and yielding the rate constant k from the slope. Each order (zero, first, second) has its own characteristic integrated equation and half-life formula, making these equations central tools in chemical kinetics.
Key facts
| Field | Physical Chemistry |
|---|---|
| Zero order | [A] = [A]0 - kt |
| First order | ln[A] = ln[A]0 - kt |
| Second order | 1/[A] = 1/[A]0 + kt |
For a first-order reaction A to products with rate = k[A], integration gives ln[A] = ln[A]0 - kt (equivalently [A] = [A]0 e^(-kt)). Plotting ln[A] against time yields a straight line with slope -k, confirming first-order behavior.
Frequently asked questions
How do integrated rate laws help determine reaction order?
By testing which concentration-time plot ([A] vs t, ln[A] vs t, or 1/[A] vs t) is linear for the experimental data; the plot that gives a straight line identifies the order, and its slope gives the rate constant.
How is an integrated rate law related to the differential rate law?
The integrated rate law is derived by separating variables and integrating the differential rate law (rate = -d[A]/dt = k[A]^n) with respect to time, converting a rate-concentration relationship into a concentration-time relationship.