Decay Rate & Remaining Quantity
Last reviewed: May 2026
Calculate exponential decay for radioactive isotopes, drug metabolism, chemical reactions, and any process where quantity decreases by a fixed fraction over equal time periods. The half-life formula N = N₀ × (1/2)^(t/t½) is one of the most elegant in science — it applies identically whether you are tracking uranium atoms over billions of years or caffeine molecules over hours.1
| Half-Lives | % Remaining | % Decayed |
|---|---|---|
| 1 | 50% | 50% |
| 2 | 25% | 75% |
| 3 | 12.5% | 87.5% |
| 5 | 3.13% | 96.87% |
| 7 | 0.78% | 99.22% |
| 10 | 0.098% | 99.9% |
| Substance | Half-Life | Field |
|---|---|---|
| Caffeine | 5–6 hours | Pharmacology |
| Ibuprofen | 2–4 hours | Pharmacology |
| Carbon-14 | 5,730 years | Archaeology |
| Uranium-238 | 4.5 billion years | Geology |
| Plutonium-239 | 24,100 years | Nuclear physics |
Half-life is the time required for a quantity to decrease to half its initial value through an exponential decay process. The concept applies across vastly different domains: radioactive atoms decaying into stable elements, medications clearing from the bloodstream, hot coffee cooling toward room temperature, and even the depreciation of trending topics on social media. The mathematical foundation is the same in every case: N(t) = N₀ × (1/2)^(t/t½), where N₀ is the initial quantity, t is elapsed time, and t½ is the half-life. After one half-life, 50% remains. After two half-lives, 25%. After three, 12.5%. After ten half-lives, less than 0.1% of the original quantity persists. This calculator computes remaining quantities, elapsed time, or the half-life itself from any two known values.
| Isotope | Half-Life | Primary Use | Decay Type |
|---|---|---|---|
| Carbon-14 | 5,730 years | Archaeological dating | Beta decay |
| Uranium-238 | 4.47 billion years | Geological dating, nuclear fuel | Alpha decay |
| Iodine-131 | 8.02 days | Thyroid treatment/imaging | Beta decay |
| Technetium-99m | 6.01 hours | Medical imaging (most common) | Gamma decay |
| Radon-222 | 3.82 days | None (health hazard in homes) | Alpha decay |
| Cobalt-60 | 5.27 years | Radiation therapy, sterilization | Beta/gamma decay |
| Plutonium-239 | 24,110 years | Nuclear weapons, fuel | Alpha decay |
| Potassium-40 | 1.25 billion years | Geological dating | Beta/electron capture |
The enormous range of half-lives — from microseconds to billions of years — demonstrates why different isotopes serve different purposes. Medical isotopes need short half-lives (hours to days) to minimize patient radiation exposure while providing diagnostic information. Dating isotopes need half-lives comparable to the timescales being measured: carbon-14's 5,730-year half-life is perfect for archaeological artifacts up to ~50,000 years old, while uranium-238's 4.47-billion-year half-life dates rocks and meteorites spanning Earth's entire geological history.
Carbon-14 dating works because living organisms continuously absorb carbon-14 from the atmosphere through eating and breathing, maintaining a constant ratio of C-14 to C-12. When the organism dies, C-14 intake stops and the existing C-14 decays with a half-life of 5,730 years. By measuring the remaining C-14 fraction, scientists calculate the time since death. A sample with 50% of the expected C-14 died approximately 5,730 years ago. A sample with 25% died about 11,460 years ago (two half-lives). The practical limit is roughly 50,000 years (about 9 half-lives), after which so little C-14 remains that measurement becomes unreliable. Calibration curves account for historical variations in atmospheric C-14 levels, which fluctuated due to solar activity, volcanic eruptions, and ocean circulation changes.
| Medication | Half-Life | Dosing Frequency | Time to Steady State |
|---|---|---|---|
| Ibuprofen | 2–4 hours | Every 4–6 hours | ~12–20 hours |
| Acetaminophen | 2–3 hours | Every 4–6 hours | ~10–15 hours |
| Amoxicillin | 1–1.5 hours | Every 8 hours | ~5–8 hours |
| Metformin | 4–8.7 hours | Twice daily | ~24–44 hours |
| Fluoxetine (Prozac) | 1–3 days | Once daily | ~1–3 weeks |
| Diazepam (Valium) | 20–100 hours | As prescribed | ~4–20 days |
Drug half-life determines dosing schedule and how long effects persist after the last dose. Medications reach steady-state concentration (where intake equals elimination) after approximately 4–5 half-lives. Ibuprofen with a 3-hour half-life reaches steady state within about 15 hours. Fluoxetine with a 2-day half-life takes roughly 10 days. This is why antidepressants take weeks to reach full effectiveness and why doctors warn against abrupt discontinuation of long-half-life medications — the drug takes many days to fully clear, and withdrawal symptoms may appear days after the last dose rather than immediately.
The half-life concept applies to any process following exponential decay. Beer foam dissipates exponentially with a half-life of about 2–3 minutes. Sound reverberation in a room decays exponentially — the RT60 measurement (time for sound to decrease by 60 dB) is essentially a "1000th-life." Atmospheric carbon dioxide following a pulse emission decays with a complex multi-component half-life: about 50% is absorbed within 30 years, but 20% persists for thousands of years. Population decline in abandoned towns, battery discharge in electronics, and even the fading of memories in cognitive science follow approximately exponential decay patterns with characteristic half-lives.
| Half-Lives Elapsed | Fraction Remaining | Percentage Remaining | Practical Example (1000 atoms) |
|---|---|---|---|
| 0 | 1 | 100% | 1,000 atoms |
| 1 | 1/2 | 50% | 500 atoms |
| 2 | 1/4 | 25% | 250 atoms |
| 3 | 1/8 | 12.5% | 125 atoms |
| 5 | 1/32 | 3.125% | 31 atoms |
| 7 | 1/128 | 0.78% | ~8 atoms |
| 10 | 1/1024 | 0.098% | ~1 atom |
The rapid exponential decline means that after just 7 half-lives, less than 1% remains, and after 10 half-lives, less than 0.1%. This has practical implications: nuclear waste with a 30-year half-life needs roughly 300 years (10 half-lives) to decay to 0.1% of its original radioactivity. Medical isotopes with 6-hour half-lives become safe within about 3 days (12 half-lives). Understanding this decay rate is essential for radiation safety protocols, environmental cleanup planning, and pharmaceutical dosing. Use our Exponent Calculator for the underlying exponential computations.
The decay constant (λ) and half-life (t½) are inversely related by the equation λ = ln(2) / t½ ≈ 0.693 / t½. The decay constant represents the probability of decay per unit time for any individual atom. For carbon-14 with t½ = 5,730 years, λ = 0.000121 per year — meaning each C-14 atom has a 0.012% chance of decaying in any given year. This probabilistic nature means decay is inherently random at the atomic level but follows precise mathematical patterns in large populations. A single atom of C-14 might decay in the next second or persist for 20,000 years — it is impossible to predict. But among a trillion C-14 atoms, the number remaining after 5,730 years will be almost exactly 500 billion, with negligible statistical variation.
Nuclear waste management is fundamentally a half-life problem. Spent nuclear fuel contains isotopes with vastly different half-lives: iodine-131 (8 days, decays to safe levels within months), strontium-90 (29 years, needs centuries of containment), and plutonium-239 (24,110 years, requires geological-timescale isolation). High-level waste must be stored until radioactivity drops below background levels — approximately 10 half-lives for the dominant isotope. For plutonium-239, that means roughly 240,000 years. This is why deep geological repositories like Finland's Onkalo facility are designed to isolate waste for 100,000+ years using multiple containment barriers. The challenge is not engineering — we can build durable containers — but ensuring institutional knowledge and warning systems persist across timescales that exceed recorded human civilization.
Environmental contaminants decay at rates characterized by their half-lives. Pesticide half-lives in soil range from days (malathion, 1–8 days) to years (DDT, 2–15 years), determining how long contamination persists after application. PFAS ("forever chemicals") have half-lives of years to decades in the human body and essentially do not degrade in the environment, which is why they are classified as persistent organic pollutants. Oil spill components degrade at different rates: volatile compounds (benzene, toluene) evaporate within days, while heavier hydrocarbons can persist in sediments for decades. Atmospheric greenhouse gases also have characteristic lifetimes: methane's atmospheric half-life is about 9 years, while nitrous oxide persists for over a century. These differences directly inform climate policy — methane reductions produce faster climate benefits than CO₂ reductions, though CO₂ accumulation causes larger long-term warming.
For radioactive substances in the body, two different half-lives operate simultaneously. The physical half-life is the time for radioactive decay regardless of location. The biological half-life is the time for the body to eliminate half of the substance through metabolism and excretion. The effective half-life combines both: 1/t_eff = 1/t_phys + 1/t_bio. Iodine-131 has an 8-day physical half-life and roughly a 120-day biological half-life in the thyroid — the effective half-life is about 7.5 days, dominated by the shorter physical decay. Cesium-137 has a 30-year physical half-life but only a 70-day biological half-life — your body eliminates it long before it decays, giving an effective half-life of about 70 days. This distinction is critical for radiation safety: the effective half-life determines actual dose exposure, which may be dramatically shorter than the physical half-life would suggest.
The half-life concept extends to financial contexts. In mean-reversion analysis, the half-life measures how quickly an asset price returns halfway to its long-term average after a shock. A stock with a mean-reversion half-life of 10 trading days corrects pricing anomalies quickly, limiting arbitrage opportunities. One with a 200-day half-life reverts slowly, potentially offering longer-duration trading strategies. Depreciation of assets follows a similar exponential pattern — a car losing 50% of its value every 5 years has a "value half-life" of 5 years, with the expected value curve matching the standard decay formula. Even viral content on social media follows half-life decay patterns: a trending tweet typically loses half its engagement rate every 18–24 minutes, while a YouTube video's daily view count decays with a half-life measured in weeks to months depending on content type.
→ 7 half-lives ≈ gone. Less than 1% remains.
→ Caffeine tip: A noon coffee still has 25% caffeine at midnight (2 half-lives).
→ Drugs and dosing: Steady state reached after ~5 half-lives of regular dosing.
→ Radiocarbon limit: C-14 dating only works to ~50,000 years (~9 half-lives).
See also: Scientific · Percentage · BAC Time · Density