What Is Half-Life?
An archaeologist excavates a bone fragment from a dig site in Arizona. She sends it to a lab for Carbon-14 dating. The lab reports that 25% of the original C-14 remains. How old is the bone? The answer depends on one number: the half-life of Carbon-14, which is approximately 5,730 years. Since 25% is two half-lives, the bone is about 11,460 years old.
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. Each radioactive isotope 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. Technetium-99m, used in medical imaging, has a half-life of just 6 hours.
The NIST Time and Frequency Division maintains the official US time standard and provides reference materials on radioactive decay measurements. NIST also publishes certified reference materials for radioactivity calibration used in laboratories nationwide.
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 x (1/2)^(t / t1/2)
Where N(t) is the remaining amount at time t, N0 is the initial amount, and t1/2 is the half-life. The exponent t / t1/2 tells you how many half-life periods have passed. After 1 half-life, 50% remains. After 2 half-lives, 25% remains. After 3 half-lives, 12.5% remains. The pattern continues indefinitely.
The Decay Constant
lambda = ln(2) / t1/2
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. For Carbon-14, lambda is approximately 0.000121 per year. For Technetium-99m, it is about 0.1155 per hour.
Finding Elapsed Time
t = -t1/2 x 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. For logarithmic calculations, try our Log Calculator.
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: Dr. Patel, an archaeologist at the University of Arizona, sends a wood fragment from a dig near Tucson to a radiocarbon lab. The lab reports 30% of original C-14 remaining. Using the calculator with N0 = 100, N = 30, and half-life = 5,730 years, the elapsed time is approximately 9,950 years. The wood dates to roughly 8,000 BC, placing it in the early Archaic period of North American prehistory.
Example 2: Dr. Nguyen, a clinical pharmacist at a hospital in Denver, needs to determine when a 200 mg dose of a medication with a 4-hour half-life drops below 10 mg. Using the calculator: after 4 hours 100 mg remains, after 8 hours 50 mg, after 12 hours 25 mg, after 16 hours 12.5 mg, after 20 hours 6.25 mg. The drug drops below 10 mg between 16 and 20 hours, approximately 17.3 hours. She uses this to set dosing intervals that avoid accumulation.
Example 3: A nuclear safety engineer at a facility in Idaho calculates storage requirements for Cesium-137, which has a half-life of about 30 years. Starting with 1,000 grams, after 90 years (3 half-lives) only 125 grams remain. After 300 years (10 half-lives), less than 1 gram remains. This guides decisions about how long storage facilities must remain secure. For exponential calculations, use our Exponent Calculator.
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. The IntCal20 calibration curve, published in 2020 and currently the standard, corrects for historical variations in atmospheric C-14 levels by cross-referencing tree ring records, coral data, and speleothem records. Recent research published in 2026 in the journal Radiocarbon continues to refine calibration curves for marine environments.
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. For example, caffeine has a half-life of about 5 hours in healthy adults. A 200 mg cup of coffee at 3 PM still has 100 mg active at 8 PM and 50 mg at 1 AM, which is enough to disrupt sleep in sensitive individuals.
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. Hospitals must order these isotopes carefully because they decay significantly during transport. A dose shipped at 6 AM has only 50% remaining by noon and 25% by 6 PM.
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. Always verify units before calculating.
- Assuming linear decay: Half-life follows exponential decay, not linear. After two half-lives, 25% remains, not 0%. After three half-lives, 12.5% remains, not a negative number. The amount approaches zero but never reaches it.
- Confusing half-life with full decay time: A substance technically never reaches zero through half-life decay. It approaches zero asymptotically. After 10 half-lives, about 0.1% remains, which is considered negligible for most practical purposes.
- Using the wrong initial amount: The initial amount should be measured at the starting reference point, not at creation or manufacture. For radiocarbon dating, N0 is the atmospheric C-14 level at the time the organism died, not the current level.
Limitations of This Calculator
This calculator assumes simple exponential decay with a single half-life value. It does not account for branched decay chains, where a substance decays through multiple intermediate isotopes. It does not handle biological half-life variations caused by individual metabolism, kidney function, liver function, or drug interactions. For radiocarbon dating, the calculator provides the raw decay calculation but does not apply calibration curve corrections, which are essential for accurate calendar age determination. The IntCal20 calibration curve accounts for variations in atmospheric C-14 production due to solar activity, geomagnetic field changes, and carbon cycle shifts. For complex scientific calculations, use the Scientific Calculator or the Log Calculator for manual decay constant computations.
Authoritative Research and Resources
- NIST Time and Frequency Division maintains the official US national time standard and provides reference materials on radioactive decay measurements, half-life values for standard isotopes, and calibration procedures used in scientific and medical laboratories.
- CDC National Center for Health Statistics publishes pharmacological reference data including drug clearance rates and biological half-life information used in clinical dosing guidelines.
- NIST Unit Conversion Resources provides official conversion factors for time units (seconds, hours, years) used in half-life calculations, ensuring consistency with the International System of Units (SI).