Half-life is the time it takes for half of a substance to decay, break down, or be eliminated. It is one of the most versatile concepts in science, appearing in nuclear physics (radioactive decay), medicine (drug metabolism), chemistry (reaction rates), archaeology (carbon dating), and even environmental science (pollutant breakdown). Once you understand the principle, the math is the same regardless of the application.
Start with 100 units of any substance with a half-life of 1 hour. After each hour, exactly half remains:
| Time Elapsed | Half-Lives | Amount Remaining | Percentage Left |
|---|---|---|---|
| 0 hours | 0 | 100 | 100% |
| 1 hour | 1 | 50 | 50% |
| 2 hours | 2 | 25 | 25% |
| 3 hours | 3 | 12.5 | 12.5% |
| 4 hours | 4 | 6.25 | 6.25% |
| 5 hours | 5 | 3.125 | 3.125% |
| 10 hours | 10 | 0.098 | ~0.1% |
The formula for any half-life calculation is: N(t) = N₀ × (1/2)t/t½, where N₀ is the starting amount, t is elapsed time, and t½ is the half-life. Use the Half-Life Calculator to compute remaining quantities for any substance.
Key insight: A substance never fully disappears through half-life decay. After 7 half-lives, about 0.78% remains. After 10 half-lives, about 0.1% remains. In practice, we consider a substance effectively gone after 7–10 half-lives, which is why this rule guides drug clearance protocols and radiation safety timelines.
Radioactive isotopes have unstable nuclei that spontaneously decay, emitting radiation in the process. Each isotope has a characteristic half-life that never changes, regardless of temperature, pressure, or chemical state. This constancy makes radioactive half-lives nature’s most reliable clock.
| Isotope | Half-Life | Primary Use |
|---|---|---|
| Fluorine-18 | 110 minutes | PET scans (medical imaging) |
| Iodine-131 | 8 days | Thyroid treatment and imaging |
| Cobalt-60 | 5.27 years | Radiation therapy, food sterilization |
| Cesium-137 | 30.2 years | Nuclear waste concern, industrial gauges |
| Carbon-14 | 5,730 years | Archaeological dating |
| Uranium-238 | 4.47 billion years | Geological dating, nuclear fuel |
All living organisms absorb carbon-14 from the atmosphere. When they die, the carbon-14 begins decaying with a half-life of 5,730 years. By measuring the ratio of carbon-14 to carbon-12 in an archaeological sample and comparing it to living tissue, scientists can determine when the organism died. A sample with half the expected carbon-14 died about 5,730 years ago. A sample with one-quarter the expected amount died about 11,460 years ago (two half-lives). This technique reliably dates organic materials up to about 50,000 years old (roughly 8–9 half-lives, after which too little carbon-14 remains to measure accurately). Use the Radioactive Decay Calculator for detailed decay calculations.
When you take medication, your body begins eliminating it through metabolism and excretion. The drug’s half-life determines how long it remains active and how frequently you need to take it.
| Medication | Half-Life | Typical Dosing |
|---|---|---|
| Ibuprofen | 2–4 hours | Every 4–6 hours |
| Aspirin | 3–5 hours | Every 4–6 hours |
| Caffeine | 5–6 hours | Morning dosing preferred |
| Amoxicillin | 1–1.5 hours | Every 8 hours |
| Metformin | 6 hours | Once or twice daily |
| Fluoxetine (Prozac) | 1–6 days | Once daily |
Drug half-lives vary by individual factors including age, liver function, kidney function, and genetic metabolism differences. These are approximate population averages.
Caffeine’s 5–6 hour half-life explains why a cup of coffee at 3 PM can keep you awake at 11 PM. A 200mg dose at 3 PM leaves about 100mg at 8–9 PM and 50mg at 1–2 AM — still enough to disrupt sleep for many people. Understanding this half-life helps optimize caffeine timing for alertness without insomnia.
When you take medication on a regular schedule, the drug accumulates in your body until it reaches “steady state” — the point where the amount being absorbed equals the amount being eliminated. Steady state occurs after approximately 5 half-lives of consistent dosing. A drug with a 12-hour half-life reaches steady state in about 60 hours (2.5 days). A drug with a 4-day half-life (like fluoxetine’s active metabolite) takes about 20 days. This is why some medications take weeks to reach full effectiveness.
Pollutants, pesticides, and chemical contaminants also have half-lives in the environment, though these are influenced by conditions like temperature, sunlight, soil type, and microbial activity. DDT has an environmental half-life of 2–15 years in soil, which is why it persists decades after being banned. Glyphosate has a much shorter soil half-life of 2–197 days depending on conditions. Understanding these half-lives informs environmental cleanup timelines and agricultural practices.
Half-life decay is exponential, not linear. This distinction matters enormously. In linear decay, a substance loses the same absolute amount each period (like draining a pool at a constant rate). In exponential decay, a substance loses the same fraction each period. The practical consequence: exponential decay starts fast and slows dramatically. In the first half-life, you lose 50% of the total. But in the fifth half-life, you only lose 3.125% of the original amount. The early losses dwarf the late ones.
If you know the starting and ending amounts and the elapsed time, you can calculate the half-life: t½ = t × ln(2) ÷ ln(N₀ ÷ N). If 1,000 grams decays to 250 grams in 10 days: t½ = 10 × 0.693 ÷ ln(4) = 10 × 0.693 ÷ 1.386 = 5 days. Conversely, if you know the half-life and want the decay constant: λ = ln(2) ÷ t½ = 0.693 ÷ t½.
Calculate decay instantly. Use the free Half-Life Calculator for any substance, the Radioactive Decay Calculator for isotope-specific calculations, and the Logarithm Calculator for the underlying math — no signup required.
Related tools: Half-Life Calculator · Radioactive Decay Calculator · Logarithm Calculator · Exponent Calculator · Scientific Notation Calculator