# 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.

Interaktive Version: https://www.calcopenly.com/de/science/half-life-radioactive-decay-calculator
Thema: Naturwissenschaftliche Rechner

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.

## Eingaben

- **Solve for** (Optionen: Amount left, Starting amount, Zeit, Halbwertszeit)
- **Isotope** (Optionen: 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)
- **Halbwertszeit**
- **Starting amount N₀**: Any unit — grams, atoms, becquerels, % — as long as N uses the same
- **Amount left N**
- **Elapsed time**

## Ergebnisse

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

## Formel

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

## Durchgerechnete Beispiele

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

### Radiocarbon age at 25% remaining

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

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

- Solve for: Halbwertszeit
- Starting amount N₀: 1000
- Amount left N: 125
- Elapsed time: 30 min
- **Halbwertszeit: 0.006944 d**
- **Half-lives elapsed: 3**
- Prüfquelle: 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
- Halbwertszeit: 10 h
- Amount left N: 10
- Elapsed time: 20 h
- **Starting amount N₀: 40**
- Prüfquelle: 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%**
- Prüfquelle: 2⁰ = 1

## Fragen

### 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.

### Wie genau arbeitet „Half-life and radioactive decay calculator“?

Die Genauigkeit hängt von Ihren Eingaben und den Annahmen der Methode ab. Die Dezimalrechnung nutzt 50 signifikante Stellen, doch Schätzungen, numerische Verfahren und Quelldaten können ungenauer sein. Die angezeigte Rundung beseitigt diese Grenzen nicht. Anhand unabhängiger Quellen geprüfte Rechenbeispiele: 8. Beispielsweise wird „Carbon-14 after one half-life“ anhand von Definition of half-life (t½ = 5700 y, NNDC) geprüft.

### Woher stammt die Methode?

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

## Quellen

- [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)