What Is Half-Life?
Half-life is the time it takes for half of a substance to decay or transform into another form. The concept originated in nuclear physics to describe radioactive decay, but it applies broadly in chemistry, pharmacology, biology, and even finance.
For radioactive isotopes, each has a unique and constant half-life. Carbon-14 has a half-life of about 5,730 years. Uranium-238 has a half-life of 4.5 billion years. These fixed rates make radioactive decay useful for dating ancient materials and for medical and industrial applications.
What This Calculator Does
This calculator solves three types of half-life problems. You can find the remaining amount after a given time has passed, calculate how much time elapsed given a starting and ending amount, or determine the half-life itself from observed decay data.
- Inputs: Initial amount, half-life duration, elapsed time (or final amount for reverse calculations)
- Outputs: Remaining amount, amount decayed, number of half-lives elapsed, and decay constant
How the Calculation Works
The Decay Formula
N(t) = N₀ × (1/2)^(t / t½)
Where N(t) is the remaining amount at time t, N₀ is the initial amount, and t½ is the half-life. The exponent t / t½ tells you how many half-life periods have passed.
The Decay Constant
λ = ln(2) / t½
The decay constant lambda represents the probability per unit time that a given atom will decay. A shorter half-life means a larger decay constant, meaning the substance decays faster.
Finding Elapsed Time
t = -t½ × log₂(N / N₀)
When you know the starting and ending amounts and the half-life, this formula rearranges the decay equation to solve for time. This is the basis of radiocarbon dating.
How to Use the Calculator
- Use the "Remaining Amount" tab to find how much substance is left after a given time
- Enter the initial amount, half-life value and unit, and elapsed time and unit
- Switch to the "Find Elapsed / Half-Life" tab to work backwards from a known final amount
- Choose whether you want to find elapsed time or the half-life itself
- Enter the required values and read the result instantly
Example Calculations
Example 1: Carbon-14 Dating
A wood sample contains 25% of its original Carbon-14. The half-life of C-14 is 5,730 years. After one half-life: 50% remains. After two half-lives: 25% remains. So two half-lives have passed: 2 × 5,730 = 11,460 years since the organism died.
Example 2: Medical Dosage
A medication has a half-life of 4 hours. If you take a 200 mg dose, after 4 hours 100 mg remains, after 8 hours 50 mg remains, and after 24 hours (6 half-lives) only 200 × (0.5)^6 = 3.125 mg remains. This helps doctors determine dosing intervals.
Example 3: Nuclear Waste
Cesium-137 has a half-life of about 30 years. Starting with 1,000 grams, after 90 years (3 half-lives) only 125 grams remain. This guides decisions about how long nuclear waste storage facilities must remain secure.
Real-World Scenarios
Archaeology and Radiocarbon Dating
Scientists use the known half-life of Carbon-14 to date organic materials up to about 50,000 years old. By measuring how much C-14 remains relative to stable Carbon-12, they calculate when the organism stopped absorbing carbon, which is when it died.
Pharmacology and Drug Clearance
Every drug has a biological half-life determining how quickly the body eliminates it. Knowing this allows clinicians to schedule doses for constant therapeutic levels, avoid toxicity from accumulation, and determine how long a substance is detectable in a drug test.
Nuclear Medicine and Imaging
Medical imaging isotopes like Technetium-99m have a half-life of about 6 hours. This is long enough to complete the imaging procedure but short enough that most of the radioactivity clears from the patient's body within a day, minimizing radiation exposure.
Why This Calculation Matters
Half-life calculations are central to nuclear safety, medicine, environmental science, and archaeology. Whether determining safe storage times for radioactive waste, scheduling drug doses, or dating ancient artifacts, the ability to model exponential decay precisely has practical consequences for human health and safety.
Common Mistakes to Avoid
- Mismatched time units: The elapsed time and the half-life must be in the same unit. Mixing years and hours gives completely wrong results
- Assuming linear decay: Half-life follows exponential decay, not linear. After two half-lives, 25% remains, not 0%
- Confusing half-life with full decay time: A substance technically never reaches zero through half-life decay. It approaches zero asymptotically, getting smaller and smaller but never disappearing completely
- Using the wrong initial amount: The initial amount should be measured at the starting reference point, not at creation or manufacture