Half-Life Calculator
N = N0 × (1/2)^(t/T). Give the half life calculator any three of the starting amount, the amount left, the time and the half-life, and it finds the fourth.
Amount remaining
- Amount remaining N = 200 × (1/2)^(2.500) = 35.36
- 35.36
- Fraction remaining 82.32% has decayed. The fraction left is (1/2) to the power of the number of half-lives, whatever the starting amount.
- 17.68%
- Half-lives elapsed 30 hours ÷ 12 hours.
- 2.500
- Decay constant (λ) λ = ln 2 / T = 0.693147 / 12 hours. That is 1.605 × 10^-5 per second. Written with it, N = N0 × e^(-λt).
- 0.05776 per hour
- Mean life (τ) τ = T / ln 2 = 1 / λ, the average time before one atom decays. After one mean life 1/e of the amount is left, about 36.79%.
- 17.31 hours
| Half-lives | Time (hours) | Amount left | Fraction left |
|---|---|---|---|
| 0 | 0 | 200.0 | 100% |
| 1 | 12.00 | 100.0 | 1/2 (50.00%) |
| 2 | 24.00 | 50.00 | 1/4 (25.00%) |
| 3 | 36.00 | 25.00 | 1/8 (12.50%) |
| 4 | 48.00 | 12.50 | 1/16 (6.250%) |
| 5 | 60.00 | 6.250 | 1/32 (3.125%) |
| 6 | 72.00 | 3.125 | 1/64 (1.563%) |
- The amount can be in any unit, because only the ratio N / N0 enters the formula. Mass, number of atoms and activity (becquerels or curies) all fall by the same fraction.
- Decay is random atom by atom, so the formula describes a large number of atoms. For a handful of atoms it gives the expected amount, and the real count scatters around it.
How the Half-Life Calculator works
The half-life formula is N = N0 × (1/2)^(t/T). N0 is the amount you start with, T is the half-life, and N is what is left after a time t. Every half-life halves what remains, so the exponent t/T counts half-lives, and it does not have to be a whole number.
The default example is 200 units with a half-life of 12 hours, after 30 hours. 30 hours is 2.5 half-lives, so the fraction left is (1/2)^2.5 = 0.1768, and 200 × 0.1768 = 35.36 units remain. 82.32% has decayed. The units of the amount do not matter: grams, atoms, becquerels and percent all fall by the same fraction.
Rearranged, the formula answers the other three questions. The time to decay from N0 to N is t = T × log2(N0 / N): from 200 down to 25 is a factor of 8, which is three half-lives, so 36 hours at a half-life of 12 hours. The half-life from two measurements is T = t × ln 2 / ln(N0 / N): a sample that falls from 200 to 50 in 10 days has a half-life of 5 days. And the starting amount is N0 = N × 2^(t/T), which is how a measurement today is run backward to the amount at an earlier date.
Two other numbers describe the same rate. The decay constant λ = ln 2 / T is the fraction that decays per unit of time in the limit of a short interval, and with it the formula becomes N = N0 × e^(-λt). For a 12-hour half-life λ is 0.05776 per hour. The mean life τ = T / ln 2 = 1 / λ is the average lifetime of a single atom: 17.31 hours here. After one mean life about 36.79% of the amount is left, a little more than after one half-life.
A rule of thumb comes out of the table. After 7 half-lives less than 1% remains (1/128, or 0.781%), and after 10 it is 1/1,024, under 0.1%. That is why material is often treated as gone after ten half-lives, though it never reaches zero in the formula.
The same arithmetic covers anything that decays by a constant fraction per unit of time: a radioactive isotope, a first-order chemical reaction, a capacitor discharging through a resistor. The half-life itself is the part that has to be measured, and this calculator takes it as an input rather than listing values for named isotopes.
Half-Life Calculator FAQ
What is the half-life formula?
N = N0 × (1/2)^(t/T), where N0 is the starting amount, T the half-life and N the amount left after time t. 200 units with a 12-hour half-life leave 35.36 after 30 hours, which is 2.5 half-lives.
How do you calculate half-life from two amounts?
T = t × ln 2 / ln(N0 / N). If a sample falls from 200 to 50 in 10 days, ln(200/50) = ln 4 = 1.386, and 10 × 0.6931 / 1.386 = 5 days.
How long does it take to decay to a given amount?
t = T × log2(N0 / N). Falling from 200 to 25 is a factor of 8, which is 3 half-lives, so with a 12-hour half-life it takes 36 hours.
What is the difference between half-life and mean life?
The half-life is the time for half the amount to decay. The mean life is the average lifetime of one atom, T / ln 2, about 1.4427 times the half-life. A 12-hour half-life has a mean life of 17.31 hours.
What is the decay constant?
λ = ln 2 / T, the rate in N = N0 × e^(-λt). For a half-life of 12 hours it is 0.05776 per hour. Its units are one over time, and it is the reciprocal of the mean life.