Half-Life Calculator
Enter known values, click Calculate, and the remaining values update automatically.
Inputs
λ is shown in per sec.
T, t, and τ use the same selected time unit.
sec
sec
per sec
sec
After the first calculation, changing a value or the time unit recalculates automatically.
Step-by-step derivation
Enter your known values, then click Calculate.
References
- Remaining quantity: N(t) = N(0) × 0.5^(t / T)
- Exponential form: N(t) = N(0) × e^(−λt) = N(0) × e^(−t / τ)
- Relations: T = ln(2) / λ = τ × ln(2)
- Rearrangements: t = ln(N(0) / N(t)) / λ, λ = ln(2) / T, τ = 1 / λ
Use this Half-Life Calculator to solve exponential decay problems using the initial quantity, remaining quantity, half-life, elapsed time, decay constant, or mean lifetime.
Half-Life Formulas
The remaining quantity after a given time is:
N(t) = N0 × (1/2)t/T
- N0: initial quantity
- N(t): remaining quantity
- t: elapsed time
- T: half-life
The same decay relationship can be written using the decay constant:
N(t) = N0 × e−λt
Half-life, decay constant, and mean lifetime are related by:
λ = ln(2) ÷ T
T = ln(2) ÷ λ
τ = 1 ÷ λ
Here, λ is the decay constant and τ is the mean lifetime.
Example
Suppose a sample starts with 100 units, has a half-life of 5 days, and decays for 15 days.
N(t) = 100 × (1/2)15/5
N(t) = 100 × (1/2)3
N(t) = 12.5 units
After 15 days, or three half-lives, approximately 12.5 units remain.
Important Notes
- Half-life, elapsed time, and mean lifetime must use compatible time units.
- The decay constant uses an inverse-time unit, such as per second or per day.
- The model assumes a constant, single exponential decay rate.
- The remaining quantity halves during each completed half-life but does not instantly become zero.
- The calculator does not model multiple decay stages, radioactive decay chains, radiation exposure, shielding, or medical dosing.