What is half-life?
Half-life is the time it takes for half of a substance to decay, react, or otherwise disappear. The idea was first measured for radium by Ernest Rutherford in 1907, and the math turned out to apply to far more than radioactive atoms — it describes drug clearance from your bloodstream, the cooling of a hot object, capacitor discharge, and any process where the rate of change is proportional to the amount currently present.
The key feature: the half-life is constant regardless of how much you start with. Begin with 1 kg of carbon-14, wait 5,730 years, you have 500 g. Begin with 1 g of carbon-14, wait 5,730 years, you have 500 mg. The ratio halves on the same schedule whether you started with a mountain or a milligram. That property is what makes half-life such a clean handle for messy real-world decay.
For example: a wood beam from an old building tests at half the carbon-14 concentration of a fresh sample. Carbon-14's half-life is 5,730 years, so the wood is roughly one half-life old. The beam was cut around 5,730 years ago. That is the entire logic of radiocarbon dating, compressed into one sentence.
How to use the Half-Life Calculator
The calculator has three solve modes, depending on which variable you don't know. Pick the mode, type the two values you have, and read the result. There's no Calculate button — the answer updates as you type.
- Find remaining amount. You know the starting amount N₀, the half-life T, and how much time has passed t. The calculator returns N(t) plus the percentage left.
- Find half-life. You measured the amount at two times and want to know the decay constant. Useful when working from lab data instead of a textbook value.
- Find elapsed time. You know the starting and current amounts plus the half-life, and you want to know how long the decay has been running. This is the radiocarbon-dating mode.
Units don't matter as long as you stay consistent. Half-life in years and time in years. Half-life in hours and time in hours. The math is a ratio, so grams or becquerels or milligrams or percent all work for the amounts.
The formula behind half-life
The canonical exponential decay equation:
N(t) = N₀ × (½)^(t/T)
N₀ is the starting amount. T is the half-life. t is elapsed time. N(t) is what's left at time t. Plug in t = T and you get N₀/2 — half remaining, as advertised. Plug in t = 2T and you get N₀/4. Plug in t = 3T, you get N₀/8. Every half-life that passes, you multiply by ½.
The same equation in exponential form, which is what physics textbooks tend to use:
N(t) = N₀ × e^(−λt), where λ = ln(2)/T ≈ 0.693/T
λ (lambda) is the decay constant — the probability per unit time that any single atom decays. The two forms give identical results; they just encode the same physics in different language. Nuclear physicists report λ; chemistry textbooks report T; both end up at the same answer.
Worked example: radiocarbon dating a wood sample. Suppose the sample contains 50% of the carbon-14 a fresh piece of wood would have. Carbon-14's half-life is 5,730 years.
- Set N(t)/N₀ = 0.5
- 0.5 = (½)^(t/5730)
- Take log of both sides: log(0.5) = (t/5730) × log(0.5)
- Therefore t/5730 = 1, so t = 5,730 years
One half-life. The sample is about 5,730 years old. That's the simplest possible case — when exactly half remains, you're exactly one half-life in. Real radiocarbon dating involves fancier corrections (atmospheric carbon-14 hasn't been constant over time), but the engine is this equation.
Common isotopes and their half-lives
Half-lives across the periodic table span 30+ orders of magnitude — from microseconds (some superheavy elements) to billions of years (uranium-238). Here are the isotopes that come up most often in homework, medical settings, and news stories about reactors and weapons.
| Isotope | Half-life | Why you might encounter it |
|---|---|---|
| Technetium-99m | 6.01 hours | The most common medical imaging isotope — used in millions of scans per year |
| Iodine-131 | 8.02 days | Thyroid imaging and treatment; also a Chernobyl/Fukushima fission product |
| Cobalt-60 | 5.27 years | External-beam radiation therapy; industrial radiography |
| Cesium-137 | 30.2 years | Major long-term contaminant from fission accidents |
| Plutonium-239 | 24,110 years | Weapons-grade plutonium; fuel in some reactors |
| Carbon-14 | 5,730 years | Radiocarbon dating of organic material up to about 50,000 years old |
| Uranium-235 | 703.8 million years | The fissile isotope used in reactors and weapons; 0.72% of natural uranium |
| Uranium-238 | 4.47 billion years | The most common uranium isotope; used for uranium-lead geological dating |
Notice the spread: technetium-99m halves in six hours; uranium-238 halves on roughly the timescale of the Earth's age. The same equation handles both — only T changes.
Drug half-life: the same math, different language
The reason your pharmacist talks about half-life is that drug concentrations in blood decay exponentially once absorption is complete. After a single dose, the concentration peaks, then declines according to N(t) = N₀ × (½)^(t/T) where T is now the biological half-life — set by your liver, kidneys, and metabolism rather than by nuclear physics.
A few common drugs and their typical half-lives in healthy adults:
- Caffeine — about 5 hours. A 200 mg coffee at 8 AM leaves about 100 mg in your system at 1 PM and 50 mg at 6 PM. Why caffeine after lunch can wreck your sleep.
- Aspirin — about 3 hours. Quick to clear, which is why you take it multiple times a day for pain.
- Warfarin (blood thinner) — about 36 hours. Long half-life is why dosing adjustments take days to show effects.
- Diazepam (Valium) — about 30 hours, with active metabolites that last days longer. Why benzodiazepines accumulate with repeated dosing.
The clinical rule of thumb: after 5 half-lives, about 97% of the drug has cleared. That's the standard answer to "how long until this is out of my system?" For caffeine that's 25 hours — about a day. For warfarin it's a week. The same exponential, just plugged into different organs.
Half-life vs. mean lifetime — the counterintuitive bit
Half-life T and mean lifetime τ (tau) are related but not equal. The mean lifetime is the average time a single atom survives before decay, and it works out to:
τ = 1/λ = T/ln(2) ≈ 1.443 × T
So the average atom lives longer than the half-life — about 44% longer. That feels backwards at first. If half the atoms decay before time T, shouldn't the average be exactly T?
No, because the other half take longer than T, and a long tail of late-decaying atoms drags the mean upward. The half-life is the median lifetime; the mean lifetime is pulled toward the slow tail. In statistics terms: for a heavily right-skewed distribution like exponential decay, the median is shorter than the mean. Half-life is the median. τ is the mean.
If you'd like to see how exponentials behave more generally, the Exponent Calculator handles arbitrary bases and powers; this calculator is the half-life-specialized version.
Edge cases and things to watch for
A few traps that catch people running these calculations for the first time.
- Mixed units. If your half-life is in years and your elapsed time is in days, the result is nonsense. Convert one to match the other before plugging in. The calculator can't tell whether your "10" is days or years.
- The first-order assumption. Pure half-life math assumes the decay rate depends only on the amount currently present. Radioactive decay obeys this exactly. Drug pharmacokinetics obeys it approximately, after the absorption phase. Chemical reactions obey it only for first-order kinetics — many real reactions are second-order or more complex and don't have a true constant half-life.
- Biological vs. physical half-life for radioactive drugs. If you take a radioactive tracer like iodine-131, you have two clocks running: physical decay (8.02 days for I-131) and biological excretion (your kidneys/thyroid). The effective half-life is faster than either alone: 1/T_eff = 1/T_phys + 1/T_bio. Whichever process is faster dominates.
- Tiny remaining fractions and detection limits. The exponential never reaches zero. After 10 half-lives, 0.098% remains. After 20, about one millionth. Mathematically there's still something there; practically your detector hits its noise floor and reads zero.
- Decay chains. Some isotopes decay into other unstable isotopes (uranium-238 → thorium-234 → protactinium-234 → … → lead-206, 14 steps). The Half-Life Calculator handles one isotope at a time. For decay chains, you'd apply the equation step by step or look up a chain-specific result.
The Half-Life Calculator is a screening tool for clean cases — single isotope, single drug, well-mixed system. That covers the overwhelming majority of homework, lab work, and back-of-envelope estimation. Anything more elaborate (multi-compartment pharmacokinetics, secular equilibrium in a decay chain) is a research problem with its own software.
Related calculations
Half-life is the most famous member of the exponential-decay family, but it's not the only one. Some related tools:
- Exponent Calculator — raises any base to any power. The Half-Life Calculator is a specialized exponent calculator with base ½; the general one lets you plug in arbitrary growth or decay rates.
- Log Calculator — solves for the exponent when you know the result. Inverting the half-life equation to "how long until X% remains?" is a logarithm problem.
- Percentage Calculator — handy when your half-life problem ends with "what percentage is left?" or "find the time when 12.5% remains."
- Compound Interest Calculator — the same exponential math running in the opposite direction. Half-life is decay; compound interest is growth.
Frequently asked questions
What's the formula again?
N(t) = N₀ × (½)^(t/T). N₀ is the initial amount, T is the half-life, t is elapsed time, N(t) is what's left. After one half-life, half remains. After two, a quarter. After n half-lives, (½)^n of the original. The exponential form N(t) = N₀ × e^(−λt) with λ = ln(2)/T gives the same numbers.
What's the decay constant λ?
λ = ln(2)/T ≈ 0.693/T. It's the probability per unit time that any given atom decays. Big λ means fast decay; small λ means slow decay. λ and T encode the same information — physicists and engineers tend to use λ; chemistry and biology stick with T.
Why is mean lifetime longer than half-life?
τ = T/ln(2) ≈ 1.443 × T, so the average atom survives about 44% longer than the half-life suggests. The reason: exponential distributions are right-skewed. Half the atoms decay before T, but the other half have a long tail of late survivors that pulls the average above the median. Half-life is the median; τ is the mean.
Does temperature affect radioactive half-life?
No, with vanishingly small exceptions. Radioactive decay rates are set by nuclear physics, not chemistry — temperature, pressure, and chemical environment have essentially no effect. The rare exceptions (electron-capture decays whose rate changes by parts per thousand under extreme chemistry) are research curiosities, not anything you'd see in a homework problem.
How does this apply to drug half-life?
Same exponential, different organ. After a single dose, blood concentration declines by N(t) = N₀ × (½)^(t/T) where T is the biological half-life. Caffeine T is about 5 hours; aspirin about 3; warfarin about 36. After 5 biological half-lives, about 97% has cleared — the medical rule for "out of your system." Real pharmacokinetics has an absorption phase too, but once the drug is fully absorbed, the decline follows this curve.
What's the difference between physical and biological half-life?
Physical half-life is set by nuclear physics and never changes for a given isotope. Biological half-life is set by your body's metabolism — kidneys, liver, etc. — and depends on health, age, and drug interactions. For a radioactive drug, both run simultaneously, and the effective half-life is shorter than either: 1/T_eff = 1/T_phys + 1/T_bio.
How do I solve for time from a percent-remaining?
Set N(t)/N₀ to the target fraction, then rearrange: t = T × log(N/N₀) / log(½). For 50% remaining: t = T. For 25%: t = 2T. For 10%: t = T × log(0.1)/log(0.5) ≈ 3.32T. For 1%: t ≈ 6.64T. Set the calculator to "find elapsed time" mode and it does this for you.