How It Works
Half-life is the time required for half the atoms in a radioactive sample to decay. It relates directly to the decay constant λ through t½ = ln(2)/λ. Each isotope has a fixed, characteristic half-life that ranges from microseconds (highly unstable nuclei) to billions of years (uranium-238 ≈ 4.47 billion years). After n half-lives, the fraction of original atoms remaining is (½)ⁿ — so after 2 half-lives 25% remains, after 3 half-lives 12.5%, and so on.
Example Problem
Cobalt-60 has a decay constant λ = 0.1318 yr⁻¹. Calculate its half-life.
- Start with the half-life formula: t½ = ln(2) / λ.
- Recall ln(2) ≈ 0.6931.
- Substitute: t½ = 0.6931 / 0.1318.
- Compute the division: t½ ≈ 5.26 years.
- Sanity check: cobalt-60's published half-life is 5.27 years — close, as expected.
Key Concepts
Half-life (t½) is the most commonly cited measure of radioactive decay speed because it has an intuitive meaning: 'wait one half-life and half the sample is gone.' It is mathematically tied to the decay constant by t½ = ln(2)/λ ≈ 0.6931/λ, and to mean lifetime by t½ = τ × ln(2). Half-life does not depend on the size of the sample, the temperature, or the chemical state of the atoms — it is a fundamental property of the nucleus.
Applications
- Radiometric dating — carbon-14 (t½ = 5,730 yr) for archaeological samples, uranium-lead and potassium-argon for rocks billions of years old.
- Nuclear medicine dose planning — technetium-99m's 6.0 hour half-life sets imaging windows and waste storage rules.
- Radiation safety — determining how long contaminated equipment or waste must be isolated before activity drops to background.
- Reactor fuel management — predicting fission-product inventory and decay-heat loads after shutdown.
- Treaty verification — distinguishing fresh fission products from aged stockpiles by isotope ratio.
Common Mistakes
- Confusing half-life with mean lifetime — τ is always longer than t½ by a factor of 1/ln(2) ≈ 1.443. They are not interchangeable.
- Forgetting to match the time unit of t½ and λ — if λ is in s⁻¹ the half-life comes out in seconds; convert before reporting.
- Assuming half-life scales linearly — after 3 half-lives only 12.5% remains, not 50% – 3 × 16.7%.
- Treating a mixture of isotopes as one — each isotope has its own t½ and must be tracked independently.
Frequently Asked Questions
How do you calculate half-life?
Divide the natural log of 2 by the decay constant: t½ = ln(2) / λ ≈ 0.6931 / λ. If you know any two of t½, λ, or mean lifetime τ, you can solve for the third.
What is the formula for half-life?
t½ = ln(2) / λ, where λ is the decay constant in inverse time units. Equivalently, t½ = τ × ln(2) where τ is the mean lifetime.
What is a half-life in radioactive decay?
It's the time required for exactly half of the radioactive nuclei in a sample to decay. After one half-life 50% remain, after two 25%, after three 12.5%, and so on — a geometric decrease, not a linear one.
Does half-life depend on the sample size?
No. Half-life is a property of the nucleus and is independent of how much material you start with, the temperature, or the chemical compound. Only the absolute number of decays per second (activity) scales with sample size.
What's the difference between half-life and mean lifetime?
Mean lifetime τ is the average time an individual atom survives before decaying; half-life is the time for half the population to decay. They are related by τ = t½ / ln(2), so τ is always ~44% longer than t½.
What's the half-life of carbon-14?
Carbon-14 has a half-life of 5,730 years, which makes it useful for dating organic material up to roughly 50,000 years old (about 9 half-lives, after which the remaining C-14 is below detection).
Reference: Lindeburg, Michael R. 1992. Engineer In Training Reference Manual. Professional Publication, Inc. 8th Edition.
Related Calculators
- Radioactive Decay Calculator — N(t) = N₀ e^(−λt) — remaining quantity over time
- Mean Lifetime Calculator — τ = 1/λ — average atom survival time
- Activity Calculator — A = λN — disintegrations per second in becquerels
- Radioactive Material Calculator — the full four-equation hub for decay problems
- Natural Log Calculator — evaluate ln(2) and other natural logarithms
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