Half-Life Calculator
Calculate remaining quantity, time elapsed, or half-life for radioactive decay or any exponential decrease. N = N₀ × (1/2)^(t/t½).
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How to use this calculator
N₀ is the initial quantity, t is the time elapsed, t½ is the half-life period, and N is the remaining quantity. Each half-life period reduces the quantity by half.
- 1
Select what you want to calculate: remaining quantity, time elapsed, or the half-life itself.
- 2
Enter the initial quantity N₀ and the other known values.
- 3
The result, percent remaining, half-lives elapsed, and the decay constant λ are shown.
Frequently asked questions
Which fields are used in each mode?
Remaining mode uses N₀, half-life and time (the N field is ignored). Time mode uses N₀, N and half-life (time is ignored). Half-life mode uses N₀, N and time (the half-life field is ignored). Fields not needed for the chosen mode can hold anything.
Do time and half-life need the same unit?
Yes. The calculator has no unit conversion, so 10,000 years against a half-life of 5,730 years works but 10,000 years against a half-life in days does not. The answer in time mode comes back in the half-life’s unit.
Why am I told “Remaining N must be between 0 and N₀”?
Time and half-life modes need 0 < N < N₀ because the formula models decay. N equal to N₀ means no time has passed, N above N₀ would be growth, and N = 0 would take infinite time. For zero elapsed time use remaining mode with t = 0.
What is the decay constant λ?
λ = ln 2 ÷ t½, the fractional rate of decay in the same time unit. For a half-life of 5 it is 0.138629. N then also equals N₀·e^(−λt). Very small values switch to exponential notation, for instance 1.20968e-4 for 5,730.
Can time be negative or zero?
In remaining mode zero is allowed and returns N₀ unchanged; negative time returns “Time must be non-negative”. In half-life mode time must be strictly positive. The half-life itself must always be positive.
Does it work for things other than radioactivity?
Yes, for any process that loses the same fraction per period, such as first-order drug elimination or a cooling excess temperature. It does not fit decay that is linear or that levels off, and it cannot model growth or doubling.
Does it predict when one atom will decay?
No. The formula gives the expected amount for a large population. For a handful of atoms, the actual count fluctuates around it, and the formula says nothing about a single nucleus.
How are tiny or huge numbers displayed?
Normal values are rounded to 6 decimals. Anything below 0.001 or above one billion is shown in exponential notation, so a long half-life or a nearly gone sample does not collapse to 0.
Half-life calculator — remaining amount, elapsed time or half-life
The decay equation
Exponential decay follows N = N₀ × (1/2)^(t ÷ t½): the amount halves every t½. The calculator rearranges that for the unknown you pick. Remaining mode finds N. Time mode finds t = t½ × ln(N/N₀) ÷ ln(1/2). Half-life mode finds t½ = t × ln(1/2) ÷ ln(N/N₀). It also reports percent remaining, half-lives elapsed and the decay constant λ = ln 2 ÷ t½.
| Half-lives elapsed | Fraction left | Percent left |
|---|---|---|
| 1 | 1/2 | 50% |
| 2 | 1/4 | 25% |
| 3 | 1/8 | 12.5% |
| 4 | 1/16 | 6.25% |
| 5 | 1/32 | 3.125% |
Worked examples
Each of these was run through the calculator. Starting at 1000 with a half-life of 5 and 15 elapsed, three halvings leave 125 (12.5%). Starting at 80 with a half-life of 8 after 20, that is 2.5 half-lives, leaves 14.142136, or 17.67767%. A sample of 500 with the commonly quoted carbon-14 half-life of 5,730 years, after 10,000 years, has 149.146218 left (29.829244%, 1.745201 half-lives).
Solving for time or half-life
To find how long 1000 units take to fall to 100 with a half-life of 8, use time mode: the answer is 26.575425 units, which is 3.321928 half-lives. Going the other way, if 200 become 150 in 10 time units, half-life mode returns t½ = 24.094208 and λ = 0.028768. Time mode also gives clean answers: a drop from 100 to 6.25 with t½ = 3 returns exactly 12, four half-lives.
Very long half-lives
When the half-life is huge compared with the time, the sample barely changes. Take 1 unit of a substance with a half-life of 4,468,000,000 years (roughly that of uranium-238) over 1,000 years: 99.999984% remains, and the decay constant appears as 1.55136e-10 per year. The tool switches to scientific notation precisely for results like this.
Assumptions and boundaries
The calculator assumes a constant fractional decay rate, one unit throughout, and an average population large enough for statistics to hold. It does not include decay chains, multiple isotopes or growth. Make sure the time and half-life use the same unit, and enter the half-life you trust for your material, as the tool does not look values up.
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