Carbon-14 after three half lives
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N₀ = 100, t = 17,190 yr, t½ = 5,730 yrt/t½ = 17,190 / 5,730 = 3. N = 100 · (½)³ = 100 / 8 = 12.5. Start with 100 g and 12.5 g is left; start with 100% and 12.5% is left.
Half Life Calculator
The amount left after radioactive decay, the time it took, or the half life.
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Want to solve these without the calculator?
The Coddy physics course teaches the method: you solve each problem in steps on an interactive board, and it tells you which step went wrong.
The half life formula, and what it says
A radioactive sample decays at a rate set by its half life, t½: the time it takes for half of the atoms to decay. Whatever amount you start with, half is left after one half life, a quarter after two, an eighth after three. The formula is N = N₀(½)^(t/t½), where N₀ is the starting amount, N the amount left and t the time that has passed.
The exponent t/t½ counts how many half lives have gone by, and it does not have to be a whole number. To go backwards to the time, take a base 2 logarithm: t = t½ · log₂(N₀/N). If 80 g has become 10 g, N₀/N = 8, log₂ 8 = 3, so three half lives have passed. The half life does not depend on how much you have: a gram and a tonne of iodine-131 both halve in 8.02 days.
The calculator solves the formula for whichever of the four you leave blank: the amount left, the starting amount, the time or the half life. The two amounts can be in any unit, grams, atoms, a percentage or becquerels of activity, as long as both use the same one. The times can be in seconds, minutes, hours, days or years (a year here is 365.25 days), and the time and the half life may use different units: both are converted to seconds first, and the answer is given in the unit you pick and in the others.
What to notice
- After 1, 2, 3, 4 and 5 half lives the fraction left is 50%, 25%, 12.5%, 6.25% and 3.125%. Each half life halves what is left, not what you started with.
- Decay never quite reaches zero in the formula. After 10 half lives about 0.1% is left; after 20, about one millionth.
- The amounts only appear as a ratio, N₀/N, so their unit cancels. Mass, number of atoms, percent and activity in becquerels all work, if both amounts use the same one.
- The time and the half life must be in the same unit before they are divided. 24 hours and a half life of 6.01 hours give 24 / 6.01 = 3.99 half lives.
- The decay constant is λ = ln 2 / t½, the fraction decaying per unit of time. N = N₀e^(−λt) is the same law written with e instead of ½.
How to calculate half life problems
Write down what you know
Three of the four: the starting amount N₀, the amount left N, the time t and the half life t½.
Put the time and the half life in the same unit
Hours with hours, years with years. 2 days is 48 hours; a year is 365.25 days.
For the amount left, raise ½ to t/t½
N = N₀ · (½)^(t/t½). 100 g of carbon-14 after 17,190 years: 17,190 / 5,730 = 3, and 100 · (½)³ = 12.5 g.
For the time, take a base 2 logarithm
t = t½ · log₂(N₀/N). From 100% to 25% is log₂ 4 = 2 half lives; for carbon-14 that is 2 · 5,730 = 11,460 years. On a calculator without log₂, use log₂ x = ln x / ln 2.
Half lives of well-known isotopes
Rounded standard values. They run from minutes to billions of years, which is why the calculator takes times from seconds to years.
| Isotope | Half life | Where you meet it |
|---|---|---|
| Fluorine-18 | 110 minutes | PET scans |
| Technetium-99m | 6.01 hours | The most used tracer in medical scans |
| Radon-222 | 3.82 days | A gas that seeps from rocks into basements |
| Iodine-131 | 8.02 days | Thyroid treatment |
| Polonium-210 | 138 days | Alpha sources |
| Cobalt-60 | 5.27 years | Radiotherapy, sterilising medical equipment |
| Hydrogen-3 (tritium) | 12.3 years | Glowing exit signs and watch dials |
| Caesium-137 | 30.1 years | Fallout from Chernobyl and Fukushima |
| Radium-226 | 1,600 years | Old luminous paint |
| Carbon-14 | 5,730 years | Radiocarbon dating |
| Plutonium-239 | 24,100 years | Nuclear reactors |
| Uranium-235 | 704 million years | Nuclear fuel |
| Potassium-40 | 1.25 billion years | Bananas, the human body, dating rocks |
| Uranium-238 | 4.47 billion years | Dating the oldest rocks on Earth |
Worked examples
The half life from two measurements
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N₀ = 80, N = 10, t = 24 dRearrange for the half life: t½ = t / log₂(N₀/N) = 24 / log₂ 8 = 24 / 3 = 8 days. That is close to iodine-131's 8.02 days. Set the half life box to days and the answer reads 8; it is also given in seconds, hours and years.
How long until a quarter is left
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N₀ = 1, N = 0.25, t½ = 8 dt = t½ · log₂(N₀/N) = 8 · log₂ 4 = 8 · 2 = 16 days. A quarter is two halvings, so the time is two half lives.
A medical tracer after one day
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N₀ = 100, t = 24 h, t½ = 6.01 hTechnetium-99m: t/t½ = 24 / 6.01 = 3.993 half lives, not a whole number, and that is fine. N = 100 · (½)^3.993 = 6.28, so 6.28% of the dose is left after a day.
Carbon dating
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N₀ = 100, N = 25, t½ = 5,730 yrA bone with 25% of the carbon-14 of living tissue: t = 5,730 · log₂(100 / 25) = 5,730 · 2 = 11,460 years old.
Common mistakes
- Thinking it is all gone after two half lives. Half goes in the first, then half of what is left: 25% remains after two, 12.5% after three.
- Subtracting instead of halving. After each half life the amount is multiplied by ½; it does not drop by the same amount each time.
- Dividing the time by the half life when the two are in different units. 24 hours over a half life of 8.02 days must become 1 day over 8.02 days first.
- Using different units for the two amounts. N in grams and N₀ in percent gives a meaningless ratio; both must be grams, both atoms or both percent.
- Taking ln or log₁₀ of N₀/N and forgetting to divide by ln 2 or log₁₀ 2. The time is t½ · log₂(N₀/N), and log₂ x = ln x / ln 2.
- Swapping N and N₀. The starting amount is the bigger one; N₀/N is more than 1 for anything that has decayed.
Half life FAQ
What is the half life formula?
N = N₀(½)^(t/t½), where N₀ is the starting amount, N the amount left after a time t, and t½ the half life. The exponent t/t½ is the number of half lives that have passed.
How do you calculate half life?
Measure the amount at the start and after a known time, then use t½ = t / log₂(N₀/N). If 80 units fall to 10 in 24 days, log₂ 8 = 3, so the half life is 24 / 3 = 8 days.
How do you find the time from the half life?
Use t = t½ · log₂(N₀/N). The logarithm counts the halvings: from 100% to 12.5% is log₂ 8 = 3 half lives, so for a half life of 5,730 years the time is 17,190 years.
How much is left after 3 half lives?
One eighth, 12.5%. Each half life halves what is left: 50% after one, 25% after two, 12.5% after three, 6.25% after four and 3.125% after five.
Can I use grams, atoms or percent for the amounts?
Yes. The formula only uses the ratio of the two amounts, so any unit works as long as the amount left and the starting amount use the same one. Activity in becquerels works too, since it falls with the same half life.
What is the decay constant?
The decay constant λ is the fraction of atoms that decay per unit of time, and λ = ln 2 / t½. For carbon-14 it is 0.693 / 5,730 = 1.21 × 10⁻⁴ per year. The law can then be written N = N₀e^(−λt).
How does carbon dating use half life?
Living things take in carbon-14 and keep the same share as the air. When they die, it decays with a half life of 5,730 years. Measure the fraction left and t = 5,730 · log₂(N₀/N) gives the age: 25% left means 11,460 years.
