CalcOpenly

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.

Updated Checked against 8 worked examples

Any unit — grams, atoms, becquerels, % — as long as N uses the same
Try
Amount left N
Amount left N: 50
Shown to 6 significant figures, half-up
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

025507510005,00010,00015,00020,00025,000Time (years)Amount left (% of N₀)2×t½3×t½4×t½50% left
How it's calculated S

  1. Half-life in seconds

    t1/2=179,875,000,000 s=5,700 yrt_{1/2} = 179{,}875{,}000{,}000\ \mathrm{s} = 5{,}700\ \mathrm{yr}

  2. Amount left

    N=N0 2−t/t1/2=100×2−1=50N = N_0\,2^{-t/t_{1/2}} = 100\times 2^{-1} = 50

  3. Decay constant and mean lifetime

    λ=ln⁡2t1/2=3.8535×10−12 s−1,τ=1λ=8,223.36 yr\lambda = \frac{\ln 2}{t_{1/2}} = 3.8535 \times 10^{-12}\ \mathrm{s^{-1}},\quad \tau = \frac1\lambda = 8{,}223.36\ \mathrm{yr}

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.

Worked examples

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)

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: Python 3.8 decimal: 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: Python 3.8 decimal: 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.

How accurate is the half-life and radioactive decay calculator?

Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 8 worked examples whose answers come from independent sources; for example, “Carbon-14 after one half-life” is checked against Definition of half-life (t½ = 5700 y, NNDC).

Where does the method come from?

NNDC NuDat 3 (ENSDF evaluated half-lives), Brookhaven National Laboratory; OpenStax University Physics Volume 3, §10.3 Radioactive decay.

About this calculator

N=N0 2−t/t1/2=N0e−λt,λ=ln⁡2t1/2,τ=1λN = N_0\,2^{-t/t_{1/2}} = N_0e^{-\lambda t},\qquad \lambda = \frac{\ln 2}{t_{1/2}},\qquad \tau = \frac1\lambda

Sources

  1. NNDC NuDat 3 (ENSDF evaluated half-lives), Brookhaven National Laboratory
  2. OpenStax University Physics Volume 3, §10.3 Radioactive decay

Checked against references

8 worked examples with independently sourced answers ship with this calculator. They run in the test suite; you can run them here too.

Related calculators

Allow optional Google Analytics to measure page visits? Calculators work either way. Privacy and choices

Optional analytics: off.