Understanding half-life decay
When a radioactive substance decays, its quantity shrinks by half during each half-life period. This calculator tells you how much material remains after a given time span. It's essential in nuclear physics, archaeology (carbon dating), medicine (radioisotope dosing), and environmental monitoring.
How it works
Half-life is the time required for a quantity to reduce to exactly 50% of its original amount. After one half-life, you have half. After two half-lives, you have a quarter. The decay follows an exponential curve—it never quite reaches zero, but gets progressively smaller.
To find what's left, you need three pieces of information:
- Initial quantity — the starting amount (in grams, milligrams, becquerels, or any unit)
- Half-life — how long one decay cycle takes (seconds, years, etc.)
- Elapsed time — how much time has actually passed
The calculator divides elapsed time by the half-life to find how many decay cycles have occurred, then applies the exponential decay formula.
The formula
Remaining = Initial × (0.5)^(Elapsed Time ÷ Half-Life)
Worked example
Imagine you have a medical sample of Technetium-99m (Tc-99m), commonly used in diagnostic imaging. You start with 50 milligrams.
Given:
- Initial quantity: 50 mg
- Half-life of Tc-99m: 6 hours
- Elapsed time: 18 hours
Step 1: Calculate the number of half-lives.
18 hours ÷ 6 hours = 3 half-lives
Step 2: Apply the formula.
Remaining = 50 × (0.5)³
Remaining = 50 × 0.125
Remaining = 6.25 mg
After 18 hours, only 6.25 mg of the original 50 mg sample remains—the rest has decayed into stable products.
To verify: After 6 hours (1 half-life), you'd have 25 mg. After 12 hours (2 half-lives), 12.5 mg. After 18 hours (3 half-lives), 6.25 mg. ✓
Things to watch
Time units must match. If your half-life is in years, express elapsed time in years too. If half-life is in hours, use hours for elapsed time. Mixing units is the most common source of error.
The remaining amount never truly reaches zero. Mathematically, the quantity approaches zero asymptotically but never equals it. In practice, after 10 half-lives, less than 0.1% remains, which is often negligible.
Fractional half-lives are valid. You don't need to wait for a complete cycle. If 3 hours pass and the half-life is 6 hours, that's 0.5 half-lives, and the formula handles it: Remaining = Initial × (0.5)^0.5 ≈ Initial × 0.707.
Real-world applications: Archaeologists use carbon-14 (half-life 5,730 years) to date ancient artifacts. Nuclear waste disposal plans account for half-lives of thousands of years. Hospitals calculate radioisotope doses based on decay rates to ensure patients receive safe, effective treatment. Environmental scientists track pollutants with long half-lives in soil and water.
This calculator assumes ideal, uninterrupted decay with no external factors. In reality, temperature, pressure, or chemical environment might slightly affect decay rates for certain substances, but for most practical purposes, the exponential model is accurate.