Carbon-14 after one half-life
- Solve for
- Amount left
- Isotope
- Carbon-14
- Starting amount N₀
- 100
- Elapsed time
- 5700 yr
- Amount left N
- 50
- Half-lives elapsed
- 1
Checked against: Definition of half-life (t½ = 5700 y, NNDC)
Half-life and radioactive decay calculator
Half-life calculator: amount left, starting amount, elapsed time or half-life from N = N₀·2^(−t/t½), with carbon-14 and other isotope presets.
By AdminDrafted with AI writing toolsUpdated Checked examples: 8
Amount left N
Amount left N: 50
- Half-life
- 5,700yr / 2,081,880d
- Fraction remaining
- 50%
- Half-lives elapsed
- 1
- Decay constant λ
- 3.8535 × 10⁻¹²1/s
- Mean lifetime τ = 1/λ
- 8,223.36yr
Years are Gregorian average years (365.2425 d). Carbon-14: t½ = 5700 years (NNDC), used for radiocarbon dating.
After 1 half-life (5,700 years), 50% of Carbon-14 remains: 50 of 100.
Decay of Carbon-14
How it's calculated S
Half-life in seconds
Amount left
Decay constant and mean lifetime
About the half-life and radioactive decay calculator
Radioactive material decays exponentially: after each half-life t½ half of the remaining nuclei are left, so N = N₀·2^(−t/t½), or N₀e^(−λt) with decay constant λ = ln 2/t½. Enter any three of starting amount, amount left, elapsed time and half-life to get the fourth, along with the fraction remaining, the number of half-lives and the mean lifetime 1/λ.
The default, 100 g of carbon-14 after 5,700 years, leaves 50 g: one half-life. Radiocarbon dating runs the same equation backwards, so a sample with 25% of the carbon-14 in living tissue is two half-lives, or 11,400 years, old. After 30 days, 7.49% of an iodine-131 dose remains.
Half-lives come from the NNDC NuDat 3 database. Amounts can be in any unit (grams, atoms, becquerels or percent) as long as both use the same one. Published radiocarbon ages use the Libby half-life of 5,568 years by convention and are then calibrated, so they differ from this raw calculation.
How to calculate radioactive decay with the half-life formula
Multiply the starting amount by ½ for each half-life that passes: N = N₀ × 2^(−t/t½). Carbon-14 has a 5,700-year half-life, so 100 g becomes 50 g after 5,700 years and 25 g after 11,400 years. It applies to any single isotope, in grams, atoms or becquerels.
How much is left after each half-life
Each half-life halves what remains, so the fraction left after n half-lives is (½)ⁿ:
- 1 half-life: 50%
- 2 half-lives: 25%
- 3 half-lives: 12.5%
- 4 half-lives: 6.25%
- 5 half-lives: 3.125%
- 10 half-lives: 0.098%
The exponent can be any number. Iodine-131 has a half-life of 8.0252 days, so 30 days is 30 ÷ 8.0252 = 3.738 half-lives, and 2^(−3.738) leaves 7.49% of a dose.
How to solve for time, starting amount or half-life
Rearrange the formula with logarithms:
- Elapsed time: t = t½ × ln(N₀/N) ÷ ln 2. A sample with 10% of its carbon-14 left is 5,700 × ln 10 ÷ ln 2 = 18,935 years old.
- Starting amount: N₀ = N × 2^(t/t½). With a 10-hour half-life, 10 units left after 20 hours means 40 units at the start.
- Half-life: t½ = t × ln 2 ÷ ln(N₀/N). An amount that falls from 100 to 25 units in 10 days has a half-life of 5 days.
Working by hand, put the half-life and the elapsed time in the same unit. The two amounts only need to share a unit with each other, because the formula uses their ratio.
Decay constant and mean lifetime
The decay constant λ = ln 2 ÷ t½ is the probability per unit time that a given nucleus decays. The same curve can be written N = N₀e^(−λt). For carbon-14, λ = 3.85 × 10⁻¹² per second.
Activity, the decay rate in becquerels, is A = λN. It falls by the same fraction as the number of nuclei, which is why a Geiger count rate can stand in for the amount.
The mean lifetime τ = 1/λ = t½ ÷ ln 2 is about 1.4427 half-lives. It is the average time a nucleus survives; after one mean lifetime 36.8% of the sample remains, against 50% after one half-life.
The OpenStax decay chapter derives these relationships.
Half-lives of common isotopes
These evaluated half-lives are from NNDC NuDat 3 at Brookhaven National Laboratory:
- Carbon-14: 5,700 years (radiocarbon dating)
- Tritium (H-3): 12.32 years (watch dials, fusion fuel)
- Cobalt-60: 1,925.28 days (radiotherapy, sterilization)
- Strontium-90: 28.79 years (fission product)
- Cesium-137: 30.08 years (fission product)
- Iodine-131: 8.0252 days (thyroid therapy)
- Technetium-99m: 6.0067 hours (medical imaging)
- Radon-222: 3.8235 days (indoor radon)
- Radium-226: 1,600 years (historic luminous paint)
- Plutonium-239: 24,110 years (reactor fuel)
- Potassium-40: 1.248 billion years (potassium-argon dating)
- Uranium-238: 4.468 billion years (uranium-lead dating)
A short half-life clears quickly: a day after an imaging dose, 6.27% of the technetium-99m remains. Cesium-137 lingers: after a century, 9.98% of a sample is still there.
Why published radiocarbon ages differ
Published radiocarbon ages use the Libby half-life of 5,568 years by convention and are then calibrated. With 25% of the carbon-14 left, the 5,700-year value gives 11,400 years and the Libby value gives 11,136 years.
Older textbooks use 5,730 years, which puts the same sample at 11,460 years. Keep the half-life with any age you report.
Decay results for carbon-14 and a measured sample
Default is 100 units of carbon-14 after 5,700 years, and two half-lives is the same sample after 11,400 years. Elapsed time finds the age when 25 of 100 units remain, and measured half-life uses 100 units falling to 25 in ten days.
| Scenario | Selected result |
|---|---|
| Default | 50 units left |
| Two half-lives | 25 units left |
| Elapsed time | 11,400 years |
| Measured half-life | 5 days |
Common mistakes
- Subtracting a fixed amount each half-life. After 50 of 100 units decay, the next half-life removes 25, not another 50.
- Mixing time units. Thirty days of iodine-131 decay leaves 7.49% of the dose; thirty years leaves essentially none.
- Including background counts. Subtract the detector's background before taking the ratio of two count rates.
- Ignoring decay chains. The formula tracks one parent isotope. Radon-222, for example, decays into daughters that are themselves radioactive.
- Expecting a sample to vanish on schedule. In the formula the amount approaches zero without reaching it; after ten half-lives about a thousandth is left, which a strong source can still make measurable.
Amount and time conventions
- Both amounts can use any unit, such as grams, atoms, becquerels or percent, as long as they use the same one.
- Half-lives and elapsed times accept seconds, minutes, hours, days and years, set separately.
- A year is the average Gregorian year of 365.2425 days.
- Choose a custom half-life when your isotope is not in the list or a problem states a different value.
The same halving rule describes how caffeine leaves the body, which the caffeine half-life calculator applies to each dose. For light from distant objects, the astronomy calculator finds travel times.
Worked examples
Radiocarbon age at 25% remaining
- Solve for
- Time
- Isotope
- Carbon-14
- Starting amount N₀
- 100
- Amount left N
- 25
- Elapsed time
- 11,400 yr
- Half-lives elapsed
- 2
Checked against: Two half-lives: 2 × 5700 y
Iodine-131 after 30 days
- Solve for
- Amount left
- Isotope
- Iodine-131
- Starting amount N₀
- 100
- Elapsed time
- 30 d
- Amount left N
- 7.49346
Checked against: 100 × 2^(−30/8.0252) = 7.4934578
Half-life from 1000 → 125 counts in 30 min
- Solve for
- Half-life
- Starting amount N₀
- 1000
- Amount left N
- 125
- Elapsed time
- 30 min
- Half-life
- 0.006944 d
- Half-lives elapsed
- 3
Checked against: t½ = 30 ln2 / ln 8 = 10 min = 0.0069444 d
Questions
How do you calculate half-life?
t½ = t·ln 2/ln(N₀/N), from a starting amount N₀ and the amount N left after time t. If a count rate falls from 1,000 to 125 in 30 minutes, N₀/N = 8 = 2³, so three half-lives have passed and t½ = 10 minutes. The logarithm handles any ratio: 1,000 falling to 300 in 30 minutes gives t½ = 17.3 minutes.
How much is left after 3 half-lives?
12.5%. Each half-life halves what remains: 50% after one, 25% after two, 12.5% after three and 6.25% after four. After n half-lives the fraction left is (1/2)ⁿ, so falling to 1% takes 6.64 half-lives and falling below 0.1% takes 10 (0.098% remains).
What is the half-life of carbon-14?
5,700 years in the evaluated nuclear data published by the NNDC; older textbooks give 5,730 years. Radiocarbon dating reaches back about 50,000 years, close to nine half-lives, after which less than 0.3% of the original carbon-14 remains, too little to measure reliably.
What is the difference between half-life and mean lifetime?
Mean lifetime τ is the average time a nucleus survives before it decays: τ = 1/λ = t½/ln 2 ≈ 1.443 t½. For carbon-14 that is 5,700/0.693 = 8,223 years. After one mean lifetime 1/e, or 36.8%, of the sample remains, compared with 50% after one half-life.
What is the decay constant?
λ = ln 2/t½, the probability per unit time that a given nucleus decays. It links amount to activity through A = λN, in becquerels when N counts atoms and λ is per second. Carbon-14's λ is 3.85 × 10⁻¹² per second, so 1 g of pure carbon-14, 4.3 × 10²² atoms, has an activity of about 1.66 × 10¹¹ Bq.
About this calculator
Sources
Checked against references
This calculator includes 8 worked examples with independently sourced answers. They run in the test suite, and you can run them here too.