Half-Life Calculator
Solve exponential decay for half-life, mean lifetime, decay constant, or remaining quantity.
Inputs
Any unit: grams, atoms, Becquerel, etc.
Must use the same time unit as elapsed time.
Remaining quantity N(t)
125.00000000
Half-life (t½)
0.00000000
Elapsed time (t)
0.00000000
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
Initial quantity N₀
= 1000
Elapsed time t
= 30
Half-life t½
= 10
Number of half-lives elapsed
t / t½ = 30 / 10
= 3
Decay constant λ
ln(2) / t½ = 0.69314718 / 10
= 0.0693147181
Remaining quantity
N₀ × (½)^(t/t½) = 1000 × 0.5^3
= 125
How it works
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)
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.