Calculator
Formula used
Exponential decay: N(t) = N0 * exp(-lambda * t)
Half life definition: t1/2 = ln(2) / lambda
From two measurements: lambda = ln(N0/N) / t, then t1/2 = ln(2) / lambda
Remaining amount with half life: N = N0 * (1/2)^(t / t1/2)
Time to reach N with half life: t = t1/2 * log2(N0/N) = t1/2 * ln(N0/N) / ln(2)
How to use this calculator
- Select a calculation mode that matches your known values.
- Enter the required fields (the tips under each field help).
- Choose time units for elapsed time and for half life.
- Optionally add uncertainties for N0, N, and time.
- Click Submit to see results above the form.
- Use the CSV and PDF buttons to export your results.
Example data table
| Scenario | N0 | N | t | Computed half life |
|---|---|---|---|---|
| Sample A (two measurements) | 100 | 25 | 10 h | 5 h |
| Sample B (two measurements) | 500 | 125 | 8 day | 4 day |
| Sample C (known lambda) | - | - | - | t1/2 = ln(2)/0.01 = 69.315 s |
Half life guide and technical notes
1) What half life represents
Half life is the time required for a decaying population to drop to half its current value. In nuclear physics, the population can represent unstable nuclei, but the same math applies to chemical reactions, capacitor discharge, and attenuation processes. Because the drop is proportional to what remains, the curve is exponential rather than linear.
2) Connecting half life and the decay constant
The decay constant λ sets the pace of decay per unit time. The relationship t1/2 = ln(2)/λ means shorter half life implies larger λ. Always keep units consistent: if λ is in 1/s, then half life is in seconds; if you convert time to days or years, the calculator converts internally before computing results.
3) Using measured amounts (N0, N) and a time interval
When you have two measurements, the calculator estimates λ from λ = ln(N0/N)/t. This method assumes the only significant change is radioactive decay, so background subtraction and detector dead-time corrections should be handled before entering values. A simple consistency check is that N must be smaller than N0 over a positive time.
4) Remaining amount and number of half lives
Many experiments are easier to reason about in half-life steps. After one half life, 50% remains; after two, 25%; after three, 12.5%. The calculator uses N = N0(1/2)^{t/t1/2}, which is numerically stable and avoids rounding issues when t is large.
5) Time-to-target planning
If you know the half life and want to reach a target fraction, solve for time with t = t1/2 ⋅ log2(N0/N). This is useful for cooling times, storage planning, and sampling schedules. For example, reaching 6.25% requires four half lives, regardless of the absolute duration.
6) Data quality and uncertainty
Small errors in N0, N, or time can produce noticeable changes in λ because the calculation uses a logarithm and division by t. If you provide 1σ uncertainties for N0, N, and time, the calculator reports an estimated half-life uncertainty using basic propagation. Treat this as a quick check, not a full metrology report.
7) Typical scales you may encounter
Half lives range from fractions of a second for short-lived isotopes to billions of years for long-lived nuclides. For short half lives, use seconds or minutes to avoid rounding; for long half lives, years help readability. Internally, all computations are performed in seconds for consistency.
8) Reporting results clearly
Include the chosen mode, units, and key inputs alongside outputs such as λ, t1/2, and percent remaining. Exported CSV is ideal for lab notebooks and spreadsheets, while the PDF format is useful for sharing a fixed report snapshot. Clear units prevent costly interpretation mistakes later.
FAQs
1) What is the difference between half life and mean life?
Half life is the time for the amount to halve. Mean life is the average lifetime of a nucleus and equals 1/λ. They are related by t1/2 = ln(2)⋅(mean life).
2) Can I use this for non-radioactive processes?
Yes. Any process that follows exponential decay can use the same equations, such as discharge, attenuation, or first-order reactions, as long as the “amount” decreases proportionally to what remains.
3) Why must N be less than N0 for some modes?
The two-measurement decay model assumes monotonic decrease from decay. If N is equal to or larger than N0, the logarithm term becomes zero or negative, which contradicts the intended decay interpretation.
4) What units should I use for N0 and N?
Any consistent unit is fine: counts, mass, concentration, or number of nuclei. The calculator uses ratios like N/N0, so the unit cancels, but you must not mix different units between N0 and N.
5) How do I interpret the activity proxy output?
Activity is typically A = λN when N is the number of unstable nuclei. If your “amount” is not atoms, treat the proxy as a relative indicator, not a calibrated activity value.
6) Why do results sometimes show scientific notation?
Very small or very large values are displayed in scientific notation to preserve readable precision. You can increase or decrease the precision setting to control the number of digits shown in tables and exports.
7) Does the uncertainty feature replace a full error analysis?
No. It provides a quick, first-order estimate using basic propagation. For publication-grade work, consider background corrections, systematic effects, repeated trials, and a fit to multiple time points.