Enter half life and measured activity values securely. Get decay time, constants, and unit conversions. Download reports, verify examples, and interpret results confidently today.
Radioactive decay is modeled with an exponential law. If N₀ is the initial amount and N is the amount after time t, then:
This tool accepts any consistent “amount” unit (activity, mass, counts). Only the ratios matter.
| Half life | N₀ | N | Computed decay time | Notes |
|---|---|---|---|---|
| 6 hours | 1000 | 125 | 18 hours | Three half lives: 1000 → 500 → 250 → 125 |
| 30 days | 2.5 | 0.625 | 60 days | Two half lives reduce to one quarter |
| 5 years | 800 | 100 | 15 years | Eight‑fold drop equals three half lives |
Half life (t1/2) is the time required for a radioactive sample to reduce to 50% of its current amount. After one half life, N becomes N/2; after two, N becomes N/4; after three, N becomes N/8. Because the process is exponential, the same proportional drop repeats regardless of the starting quantity.
Many lab models use the decay constant, λ, which has units of 1/time. It is linked to half life by λ = ln(2) / t1/2. For example, a 6 hour half life corresponds to λ ≈ 0.1155 h−1 (or 3.209×10−5 s−1). Smaller half life values produce larger λ and faster decay.
If you know the initial amount N₀ and a later amount N, the elapsed time is t = (1/λ) ln(N₀/N). A useful data check is the ratio N/N₀. A ratio of 0.125 equals 1/8, which indicates three half lives. With t1/2 = 6 h, the expected time is 3 × 6 = 18 h, matching the example table on this page.
To forecast activity or mass after an elapsed time, use N = N₀ e−λt. This form is convenient for continuous time steps and simulation. For instance, if t equals one half life, e−λt becomes 0.5. If t equals 10 half lives, the fraction is 2−10 ≈ 0.0009766, meaning less than 0.1% remains.
This calculator converts all time inputs to seconds internally, then converts outputs to your selected unit. Supported units include seconds, minutes, hours, days, and years (using 365.25 days per year). Keep N₀ and N in the same “amount” unit (Bq, Ci, counts, grams, or concentration). Only consistent ratios are physically meaningful.
In radiation protection work, half lives may span microseconds to thousands of years. Short half lives imply rapid activity changes during measurement; long half lives may appear nearly constant over a lab session. Reporting λ in 1/s helps when instrument timestamps are in seconds, while hours or days may be clearer for operational planning.
Real measurements include detector efficiency, background subtraction, and counting statistics. Because ln(N₀/N) amplifies small ratio errors, use well-separated measurements when estimating half life. If N is larger than N₀, the implied time is negative, indicating growth or inconsistent inputs rather than decay.
The equations assume a single radionuclide with constant λ and no additional production, branching, or chemical losses. For decay chains or mixed isotopes, use multi-exponential models. For high-dose systems, verify that detector dead time and saturation do not bias N readings.
Yes. N₀ and N can represent activity, mass, counts, or concentration. Keep both in the same unit so the ratio N/N₀ stays correct and the time result remains valid.
A negative time occurs when N is greater than N₀, so ln(N₀/N) becomes negative. That indicates growth, replenishment, or swapped values. Recheck inputs and measurement direction.
It is t / t1/2. A value of 3 means three half lives have passed and the remaining fraction should be about 1/8. It helps you quickly sanity-check results.
Pick a unit that matches your workflow. Seconds are useful for instrument logs; hours or days work well for lab planning; years are better for long-lived isotopes and storage timelines.
λ is calculated from half life using λ = ln(2) / t1/2. The calculator converts the half life to seconds first, so λ is reported in 1/s for consistency.
This tool assumes a single exponential decay. Decay chains and branching require multiple coupled equations or sums of exponentials. Use specialized models when more than one nuclide contributes.
Use longer counting times, subtract background, and avoid very small differences between N₀ and N. When N is close to N₀, the logarithm term is small and relative uncertainty becomes large.