☢️ Radioactive Decay Calculator
Calculate radioactive decay using the exponential decay law N(t) = N₀ · e^(−λt). Solve for remaining quantity, initial quantity, half-life, decay constant, or elapsed time. Includes common isotope data and step-by-step solutions.
What Is Radioactive Decay?
Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation (alpha particles, beta particles, or gamma rays). This transformation changes the nucleus into a different nuclide (the daughter product), which may itself be radioactive, leading to a decay chain.
Radioactive decay is a statistical process: it is impossible to predict when a specific atom will decay, but the behavior of a large sample is governed by a precise mathematical law.
Key concepts of radioactive decay:
- Half-life (t₁/₂): The time required for half of the radioactive nuclei in a sample to decay. This is constant for each isotope and independent of external conditions.
- Decay constant (λ): The probability per unit time that a given nucleus will decay. Related to half-life by λ = ln 2 / t₁/₂.
- Activity (A): The rate of decay, measured in Becquerels (Bq) or Curies (Ci). Activity is proportional to the number of radioactive nuclei: A = λN.
- Exponential law: The number of remaining nuclei decreases exponentially with time.
The Radioactive Decay Law
The fundamental equation of radioactive decay is:
N(t) = N₀ · e^(−λt)
where N(t) = remaining nuclei at time t, N₀ = initial nuclei, λ = decay constant, t = elapsed time
The relationship between decay constant and half-life is:
λ = ln 2 / t₁/₂ = 0.693 / t₁/₂
The mean lifetime (τ) is the average time a nucleus exists before decaying:
τ = 1 / λ = t₁/₂ / ln 2 = 1.4427 · t₁/₂
Common Radioisotopes and Their Half-Lives
Different isotopes decay at vastly different rates, ranging from fractions of a second to billions of years:
- Technetium-99m (Tc-99m): 6.01 hours — used in medical imaging (SPECT scans).
- Iodine-131 (I-131): 8.04 days — used in thyroid treatment and imaging.
- Cobalt-60 (Co-60): 5.27 years — used in radiotherapy and industrial radiography.
- Cesium-137 (Cs-137): 30.1 years — a fission product; used in level gauges and brachytherapy.
- Carbon-14 (C-14): 5,730 years — used in radiocarbon dating of organic materials.
- Uranium-238 (U-238): 4.47 × 10⁹ years — the most common uranium isotope; the start of the uranium decay series.
- Potassium-40 (K-40): 1.25 × 10⁹ years — used in potassium-argon dating of rocks.
Why Use This Calculator?
- Solve for any variable: Calculate N(t), N₀, half-life, decay constant, or elapsed time.
- Isotope presets: Quickly load half-life data for common isotopes.
- Step-by-step solution: See exactly how the calculation is performed.
- Decay curve visualization: See how the sample decays over multiple half-lives.
- Free & private: No registration, no data storage — all calculations are done in your browser.
❓ Radioactive Decay Calculator FAQ
What is the radioactive decay formula?
The formula is N(t) = N₀ · e^(−λt), where N(t) is the remaining quantity, N₀ is the initial quantity, λ is the decay constant, and t is elapsed time.
How is half-life related to the decay constant?
They are related by λ = ln 2 / t₁/₂ = 0.693 / t₁/₂. The decay constant is inversely proportional to half-life: shorter half-life means larger decay constant (faster decay).
What is the difference between half-life and mean lifetime?
Half-life is the time for half the sample to decay. Mean lifetime (τ) is the average time a nucleus exists before decaying. They are related by τ = 1/λ = t₁/₂ / ln 2 ≈ 1.4427 · t₁/₂.
Does temperature affect radioactive decay?
No. Radioactive decay is a nuclear process and is essentially unaffected by temperature, pressure, chemical state, or other external conditions. This is one reason it is so useful for dating geological and archaeological samples.
What is the difference between activity and quantity?
Quantity (N) is the number of radioactive nuclei. Activity (A) is the rate of decay: A = λN. Activity is measured in Becquerels (Bq) or Curies (Ci).
How do I convert between Bq and Ci?
1 Ci = 3.7 × 10¹⁰ Bq, so 1 Bq = 2.7 × 10⁻¹¹ Ci. For example, 1 mCi = 3.7 × 10⁷ Bq = 37 MBq.
What happens after 10 half-lives?
After 10 half-lives, the fraction remaining is (1/2)¹⁰ = 1/1024 ≈ 0.1%. The sample is considered essentially decayed for most practical purposes.
Why is carbon-14 used for dating?
Carbon-14 has a half-life of 5,730 years, which is well-suited for dating organic materials up to about 50,000 years old. It is continuously produced in the atmosphere and incorporated into living organisms.
How accurate is this calculator?
This calculator uses the exact exponential decay formula. Results are accurate to the precision you select. For very large or very small numbers, scientific notation is used to maintain precision.
Is this calculator free?
Yes, this calculator is completely free to use. No registration or personal data storage is required.