Integrated Rate Law
Related Examples and Practice Problems
Topic Summary & Highlights
and Help Videos
Core Concept
Integrated rate laws provide a mathematical relationship between the concentration of reactants and time.
Purpose: To determine the concentration of a reactant at any given time or to find the time required for a reaction to reach a specific concentration.
Practice Tips
Understand the physical meaning: What does each order tell you about how reactants affect the rate?
Memorize key formulas: Focus on integrated rate laws, half-life equations, and graphing criteria.
Practice with graphs: Use experimental data to plot and determine the reaction order.
Solve varied problems: Ensure you can switch between mathematical, graphical, and conceptual approaches.
Core Concept
| Zero-Order | First-Order | Second-Order | |
|---|---|---|---|
| Rate Law | Rate=k | Rate=k[A] | Rate=k[A]2 |
| Integrated Rate Law | [A]t=[A]0−kt | ln[A]t=ln[A]0−kt | 1[A]t=1[A]0+kt |
| Graph for Linearity | [A] vs. t | ln[A] vs. t | 1[A] vs. t |
| Slope | −k | −k | k |
| Half-Life (t1/2) | t1/2=[A]02k | t1/2=ln2k | t1/2=1k[A]0 |
| Units of k | M/s | s−1 | M−1s−1 |
| Memorization Suggestion | Zero slope is constant decline; concentration decreases linearly. | First follows natural logs; think exponential decay. | Second order is reciprocal; graphing 1/[A] makes it linear. |
Where the differential rate law expresses rate as a function of reactant concentration(s) at an instant in time (hence instantaneous rate), integrated rates express the reactant concentrations as a function of time.
To solve integrated rate problems, construct a graph with time on the x-axis and then make 3 plots where the y-axis is
Concentration of A [A] vs. t
Natural log of the concentration of A ln [A] vs. t
Reciprocal of [A]
