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 | \(\text{Rate} = k\) | \(\text{Rate} = k[A]\) | \(\text{Rate} = k[A]^2\) |
| Integrated Rate Law | \([A]_t = [A]_0 - kt\) | \(\ln[A]_t = \ln[A]_0 - kt\) | \(\frac{1}{[A]_t} = \frac{1}{[A]_0} + kt\) |
| Graph for Linearity | \([A]\) vs. \(t\) | \(\ln[A]\) vs. \(t\) | \(\frac{1}{[A]}\) vs. \(t\) |
| Slope | \(-k\) | \(-k\) | \(k\) |
| Half-Life (\( t_{1/2} \)) | \(t_{1/2} = \frac{[A]_0}{2k}\) | \(t_{1/2} = \frac{\ln 2}{k}\) | \(t_{1/2} = \frac{1}{k[A]_0}\) |
| Units of \( k \) | \(\text{M/s}\) | \(\text{s}^{-1}\) | \(\text{M}^{-1}\text{s}^{-1}\) |
| 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]
