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Radioactive Decay Calculator

Chemistry

Calculate radioactive decay using N(t) = N₀ × e^(-λt). Find remaining quantity, activity (Bq and Ci), and decay constant from half-life for any radioactive isotope.

1,000
0
180,000,000,000

Remaining Amount N(t)

500.975
Amount Decayed
499.025
Percent Remaining (%)
50.097
Half-Life (years)
5,719.92

This calculator computes your Remaining Amount N(t), Amount Decayed, Percent Remaining (%), Half-Life (years) from the values you enter.

Inputs
Initial Amount N₀ (atoms or mass units)Decay Constant λ (per second)Time (seconds)
Outputs
Remaining Amount N(t)Amount DecayedPercent Remaining (%)Half-Life (years)

What is a Radioactive Decay?

The Radioactive Decay Calculator computes the remaining quantity of a radioactive substance at any time using the fundamental decay law N(t) = N₀ × e^(−λt), where λ is the decay constant (per second) and t is time in seconds. It also returns the amount decayed, percent remaining, and the half-life in years derived from the entered decay constant.

Radioactive decay follows first-order kinetics — the decay rate is always proportional to the number of undecayed atoms present. This produces an exponential decrease: N falls by a constant fraction in each equal time interval. The decay constant λ determines how rapidly this decrease occurs: large λ means fast decay (short-lived isotope); small λ means slow decay (long-lived isotope).

This calculator uses the decay constant formulation (λ in s⁻¹), which is the form used in nuclear physics and radiochemistry. For the equivalent half-life formulation (t₁/₂ directly), use the Half-Life Calculator. For dating applications using carbon-14's known decay constant, see the Radiocarbon Dating Calculator.

How to use this Radioactive Decay calculator

  1. Enter the Initial Amount N₀ — atoms, grams, or activity (Bq or Ci, as long as units are consistent).
  2. Enter the Decay Constant λ in per second (s⁻¹). Convert from half-life: λ = 0.6931 / t₁/₂(s). Common values: ¹⁴C: 3.84 × 10⁻¹² s⁻¹; ¹³¹I: 1.00 × 10⁻⁶ s⁻¹; ⁹⁹ᵐTc: 3.21 × 10⁻⁵ s⁻¹.
  3. Enter the Time in seconds. Convert: 1 minute = 60 s; 1 hour = 3,600 s; 1 day = 86,400 s; 1 year = 31,557,600 s.
  4. Read Remaining Amount N(t) and Percent Remaining.
  5. Check Half-Life (years) against the known isotope half-life to verify the decay constant is entered correctly.

Formula & Methodology

Radioactive decay law:

N(t) = N₀ × e^(−λt) A(t) = λ × N(t)     [activity]

Derived half-life:

t₁/₂ = ln(2) / λ = 0.6931 / λ     [in same time units as λ⁻¹]

Worked example — Cs-137 contamination:

An area was contaminated with Cs-137 (t₁/₂ = 30.17 years = 9.514 × 10⁸ s) at N₀ = 1.0 × 10¹² atoms. λ = 0.6931 / (9.514 × 10⁸) = 7.284 × 10⁻¹⁰ s⁻¹. Time elapsed = 50 years = 1.578 × 10⁹ s.

N(t) = 1.0 × 10¹² × e^(−7.284 × 10⁻¹⁰ × 1.578 × 10⁹)       = 1.0 × 10¹² × e^(−1.149)       = 1.0 × 10¹² × 0.3171       = 3.17 × 10¹¹ atoms  Percent remaining = 31.7% Amount decayed = 6.83 × 10¹¹ atoms

After 50 years — 1.66 half-lives of Cs-137 — approximately 31.7% of the original contamination remains, and 68.3% has decayed to barium-137m (stable). This calculation is directly applicable to the remediation timeline assessment after the 1957 Kyshtym disaster (USSR) and is used in planning long-term monitoring at contaminated sites worldwide.

Frequently Asked Questions

Radioactive decay is the spontaneous transformation of an unstable atomic nucleus into a more stable configuration by emitting particles or energy. The main decay modes are alpha decay (emission of a ⁴He nucleus), beta decay (emission of an electron or positron), and gamma decay (emission of high-energy photons). The decay rate at any instant is proportional to the number of radioactive nuclei present: dN/dt = −λN, where λ is the decay constant. This gives the exponential decay law N(t) = N₀ × e^(−λt).
The fundamental decay law is N(t) = N₀ × e^(−λt), where N₀ is the initial number of atoms, N(t) is the number remaining at time t, λ is the decay constant (per unit time), and e is Euler's number (≈2.71828). The decay constant λ = ln(2)/t₁/₂, where t₁/₂ is the half-life. Activity (decay rate) is A(t) = λ × N(t) = A₀ × e^(−λt), measured in Becquerels (Bq) or Curies (Ci).
The decay constant λ and half-life t₁/₂ are related by λ = ln(2)/t₁/₂ ≈ 0.6931/t₁/₂. The decay constant is the probability that any single nucleus decays per unit time — it has units of (time)⁻¹ (e.g., s⁻¹, yr⁻¹). The half-life is the time for half the nuclei to decay — it has units of time. Both contain the same information; chemists often use t₁/₂ while nuclear physicists frequently use λ. The [Half-Life Calculator](/half-life-calculator/) provides the t₁/₂ formulation.
Amount N is the number of radioactive atoms present; activity A is the rate of decay events per unit time: A = λN. Activity is what radiation detectors measure and what determines biological dose. A = λN means a short-lived isotope (large λ) with fewer atoms can have the same or higher activity than a long-lived isotope with many atoms. For example, 1 mg of ²²⁶Ra (t₁/₂ = 1600 years) has the same activity (1 mCi = 3.7 × 10⁷ Bq) as 1 mg of several nanograms of ²¹⁰Po (t₁/₂ = 138 days).
Enter the Initial Amount N₀ (atoms, grams, or any proportional unit), the Decay Constant λ in per second, and the Time elapsed in seconds. The calculator returns N(t), the amount decayed, percent remaining, and the half-life in years derived from λ. To use with a known half-life: λ = ln(2)/t₁/₂ — convert t₁/₂ to seconds first.
Carbon-14 has a half-life of 5,730 years = 5,730 × 365.25 × 24 × 3600 = 1.807 × 10¹¹ seconds. Its decay constant is λ = ln(2) / (1.807 × 10¹¹) = 3.83 × 10⁻¹² per second. This calculator's default values use this λ (3.84 × 10⁻¹² per second) and a default time of 1.8 × 10¹¹ seconds (approximately 5,700 years), giving remaining ≈ 50% — one half-life of carbon-14 decay. The [Radiocarbon Dating Calculator](/radiocarbon-dating-calculator/) specialises in age-from-activity calculations.
The Becquerel (Bq) is the SI unit of radioactivity: 1 Bq = 1 decay per second. The Curie (Ci) is the older unit: 1 Ci = 3.7 × 10¹⁰ Bq (based on the activity of 1 gram of radium-226). Common conversions: 1 μCi = 37,000 Bq; 1 MBq = 27 μCi. Medical doses in India are typically specified in MBq (megabecquerels): Tc-99m bone scan ≈ 740 MBq; I-131 thyroid ablation ≈ 1.1–3.7 GBq. AERB (India's nuclear regulator) specifies radiation limits in mSv (dose) and activity in Bq.
India operates 22 nuclear reactors (primarily pressurised heavy water reactors, PHWRs) generating about 7 GW of power. The fuel cycle involves radioactive decay at multiple stages: U-235 fission in the reactor; decay of fission products (Cs-137, Sr-90, I-131 among hundreds of fission products) in spent fuel; decay of actinides (Pu-239, Am-241) in long-lived waste. BARC (Bhabha Atomic Research Centre) and DAE (Department of Atomic Energy) manage fuel fabrication, reprocessing, and waste decay calculations for India's nuclear fleet.
In a decay chain where a long-lived parent (λ₁ ≪ λ₂) produces a short-lived daughter, secular equilibrium is reached when the daughter activity equals the parent activity: A₁ = A₂, meaning N₁λ₁ = N₂λ₂. After establishment of secular equilibrium (approximately 5–7 daughter half-lives after isolation of the parent), the daughter's activity mimics the parent's slow decline. This is exploited in technetium-99m generators (⁹⁹Mo parent, t₁/₂ = 65.9 hours; ⁹⁹ᵐTc daughter, t₁/₂ = 6.01 hours): the generator reaches secular equilibrium every 20–24 hours, allowing daily elution.
Nuclear decay rates are essentially independent of temperature, pressure, and chemical environment — this is what makes them reliable for dating and for predicting long-term behaviour. The decay constant λ is a nuclear property, not a chemical one. There is a tiny (< 1%) effect from electronic environments on some electron-capture decays (e.g., ⁷Be) due to changes in electron density at the nucleus, but for all practical purposes, radioactive decay is a constant-rate process unaffected by physical or chemical conditions.