Half-Life Calculator | Radioactive Decay & Exponential Decay

📖 How to Use This Half-Life Calculator

Select what you want to calculate - Remaining Quantity, Time Elapsed, Half-Life, or Initial Quantity.

Enter the known values - For remaining quantity: initial quantity, half-life, and time. For time: initial quantity, remaining quantity, and half-life, etc.

Click "Calculate" - Get your result with a complete step-by-step solution.

☢️ What is Half-Life?

Half-life (t½) is the time required for a quantity to reduce to half its initial value. The term is most commonly used in nuclear physics to describe radioactive decay, but it also applies to any exponential decay process like drug metabolism in the body or capacitor discharge.

The half-life formula is: N(t) = N₀ × (1/2)^(t / t½), where N(t) is the remaining quantity after time t, N₀ is the initial quantity, and t½ is the half-life.

📊 Half-Life Formulas

Remaining Quantity: N(t) = N₀ × (1/2)^(t / t½)
Time Elapsed: t = t½ × log₂(N₀ / N)
Half-Life: t½ = t × ln(2) / ln(N₀ / N)
Initial Quantity: N₀ = N / (1/2)^(t / t½)

Decay Constant: λ = ln(2) / t½ ≈ 0.693 / t½

🔬 Common Half-Life Examples

Carbon-14 (¹⁴C): t½ = 5,730 years - Used for radiocarbon dating of organic materials
Uranium-238 (²³⁸U): t½ = 4.5 billion years - Used for dating Earth's age
Plutonium-239 (²³⁹Pu): t½ = 24,100 years - Nuclear waste hazard
Iodine-131 (¹³¹I): t½ = 8 days - Medical thyroid treatment
Caffeine in body: t½ ≈ 5 hours - How long effects last
Radon-222 (²²²Rn): t½ = 3.8 days - Indoor air hazard

💡 Applications of Half-Life

Carbon Dating: Determining age of archaeological artifacts up to 60,000 years
Medical Imaging: Radioactive tracers (Technetium-99m, t½ = 6 hours)
Cancer Treatment: Radiation therapy using cobalt-60 (t½ = 5.27 years)
Nuclear Waste Management: Understanding storage requirements
Geological Dating: Determining rock and fossil ages
Pharmacology: Drug dosing intervals based on elimination half-life

❓ Frequently Asked Questions (FAQ)

How many half-lives until a substance is gone?

After 1 half-life: 50% remains; 2 half-lives: 25%; 3: 12.5%; 4: 6.25%; 5: 3.125%; 10: ~0.1%. Technically, it never reaches zero, but after about 10 half-lives (<0.1%), it's considered negligible.

What is the difference between half-life and mean life?

Half-life is the time for half to decay. Mean lifetime (τ) is the average time an atom exists before decaying. τ = t½ / ln(2) ≈ 1.44 × t½. For carbon-14, half-life = 5,730 years, mean life = 8,267 years.

What is the decay constant (λ)?

The decay constant (λ) represents the probability of decay per unit time. λ = ln(2) / t½ ≈ 0.693 / t½. It is used in the exponential decay equation: N(t) = N₀ × e^(-λt).

How accurate is carbon-14 dating?

Carbon-14 dating is accurate up to about 50,000-60,000 years with an error margin of ±40-100 years for younger samples. Calibration curves improve accuracy. Beyond 60,000 years, very little C-14 remains to measure.

Can half-life be used for non-radioactive processes?

Yes! Half-life applies to any exponential decay: drug elimination, chemical reactions, capacitor discharge, population decline, and financial depreciation.

What is the half-life of uranium-235?

Uranium-235 has a half-life of 704 million years. It's used in nuclear reactors and weapons. Natural uranium is about 0.7% U-235 and 99.3% U-238 (4.5 billion years half-life).