Solve a constant half-life decay model in either direction, or convert between half-life, decay constant and mean lifetime using your chosen time unit.
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Check your inputs—Complete valid inputs to calculate a current result.
amount units
1e-100 – 1e+100
selected time unit
0 – 1e+100
selected time unit
1e-100 – 1e+100
Amount remaining
25 amount units
Starting amount100
Amount remaining25
Remaining percentage25 %
Half-lives elapsed2
Decay constant λ0.0693147181 per years
Mean lifetime τ14.42695041 years
Assumes a single exponential decay process with a constant half-life. Use identical amount units and one time unit. The unit selector labels your inputs; it does not convert existing values. This is not a medication dosing model.
How this was calculated
N = N₀ × 2^(−t / h) = 100 × 2^(−20 / 10) = 25.
For inverse calculations: t = h × ln(N₀ / N) ÷ ln(2); h = t × ln(2) ÷ ln(N₀ / N); N₀ = N × 2^(t / h).
The exponential decay model
Half-life is the interval over which the modeled quantity falls by half. With starting amount N₀, elapsed time t and half-life h, the remaining amount is N = N₀ × 2^(−t / h). The same time unit must be used for t and h.
For 100 starting units, a half-life of 10 years and 20 years elapsed, two half-lives have passed. The remaining amount is 100 × 2^-2 = 25 units, or 25% of the start.
Solve for a missing quantity
Rearrange the model to find time: t = h × ln(N₀ / N) / ln(2). To find half-life, use h = t × ln(2) / ln(N₀ / N). To recover the starting amount, use N₀ = N × 2^(t / h).
A unique finite half-life requires a positive elapsed time and a strictly smaller positive remaining amount. Equal amounts imply no observed decay and do not determine a finite half-life. Exponential decay does not reach exactly zero at a finite time.
Decay constant, mean lifetime and limits
The decay constant is λ = ln(2) / h and mean lifetime is τ = 1 / λ = h / ln(2). For a 10-year half-life, λ is about 0.069314718 per year and τ is about 14.426950409 years.
The time selector labels the values you enter; it does not convert them. Amount units must match. The model assumes one exponential process with a constant rate and does not model multiple isotopes, replenishment, or medication dosing. Extreme inputs that overflow or underflow are rejected.
How to use the Half-Life Calculator
Solve a constant half-life decay model in either direction, or convert between half-life, decay constant and mean lifetime using your chosen time unit.
Choose the quantity to solve
Select remaining amount, starting amount, time, half-life or a decay-parameter conversion.
Use consistent units
Choose the time label and enter the known values in that unit. Use matching units for amounts.
Read the working
Check the exponential decay equation, remaining percentage and decay parameters.