Half Life Calculator
Work out how much of a decaying sample is left, or how long it takes to reach a target amount.
What Is the Half Life Calculator?
Radioactive decay is beautifully simple in its arithmetic and subtle in its consequences. A sample does not decay gradually, it halves repeatedly, each halving taking exactly the same amount of time. After one half life half the sample is gone, after two half lives a quarter is left, and after ten half lives a thousandth remains. Because every halving is identical, the same formula works for a medical tracer clearing from a body, a carbon 14 date for an archaeological artefact, or the decay of a waste isotope in a storage facility. This calculator works in both directions. Enter an initial amount, a half life and a time to find what is left, or enter an initial amount, a half life and a target to find how long the decay takes to get there.
Key Statistics
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5,730 years
The half life of carbon 14, the isotope that made radiocarbon dating possible. Living things constantly exchange carbon with the atmosphere, so the ratio of carbon 14 to carbon 12 starts to fall the moment an organism dies, and the remaining ratio dates the specimen.
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4.468 x 10 to the power of 9 years
The half life of uranium 238, the most common isotope of uranium in natural uranium. At roughly 4.5 billion years it is longer than the current age of the Earth, which is why long lived isotopes like this remain a lasting storage consideration.
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1.38 x 10 to the power of 10 years
The half life of carbon 14 measured by accelerator mass spectrometry, a direct count method far more precise than the older decay counting approach. Counting individual carbon 14 atoms directly made radiocarbon dates usable over much shorter samples.
Source: Libby half life and AMS dating background, Wikipedia
The Formula
Amount remaining = Initial amount x (1/2) raised to the power of (time elapsed divided by half life). The number of half lives elapsed is time elapsed divided by half life. The decay constant lambda is ln(2) divided by the half life. Working backwards, time = half life x ln(initial divided by target) divided by ln(2).
Worked Examples
A sample of 100 grams of carbon 14 with a half life of 5,730 years is measured after 11,460 years. That is two half lives. Two half lives leaves one quarter, so 100 multiplied by a quarter is 25 grams remaining. Percent remaining is 25 percent and percent decayed is 75 percent.
Dating an artefact with carbon 14
- Take a starting amount of 100 percent of carbon 14 at the moment of death.
- The half life of carbon 14 is 5,730 years.
- After 5,730 years, 50 percent remains.
- After 11,460 years, 25 percent remains.
- After 17,190 years, 12.5 percent remains.
- If a specimen shows 25 percent remaining, it is 11,460 years old.
25 percent remaining means 11,460 years since death
Working out how long until a target amount remains
- Take an initial amount of 100 and a half life of 5,730 years.
- You want to know when 10 percent remains.
- Use t = half life x ln(initial divided by target) divided by ln(2).
- ln(100 divided by 10) is ln(10), which is about 2.3026.
- Multiply by 5,730 and divide by ln(2), which is about 0.6931.
- The result is about 19,030 years.
About 19,030 years to reach 10 percent remaining
A medical tracer cleared from the body
- A tracer with a half life of 6 hours is administered, with 100 percent in the blood at the moment of injection.
- After 6 hours, 50 percent remains.
- After 12 hours, 25 percent remains.
- After 18 hours, 12.5 percent remains.
- After 24 hours, 6.25 percent remains, so nearly 94 percent has cleared.
About 6.25 percent remains after 24 hours
Real World Use Cases
Radiocarbon dating
Measuring the carbon 14 remaining in a specimen gives its age directly, because the ratio falls in exact half life steps from the moment an organism stops exchanging carbon with the atmosphere. Ages beyond roughly 50,000 years become unreliable because the remaining signal falls below the detection threshold of any instrument.
Medical imaging tracers
Diagnostic tracers such as technetium 99m have short half lives of around six hours, chosen so the isotope is gone well before the patient leaves. Understanding the decay curve tells clinicians how long a scan stays valid and when background counts start to interfere.
Nuclear waste and isotope storage
Storage requirements for long lived isotopes depend on how many half lives have to pass before the material is safe. An isotope with a half life measured in millions of years still has a meaningful fraction remaining after a human lifetime, which is why disposal is engineered on far longer timescales.
Biology and pharmacology
Drug clearance in the body follows exponential decay, and the half life determines dosing intervals. A drug with a six hour half life is dosed roughly every six hours, while a thirty hour half life supports once daily dosing.
Expert Tips
- ✓ Always express a half life in the same time units as the time you are comparing it against. Mixing years and hours is the single most common arithmetic error with this calculation.
- ✓ Think in half lives rather than percentages. After one half life it is 50 percent, after two 25 percent, after ten it is 0.1 percent, and the pattern is far easier to hold in your head than a continuous decay.
- ✓ Be honest about measurement limits. Below roughly 0.1 percent remaining, background radiation makes the measurement unreliable no matter how good the arithmetic is.
- ✓ A decaying sample never grows. If you are asked for a time at which the amount is higher than the starting amount, the answer is that no such time exists rather than a negative number.
- ✓ Carbon 14 dating should be treated as a probabilistic estimate, not a date to the exact year. Calibration curves, contamination and sample handling all shift the result by decades or more.
Frequently Asked Questions
How do I calculate the amount remaining after radioactive decay? ▼
Divide the time elapsed by the half life to get the number of half lives, then multiply the initial amount by a half raised to that power. For an initial 100, a half life of 5,730 years and 11,460 years elapsed, that is 100 x (1/2) to the power of 2, which is 25.
What is the half life of carbon 14? ▼
5,730 years. That is why radiocarbon dating works over a range of roughly the last 50,000 years, since after about nine half lives the remaining carbon 14 is too small to measure reliably against background radiation.
How long until half of a sample decays? ▼
Exactly one half life. That is the definition. After two half lives half of that half has gone, leaving a quarter, and the pattern continues indefinitely, with the sample approaching zero but never reaching it.
What is the half life of carbon 14 used for? ▼
Radiocarbon dating. Living organisms maintain a known ratio of carbon 14 to carbon 12 while alive. Once an organism dies the exchange stops and carbon 14 decays away at its 5,730 year half life, so the remaining ratio gives the time since death.
Why does the amount never reach zero? ▼
Because each halving halves a positive number rather than subtracting a fixed amount. The remaining amount is always initial multiplied by a half raised to a positive power, which is always greater than zero. It approaches zero as the number of half lives grows without bound.
How do I find the time it takes to reach a certain amount? ▼
Use the working backwards button, which applies t = half life x ln(initial divided by target) divided by ln(2). For an initial 100 with a 5,730 year half life and a target of 10 percent, that gives about 19,030 years.
What happens if the target amount is greater than the initial amount? ▼
There is no such time. A decaying sample only ever gets smaller, so no amount of waiting produces a larger quantity. The calculator reports this rather than returning a negative time, which would be meaningless.
What is the decay constant lambda? ▼
Lambda is ln(2) divided by the half life, and it appears in the exponential form of the decay equation N = N0 multiplied by e raised to minus lambda t. It describes the rate of decay per unit time and has units of the reciprocal of the time unit you are using.
How accurate is radiocarbon dating? ▼
For samples under about 50,000 years the method is generally reliable, but the result is an estimate expressed with a range rather than an exact year. Calibration curves, contamination and the material sampled all introduce uncertainty that a reputable laboratory will report rather than hide.
Can I use this for a half life in hours or minutes? ▼
Yes, as long as the half life and the time you are comparing it against use the same unit. A tracer with a six hour half life measured after 18 hours gives three half lives, leaving 12.5 percent. Converting one of the two to a different unit breaks the calculation.
Common Mistakes to Avoid
- ⚠ Treating decay as linear rather than exponential, so assuming 10 percent is lost every half life instead of half of whatever remains.
- ⚠ Mixing time units, such as entering a half life in years and a time elapsed in hours.
- ⚠ Subtracting the half life repeatedly from a fixed starting amount instead of halving the remainder each time.
- ⚠ Expecting the sample to reach exactly zero, which never happens because it only approaches zero.
- ⚠ Using the half life directly in the exponential as if it were the decay constant, forgetting the ln(2) term.
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