Half-Life Calculations

Why This Matters

Half-life is one of the most testable concepts in chemical kinetics because it bridges mathematical calculations with real-world applications. You're being tested on your ability to connect rate constants to measurable time intervals, distinguish how different reaction orders behave, and apply these principles to everything from drug metabolism to radioactive dating. The AP exam loves half-life problems because they require you to demonstrate both computational skills and conceptual understanding of reaction kinetics.

The key insight here is that half-life behavior reveals the underlying mathematics of a reaction. First-order reactions have constant half-lives—a unique property that shows up repeatedly in multiple-choice questions and FRQs. Don't just memorize the formula $t_{1/2} = \frac{0.693}{k}$—understand why it works and how it differs from zero-order and second-order kinetics. That conceptual grasp is what separates a 3 from a 5.

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The Core First-Order Relationship

First-order reactions dominate half-life questions on the AP exam because their mathematics are elegant and their applications are everywhere. The half-life of a first-order reaction depends only on the rate constant, not on how much reactant you start with.

The Half-Life Equation

• $t_{1/2} = \frac{0.693}{k}$—this equation is derived from the integrated rate law $\ln[A] = -kt + \ln[A]_0$ by setting $[A] = \frac{1}{2}[A]_0$
• The constant 0.693 is $\ln(2)$—understanding this derivation helps you reconstruct the formula if you forget it
• Units of k must be $s^{-1}$ (or $min^{-1}$, etc.)—first-order rate constants always have inverse time units

Rate Constant from Half-Life

• $k = \frac{0.693}{t_{1/2}}$—simply rearranging the half-life equation lets you find reaction speed from decay time
• This conversion appears frequently in multi-step problems—you might be given half-life and asked to find concentration at a specific time
• Check unit consistency—if half-life is in minutes, your rate constant will be in $min^{-1}$

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The Concentration-Independence Principle

What makes first-order kinetics special—and highly testable—is that half-life doesn't care about starting concentration. This counterintuitive property distinguishes first-order from other reaction orders.

Constant Half-Life Behavior

• Half-life remains the same whether you start with 1.0 M or 0.001 M—this is the defining characteristic of first-order kinetics
• Each successive half-life reduces concentration by exactly 50%—after 3 half-lives, you have $\frac{1}{8}$ of your original amount
• Zero-order and second-order reactions don't share this property—their half-lives depend on $[A]_0$, making calculations more complex

Calculating Remaining Quantity

• Remaining = Initial $\times \left(\frac{1}{2}\right)^n$—where $n$ is the number of half-lives elapsed
• For non-integer half-lives, use the integrated rate law—this formula only works cleanly when $n$ is a whole number
• Quick mental math: after 4 half-lives, ~6% remains—useful for estimation on multiple-choice questions

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Graphical Analysis Methods

The AP exam frequently presents kinetic data graphically and asks you to extract half-life or rate constant information. Knowing which plot to use and how to read it is essential.

Determining Half-Life from Plots

• Plot $\ln[A]$ vs. time for first-order reactions—a straight line with slope $= -k$ confirms first-order behavior
• Read half-life directly from a concentration vs. time curve—find where $[A]$ drops to half its initial value and note the elapsed time
• Verify constant half-life by checking multiple intervals—if each halving takes the same time, you've confirmed first-order kinetics

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Real-World Applications

Half-life calculations connect abstract kinetics to tangible applications—exactly the kind of interdisciplinary thinking the AP exam rewards. These applications demonstrate why mastering half-life matters beyond the test.

Radioactive Decay

• Each isotope has a characteristic, unchanging half-life—Carbon-14's 5,730-year half-life enables archaeological dating
• Radioactive decay follows first-order kinetics—the same equations apply whether you're studying reactions or nuclear processes
• Half-lives range from microseconds to billions of years—Uranium-238's 4.5-billion-year half-life helps date geological formations

Pharmacokinetics and Drug Dosing

• Drug elimination often follows first-order kinetics—half-life determines dosing intervals to maintain therapeutic levels
• Steady-state concentration requires ~5 half-lives—this principle guides how doctors design medication schedules
• Environmental pollutant degradation uses identical mathematics—predicting how long contaminants persist requires half-life calculations

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Quick Reference Table

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Self-Check Questions

1. A first-order reaction has a rate constant of $0.0231 \, min^{-1}$. Calculate the half-life and determine what fraction of reactant remains after 2 hours.

2. Two reactions have half-lives of 10 minutes and 30 minutes. Without calculating, which has the larger rate constant, and by what factor?

3. How would you experimentally distinguish a first-order reaction from a second-order reaction using half-life measurements at different initial concentrations?

4. Carbon-14 has a half-life of 5,730 years. If an artifact contains 25% of its original Carbon-14, approximately how old is it? Explain your reasoning.

5. Compare and contrast how half-life depends on initial concentration for zero-order, first-order, and second-order reactions. Which reaction order would show decreasing half-life as the reaction progresses?