Every radioactive nucleus has a fixed probability per second of decaying, the decay constant λ. In a sample of N nuclei, λN of them decay each second on average – that number is the activity A, measured in becquerels (1 Bq = one decay per second) or in the older unit curie (1 Ci = 3.7 × 10¹⁰ Bq, roughly the activity of one gram of radium-226).
To find the activity of a real sample you need to know how many radioactive atoms it contains, which follows from its mass and molar mass via Avogadro’s number. Short-lived isotopes are fiercely active per gram: one gram of iodine-131 (half-life 8 days) emits about 4.6 × 10¹⁵ decays per second, while a gram of uranium-238 (4.5 billion years) manages only 12 400. That ratio – the specific activity – is why medical isotopes are used in nanogram amounts while uranium can be handled with gloves.
Because activity is proportional to the number of atoms left, it falls exponentially with the same half-life. The calculator therefore also projects the activity and remaining mass after a chosen time, the number of decays in between, and – run in reverse – the mass you need for a target activity, which is how radiopharmacies and calibration-source makers plan their sources.
How to use the Radioactive Decay Calculator
- Isotope
- Half-life
- Molar (atomic) mass
- Sample mass
- Time later
- Want to work backwards? Type a value into Activity A and Sample mass is calculated for you.
Results update as you type, every field has its own unit menu, and everything is calculated in your browser.
How it works
Choose an isotope from the list (half-life and atomic mass are filled in) or enter your own, then enter the sample mass and a time span. The calculator converts mass to atoms, multiplies by λ to get the activity, and applies e^(−λt) for the later activity and mass. Type a target activity into the activity result to find the mass that gives it.
Worked example
1 mg of caesium-137 (T½ = 30.08 years, M = 136.907 g/mol): N = 0.001 / 136.907 × 6.02214 × 10²³ = 4.399 × 10¹⁸ atoms; λ = 0.69315 / (30.08 × 31 557 600 s) = 7.302 × 10⁻¹⁰ s⁻¹; A = λN = 3.212 × 10⁹ Bq (3.21 GBq or 86.8 mCi). After 10 years the activity is 3.212 × (½)^(10/30.08) = 2.55 GBq.
Isotope data used
| Isotope | Half-life | Atomic mass (g/mol) | Specific activity |
|---|---|---|---|
| Carbon-14 (¹⁴C) | 5730 years | 14.0032 | 1.649e+11 Bq/g |
| Tritium (³H) | 12.32 years | 3.0160 | 3.56e+14 Bq/g |
| Fluorine-18 (¹⁸F) | 1.829 h | 18.0009 | 3.521e+18 Bq/g |
| Phosphorus-32 (³²P) | 14.27 days | 31.9739 | 1.059e+16 Bq/g |
| Potassium-40 (⁴⁰K) | 1.248 × 10^9 years | 39.9640 | 2.652e+05 Bq/g |
| Cobalt-60 (⁶⁰Co) | 5.271 years | 59.9338 | 4.187e+13 Bq/g |
| Technetium-99m (⁹⁹ᵐTc) | 6.007 h | 98.9063 | 1.952e+17 Bq/g |
| Iodine-131 (¹³¹I) | 8.025 days | 130.9061 | 4.599e+15 Bq/g |
| Caesium-137 (¹³⁷Cs) | 30.08 years | 136.9071 | 3.212e+12 Bq/g |
| Polonium-210 (²¹⁰Po) | 138.4 days | 209.9829 | 1.663e+14 Bq/g |
| Radon-222 (²²²Rn) | 3.824 days | 222.0176 | 5.691e+15 Bq/g |
| Radium-226 (²²⁶Ra) | 1600 years | 226.0254 | 3.658e+10 Bq/g |
| Thorium-232 (²³²Th) | 1.4 × 10^10 years | 232.0381 | 4072 Bq/g |
| Uranium-235 (²³⁵U) | 7.04 × 10^8 years | 235.0439 | 7.994e+04 Bq/g |
| Uranium-238 (²³⁸U) | 4.468 × 10^9 years | 238.0508 | 1.244e+04 Bq/g |
| Plutonium-239 (²³⁹Pu) | 2.411 × 10^4 years | 239.0522 | 2.295e+09 Bq/g |
| Americium-241 (²⁴¹Am) | 432.6 years | 241.0568 | 1.268e+11 Bq/g |
Inputs explained
| Field | What to enter |
|---|---|
| Isotope | Choose one: Carbon-14 (¹⁴C) – T½ 5730 years, Tritium (³H) – T½ 12.32 years, Fluorine-18 (¹⁸F) – T½ 1.829 h, Phosphorus-32 (³²P) – T½ 14.27 days … |
| Half-life | Units: years, s, min, h, days, ms |
| Molar (atomic) mass | Units: g/mol |
| Sample mass | Units: g, mg, µg, kg, oz, lb |
| Time later | Units: years, s, min, h, days, ms |
What each result means
- Decay constant λ = ln(2)÷([Half-life used]×31557600)
- Number of radioactive atoms N = [Sample mass]÷[Molar mass used]×6.02214076e+23
- Activity A = [Decay constant λ]×[Number of radioactive atoms N]
- Specific activity = [Decay constant λ]×6.02214076e+23÷[Molar mass used]
- Activity after the chosen time = [Activity A]×exp(-[Decay constant λ]×[Time later])
- Mass remaining after the chosen time = [Sample mass]×exp(-[Decay constant λ]×[Time later])
- Atoms decayed in that time = [Number of radioactive atoms N]×(1 – exp(-[Decay constant λ]×[Time later]))
- Fraction of activity left = (exp(-[Decay constant λ]×[Time later]))×100
Formula
N = m·N_A / M · λ = ln 2 / T½ · A = λ·N · specific activity = λ·N_A / M · A(t) = A₀·e^(−λt) · 1 Ci = 3.7 × 10¹⁰ Bq
Worked example
Inputs — Isotope: Caesium-137 (¹³⁷Cs) – T½ 30.08 years; Sample mass: 0.001 g; Time later: 10 years.
Results — Half-life used = 30.08 years; Molar mass used = 136.9071 g/mol; Decay constant λ = 7.3020e-10 1/s; Number of radioactive atoms N = 4.3987e18 atoms; Activity A = 3,211,948,894.003579 Bq; Specific activity = 3,211,948,894,003.579 Bq/g; Activity after the chosen time = 2,550,892,548.620575 Bq; Mass remaining after the chosen time = 0.000794 g; Atoms decayed in that time = 9.0530e17; Fraction of activity left = 79.4188 %.
Key terms
- Activity A
- Number of nuclear decays per second, in becquerels (Bq).
- Becquerel (Bq)
- SI unit of activity: one decay per second.
- Curie (Ci)
- Older unit of activity: 3.7 × 10¹⁰ Bq.
- Specific activity
- Activity per unit mass of a radionuclide.
- Decay constant λ
- Probability per second that a given nucleus decays.
Frequently asked questions
How do you calculate activity from mass?
A = λN = (ln 2 / T½) × (m/M) × N_A. For 1 mg of Cs-137: N = 4.40 × 10¹⁸ atoms, λ = 7.30 × 10⁻¹⁰ s⁻¹, A = 3.21 × 10⁹ Bq (3.21 GBq, 86.8 mCi).
What is specific activity?
Activity per gram of the pure isotope, λN_A/M. It is about 3.66 × 10¹⁰ Bq/g for Ra-226 – close to 1 Ci/g, the original basis of the curie – and 1.24 × 10⁴ Bq/g for U-238.
How do I convert becquerels to curies?
Divide by 3.7 × 10¹⁰: 1 MBq = 27.03 µCi, 1 GBq = 27.03 mCi, and 1 mCi = 37 MBq.
How radioactive is a banana?
A banana holds about 0.4 g of potassium; 0.0117% of it is K-40, about 47 µg, giving roughly 12–15 Bq.
How long until a source falls to 1% of its activity?
log₂(100) = 6.64 half-lives – 200 years for Cs-137, 53 days for I-131.
What is the difference between this and the half-life calculator?
This page works with real isotopes and absolute numbers (atoms, becquerels, curies); the half-life calculator is a general first-order decay solver for any amount.
Why are short-lived isotopes so active?
Activity per atom is λ, which is inversely proportional to the half-life. Tc-99m (6 h) is about 44 000 times more active per atom than Cs-137 (30 years).
Can I work backwards from the result?
Yes. Type the result you want into Activity A and the calculator finds the matching sample mass.