Radioactive Decay & Half-Life Calculator

Nuclear Chemistry & Isotope Decay Tool

Calculate remaining isotope, half-life, decay constant, and decayed amount for any radioactive element or isotope. Supports all major units and real-world scenarios.

Example Calculations

Try these real-world radioactive decay scenarios

Carbon-14 Dating of Ancient Artifact

Radiocarbon Dating

Estimate the age of an artifact using C-14 decay. Initial amount: 1000 atoms, half-life: 5730 years, elapsed time: 11460 years.

Initial Amount: 1000

Half-Life: 5730 Years (y)

Elapsed Time: 11460 Years (y)

Amount Unit: Atoms

Iodine-131 in Thyroid Treatment

Medical Isotope

Calculate remaining I-131 after 16 days. Initial: 5 mg, half-life: 8 days, elapsed: 16 days.

Initial Amount: 5

Half-Life: 8 Days (d)

Elapsed Time: 16 Days (d)

Amount Unit: Grams (g)

Uranium-238 Decay in Geology

Nuclear Power

Determine remaining U-238 after 4.5 billion years. Initial: 1 mol, half-life: 4.468e9 years, elapsed: 4.5e9 years.

Initial Amount: 1

Half-Life: 4468000000 Years (y)

Elapsed Time: 4500000000 Years (y)

Amount Unit: Moles (mol)

Cobalt-60 for Sterilization

Industrial Source

Find remaining Co-60 after 10 years. Initial: 2 kg, half-life: 5.27 years, elapsed: 10 years.

Initial Amount: 2

Half-Life: 5.27 Years (y)

Elapsed Time: 10 Years (y)

Amount Unit: Kilograms (kg)

Other Titles
Understanding Radioactive Decay: A Comprehensive Guide
Master nuclear decay, half-life, and isotope calculations with this advanced tool

What is Radioactive Decay?

  • Fundamentals of Nuclear Decay
  • Types of Radioactive Decay
  • Decay Law and Exponential Behavior
Radioactive decay is the spontaneous transformation of an unstable atomic nucleus into a more stable one, accompanied by the emission of radiation. This process follows a predictable exponential law, making it possible to calculate the remaining amount of a substance over time.
Types of Decay
The most common types of radioactive decay are alpha, beta, and gamma decay. Each type involves different particles and energy emissions, but all follow the same mathematical decay law.
Exponential Decay Law
The decay law is expressed as N(t) = N0 * exp(-λt), where N0 is the initial amount, λ is the decay constant, and t is time. This formula allows precise calculation of remaining isotope, decayed amount, and more.

Decay Law Examples

  • C-14 decays with a half-life of 5730 years
  • I-131 used in medicine decays rapidly (half-life: 8 days)
  • U-238 decays over billions of years (half-life: 4.468e9 years)

Step-by-Step Guide to Using the Radioactive Decay Calculator

  • Input Data
  • Select Units
  • Interpret Results
This calculator allows you to compute the remaining amount, decayed amount, and decay constant for any radioactive substance. Follow these steps for accurate results.
Entering Initial Data
Input the initial amount, half-life, and elapsed time. Choose the correct units for each value to ensure accurate calculations.
Selecting Units
You can select from atoms, moles, grams, or kilograms for amount, and seconds, minutes, hours, days, or years for time. The calculator automatically converts and computes based on your selections.
Interpreting Results
The results section displays the remaining amount, decayed amount, and decay constant. All values are shown in the units you selected.

Step-by-Step Examples

  • Calculate remaining C-14 after 2 half-lives (25% remains)
  • Find decay constant for I-131 (λ = 0.0866 1/d)
  • Determine decayed mass of Co-60 after 10 years

Real-World Applications of Radioactive Decay Calculations

  • Radiometric Dating
  • Medical Diagnostics & Therapy
  • Nuclear Power & Industry
Radioactive decay calculations are essential in many fields, from archaeology to medicine and energy production. Understanding decay helps date ancient artifacts, treat diseases, and manage nuclear materials.
Radiometric Dating
By measuring the remaining amount of a radioactive isotope, scientists can determine the age of rocks, fossils, and archaeological finds. Carbon-14 dating is a famous example used in archaeology.
Medical Applications
Radioactive isotopes are used in diagnostics (e.g., PET scans) and therapy (e.g., cancer treatment). Calculating decay ensures correct dosing and safety for patients.
Nuclear Power & Industry
Decay calculations are vital for managing nuclear fuel, waste, and industrial sources like Cobalt-60 used for sterilization and radiography.

Application Examples

  • Dating a fossil with C-14
  • Calculating I-131 dose for thyroid therapy
  • Managing spent nuclear fuel

Common Misconceptions and Correct Methods

  • Half-Life vs. Total Decay
  • Unit Confusion
  • Decay Constant Calculation
Many misunderstandings exist about radioactive decay, especially regarding half-life and decay constant. This section clarifies common errors and best practices.
Half-Life Does Not Mean Total Disappearance
Half-life is the time for half the substance to decay, not for it to disappear completely. After each half-life, half of the remaining substance decays, so some always remains.
Unit Consistency is Critical
Always use consistent units for half-life and elapsed time. Mixing units leads to incorrect results. The calculator helps by letting you select units for each value.
Decay Constant Calculation
The decay constant (λ) is calculated as λ = ln(2) / half-life. It represents the probability per unit time that a nucleus will decay.

Best Practice Guidelines

  • After 3 half-lives, 12.5% remains, not zero
  • Mixing years and days gives wrong results
  • λ for C-14: 0.000121 1/y

Mathematical Derivation and Examples

  • Exponential Decay Formula
  • Half-Life and Decay Constant Relationship
  • Worked Calculation Examples
The mathematics of radioactive decay is based on exponential functions. Understanding the derivation helps apply the formulas correctly in any context.
Exponential Decay Formula
N(t) = N0 * exp(-λt), where N0 is the initial amount, λ is the decay constant, and t is time. This formula describes the decrease of radioactive material over time.
Half-Life and Decay Constant
The half-life (t1/2) and decay constant (λ) are related by λ = ln(2) / t1/2. Knowing one allows calculation of the other.
Worked Examples
Example: If N0 = 1000 atoms, t1/2 = 10 days, t = 30 days, then λ = 0.0693 1/d, N = 1000 exp(-0.069330) ≈ 125 atoms remain.

Calculation Examples

  • Calculate remaining after 5 half-lives: 3.125% remains
  • Find λ for t1/2 = 8 days: λ = 0.0866 1/d
  • N(t) for N0=500, t1/2=2h, t=6h: N=62.5