# 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.

Version interactive : https://www.calcopenly.com/fr/science/half-life-radioactive-decay-calculator
Sujet : Calculatrices scientifiques

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.

## Données

- **Solve for** (options : Amount left, Starting amount, Temps, Demi-vie)
- **Isotope** (options : Carbon-14, Tritium (H-3), Cobalt-60, Strontium-90, Caesium-137, Iodine-131, Technetium-99m, Radon-222, Radium-226, Plutonium-239, Potassium-40, Uranium-238, Custom half-life)
- **Demi-vie**
- **Starting amount N₀**: Any unit — grams, atoms, becquerels, % — as long as N uses the same
- **Amount left N**
- **Elapsed time**

## Résultats

- Amount left N — résultat principal
- Starting amount N₀
- Elapsed time (yr)
- Elapsed time (d)
- Demi-vie (yr)
- Demi-vie (d)
- Fraction remaining
- Half-lives elapsed
- Decay constant λ (1/s)
- Mean lifetime τ = 1/λ (yr)

## Formule

$$
N = N_0\,2^{-t/t_{1/2}} = N_0e^{-\lambda t},\qquad \lambda = \frac{\ln 2}{t_{1/2}},\qquad \tau = \frac1\lambda
$$

## Exemples détaillés

### 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**
- Source de vérification : Definition of half-life (t½ = 5700 y, NNDC)

### Radiocarbon age at 25% remaining

- Solve for: Temps
- Isotope: Carbon-14
- Starting amount N₀: 100
- Amount left N: 25
- **Elapsed time: 11,400 yr**
- **Half-lives elapsed: 2**
- Source de vérification : 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**
- Source de vérification : Python 3.8 decimal: 100 × 2^(−30/8.0252) = 7.4934578

### Half-life from 1000 → 125 counts in 30 min

- Solve for: Demi-vie
- Starting amount N₀: 1000
- Amount left N: 125
- Elapsed time: 30 min
- **Demi-vie: 0.006944 d**
- **Half-lives elapsed: 3**
- Source de vérification : Python 3.8 decimal: t½ = 30 ln2 / ln 8 = 10 min = 0.0069444 d

### Starting activity for a custom 10 h half-life

- Solve for: Starting amount
- Isotope: Custom half-life
- Demi-vie: 10 h
- Amount left N: 10
- Elapsed time: 20 h
- **Starting amount N₀: 40**
- Source de vérification : Two half-lives: N₀ = 10 × 2²

### No time elapsed

- Solve for: Amount left
- Isotope: Caesium-137
- Starting amount N₀: 100
- Elapsed time: 0 yr
- **Amount left N: 100**
- **Fraction remaining: 100%**
- Source de vérification : 2⁰ = 1

## 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.

### Quelle est la précision de « Half-life and radioactive decay calculator » ?

La précision dépend de vos données et des hypothèses de la méthode. Le calcul décimal utilise 50 chiffres significatifs, mais les estimations, méthodes numériques et données sources peuvent être moins précises ; l’arrondi affiché ne supprime pas ces limites. Exemples résolus vérifiés à partir de sources indépendantes : 8. Par exemple, « Carbon-14 after one half-life » est vérifié à l’aide de Definition of half-life (t½ = 5700 y, NNDC).

### D’où vient cette méthode ?

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

## Sources

- [NNDC NuDat 3 (ENSDF evaluated half-lives), Brookhaven National Laboratory](https://www.nndc.bnl.gov/nudat3/)
- [OpenStax University Physics Volume 3, §10.3 Radioactive decay](https://openstax.org/books/university-physics-volume-3/pages/10-3-radioactive-decay)