Half-Life Calculator

How to use this calculator

The calculator has four modes, each solving for a different unknown.

Find remaining amount (N) — Enter initial amount (N₀), half-life (t½), and elapsed time (t). Returns how much is left after that time.

Find elapsed time (t) — Enter initial amount, current amount, and half-life. Returns how much time has passed based on how much has decayed.

Find half-life (t½) — Enter initial amount, current amount, and elapsed time. Calculates the half-life of the substance from two measurements.

Find initial amount (N₀) — Enter current amount, half-life, and elapsed time. Works backwards to tell you how much you started with.

The units of t½ and t must match. If the half-life is in years, elapsed time must also be in years. The amount can be in any unit: grams, becquerels, atoms, or percentage remaining — the formula doesn’t care, as long as N and N₀ use the same unit.

What problem this calculator solves

Radioactive decay is one of many processes where a quantity decreases at a rate proportional to how much is left. You can’t subtract a fixed amount each period — the amount that decays each period depends on the current amount, which is shrinking. That self-referential structure produces an exponential curve, and the half-life is the most intuitive way to describe how fast that curve falls.

The same mathematical structure appears in drug clearance, population decline, capacitor discharge, and atmospheric pressure with altitude. The formula is the same in all cases. The domain changes.

The calculator converts between the four unknowns in the half-life relationship so you can answer: how much radioactive waste remains after 300 years, how old is this organic sample, or how long until a medical isotope clears the body.

The formulas

All four arrangements come from the same underlying relationship.

Both the (½)^(t/t½) form and the e^(−λt) form are mathematically identical. The half-life form is easier to interpret directly — “how many half-lives have passed?” The exponential form with λ is preferred in physics and engineering because it connects naturally to differential equations and activity calculations.

How much remains after multiple half-lives

Half-lives elapsed	Fraction remaining	Percentage remaining
0	1	100%
1	1/2	50%
2	1/4	25%
3	1/8	12.5%
4	1/16	6.25%
5	1/32	3.125%
7	1/128	0.78%
10	1/1,024	0.098%

Ten half-lives reduces any substance to under 0.1% of the original amount. This is why “10 half-lives” is the standard threshold for practical safety in both nuclear waste management and pharmaceutical clearance. After 5 half-lives, less than 3.2% of a drug remains. Most pharmacokinetic models consider a drug effectively eliminated at that mark.

Half-lives of notable isotopes

Isotope	Half-life	Decay type	Application
Carbon-14 (C-14)	5,730 years	Beta	Archaeological dating (up to ~50,000 years)
Iodine-131 (I-131)	8.02 days	Beta/gamma	Thyroid cancer treatment
Technetium-99m (Tc-99m)	6.01 hours	Gamma	SPECT imaging (bone, heart, lung scans)
Cesium-137 (Cs-137)	30.2 years	Beta/gamma	Chernobyl/Fukushima contamination monitoring
Strontium-90 (Sr-90)	28.8 years	Beta	Nuclear waste management
Radium-226 (Ra-226)	1,600 years	Alpha	Historical radiotherapy (now replaced)
Plutonium-239 (Pu-239)	24,100 years	Alpha	Nuclear weapons; 241,000 years to decay safely
Uranium-238 (U-238)	4.47 billion years	Alpha	Geological and meteorite age dating

Real-world examples

Iodine-131 in clinical use

A hospital receives 200 mg of iodine-131 (t½ = 8.02 days). How much remains after 24 days? After 40?

Carbon-14 dating

Every living organism absorbs carbon dioxide from the atmosphere. A fixed fraction of that carbon is C-14, a radioactive isotope that decays with a half-life of 5,730 years. While alive, the organism maintains the atmospheric C-14 ratio. After death, the C-14 decays and C-12 stays constant. Measuring the ratio tells you how many half-lives have passed since death.

The practical limit of carbon dating is around 50,000 years — about 8.7 half-lives — when less than 0.3% of the original C-14 remains. For older material, geologists use uranium-lead dating (U-238 half-life: 4.47 billion years) or potassium-argon (K-40 half-life: 1.25 billion years).

Drug clearance: ibuprofen

Ibuprofen has a plasma half-life of about 2 hours. Take 400 mg at 8:00 AM:

Long half-life drugs like diazepam (t½ = 20–100 hours, highly variable by individual) accumulate with repeated dosing because each new dose adds to the still-substantial amount from previous doses. After 5–7 days of daily dosing, concentrations stabilise at a plateau called steady state.

Nuclear waste: the two timelines

Spent nuclear fuel contains a spectrum of isotopes. Some are short-lived and decay to safe levels within decades. Others persist for geological time scales.

Cesium-137 (t½ = 30.2 years) requires about 300 years — 10 half-lives — to reach 0.1% of original activity. Cesium deposited by the Chernobyl accident in 1986 was still at roughly 40% of original levels in 2026.

Plutonium-239 (t½ = 24,100 years) requires 241,000 years for the same 10-half-life reduction. The engineering challenge of safely isolating this material outlasts every human institution that has ever existed.

The decay constant λ and when to use it

The decay constant λ (lambda) represents the probability that any given atom will decay per unit time.

Activity is measured in becquerels — 1 Bq = 1 decay per second. Since both activity and atom count decrease by the same factor over time, activity follows the same half-life as the mass.

The λ form is preferred when working with differential equations (dN/dt = −λN), activity measurements in Bq, or when combining multiple isotopes in decay chain calculations. The (½)^(t/t½) form is more direct for “how much remains after X time” questions.

Common mistakes

Mixing units for t and t½. If the half-life is in days and elapsed time is in hours, the exponent (t ÷ t½) is wrong by a factor of 24. Always confirm both values use the same time unit before calculating.

Expecting the substance to reach zero. Exponential decay asymptotically approaches zero. After 10 half-lives, 0.098% remains. After 20, about 9.5 × 10⁻⁵ % remains. Mathematically, it never hits zero — though practically it becomes undetectable. This is why nuclear waste doesn’t “disappear” after a fixed date; it declines continuously toward but never reaches zero.

Using the formula for decay chains. N(t) = N₀ × (½)^(t/t½) describes single-isotope decay of a pure sample. If U-238 decays to Th-234, which decays to Pa-234, and so on, each isotope in the chain has its own population dynamics. The amount of U-238 decreasing doesn’t equal the amount of Th-234 increasing unless secular equilibrium has been established. Decay chains need the Bateman equations, not a single half-life formula.

Assuming activity and mass decay at different rates. Activity (in Bq) and mass both follow the same half-life for a single isotope. One gram of C-14 and one gram of U-238 don’t have the same activity per gram — U-238’s half-life is 2.8 × 10¹³ times longer, so its activity per gram is proportionally lower — but for a given isotope, activity and mass decline at identical rates.

When to use which formula — a practical guide

Knowing which of the four calculator modes to reach for saves time. Here’s how to match the question to the mode.

You know t½ and t, want to know what’s left — use Find Remaining Amount. This is the most common mode: given an isotope’s known half-life and a known time period, how much of a sample survives?

You have two measurements of the same sample at different times — use Find Half-Life. This is how half-lives are experimentally determined. Measure activity at time zero and again at time t, then solve for t½. This mode also applies to any first-order decay process where the half-life is unknown, including chemical reactions and biological clearance.

You have a sample and know its half-life, but not when it started decaying — use Find Elapsed Time. Carbon dating works exactly this way: you know the current C-14 fraction and the half-life, so you solve for t.

You’re working backwards from a remnant to the original quantity — use Find Initial Amount. Useful in forensic and environmental contexts where you’ve measured what’s left and want to reconstruct the original release or starting mass.

The four modes are four rearrangements of the same equation. If you know any three of the four variables (N, N₀, t, t½), you can find the fourth. The calculator just picks the right rearrangement for you.

What to do with the result

For medical isotope dosing — the remaining amount tells you the current activity of your sample and whether it’s still above the therapeutic threshold. Divide the remaining mass by the original mass to get fraction remaining, then compare that to the minimum effective dose for the treatment. Below that threshold, the sample is clinically depleted even if material remains.

For archaeological dating — the elapsed time result is your age estimate. Report it with the appropriate uncertainty range, which typically widens as age increases (less C-14 remaining means measurement error has a proportionally larger effect). Cross-reference with stratigraphy or other dating methods when possible.

For pharmaceutical clearance — 5 half-lives is the conventional threshold for “effectively eliminated.” Use Find Elapsed Time and set N = N₀ × 0.03125 (i.e., 3.125% = 1/32 remaining) to find when a drug crosses that threshold. For drugs with variable half-lives (those metabolised by CYP450 enzymes that are subject to genetic variation), treat the result as a population average rather than a precise individual prediction.

For nuclear waste and contamination monitoring — the remaining amount after a given time tells you what fraction of the original release is still active. For Chernobyl Cs-137 (t½ = 30.2 years), entering 1986 to 2026 as elapsed time = 40 years gives (0.5)^(40/30.2) = 40.4% remaining. That aligns with field measurements in the most contaminated zones.

Things the calculator can’t do — and what to use instead

Decay chains. Real radioactive materials decay into other radioactive isotopes. U-238 produces 13 intermediate isotopes before reaching stable Pb-206. The single half-life formula handles only one isotope at a time. For full chain modelling, use the Bateman equations or dedicated nuclear decay software like ORIGEN or Radiological Toolbox from ORNL.

Variable biological half-lives. The pharmaceutical half-life of a drug isn’t always a fixed number. Drugs metabolised by CYP450 liver enzymes are affected by genetic polymorphisms, other drugs (enzyme inhibitors or inducers), liver function, and body composition. The half-life calculator gives you a population-average result for a given t½ input. Individual clearance can differ by a factor of 2–5 for drugs with known pharmacogenomic variability (warfarin, codeine, clopidogrel).

Non-first-order decay. Some radioactive decay processes at extreme densities or in exotic matter don’t follow the standard exponential model. In neutron star interiors or during supernovae, electron capture rates change dramatically under extreme pressure, effectively altering decay rates. For any terrestrial or medical application, the standard formula holds perfectly. For astrophysics, it may not.

Activity in becquerels from mass alone. The calculator works with fractional amounts — how much remains relative to how much you started with. To convert mass to activity in Bq, you need the atomic mass and Avogadro’s number to get the atom count, then multiply by λ. The decay constant section above walks through this for Cs-137.

The bottom line

One formula. Four unknowns. Whichever three you know, the calculator finds the fourth.

The half-life is the variable that defines the substance — it’s a physical constant for each isotope, unchanged by how much you have, what temperature it’s at, or what chemical compound it’s in. That constancy is what makes the formula reliable across such wildly different applications: a 5,730-year archaeological clock and a 6-hour medical isotope and a 2-hour drug clearance curve all obey identical mathematics.

Match your time units. Confirm your formula covers a single isotope. Use the result to drive the next decision — whether that’s scheduling isotope delivery, reporting a radiocarbon date, or calculating a redosing interval.