Radioactive decay, half-life, activity and dating
N(t) = N₀ e^(−λt)
Two readings of one law: what is left of a source after a given time, and the time elapsed when you know what is left, which is dating. The tool returns the remaining and decayed quantities, the matching fractions, the number of half-lives elapsed, the decay constant λ and the mean lifetime τ, often confused with the half-life although it is 44% longer. Half-life and time are entered in independent units, and the year used is the Julian year of the tables, declared rather than assumed.
12.565
- Calculation
- N = N₀ e^(−λt) = 100 × e^(−1.00032E-6 × 2073600) = 12.565
- Decay constant
- λ = ln 2 / T½ = 0.693147 / 692928 = 1.00032E-6 s⁻¹
12.565% of the starting quantity remains, after 2.99252 half-life/lives.
The mean lifetime τ is not the half-life: it equals T½ / ln 2, that is 1.4427 times longer. It is the average lifetime of a nucleus, and the time after which 1/e ≈ 36.8% of the sample is left, not half.
The quantity may be a mass, a number of nuclei or an activity in becquerels: all three are proportional, decay with the same exponential, and the ratio N/N₀ does not depend on which one was chosen.
The exponential law describes a very large number of nuclei: it gives the average of a random process, not the trajectory of one given sample. On a few dozen nuclei, statistical fluctuation dominates. It also assumes the decay product does not decay back into the starting nucleus, and that nothing supplies new ones.
Scientific dossier
What the tool computes, what it assumes, where it stops being valid, and where its data comes from.
Method & formulasN(t) = N₀ e^(−λt)
N(t) = N₀ e^(−λt)
λ = ln 2 / T½
τ = 1 / λ = T½ / ln 2
t = −ln(N/N₀) / λ
N/N₀ = 2^(−t/T½)
The decay of a nucleus is a random event whose probability per unit time, λ, depends neither on the age of the nucleus nor on its surroundings. That absence of memory is what produces the exponential, and what makes dating possible: the remaining fraction depends only on elapsed time, never on the starting quantity.
- 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. LONGER than the half-life by a factor 1/ln 2 ≈ 1.4427: after τ, 36.8% is left, not 50%.
- 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 of a random process over a very large number of nuclei.
The exponential law describes the average of a random process over a very large number of nuclei. On a few dozen nuclei, statistical fluctuation dominates and the curve no longer holds. The model also assumes simple decay: no chain feeding the starting nucleus back, no external supply. Carbon-14 dating further assumes the starting atmospheric content matched today’s, an assumption that calibration curves correct and this tool does not apply. On times very short against the half-life, the decayed quantity is computed with expm1 rather than 1 − e^(−λt), whose subtraction would return exactly zero.
Reading the resultThe number of half-lives elapsed is the most telling reading: after n half-lives, 2⁻ⁿ is left, that is a half, a quarter, an eighth.
The number of half-lives elapsed is the most telling reading: after n half-lives, 2⁻ⁿ is left, that is a half, a quarter, an eighth. The usual ten-half-lives rule comes from there, 2⁻¹⁰ being less than a thousandth. The quantity entered may be a mass, a number of nuclei or an activity: all three decay with the same exponential and the ratio N/N₀ does not depend on which was chosen.