Half-Life Calculator
Calculate the remaining quantity, initial quantity, elapsed time or half-life for exponential decay. Results also show the decay constant, mean lifetime, number of half-lives and percentages remaining and decayed.
Half-life formulas
The calculator uses first-order exponential decay with a constant half-life:
N = N₀ × (1/2)t/T½
Equivalent exponential form:
N = N₀ × e−λt, λ = ln(2) / T½, τ = 1/λ
- N₀ is the initial quantity.
- N is the remaining quantity after elapsed time t.
- T½ is the half-life, λ the decay constant and τ the mean lifetime.
To solve for time, use t = T½ × log₂(N₀/N). To solve for half-life, use T½ = t / log₂(N₀/N).
How to use the calculator
- Choose the unknown you want to calculate.
- Enter the other three positive values. When solving for time or half-life, the remaining quantity must be less than the initial quantity.
- Select one quantity unit and one time unit. Both time values must use the selected time unit.
- Select Calculate and review the solved variable, decay constant, mean lifetime and percentage breakdown.
Checked examples
| Inputs | Solve for | Result |
|---|---|---|
| N₀ = 100, T½ = 8 days, t = 16 days | Remaining | 25 units |
| N = 12.5 g, T½ = 10 h, t = 30 h | Initial | 100 g |
| N₀ = 80 mg, N = 10 mg, T½ = 6 h | Time | 18 h |
| N₀ = 160, N = 20, t = 15 years | Half-life | 5 years |
| N₀ = 64 atoms, T½ = 2 s, t = 1 s | Remaining | ≈ 45.2548 atoms |
| N₀ = 1, T½ = 5730 years, t = 5730 years | Remaining | 0.5 units |
| N₀ = 200 Bq, N = 50 Bq, T½ = 4 days | Time | 8 days |
| N₀ = 500 mg, N = 400 mg, t = 3 h | Half-life | ≈ 9.319 h |
Scope and limits
- This model assumes a single first-order decay process with a constant half-life. It does not model decay chains, daughter products, changing environmental rates or biological redistribution.
- All inputs must be finite positive decimal numbers no greater than 10300. Commas are accepted as decimal separators only when no period is present.
- For time and half-life calculations, N must be below N₀. Equal quantities imply zero elapsed time and cannot determine a finite half-life.
- “Atoms” may produce a non-integer expected value because exponential decay describes an average over many identical systems.
- A year is treated as 365.2425 days for the supplementary time conversions.