Half-Life Calculator
Choose what you need — the amount remaining, the starting amount, the elapsed time or the half-life — enter the other three, and get the answer with every step of the working, the decay constant, the mean lifetime and a table of how much is left after each half-life. Free, no sign-up.
Plain, decimal-comma or scientific input: 5730, 5,73 or 4.468e9.
Amount remaining (N) is calculated below.
Rounded values; evaluated half-lives differ slightly between data sources.
Any consistent unit — g, Bq, atoms, mCi. Only the ratio N ÷ N₀ matters.
Results
25.00 g
Amount remaining (N)
2.000
Half-lives elapsed (k)
25.00%
Percent remaining
1.210 × 10⁻⁴ per y
Decay constant (λ)
8267 y
Mean lifetime (τ)
The working
k = t ÷ T½ = 11460 y ÷ 5730 y = 2.000
N = N₀ × (½)^k = 100.0 × 0.2500 = 25.00
λ = ln 2 ÷ T½ = 0.693147 ÷ 5730 y = 1.210 × 10⁻⁴ per y
τ = 1 ÷ λ = T½ ÷ ln 2 = 8267 y
Amount left after each half-life
| Half-lives (k) | Fraction left | Percent left | Amount | Elapsed time |
|---|---|---|---|---|
| 0 | 1 | 100.0% | 100.0 g | 0 y |
| 1 | 1/2 | 50.00% | 50.00 g | 5730 y |
| 2 | 1/4 | 25.00% | 25.00 g | 11460 y |
| 3 | 1/8 | 12.50% | 12.50 g | 17190 y |
| 4 | 1/16 | 6.250% | 6.250 g | 22920 y |
| 5 | 1/32 | 3.125% | 3.125 g | 28650 y |
| 6 | 1/64 | 1.563% | 1.563 g | 34380 y |
| 7 | 1/128 | 0.7813% | 0.7813 g | 40110 y |
| 8 | 1/256 | 0.3906% | 0.3906 g | 45840 y |
| 9 | 1/512 | 0.1953% | 0.1953 g | 51570 y |
| 10 | 1/1024 | 0.09766% | 0.09766 g | 57300 y |
How to Use the Half-Life Calculator
- 1. Choose what to solve for. Pick the amount remaining, the initial amount, the elapsed time or the half-life. The box for that quantity is replaced by a note, because it becomes the answer.
- 2. Enter the other three values. Amounts can be in any one unit — grams, becquerels, atoms — and you can name it so it appears in the answer. Time and half-life each have their own unit menu, from microseconds to billions of years, so a half-life in days can go with a time in years. Scientific input such as 4.468e9 works.
- 3. Or start from a nuclide. The buttons set the half-life to a common radionuclide such as ¹⁴C, ¹³¹I or ¹³⁷Cs, in its usual unit.
- 4. Read the result. You get the answer, the number of half-lives elapsed, the percentage left, the decay constant and the mean lifetime, with the working written out and a table running from 0 to 10 half-lives.
The Half-Life Equation
Radioactive decay is a first-order process: in any short interval each nucleus has the same chance of decaying, whatever its age. So the amount falls by the same factor in every interval of equal length, and the half-life T½ is the time for the amount to fall to one half.
Half-life form
N = N₀ × (½)^(t / T½)
Exponential form
N = N₀ × e^(−λt)
Decay constant
λ = ln 2 / T½ ≈ 0.693147 / T½
Mean lifetime
τ = 1 / λ = T½ / ln 2 ≈ 1.4427 × T½
Elapsed time
t = T½ × log₂(N₀ / N)
Half-life
T½ = t / log₂(N₀ / N)
Here N₀ is the initial amount, N the amount at time t, and k = t / T½ is the number of half-lives elapsed. The two forms are the same equation, because (½)^(t/T½) = e^(−λt) when λ = ln 2 / T½. The mean lifetime τ is the average time a nucleus survives; it is about 44% longer than the half-life, which is why a sample is not gone after one mean lifetime but has fallen to 1/e, about 36.8%.
Worked Example: Iodine-131
A hospital receives 80.0 g of a compound labelled with iodine-131, whose half-life is 8.02 days (rounded). How much is left after 30.0 days? Choose “Amount remaining”, set the half-life to ¹³¹I and enter the other values to get:
- k = t ÷ T½ = 30.0 d ÷ 8.02 d = 3.741 half-lives
- N = N₀ × (½)^k = 80.0 × 0.07481 = 5.985 g, which is 7.481% of the original
- λ = ln 2 ÷ 8.02 d = 0.08643 per day, and τ = 11.57 days
To go the other way, choose “Elapsed time”. If a carbon-14 sample has 30% of its original activity left and T½ = 5,730 y, then k = log₂(100 ÷ 30) = 1.737 and t = 1.737 × 5,730 = 9,953 years. When the fraction left is exactly 12.5%, k is exactly 3 and the time is 17,190 years; for dating work, use the Radiocarbon Dating Calculator, which also covers the conventions radiocarbon laboratories report.
How Much Is Left After n Half-Lives?
| Half-lives | Fraction left | Percent left |
|---|---|---|
| 0 | 1 | 100% |
| 1 | 1/2 | 50% |
| 2 | 1/4 | 25% |
| 3 | 1/8 | 12.5% |
| 5 | 1/32 | 3.125% |
| 7 | 1/128 | 0.78% |
| 10 | 1/1024 | 0.098% |
A common rule of thumb is that a radioactive source is effectively gone after about seven to ten half-lives, when less than 1% to 0.1% remains. How safe that is depends on the starting activity and the nuclide, so for waste handling follow the regulations rather than the rule.
Limits to Keep in Mind
- One decay step only. The equation assumes the nuclide decays to something that is not itself tracked. Parent–daughter chains such as ²³⁸U → … → ²⁰⁶Pb need the Bateman equations.
- Statistics. Half-life describes a large number of atoms. A single atom has no predictable lifetime, and a very small sample will scatter around the curve.
- Biological half-life. A radionuclide in the body also leaves by excretion. The effective half-life follows 1/T_eff = 1/T_physical + 1/T_biological, which is always shorter than either.
- Rounded presets. The half-lives on the buttons are rounded to three or four significant figures. Evaluated values (ENSDF, DDEP) differ slightly between editions, and some, like ¹³⁷Cs, are quoted to different last digits by different evaluations; use the value your data sheet gives when the digits matter.
- Not only nuclei. Any first-order process follows the same equation: a drug cleared from blood, a first-order chemical reaction, a decaying signal. Use whatever half-life applies.
Frequently Asked Questions
- How do you calculate half-life?
- Divide the elapsed time by the number of half-lives. The number of half-lives is log₂(N₀ ÷ N), so T½ = t ÷ log₂(N₀ ÷ N). If a sample falls from 100 to 6.25 in 2 hours, N₀ ÷ N = 16, which is 4 half-lives, so T½ = 2 h ÷ 4 = 30 minutes.
- What is the half-life formula?
- N = N₀ × (½)^(t ÷ T½), where N₀ is the starting amount, N the amount left, t the elapsed time and T½ the half-life. It is equivalent to N = N₀e^(−λt) with the decay constant λ = ln 2 ÷ T½. Time and half-life must be in the same unit.
- How many half-lives until a sample is gone?
- Never exactly, because each half-life removes half of what is left. After 7 half-lives, 0.78% remains; after 10, about 0.098%. For a given number of atoms the sample does run out in practice, once the expected number left falls below one.
- What is the difference between half-life and mean lifetime?
- The half-life is the time for half the nuclei to decay. The mean lifetime τ is the average survival time and equals T½ ÷ ln 2, about 1.4427 times longer. The decay constant λ is its reciprocal, the decay probability per unit time.
- Does temperature or pressure change a half-life?
- Practically never. Nuclear decay rates are set by the nucleus, not by the chemistry around it. The small exceptions are decays that capture an atom's own electrons, such as ⁷Be, where the chemical environment shifts the half-life by a fraction of a percent to about 1%.
- Can I use this for non-radioactive decay, such as drug elimination?
- Yes, as long as the process is first-order: the rate is proportional to the amount present. Enter the drug's elimination half-life, a time and an amount and the same equation applies. For processes that are not first-order, such as zero-order elimination, the half-life depends on the starting amount and this calculator does not apply.