Summarizing Integrated Rate Laws

Key Concepts and Summary

Integrated rate laws are determined by integration of the corresponding differential rate laws. Rate constants for those rate laws are determined from measurements of concentration at various times during a reaction.

The half-life of a reaction is the time required to decrease the amount of a given reactant by one-half. The half-life of a zero-order reaction decreases as the initial concentration of the reactant in the reaction decreases. The half-life of a first-order reaction is independent of concentration, and the half-life of a second-order reaction decreases as the concentration increases.

Key Equations

• integrated rate law for zero-order reactions: \(\left[A\right]=\text{−}kt+{\left[A\right]}_{0}, \)\({t}_{1\text{/}2}=\phantom{\rule{0.1em}{0ex}}\cfrac{{\left[A\right]}_{0}}{2k}\)
• integrated rate law for first-order reactions: \(\text{ln}\left[A\right]=\text{−}kt+\text{ln}{\left[A\right]}_{0}, \phantom{\rule{0.2em}{0ex}}\text{}{t}_{1\text{/}2}=\phantom{\rule{0.1em}{0ex}}\cfrac{0.693}{k}\)
• integrated rate law for second-order reactions: \(\cfrac{1}{\left[A\right]}\phantom{\rule{0.1em}{0ex}}=kt+\phantom{\rule{0.2em}{0ex}}\cfrac{1}{{\left[A\right]}_{0}}, \)\({t}_{1\text{/}2}=\phantom{\rule{0.1em}{0ex}}\cfrac{1}{{\left[A\right]}_{0}k}\)

Glossary

half-life of a reaction (t_l/2)

time required for half of a given amount of reactant to be consumed

integrated rate law

equation that relates the concentration of a reactant to elapsed time of reaction