Half-Life Calculator
Solve exponential decay for half-life, mean lifetime, decay constant or remaining quantity.
About Half-Life Calculator
Half-life t½ is the time for a quantity to fall to half its value. It describes any exponential decay: radioactive atoms, drug concentration in blood, carbon-14 dating, population decline. The decay constant λ = ln(2)/t½ gives the instantaneous fractional decay rate per unit time. The mean lifetime τ = 1/λ = t½/ln(2) is the average time a particle survives.
Formulas
Remaining quantity
N(t) = N₀ × (1/2)^(t/t½) = N₀ × e^(−λt)
- N₀
- Initial quantity
- N(t)
- Remaining quantity at time t
- t½
- Half-life
- λ
- Decay constant
Decay constant
λ = ln(2) / t½ ≈ 0.693 / t½
Mean lifetime
τ = 1/λ = t½ / ln(2)
Worked example
With the default values below, Half-Life Calculator returns Remaining quantity N(t) of 125.00000000. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Solve forone of Remaining quantity N(t), Half-life (t½), Elapsed time (t), Decay constant (λ)
- Remaining quantity N(t)
- Initial quantity N₀minimum 0
- 1,000
- Remaining quantity N(t)minimum 0
- 125
- Elapsed time (t)minimum 0
- 30
- Half-life (t½)minimum 0
- 10
Results
- Remaining quantity N(t)
- 125.00000000
- Decay constant (λ)
- 0.0693147181
- Mean lifetime (τ)
- 14.42695000
- Fraction remaining
- 0.12500000
- Percent remaining
- 12.5000%
- Half-lives elapsed
- 3.000000
Step by step
- 1Initial quantity N₀= 1000
- 2Elapsed time t= 30
- 3Half-life t½= 10
- 4Number of half-lives elapsed= 3
- 5Decay constant λ= 0.0693147181
- 6Remaining quantity= 125
Assumptions and limits
- The result is derived only from the 5 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 10 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
What is the difference between half-life and mean lifetime?
Half-life t½ is the time for exactly half the material to decay. Mean lifetime τ is the average time a single atom survives before decaying. They are related by τ = t½ / ln(2) ≈ 1.4427 × t½. The mean lifetime is always longer because a small fraction of atoms survive much longer than average.
After 3 half-lives, how much is left?
(1/2)³ = 1/8 = 12.5% of the original. After n half-lives, the fraction remaining is (1/2)ⁿ.
Can half-life be used for things other than radioactivity?
Yes. Drug elimination in pharmacokinetics, attenuation of a signal, population decline, and compound interest all follow the same exponential model. The calculator works with any unit of time and any unit of quantity.