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Chemistry

Integrated Rate Laws & Half-Life

Zero, First, and Second Order: Graphs, Equations, and Exponential Decay — A TLDR Primer

Integrated rate laws are one of the most calculation-heavy topics in general and AP chemistry — and one of the most reliably tested. If you can't tell a first-order ln[A] plot from a second-order 1/[A] plot, or you freeze up when a problem asks for half-life without telling you the order, this guide is for you.

This TLDR primer covers zero-, first-, and second-order integrated rate laws from the ground up: where the equations come from, how to use them to find concentration at any point in time, how to linearize data to identify reaction order, and how to calculate and interpret half-life for each case. Worked examples walk through the algebra step by step, and every graph type is explained in plain language so you can recognize it on sight.

Topics include: the difference between differential and integrated rate laws, the exponential decay equation for first-order reactions and its connection to carbon dating, the constant half-life property that makes first-order reactions unique, second-order and zero-order behavior and their telltale linear plots, a practical workflow for taking raw concentration-time data and extracting order and rate constant, and real-world applications in pharmacokinetics and radiometric dating.

Written for high school chemistry students, AP Chemistry test-takers, and anyone in a first-semester college course who needs integrated rate laws chemistry concepts explained without the bloat. Short by design, no filler, and built around the problems you'll actually face on an exam.

If chemical kinetics has been giving you trouble, start here.

What you'll learn
  • Distinguish differential rate laws from integrated rate laws and know when to use each
  • Apply the integrated rate equations for zero-, first-, and second-order reactions
  • Determine reaction order from concentration-vs-time data using linearized plots
  • Calculate half-lives for each order and explain why first-order half-life is constant
  • Solve quantitative problems involving radioactive decay, drug clearance, and reactant depletion
What's inside
  1. 1. From Rate Laws to Integrated Rate Laws
    Sets up the difference between a differential rate law and its integrated form, and motivates why we need the integrated version.
  2. 2. First-Order Reactions and the Exponential Decay Equation
    Derives and applies the first-order integrated rate law, including its linearized ln[A] plot and worked examples.
  3. 3. Second-Order and Zero-Order Reactions
    Covers the integrated rate laws for second- and zero-order reactions, their linear plots, and how to recognize each from data.
  4. 4. Half-Life for Each Reaction Order
    Defines half-life and derives the half-life formulas for zero-, first-, and second-order reactions, emphasizing why only first-order has a constant half-life.
  5. 5. Determining Reaction Order from Experimental Data
    A practical workflow for taking concentration-time data and figuring out the order, k, and half-life through linearization.
  6. 6. Why It Matters: Pharmacokinetics, Carbon Dating, and Beyond
    Connects integrated rate laws and half-life to real applications students will see in biology, medicine, and geology.
Published by Solid State Press
Integrated Rate Laws & Half-Life cover
TLDR STUDY GUIDES

Integrated Rate Laws & Half-Life

Zero, First, and Second Order: Graphs, Equations, and Exponential Decay — A TLDR Primer
Solid State Press

Integrated Rate Laws & Half-Life

Zero, First, and Second Order: Graphs, Equations, and Exponential Decay — A TLDR Primer

Copyright © 2026 Solid State Press.

Published by Solid State Press.

All rights reserved. No part of this book may be reproduced or transmitted in any form without prior written permission, except for brief quotations in critical reviews and educational use.

Part of the TLDR Study Guides series.

Contents

  1. 1 From Rate Laws to Integrated Rate Laws
  2. 2 First-Order Reactions and the Exponential Decay Equation
  3. 3 Second-Order and Zero-Order Reactions
  4. 4 Half-Life for Each Reaction Order
  5. 5 Determining Reaction Order from Experimental Data
  6. 6 Why It Matters: Pharmacokinetics, Carbon Dating, and Beyond
Chapter 1

From Rate Laws to Integrated Rate Laws

Suppose you're watching a chemical reaction and you want to know how fast it's going. The tool chemists reach for first is the rate law — an equation that says the reaction rate depends on the concentration of one or more reactants. For a reaction where a single reactant A converts to products, a rate law looks like this:

$\text{rate} = k[\text{A}]^n$

Here, $k$ is the rate constant, a number that sets the overall speed of the reaction at a given temperature. The exponent $n$ is the reaction order — it tells you how sensitively the rate responds to changes in concentration. If $n = 1$, doubling $[\text{A}]$ doubles the rate. If $n = 2$, doubling $[\text{A}]$ quadruples it. If $n = 0$, the rate doesn't depend on $[\text{A}]$ at all.

This form of the rate law is also called the differential rate law, because "rate" is technically a derivative — specifically, the rate equals $-d[\text{A}]/dt$, the rate of decrease of $[\text{A}]$ over time. Written that way:

$-\frac{d[\text{A}]}{dt} = k[\text{A}]^n$

That expression is exact and powerful, but it has a practical limitation: it tells you the instantaneous speed of the reaction, not where the concentration will be at some later time. If someone asks "how much of reactant A is left after 30 minutes?", the differential rate law alone cannot answer that directly. It describes the slope of the concentration-time curve at every moment, but it doesn't hand you the curve itself.

To get the curve — to find $[\text{A}]$ as an explicit function of time — you have to integrate both sides of that equation with respect to time. The result is the integrated rate law. Think of it this way: if the differential rate law is the speedometer, the integrated rate law is the odometer. One tells you how fast you're going right now; the other tells you how far you've traveled.

About This Book

If you're staring down a kinetics unit in AP Chemistry and need focused exam prep, this is the book. It's also for the general chemistry student who wants genuine rate law help — the kind that turns confusion about reaction orders into actual problem-solving ability. Parents supporting a student through a tough chapter and tutors who need a clean, teachable framework will find it equally useful.

This guide covers the core of chemical kinetics: integrated rate laws for zero-, first-, and second-order reactions, the half-life formula for each order, exponential decay, and how to determine reaction order from experimental data — including graphs and the linearization tricks that show up on every exam. It works as a chemical kinetics textbook supplement or as a standalone quick review. Concise by design, with no filler.

Read it straight through, work every example as you go, then test yourself with the practice problems at the end. The problems are where the concepts lock in.

Keep reading

You've read the first half of Chapter 1. The complete book covers 6 chapters in roughly fifteen pages — readable in one sitting.

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