Half-Life Calculator
Calculate the half-life, remaining quantity, elapsed time, or decay constant of a radioactive substance using the exponential decay formula.
☢️ Radioactive Decay Calculator
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What Is Half-Life?
Half-life (t½) is the time required for a quantity to reduce to half of its initial value. The term is most commonly used in nuclear physics to describe the decay of radioactive isotopes, but it also applies to any process that follows exponential decay, including chemical reactions, pharmacokinetics (drug elimination), and even the decay of internet memes.
Half-life should not be confused with mean lifetime (τ), which is the average time a single atom or particle survives before decaying. The relationship between them is: τ = t½ / ln(2) ≈ 1.4427 × t½.
The Exponential Decay Formula
Radioactive decay follows an exponential pattern:
Or equivalently, using the decay constant λ:
Where:
- N(t) — Remaining quantity at time t
- N0 — Initial quantity
- t½ — Half-life
- λ — Decay constant = ln(2) / t½
- t — Elapsed time
Rearranged Forms
| Solve For | Formula |
|---|---|
| Remaining quantity (N) | N = N0 × (1/2)t/t½ |
| Half-life (t½) | t½ = −t × ln(2) / ln(N/N0) |
| Time elapsed (t) | t = −t½ × ln(N/N0) / ln(2) |
| Initial quantity (N0) | N0 = N / (1/2)t/t½ |
| Decay constant (λ) | λ = ln(2) / t½ |
Half-Lives of Common Isotopes
| Isotope | Half-Life | Application |
|---|---|---|
| Carbon-14 | 5,730 years | Radiocarbon dating of organic materials |
| Uranium-238 | 4.5 billion years | Geological dating of rocks |
| Iodine-131 | 8.02 days | Thyroid cancer treatment |
| Cobalt-60 | 5.27 years | Radiation therapy, food sterilization |
| Cesium-137 | 30.17 years | Nuclear fallout monitoring |
| Tritium (H-3) | 12.32 years | Luminous paints, fusion research |
| Radon-222 | 3.82 days | Indoor air quality concern |
| Plutonium-239 | 24,110 years | Nuclear weapons, reactors |
| Americium-241 | 432.2 years | Smoke detectors |
| Polonium-210 | 138.4 days | Static eliminators, historic poisoning cases |
How to Calculate Half-Life
- Identify known values: You need any 3 of the 4 variables (N0, N, t, t½).
- Apply the formula: Use the appropriate rearranged form.
- Verify units: Ensure time elapsed and half-life use the same units.
A fossil has 25% of its original C-14. How old is it?
Given: N/N0 = 0.25, t½ = 5730 years
Formula: t = −5730 × ln(0.25) / ln(2) = −5730 × (−1.3863) / 0.6931 = 11,460 years
The fossil is approximately 11,460 years old (exactly 2 half-lives, since 0.25 = (1/2)2).
Applications of Half-Life
- Radiocarbon dating: Carbon-14 allows dating of organic materials up to ~50,000 years old.
- Nuclear medicine: Short-lived isotopes like Technetium-99m (6 hours) are used for diagnostic imaging.
- Pharmacology: Drug half-life determines dosing frequency. A drug with a 4-hour half-life needs more frequent dosing than one with a 24-hour half-life.
- Nuclear waste management: Long half-lives (Pu-239: 24,110 years) dictate how long waste must be stored safely.
- Geology: Uranium-lead dating uses the 4.5-billion-year half-life of U-238 to date ancient rocks.
Frequently Asked Questions
What happens after 10 half-lives?
After 10 half-lives, only (1/2)10 = 1/1024 ≈ 0.098% of the original substance remains. This is why 10 half-lives is often used as a practical threshold for considering a radioactive source "decayed."
Can half-life be changed?
Under normal conditions, radioactive half-life is a fixed nuclear property that cannot be altered by temperature, pressure, or chemical reactions. However, extreme conditions (like the interiors of stars or electron capture decay) can slightly affect decay rates.
What is the relationship between half-life and decay constant?
They are inversely proportional: λ = ln(2) / t½ ≈ 0.6931 / t½. A shorter half-life means a larger decay constant (faster decay).