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Radioactive Decay Calculator

Chemistry

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

Reviewed by the thecalcu.com team · Last updated February 18, 2025

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.

Why Use a Radioactive Decay Calculator?

The exponential function e^(−λt) produces numbers that span many orders of magnitude, from near 1 (almost no decay) to essentially 0 (essentially complete decay), depending on how λt compares to 1. Manual calculation requires careful handling of exponents. The calculator handles this directly, and also derives the half-life from the entered λ, allowing cross-checking between the two representations.

For chemistry students, the decay law and its connections to first-order kinetics, activity (A = λN), and half-life appear across nuclear chemistry (NCERT Class 12, Chapter 15), chemical kinetics (rate constants), and JEE problems requiring conversion between different representations of decay rates.

Who Should Use This Calculator?

Nuclear chemistry students at Class 12 and undergraduate level studying the radioactive decay law and its connections to activity, half-life, and first-order kinetics.

Nuclear medicine physicists and technologists computing remaining isotope activity from initial calibrated activity and decay constant for dosimetry and quality assurance.

Radiation safety officers at nuclear facilities, hospitals, and research institutions computing decay of radioactive waste to determine appropriate storage times before disposal.

Environmental scientists computing long-term decay of radioactive contaminants in soil, water, and biological samples for risk assessment and remediation planning.

What Insights Does the Radioactive Decay Calculator Give You?

Remaining Amount N(t) is the primary output, the number of radioactive atoms (or proportional mass/activity) remaining after time t. Comparing N(t) to N₀ immediately shows the fractional survival.

Amount Decayed is N₀ − N(t), how much has decayed. Relevant for computing cumulative radiation dose (integrating decay over time) and for calculating the number of decay events for safety purposes.

Percent Remaining expresses the fractional survival as a percentage, directly comparing the remaining radiation source to its initial strength.

Half-Life (years) is back-calculated from the entered λ as t₁/₂ = ln(2)/λ, converted to years from seconds. This cross-check allows verification that the decay constant entered corresponds to the expected isotope.

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.
Show formula & methodology ↓Show less ↑

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

What is radioactive decay?
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).
What is the radioactive decay formula?
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).
What is the difference between the decay constant and half-life?
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.
What is the difference between activity and amount in radioactive decay?
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).
How do I use the Radioactive Decay Calculator?
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.
What is the decay constant of carbon-14 used in radiocarbon dating?
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.
What are Becquerels and Curies and how do they relate?
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.
How does radioactive decay relate to nuclear power in India?
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.
What is secular equilibrium in radioactive decay chains?
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.
Is radioactive decay affected by temperature, pressure, or chemical environment?
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.