Half-Life Calculations

Why This Matters

Half-life is one of the most testable concepts in chemical kinetics because it connects rate constants to measurable time intervals. You need to distinguish how different reaction orders behave and apply these principles to problems ranging from drug metabolism to radioactive dating.

Half-life behavior reveals the underlying mathematics of a reaction. First-order reactions have constant half-lives, and this unique property shows up repeatedly on exams. Don't just memorize $t_{1/2} = \frac{0.693}{k}$. Understand why it works and how it differs from zero-order and second-order kinetics.

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The Core First-Order Relationship

First-order reactions dominate half-life questions because their math is clean 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 comes from the integrated rate law $\ln[A] = -kt + \ln[A]_0$. Here's how to derive it:

1. Set the concentration at time $t_{1/2}$ equal to half the initial concentration: $[A] = \frac{1}{2}[A]_0$
2. Substitute into the integrated rate law: $\ln\left(\frac{1}{2}[A]_0\right) = -kt_{1/2} + \ln[A]_0$
3. Simplify the left side using log rules: $\ln\left(\frac{1}{2}\right) + \ln[A]_0 = -kt_{1/2} + \ln[A]_0$
4. The $\ln[A]_0$ terms cancel, leaving: $-\ln(2) = -kt_{1/2}$
5. Solve for $t_{1/2}$: $t_{1/2} = \frac{\ln(2)}{k} = \frac{0.693}{k}$

Notice that $[A]_0$ cancels out entirely. That's why first-order half-life is independent of starting concentration. The constant 0.693 is just $\ln(2)$, so if you forget the formula, you can re-derive it from the integrated rate law.

Units of $k$ for first-order reactions are always inverse time ($s^{-1}$, $min^{-1}$, etc.).

Rate Constant from Half-Life

Rearranging gives you $k = \frac{0.693}{t_{1/2}}$. This conversion shows up often in multi-step problems where you're given a half-life and asked to find the concentration at some specific time. Just watch your units: 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 is that half-life doesn't depend on starting concentration. This counterintuitive property is what distinguishes first-order from other reaction orders.

Constant Half-Life Behavior

• Half-life stays the same whether you start with 1.0 M or 0.001 M. This is the defining feature 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. After 4, about 6.25% remains.
• Zero-order and second-order reactions don't behave this way. Their half-lives change as the reaction progresses because their half-life formulas include $[A]_0$.

Calculating Remaining Quantity

For a whole number of half-lives, use:

$\text{Remaining} = \text{Initial} \times \left(\frac{1}{2}\right)^n$

where $n$ is the number of half-lives elapsed. This is fast and works great for mental math. But if the elapsed time isn't a clean multiple of the half-life, you'll need the full integrated rate law: $\ln[A] = -kt + \ln[A]_0$.

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Graphical Analysis Methods

Kinetic data is frequently presented as graphs, and you need to know how to extract half-life or rate constant information from them.

Determining Half-Life from Plots

• Plot $\ln[A]$ vs. time for first-order reactions. A straight line confirms first-order behavior, and the slope equals $-k$.
• Read half-life directly from a $[A]$ vs. time curve. Find where the concentration drops to half its initial value and read the elapsed time off the x-axis.
• Verify constant half-life by checking multiple intervals. Pick any concentration on the curve, then find how long it takes to halve. Repeat at a different starting point. If the time interval is the same both times, you've confirmed first-order kinetics.

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Real-World Applications

Half-life calculations connect abstract kinetics to real scenarios. These applications also show up in exam problems as context for calculations.

Radioactive Decay

Radioactive decay follows first-order kinetics, so the same equations apply. Each isotope has a characteristic, unchanging half-life. Carbon-14's half-life of 5,730 years enables archaeological dating, while Uranium-238's half-life of 4.5 billion years is used for geological dating. Half-lives across different isotopes range from microseconds to billions of years, but the math is identical.

Pharmacokinetics and Drug Dosing

Drug elimination from the body often follows first-order kinetics. A drug's half-life determines how frequently doses need to be given to maintain effective levels. It takes roughly 5 half-lives to reach steady-state concentration, which is why doctors space out loading doses accordingly. The same math applies to predicting how long environmental pollutants persist in soil or water.

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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 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?