Calculate decay factors, remaining amounts, or original amounts for radioactive decay from rate, half-life, λ, time, or isotope inputs.
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Decay Factor Formula
The calculator uses one of three core relationships depending on the selected calculation.
Basic mode converts a known decay rate into a decay factor:
Amount Over Time mode applies the factor across multiple periods, or uses the continuous exponential form:
Half-Life / λ mode converts between half-life, decay constant, and the per-period factor:
- f — decay factor (fraction remaining after one period)
- r — decay rate per period (as a decimal)
- N0 — initial amount or activity
- N(t) — amount remaining after time t
- t — elapsed time, in the same units as the period or λ
- T_half — half-life
- lambda (λ) — decay constant
The formulas assume first-order decay with a constant rate. They describe a single parent quantity without replenishment or daughter contributions. The isotope presets are approximate physical half-lives; they do not estimate biological clearance, absorbed dose or clinical effects. Time units must match across inputs. If you mix percent and decimal entries, the calculator handles the conversion based on the unit you pick.
Rate to factor returns f from r. Amount over time calculates the remaining or original amount. Half-life / λ to factor calculates the remaining fraction for the selected interval. Rate/factor inputs use elapsed period counts; half-life, isotope and λ inputs use physical duration units. Date-only intervals use ordered UTC calendar dates, with each day equal to 24 hours. Original amounts cannot be uniquely recovered when the factor is zero.
Reference Tables
Common decay rates and their matching factors:
| Decay rate per period | Decay factor (f) | Remaining after 10 periods |
|---|---|---|
| 1% | 0.99 | 90.4% |
| 5% | 0.95 | 59.9% |
| 10% | 0.90 | 34.9% |
| 20% | 0.80 | 10.7% |
| 50% | 0.50 | 0.098% |
Half-life and decay constant for common isotopes:
| Isotope | Half-life | λ | Daily factor |
|---|---|---|---|
| Tc-99m | 6.01 hours | 0.1153 / hr | 0.06279 |
| I-131 | 8.02 days | 0.0864 / day | 0.9172 |
| Co-60 | 5.271 years | 0.1315 / yr | 0.99964 |
| Cs-137 | 30.04 years | 0.02307 / yr | 0.999937 |
| C-14 | 5700 years | 1.22e-4 / yr | ≈ 1.000 |
Worked Examples and FAQ
Example 1: Rate to factor. A population drops 7% per year. The decay factor is 1 − 0.07 = 0.93. After 5 years, the share remaining is 0.93^5 = 0.696, or 69.6%.
Example 2: Isotope activity. An I-131 sample reads 200 mCi today. After 16 days (about two half-lives), the activity is 200 × 0.5^(16/8.02) = 200 × 0.25087 ≈ 50.17 mCi.
Example 3: Back-solving. A Tc-99m vial reads 12 mCi at 2:00 PM and you want the activity at 8:00 AM that morning (6 hours earlier, about one half-life). Using the approximate 6.01-hour physical half-life, original = 12 / 0.5^(6/6.01) ≈ 23.97 mCi.
Is decay factor the same as decay rate? No. The rate is the fraction lost each period. The factor is the fraction kept. They sum to 1.
When should you use λ instead of half-life? Use λ when your data comes from continuous exponential models or when the time step does not divide evenly into the half-life. The two are interchangeable through λ = ln(2) / T_half.
Can the factor be greater than 1? Not for decay. A factor above 1 describes growth, which this calculator does not model.
Why does my answer use scientific notation? Very long elapsed times produce factors that round to extremely small numbers. Scientific notation displays representable small values. Intervals or inverse amounts outside floating-point range are rejected rather than reported as misleading zeros or infinite values.
