Calculate half-life, remaining quantity, initial amount, elapsed time, or the time to reach any target level for radioactive isotopes and drugs.
Half Life Formula
Every mode of the calculator is a rearrangement of the same exponential decay equation. The remaining quantity after a given time is:
Solving that equation for the half-life gives:
Solving for the elapsed time gives:
Solving for the initial quantity gives:
The time needed to fall to a chosen percentage P of the starting amount is:
The decay constant and the mean lifetime come from the half-life directly:
- N = quantity remaining after time t (grams, atoms, mg, counts per minute, or percent)
- N0 = initial quantity, in the same unit as N
- t = elapsed time
- T = half-life, the time for the quantity to fall by half
- P = target percentage of the original amount still present
- lambda = decay constant, the fraction decaying per unit time
- tau = mean lifetime, the average survival time of one particle
- ln = natural logarithm
The five solve-for modes map onto those formulas. Remaining quantity uses the first equation and needs the starting amount, the half-life, and the elapsed time. Half-life uses the second equation and needs two measurements of the same sample plus the time between them. Elapsed time uses the third equation and is the mode used for radiometric dating, since it converts a measured ratio N/N0 into an age. Initial quantity uses the fourth equation and works backward from what is left today. Time to reach a target level uses the fifth equation and answers the practical question of how long until only 10 percent, 1 percent, or some specific amount is left. Only the ratio N0/N matters in every mode, so you can enter grams, atoms, or activity readings as long as both quantities use the same unit.
Decay Progress and Practical Clearance Times
The first table converts half-lives elapsed into how much is left. Because the loss is proportional rather than fixed, the amount never reaches exactly zero, but it drops below any measurable threshold after enough half-lives.
| Half-lives elapsed | Fraction left | Percent left | Percent decayed | Common interpretation |
|---|---|---|---|---|
| 1 | 1/2 | 50% | 50% | Peak effect gone for most drugs |
| 2 | 1/4 | 25% | 75% | Typical textbook dating example |
| 3 | 1/8 | 12.5% | 87.5% | Roughly one working shift for a 6 hour tracer |
| 4 | 1/16 | 6.25% | 93.75% | Approaching steady state on repeat dosing |
| 5 | 1/32 | 3.125% | 96.875% | Standard pharmacology rule for “cleared” |
| 6.64 | 1/100 | 1% | 99% | Common radiation safety benchmark |
| 7 | 1/128 | 0.78% | 99.22% | Below most routine detection limits |
| 9.97 | 1/1000 | 0.1% | 99.9% | Effectively removed for practical purposes |
| 20 | 1/1,048,576 | 0.000095% | >99.9999% | Upper limit of most measurement techniques |
The second table pairs published half-lives with two numbers that decide how the value is actually used: the time to drop below 1 percent of the starting amount, which is 6.64 half-lives, and the working range over which the isotope or substance is useful. An isotope is only useful for dating while enough of it survives to measure, which is why carbon-14 cannot date a fossil older than roughly 50,000 years even though the math will happily return an answer.
| Substance | Half-life | Time to below 1% left | Practical working range or use |
|---|---|---|---|
| Fluorine-18 | 109.8 minutes | 12.2 hours | PET tracer, must be used within hours of production |
| Technetium-99m | 6.01 hours | 1.7 days | Imaging scans same day, dose gone in about two days |
| Radon-222 | 3.82 days | 25.4 days | Indoor air testing over 2 to 7 day windows |
| Iodine-131 | 8.02 days | 53 days | Thyroid therapy, contact precautions for days after |
| Cobalt-60 | 5.27 years | 35 years | Sterilization sources replaced every few years |
| Cesium-137 | 30.08 years | 200 years | Dominant contaminant after reactor accidents |
| Carbon-14 | 5,730 years | 38,000 years | Dating organic material from about 300 to 50,000 years |
| Plutonium-239 | 24,110 years | 160,000 years | Long term waste isolation planning |
| Potassium-40 | 1.248 billion years | 8.3 billion years | Potassium-argon dating from 100,000 years upward |
| Uranium-238 | 4.468 billion years | 29.7 billion years | Dating rocks and meteorites over billions of years |
| Caffeine | about 5 hours | 33 hours | Varies 2 to 10 hours with liver enzyme activity |
| Ibuprofen | about 2 hours | 13 hours | Explains 4 to 6 hour dosing intervals |
| Fluoxetine | about 4 days | 27 days | Long tail is why missed doses matter less |
Biological half-lives in the lower rows are population averages, not fixed constants. A nuclear half-life is a property of the nucleus and does not change, while a drug half-life shifts with age, liver and kidney function, genetics, and other medications.
Example Problems
Example 1: dating a sample with carbon-14. A bone fragment contains 25 units of carbon-14 for every 100 units expected in a living sample of the same size. The half-life of carbon-14 is 5,730 years. Set the mode to elapsed time, enter an initial quantity of 100, a remaining quantity of 25, and select the carbon-14 preset.
t = 5730 * ln(100 / 25) / ln(2) = 5730 * 1.3863 / 0.6931 = 11,460 years
The ratio is one quarter, so exactly two half-lives have passed and the sample is about 11,460 years old.
Example 2: how long until caffeine drops to a target level. You drink 200 mg of caffeine and want to know when only 25 mg is left, using a 5 hour half-life. Set the mode to time to reach a target level, choose a specific remaining amount, enter 200 as the initial quantity and 25 as the target.
t = 5 * ln(200 / 25) / ln(2) = 5 * 2.0794 / 0.6931 = 15 hours
That is three half-lives, so a 2 p.m. coffee still leaves about 25 mg in your system at 5 a.m. the next morning. Dropping to 1 percent of the dose, or 2 mg, would take 6.64 half-lives, which is about 33 hours.
Frequently Asked Questions
How many half-lives until a substance is completely gone?
Mathematically it never reaches zero, because each half-life removes a proportion rather than a fixed amount. In practice a threshold is used. Five half-lives leave about 3 percent, which is the standard pharmacology cutoff for a drug being cleared. Roughly 6.64 half-lives leave 1 percent, and about 10 half-lives leave 0.1 percent, which is below the detection limit of most routine measurements. Set the calculator to time to reach a target level and enter your own threshold to get the exact figure for your substance.
What is the difference between half-life, decay constant, and mean lifetime?
They describe the same decay at different angles. The half-life T is the time for the amount to fall by half. The decay constant lambda = ln(2) / T is the fraction of the remaining material that decays per unit time, and it is the quantity that appears in the equation N = N0 * e^(-lambda * t). The mean lifetime tau = T / ln(2) is the average time an individual atom survives, and it is always about 1.4427 times the half-life, or the time it takes to fall to about 36.8 percent of the original amount. The calculator reports all three from whichever one you supply.
Does temperature, pressure, or sample size change the half-life?
For radioactive decay, no. The half-life is a property of the nucleus itself and is unaffected by temperature, pressure, chemical bonding, or how much material you have, which is what makes radiometric dating reliable. The only meaningful exceptions are a few electron-capture isotopes where extreme chemical or pressure environments shift the rate by a fraction of a percent. Biological half-lives are different: they describe elimination by the liver and kidneys, so they vary between people and can change with dose, illness, age, and drug interactions.