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Radioactive Decay Calculator

Ready to calculate
A₀ = λ · N₀ · N(t) = N₀·e^(−λt).
13 Reference Isotopes.
Bq · Ci · Auto Best Unit.
100% Free.
No Data Stored.

How it Works

01Enter Sample Mass & M

Mass in 7 units (μg → metric tons + oz/lb); molar mass in g/mol or kg/mol

02Enter Half-life

9 time units from seconds to billion years — covers Tc-99m to U-238

03Compute N₀, λ, A₀

Atoms via N_A · moles · λ = ln(2)/t½ · activity A₀ = λN₀ in Bq and Ci

04(Optional) Decay Over Time

Toggle advanced for N(t) = N₀·e^(−λt) — remaining mass and activity at any t

What is a Radioactive Decay Calculator?

Radioactive decay is the spontaneous transformation of unstable atomic nuclei into more stable forms — emitting alpha particles, beta particles, gamma rays, or other ionizing radiation in the process. Every radioactive isotope has its own characteristic half-life (the time for half the atoms to decay), and the math that governs the process is one of the cleanest first-order kinetic relations in all of physics. Our Radioactive Decay Calculator computes the foundational quantities — initial atom count, decay constant, and activity — from just three inputs: sample mass, molar mass of the substance, and half-life.

Enter the mass in any of 7 units (μg, mg, g, kg, metric tons, ounces, pounds), the molar mass in g/mol or kg/mol, and the half-life in any of 9 time units (seconds through billion years — covering medical isotopes like Tc-99m at 6 hours all the way up to U-238 at 4.5 billion years). The calculator returns: the total atom count via Avogadro's number, the decay constant λ = ln(2)/t½, the initial activity A₀ = λN₀ in becquerels (SI) and curies (legacy / medical), with auto-best-magnitude display from Bq through PBq. Optionally enter an elapsed time and the calculator additionally returns the remaining mass, atom count, and activity at that moment via N(t) = N₀·e^(−λt).

A built-in 13-isotope reference library (Tc-99m, I-131, P-32, Co-60, Cs-137, Sr-90, Po-210, Pu-239, Ra-226, C-14, U-235, U-238, K-40) lets you load any well-known radioisotope with one click — molar mass and half-life auto-populate so you only need to enter the sample mass.

Pro Tip: Pair this with our Molecular Weight Calculator if you need to compute the molar mass first, or the Nernst Equation Calculator for related electrochemistry work.

How to Use the Radioactive Decay Calculator?

Enter the Sample Mass: 7 supported units — micrograms (μg), milligrams (mg), grams (g, default), kilograms (kg), metric tons (t), ounces (oz), pounds (lb). The tool normalizes to grams internally.
Enter the Molar Mass: In g/mol (standard) or kg/mol. For elemental isotopes use the isotope mass (e.g., U-235 = 235.04 g/mol, Tc-99m = 98.91 g/mol). Or use the natural-abundance atomic weight if working with a natural isotope mixture.
Enter the Half-life: 9 time units from seconds (Tc-99m short-lived medical isotope) up to billion years (U-238). Pick the unit that matches your reference source — the tool converts internally.
(Optional) Open the Time Elapsed Section: Enter how much time has passed since t = 0, in any of the 9 time units. The calculator additionally returns remaining mass, atoms, and activity at that time using the exponential decay law.
Press Calculate: Get N₀ (initial atoms via Avogadro), λ (decay constant), A₀ (activity in Bq, Ci, and best-magnitude unit from kBq to PBq), life-cycle benchmarks (50/75/99/99.9% decay times), and a 5-band activity classification (trivial → very-high). The reference-isotope table is shown alongside, with the closest-half-life isotope auto-highlighted.

How do I calculate radioactive decay?

Radioactive decay follows first-order kinetics — the rate of decay is proportional to the number of remaining atoms. From this single starting point, the entire suite of decay equations unfolds:

Think of it like compound interest in reverse: each atom independently has a fixed probability per unit time of decaying. With many atoms, the population shrinks exponentially — half are gone after one half-life, three-quarters after two, and so on.

Step 1: Number of Atoms

N₀ = (mass / molar mass) × N_A

where N_A = 6.022 × 10²³ /mol is Avogadro's number. For 1 g of U-238 (M = 238 g/mol): N₀ = (1/238) × 6.022e23 ≈ 2.53 × 10²¹ atoms.

Step 2: Decay Constant

λ = ln(2) / t½ ≈ 0.6931 / t½

λ is the probability per unit time that any given atom decays. Units: 1/seconds (s⁻¹). Short half-life → large λ → fast decay. For U-238 (t½ ≈ 4.5 billion years ≈ 1.4 × 10¹⁷ s): λ ≈ 4.9 × 10⁻¹⁸ s⁻¹ — extremely small, hence U-238's billions-of-years stability.

Step 3: Initial Activity

A₀ = λ · N₀

Activity is the number of decays per second — units of becquerels (Bq), where 1 Bq = 1 decay/second. The legacy unit is the curie (Ci), where 1 Ci = 3.7 × 10¹⁰ Bq (originally chosen as the activity of 1 gram of Ra-226). Modern radiology uses Bq; legacy literature and the US still use Ci.

Step 4: Decay Over Time

N(t) = N₀ · e^(−λt)

The number of remaining atoms decreases exponentially. Equivalently, mass and activity follow the same form: m(t) = m₀ · e^(−λt) and A(t) = A₀ · e^(−λt). After one half-life, half remains; after 10 half-lives, ~0.1%; after 20 half-lives, ~0.0001%.

Step 5: Time-to-Decay Benchmarks

Inverting the decay equation: t = −ln(N/N₀) / λ = (t½ × log₂(N₀/N)). Common benchmarks:

  • 50% decayed: 1 half-life
  • 75% decayed: 2 half-lives
  • 90% decayed: 3.32 half-lives
  • 99% decayed: 6.64 half-lives
  • 99.9% decayed: 9.97 half-lives
  • 99.99% decayed: 13.29 half-lives

The often-cited "10 half-lives = essentially gone" rule comes from the 99.9% benchmark. Useful for waste-decay planning: a Tc-99m hospital sample (t½ = 6 hours) is below detection after ~60 hours; a Cs-137 contamination (t½ = 30 years) takes ~300 years.

Real-World Example

Radioactive Decay Calculator – Activity & Decay In Practice

Consider a 10 mg sample of Tc-99m, the most-used medical imaging isotope (responsible for ~80% of nuclear-medicine procedures). Half-life: 6.01 hours. Molar mass: 98.91 g/mol.
  • Step 1: Convert inputs. Mass = 10 mg = 0.01 g. M = 98.91 g/mol. t½ = 6.01 hr = 21,636 s.
  • Step 2: Compute moles. n = 0.01 / 98.91 = 1.011 × 10⁻⁴ mol.
  • Step 3: Compute atoms. N₀ = 1.011 × 10⁻⁴ × 6.022 × 10²³ = 6.09 × 10¹⁹ atoms.
  • Step 4: Compute decay constant. λ = ln(2) / 21,636 = 3.20 × 10⁻⁵ s⁻¹.
  • Step 5: Compute initial activity. A₀ = λ × N₀ = 3.20 × 10⁻⁵ × 6.09 × 10¹⁹ = 1.95 × 10¹⁵ Bq = 1.95 PBq. Equivalent to ~52,700 Ci.
  • Step 6: "Very High Activity" band — far beyond any clinical dose. (A typical clinical Tc-99m injection is ~1 GBq, requiring ~5 ng of pure Tc-99m. 10 mg is enough for ~2 million patient doses.)

Now consider 1 kg of U-238 (depleted uranium): M = 238 g/mol, t½ = 4.468 × 10⁹ years. N₀ = (1000/238) × 6.022 × 10²³ ≈ 2.53 × 10²⁴ atoms. λ = ln(2) / (1.41 × 10¹⁷ s) ≈ 4.92 × 10⁻¹⁸ s⁻¹. A₀ = λN₀ ≈ 1.24 × 10⁷ Bq = 12.4 MBq. Despite having far more atoms than the Tc-99m example, U-238's enormous half-life makes it ~100 million times less active — which is why depleted uranium is handled as a heavy-metal toxicity hazard rather than a radiological one.

Who Should Use the Radioactive Decay Calculator?

1
Nuclear Medicine Technologists: Calculate dose decay between calibration time and patient injection time. Tc-99m loses ~10% activity per hour, so dose timing matters.
2
Radiation Safety Officers: Plan waste-storage decay times. The "10 half-lives → essentially gone" rule lets you predict when contaminated lab waste falls below regulatory thresholds.
3
Radiometric Dating: Compute remaining C-14 fraction in archaeological samples (5,730-year half-life), or U-Pb ratios in zircons for geological dating.
4
Nuclear Chemistry Students: Solve coursework problems on activity, decay constants, and time-to-decay — the most-tested topic in nuclear chemistry.
5
Health Physicists: Quick activity estimation from mass — useful for spill assessment, source inventory, transport documentation.
6
Geologists & Cosmologists: K-40 dating of igneous rocks, U-Pb dating of zircons, Th-232 dating — radioactive decay is the geological clock.

Technical Reference

Constants used:

  • N_A = 6.02214076 × 10²³ /mol (Avogadro's number, exact since 2019 SI redefinition)
  • ln(2) = 0.693147... (used in λ = ln(2) / t½)
  • 1 Ci = 3.7 × 10¹⁰ Bq (definition: activity of 1 g of Ra-226)
  • 1 Bq = 1 disintegration per second

Specific Activity (activity per unit mass): A different way to characterize radioactivity. SA = A/m = (λN_A)/M. Pure isotopes vary widely: Tc-99m has SA ≈ 1.95 × 10¹⁷ Bq/g (incredibly hot, short-lived), U-238 has SA ≈ 1.24 × 10⁴ Bq/g (cold, long-lived). The calculator computes total activity given mass — divide by mass to get SA.

Decay Modes. The calculator handles total activity regardless of decay mode:

  • α decay: emission of He-4 nucleus; common for heavy nuclei (Po-210, U-238, Pu-239).
  • β⁻ decay: neutron → proton + electron + antineutrino (P-32, C-14, Cs-137).
  • β⁺ decay / electron capture: proton → neutron + positron + neutrino (PET imaging tracers).
  • γ emission: excited nuclear state → ground state + photon (Tc-99m → Tc-99 + γ).
  • Spontaneous fission: heavy nucleus splits (Cf-252; minor branch in U-238).

Decay Chains. Many isotopes decay through multiple intermediate isotopes before reaching a stable end-product. U-238 decays through 14 intermediate isotopes ending at stable Pb-206. This calculator handles single-isotope decay only; for chains, daughter activities accumulate via Bateman equations. In secular equilibrium (parent t½ ≫ daughter t½), all isotopes in the chain have equal activity.

Why "10 half-lives ≈ done"? Pure exponential decay never reaches zero, but the fraction remaining drops below 0.1% after 10 half-lives (since (1/2)¹⁰ = 1/1024 ≈ 0.001). Common regulatory and operational rule: store radioactive waste for 10 half-lives, then handle as conventional waste. For Tc-99m (6 hr half-life), 60 hours suffices; for Cs-137 (30 yrs), 300 years.

Selected isotope half-lives:

  • Tc-99m: 6.01 hours · I-131: 8.02 days · F-18 (PET): 109.8 min · P-32: 14.29 days
  • Co-60: 5.27 years · Ir-192: 73.8 days · Cs-137: 30.05 years · Sr-90: 28.79 years
  • Po-210: 138.4 days · Ra-226: 1,600 years · C-14: 5,730 years · K-40: 1.25 × 10⁹ years
  • U-235: 7.04 × 10⁸ years · U-238: 4.468 × 10⁹ years · Pu-239: 24,110 years · Th-232: 1.405 × 10¹⁰ years

Key Takeaways

Radioactive decay is the cleanest first-order kinetic process in nature — every atom decays independently with a constant probability per unit time, leading to exponential decline of any radioactive sample. The three master equations are A₀ = λN₀, λ = ln(2)/t½, and N(t) = N₀·e^(−λt). Use the ToolsACE Radioactive Decay Calculator to compute initial activity from mass and half-life, predict remaining activity at any future time, and validate your inputs against a 13-isotope reference library covering medical, industrial, fission-product, and natural-background isotopes. The activity classification (trivial → very-high) places your result in regulatory and safety context. Bookmark it for nuclear medicine, radiation safety, radiometric dating, and nuclear chemistry coursework.

Frequently Asked Questions

What is the Radioactive Decay Calculator?
The calculator computes the foundational quantities of radioactive decay from three simple inputs: sample mass, molar mass, and half-life. Output includes initial atom count N₀, decay constant λ = ln(2)/t½, and initial activity A₀ = λN₀ in becquerels (SI) and curies (legacy/medical). With an optional elapsed-time input, it also returns remaining mass and activity at that time via N(t) = N₀·e^(−λt).

A built-in 13-isotope reference library covers medical isotopes (Tc-99m, I-131, P-32), reactor fuels (U-235, U-238, Pu-239), legacy contamination (Cs-137, Sr-90, Po-210), and natural background (C-14, K-40). Click any reference isotope to auto-load its molar mass and half-life. The activity result is auto-displayed in the most readable magnitude unit (from Bq up to PBq) and classified into 5 reference bands (trivial → very high).

Pro Tip: For more chemistry tools, try our Molecular Weight Calculator.

What's the formula for radioactive decay?
Three core equations: λ = ln(2)/t½ (decay constant from half-life), A = λN (activity from atom count), and N(t) = N₀·e^(−λt) (atoms remaining after time t). For initial conditions, N₀ comes from mass via N₀ = (mass/molar mass) × N_A. All three follow from the first-order kinetic assumption that each atom has a fixed decay probability per unit time, independent of others.
What's the difference between becquerel and curie?
Both measure activity (decays per second). 1 Bq = 1 decay/second — the SI unit, used in modern science and most countries. 1 Ci = 3.7 × 10¹⁰ Bq — the legacy unit, originally defined as the activity of 1 gram of Ra-226. The US, nuclear medicine, and older literature still use curies (mCi, μCi for clinical doses). The calculator reports both for compatibility.
What is a 'half-life'?
The time required for half of the radioactive atoms in a sample to decay. After 1 half-life, 50% remains; after 2, 25%; after 3, 12.5%; etc. The half-life is independent of starting amount, temperature, pressure, or chemical form — it's a fundamental property of each isotope. Half-lives span 30 orders of magnitude across known isotopes: from microseconds (Po-214: 164 μs) to billions of years (U-238: 4.5 Gyr).
Why is decay exponential and not linear?
Because each atom decays independently with a constant probability per unit time. With N atoms remaining, the rate of decay is dN/dt = −λN — proportional to N, not constant. Solving this differential equation gives N(t) = N₀·e^(−λt). The 'half-life' is just the time for the exponential to halve. If decay were linear, you'd lose more atoms early and have nothing left after 2 half-lives — but the actual physics gives the asymptotic exponential approach to zero.
How long until my sample is 'safe'?
The conventional rule: 10 half-lives reduces activity to ~0.1% of original (since 2¹⁰ = 1024). For most laboratory and medical contexts, this is 'essentially gone'. For high-stakes applications (nuclear waste, radiocesium contamination), you might wait 20+ half-lives (~1 ppm remains). For Tc-99m (6 hour half-life): 10 half-lives = 60 hours, 20 = 5 days. For Cs-137 (30 years): 10 half-lives = 300 years.
Does the calculator handle decay chains?
No — only single-isotope decay. Real-world isotopes like U-238 decay through a chain of 14 intermediate isotopes before reaching stable Pb-206. Daughter products contribute their own activity once they accumulate (within ~10 of their own half-lives). For chains, you'd use the Bateman equations or specialized software like ICRP databases. For routine work with a single dominant isotope, this calculator suffices.
What's 'specific activity'?
Activity per unit mass: SA = A/m, in units of Bq/g or Ci/g. Pure radioisotopes have very high SA (Tc-99m: ~1.9 × 10¹⁷ Bq/g; perfectly hot). Naturally occurring isotopes diluted in stable mass have low SA (natural uranium: ~12.4 kBq/g of total U; natural K with 0.012% K-40: ~31 Bq/g of K). Compute it from the calculator's output as A₀ / m.
How is the activity 'classification' band determined?
Five bands by total activity in becquerels: Trivial (< 1 kBq, comparable to background), Minor (1 kBq – 1 MBq, consumer items / lab tracers), Moderate (1 MBq – 1 GBq, research-scale), Significant (1 GBq – 1 TBq, industrial / clinical therapy), Very High (≥ 1 TBq, reactor-scale or industrial irradiation). These boundaries align loosely with regulatory exemption thresholds and operational practice — but specific licensing thresholds vary by jurisdiction.
What activity is in a banana? Or a person?
Bananas have ~15 Bq from natural K-40 (about 0.012% of all potassium is the radioactive K-40 isotope). A 70 kg human contains ~140 g K → ~4 kBq of K-40 plus another ~3 kBq from C-14 — total ~7 kBq of natural radioactivity. These are 'trivial' on the activity-band scale and contribute roughly 0.1 mSv/year of internal dose, far below the 1 mSv/year regulatory limit for the public.
Can I use this for radiocarbon dating?
Yes, indirectly. C-14 has a half-life of 5,730 years. Living organisms maintain a constant ratio of C-14/C-12 ≈ 1.2 × 10⁻¹². At death, C-14 begins to decay. Measure the residual ratio and solve for t using N(t) = N₀·e^(−λt). The calculator computes activity given the assumed initial; you'd compare to a measured remaining activity to back out the age. Effective dating range: 0 – 50,000 years (after which residual C-14 is below detection).

Author Spotlight

The ToolsACE Team - ToolsACE.io Team

The ToolsACE Team

Our chemistry tools team implements the foundational radioactive-decay equations: A₀ = λN₀ for initial activity, λ = ln(2)/t½ for the decay constant, and N(t) = N₀·e^(−λt) for time evolution. The 13-isotope reference library covers the most-encountered species in nuclear medicine (Tc-99m, I-131), reactor fuels (U-235, U-238, Pu-239), legacy contamination (Cs-137, Sr-90, Po-210), and natural background (C-14, K-40). Activity is reported in both becquerels (SI) and curies (legacy/medical) with auto-best-magnitude display from Bq up to PBq.

Nuclear ChemistryRadiometric DatingSoftware Engineering Team

Disclaimer

The calculator handles pure single-isotope decay only. For decay chains (e.g., U-238 → Pb-206 through 14 intermediates), daughter products contribute additional activity not captured here — use Bateman equations or specialized nuclear-chemistry software. For radiation safety, dosimetry, or licensing decisions, consult a qualified health physicist.