Library / Foundations: Physics / FPH-40
Foundations: Physics
Radioactive decay and half-life
Activity remaining after a given time from the half-life.
- Ref
- FPH-40
- Rev
- v1
- Field
- Foundations: Physics
- Result
- A [kBq]
λ is the decay constant in s⁻¹. Activity A is in Bq. Subtract background from measured count rates before comparing decay fractions. The calculator uses A = A₀ × 2^(−t/t½).
Radioactive material halves in activity every half-life, so activity falls fast at first and then slowly.
Calculator
- Decay constant λ0.1155 h⁻¹
- Fraction remaining6.25 %
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- After 10 half-lives less than 0.1% remains.
- Decay is random for each nucleus but follows this smooth rule for large numbers of atoms.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (kBq) | Local step | Change in result |
|---|---|---|---|---|---|---|
| A₀ Initial activity | 1 to 100,000 kBq | 0.0625 | 6,250 | 6,250 | +10% | +10 % |
| t½ Half-life | 0.1 to 1,000 h | 5.66e-71 | 98.35 | 98.35 | +10% | +28.7 % |
| t Elapsed time | 0.5 to 200 h | 9.24e-9 | 94.39 | 94.39 | +10% | −24.2 % |
Sweep of Elapsed time
| Elapsed time (h) | Activity remaining (kBq) |
|---|---|
| 0.5 | 94.39 |
| 1.057 | 88.5 |
| 2.236 | 77.23 |
| 4.729 | 57.91 |
| 10 | 31.5 |
| 21.15 | 8.69 |
| 44.72 | 0.5705 |
| 94.57 | 0.001799 |
| 200 | 9.24e-9 |
Sweep one variable
Variables and how each one contributes
Activity remaining
What the equation gives you. Everything else feeds it.
Initial activity
Activity at t = 0. Directly proportional.
Half-life
Time for the activity to halve. A longer half-life slows the decay.
Elapsed time
Time since t = 0. Each half-life halves the activity.
Worked example with the values above
- Number of half-lives: t/t½ = 24 / 6 = 4.
- Remaining fraction: 2−4 = 0.0625.
- A = 100 × 0.0625 = 6.25 kBq.
Assumptions and limits
- A single isotope with no daughter activity.
- Background subtracted.
Common mistakes
- Using minutes for t but hours for the half-life.
- Subtracting instead of multiplying by 2^(−n).
Statistics and uncertainty
Radioactive counts follow Poisson statistics: a count of N has an uncertainty of about √N, so longer counting times improve accuracy.
Where it is used
- Radiological safety and source storage.
- Dating and tracer work.
Maths you need
- Exponentials: e^(−x) and 2^(−n)
- Half-life: time to halve