Radioactive decay calculator: half-life, activity and dating
Remaining quantity or elapsed time (dating), decay constant λ and mean lifetime, in Julian years, including times much shorter than the half-life.
Result
12.565
- Calculation
- N = N₀ e^(−λt) = 100 × e^(−1.00032 × 10⁻⁶ × 2 073 600) = 12.565
- Decay constant
- λ = ln 2 / T½ = 0.693147 / 692 928 = 1.00032 × 10⁻⁶ s⁻¹
- Half-lives elapsed
- 2.99252
- Remaining fraction
- 12.565 %
- Decayed fraction
- 87.435 %
- Decayed quantity
- 87.435
- Mean lifetime
- 11.5704 d
- Decay constant
- 1.00032 × 10⁻⁶ s⁻¹
Note
12.565% of the starting quantity remains, after 2.99252 half-lives.
Calculation, assumptions and sources
OverviewWhat remains of a source after a given time, or the time elapsed knowing what remains (dating).
What remains of a source after a given time, or the time elapsed knowing what remains (dating). The calculation gives the remaining and decayed quantities and fractions, the number of half-lives elapsed, the decay constant λ and the mean lifetime τ, 44% longer than the half-life. Half-life and time are entered in independent units; the year is the Julian year of the tables, 365.25 days.
Method & formulasN(t) = N₀ e^(−λt)
The decay of a nucleus is random, with a probability per unit time λ independent of the nucleus’s age and environment. This lack of memory produces the exponential: the remaining fraction depends on the elapsed time and not on the initial quantity. Dating relies on this property.
- T½
- half-life: the time after which half is left.
- λ
- decay constant, probability of decay per unit time, in s⁻¹.
- τ
- mean lifetime, 1/λ = T½/ln 2 ≈ 1.4427 T½. After τ, 1/e ≈ 36.8% remains.
- Activity
- number of decays per second, A = λN, in becquerels. It decays with the same exponential as N.
- Julian year
- exactly 365.25 days, that is 31,557,600 s: the unit of half-life tables.
Validity domainThe exponential law describes the average over a very large number of nuclei; on a few dozen nuclei, statistical fluctuation dominates.
The exponential law describes the average over a very large number of nuclei; on a few dozen nuclei, statistical fluctuation dominates. The model assumes a simple decay, with no parent chain feeding the starting nucleus and no external supply. A carbon-14 dating assumes the starting atmospheric content was today’s; calibration curves correct this assumption, and this tool does not apply them. Over times very short compared with the half-life, the decayed quantity is computed with expm1, without loss of precision.
Reading the resultAfter n half-lives, 2⁻ⁿ remains; the ten-half-life rule comes from this, 2⁻¹⁰ being less than one thousandth.
After n half-lives, 2⁻ⁿ remains; the ten-half-life rule comes from this, 2⁻¹⁰ being less than one thousandth. The quantity entered may be a mass, a number of nuclei or an activity; the ratio N/N₀ is the same for all three.