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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.

Aggiornato Esempi verificati: 8

Any unit — grams, atoms, becquerels, % — as long as N uses the same
Prova
Amount left N
Amount left N: 50
Cifre significative: 6; Al più vicino; a parità lontano da zero
Tempo di dimezzamento
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
Come si calcola 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}

Informazioni su 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.

Esempi svolti

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

Fonte di verifica: Definition of half-life (t½ = 5700 y, NNDC)

Radiocarbon age at 25% remaining

Solve for
Tempo
Isotope
Carbon-14
Starting amount N₀
100
Amount left N
25
Elapsed time
11,400 yr
Half-lives elapsed
2

Fonte di verifica: 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

Fonte di verifica: Python 3.8 decimal: 100 × 2^(−30/8.0252) = 7.4934578

Half-life from 1000 → 125 counts in 30 min

Solve for
Tempo di dimezzamento
Starting amount N₀
1000
Amount left N
125
Elapsed time
30 min
Tempo di dimezzamento
0.006944 d
Half-lives elapsed
3

Fonte di verifica: Python 3.8 decimal: t½ = 30 ln2 / ln 8 = 10 min = 0.0069444 d

Domande

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.

Quanto è preciso «Half-life and radioactive decay calculator»?

La precisione dipende dai dati inseriti e dalle ipotesi del metodo. Il calcolo decimale usa 50 cifre significative, ma stime, metodi numerici e dati di origine possono essere meno precisi; l’arrotondamento visualizzato non elimina questi limiti. Esempi svolti verificati con fonti indipendenti: 8. Per esempio, «Carbon-14 after one half-life» viene verificato con Definition of half-life (t½ = 5700 y, NNDC).

Da dove proviene il metodo?

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

Informazioni su questa calcolatrice

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

Fonti

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

Verificato con le fonti

Questa calcolatrice include 8 esempi svolti con risposte da fonti indipendenti. Fanno parte della suite di test e puoi eseguirli anche qui.

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