Half-life is the standard way to describe the speed of any first-order decay process. It applies identically to radioactive isotopes, drugs in the bloodstream, fluorescent intensity and even some chemical reactions. The decay equation N(t) = N₀ × (1/2)^(t / t½) only needs the half-life and starting amount. The same shrink-by-a-fixed-proportion pattern is covered in more general form by the exponential decay calculator, which uses a percent rate instead of a half-life period.
Half-life formula
t½ = ln(2) / λ ≈ 0.693 / λ, where λ is the decay constant. Equivalently λ = 0.693 / t½. After n half-lives the remaining fraction is (1/2)ⁿ: 50%, 25%, 12.5%, 6.25%, 3.125%.
Radiocarbon dating
Living matter maintains a fixed ratio of carbon-14 to carbon-12. After death the carbon-14 decays with a 5,730-year half-life. Measuring the remaining ratio and applying t = -5730 × log₂(fraction) gives the sample age, the basis of every published radiocarbon date up to about 50,000 years. A sample from roughly 20,000 years ago retains about 8.9% of its original carbon-14, since (1/2)^(20000/5730) ≈ 0.089, a small enough fraction that dating gets progressively less precise the further back it reaches.
Drug elimination and dosing
Caffeine half-life is roughly 5 hours, ibuprofen 2 hours, fluoxetine 1–3 days, amiodarone 25–60 days. The half-life sets the dosing interval and how long it takes to reach steady state. Both reaching and clearing steady-state plasma concentrations take about 5 half-lives. Fluoxetine is a special case: its active metabolite, norfluoxetine, has its own half-life of 4 to 16 days, well beyond fluoxetine's own 1 to 3 days. Because of this, doctors taper the dose slowly rather than stopping it outright.
Worked example
A 400 mg dose of ibuprofen (t½ = 2 hours) leaves 400 × (1/2)^(8/2) = 400 × 1/16 = 25 mg after 8 hours.
Half-life vs mean lifetime
Half-life is not the same number as mean lifetime (τ), the average time a single atom or molecule survives before decaying. The two relate by τ = t½ / ln(2) ≈ 1.4427 × t½. Caffeine, with a 5-hour half-life, has a mean lifetime of about 5 / 0.693 ≈ 7.2 hours, longer than the half-life because a few molecules linger well past the point where half the batch has already cleared. Mean lifetime is the number that shows up directly inside the exponential decay formula N(t) = N₀ × e^(−t/τ).
Effective half-life for medical imaging
Radioactive tracers used in nuclear medicine clear the body through two routes at once: physical decay of the isotope itself and biological elimination through the kidneys and liver. The effective half-life combines both using 1/t_eff = 1/t_bio + 1/t_phys, and it is always shorter than either route alone. Technetium-99m, the most widely used imaging isotope, has a 6-hour physical half-life and roughly a 24-hour biological half-life from typical renal clearance. That gives an effective half-life of (6 × 24) / (6 + 24) = 144 / 30 = 4.8 hours, the figure a physicist actually uses when timing a scan and estimating patient radiation dose. Anyone solving for a decay constant from raw measurements rather than a textbook value can use the decay rate calculator to back out λ from two data points.
Common half-lives
Half-lives span an enormous range, from hours to billions of years, so the same formula describes a dose of caffeine and the age of a meteorite.
| Substance | Half-life |
|---|---|
| Technetium-99m (medical imaging) | 6 hours |
| Caffeine (adult metabolism) | ~5 hours |
| Iodine-131 (thyroid treatment) | 8 days |
| Cobalt-60 | 5.27 years |
| Carbon-14 | 5,730 years |
| Uranium-238 | 4.47 billion years |
Because uranium-238 has a half-life of 4.47 billion years, close to the age of the Earth itself, only about half of the uranium-238 present when the planet formed has decayed so far. That slow, steady clock is exactly why it remains useful for dating the oldest rocks and meteorites on the planet, far outside the range where carbon-14 dating still works.
FAQ
What is half-life?
Half-life is the time required for half of a quantity to decay or disappear, and it stays constant no matter how much material remains. A substance with a 6-hour half-life falls to 50% of its starting amount in 6 hours, 25% in 12 hours, 12.5% in 18 hours, and 6.25% in 24 hours.
What is the half-life formula?
The half-life formula is t½ = ln(2) / λ, approximately 0.693 divided by the decay constant λ. Rearranged, λ = 0.693 / t½, and the amount remaining at any time t is N(t) = N₀ × (1/2)^(t / t½). A substance with λ = 0.1155 per hour, for example, has a half-life of 0.693 / 0.1155, or 6 hours.
How is half-life used in radiocarbon dating?
Radiocarbon dating uses the 5,730-year half-life of carbon-14 to estimate the age of organic material. Measuring the fraction of carbon-14 remaining in a sample and solving t = -5730 × log₂(fraction) returns the age. A sample that retains 25% of its original carbon-14 is about 11,460 years old, exactly two half-lives, since 25% equals (1/2) squared.
How does half-life apply to drugs?
Drug half-life determines how often a dose must be repeated to keep blood levels steady. Steady-state plasma concentration is reached after about 5 half-lives, and the drug is considered fully cleared after another 5 half-lives once dosing stops. Ibuprofen, with a 2-hour half-life, reaches steady state in about 10 hours; amiodarone, with a 40-day half-life, takes roughly 200 days.
What is biological vs effective half-life?
Biological half-life measures elimination by the body alone, while effective half-life combines biological elimination with physical radioactive decay for tracers used in medical imaging. Effective half-life is always shorter than either component, calculated as 1/t_eff = 1/t_bio + 1/t_phys. Technetium-99m, with a 6-hour physical half-life and a 24-hour biological half-life, has an effective half-life of about 4.8 hours.
Is half-life constant?
Yes, for first-order processes such as radioactive decay and most drug elimination, half-life stays constant regardless of the starting amount. For zero-order or saturable processes, like alcohol metabolism above roughly 0.02% blood alcohol concentration, the elimination rate is fixed per hour rather than proportional to what remains, so half-life shortens as concentration falls.