Decay Calculator

Radioactive Decay & Half-Life | N = N₀ × e^−λt | Nuclear Physics Tool

Result

📝 Step-by-step solution

Select what to calculate and enter values

💡 N = remaining amount | N₀ = initial amount | λ = decay constant (per unit time) | t = elapsed time | T½ = half-life

📖 What is Radioactive Decay?

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. The rate of decay is exponential — meaning the amount of radioactive material decreases by a constant fraction over equal time intervals. This calculator uses the standard exponential decay equation:

N = N₀ × e^−λt

Where:
• N = Remaining amount after time t
• N₀ = Initial amount
• λ (lambda) = Decay constant (per unit time)
• t = Elapsed time
• e = Euler's number (≈ 2.71828)

🔁 Related Formulas

Decay Constant: λ = ln(2) / T½
Half-Life: T½ = ln(2) / λ
Elapsed Time: t = [ln(N₀/N)] / λ
Remaining Fraction: N/N₀ = e^−λt

📝 Worked Example

Example: Carbon-14 Dating (Half-Life = 5730 years)

Given: Half-life T½ = 5730 years | Initial C-14 N₀ = 100% | Remaining N = 25%

Step 1: λ = ln(2) / 5730 = 0.00012097 per year
Step 2: t = [ln(100/25)] / 0.00012097
Step 3: t = [1.3863] / 0.00012097
Step 4: t ≈ 11,460 years (two half-lives)

💡 Explanation: If 25% of C-14 remains, the sample has undergone 2 half-lives (100% → 50% → 25%), which equals 2 × 5730 = 11,460 years.

🎯 Common Half-Life Values

Carbon-14 (¹⁴C): 5,730 years — archaeology / carbon dating
Uranium-238 (²³⁸U): 4.47 billion years — geology / rock dating
Potassium-40 (⁴⁰K): 1.25 billion years — geological dating
Iodine-131 (¹³¹I): 8.02 days — medical imaging
Technetium-99m (⁹⁹ᵐTc): 6.01 hours — medical scans
Fluorine-18 (¹⁸F): 109.7 minutes — PET scans

💡 Real-World Applications

Carbon Dating: Determining the age of archaeological artifacts up to ~50,000 years.
Medical Imaging: Radioisotopes with short half-lives used in PET and SPECT scans.
Nuclear Waste Management: Calculating safe storage times for radioactive materials.
Geological Dating: Determining the age of rocks and Earth's formations.
Radiometric Dating: Using decay chains to date fossils, minerals, and meteorites.
Radiation Safety: Estimating dose decay over time for medical and industrial use.

⚠️ Limitations & Assumptions

📌 Important Notes

Assumes a constant decay constant λ throughout the time period.
Assumes no external source of the radioactive isotope (closed system).
Does not account for radiogenic daughter products that may also decay.
Carbon-14 dating is only reliable up to ~50,000 years.
Results are theoretical estimates — real-world samples may have contamination.
Units of λ and t must match (e.g., both in years, or both in seconds).

📚 Sources & References

Authoritative Sources

NIST — Atomic and Nuclear Physics Data
IAEA — Radioactive Decay and Half-Life Tables
OpenStax — University Physics, Volume 3 (Nuclear Physics)
US EPA — Radioactive Decay Basics
IAEA — Nuclear Data Services (NuDat)

❓ Frequently Asked Questions

What is half-life?

Half-life (T½) is the time required for half of a radioactive substance to decay. It's a constant for each isotope and is independent of the initial amount or external conditions (temperature, pressure, chemical state).

What is the difference between decay constant and half-life?

Decay constant (λ) is the probability per unit time that a given nucleus will decay. Half-life (T½) is the time for half the sample to decay. They are related by: λ = ln(2) / T½.

How does carbon dating work?

Living organisms continuously absorb carbon-14 (¹⁴C) from the atmosphere. When they die, the ¹⁴C begins to decay with a half-life of 5,730 years. By measuring the remaining ¹⁴C, scientists can estimate the age of the sample — up to about 50,000 years.

Why does decay follow an exponential curve?

Because each nucleus has a fixed probability of decaying per unit time, the number of decays per second is proportional to the current number of nuclei. This leads to an exponential decay equation N = N₀ × e^−λt.

Can I use this for medical radioisotopes?

Yes — the same formula applies to medical isotopes like Iodine-131 (8.02 days) or Technetium-99m (6.01 hours). Simply enter the half-life in the same units as your time.

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⚠️ Disclaimer

This Decay Calculator is provided for informational and educational purposes only. Results are based on the standard exponential decay formula and assumed constant decay constants. Real-world samples may be affected by contamination, external radiation sources, or measurement uncertainty. For medical, industrial, or safety-critical applications, always consult a qualified nuclear physicist, radiation safety officer, or authoritative data source.