Half-Life Calculator
Calculate the remaining quantity of a substance after radioactive or exponential decay, given its half-life.
Use the same time unit for half-life and elapsed time (e.g. both in years, or both in days).
Half-Life Calculator: Radioactive Decay Formula & Complete Guide
Written by Mathew | Financial Tools & Calculation Specialist · Last updated August 21, 2026
In two sentences: A half-life calculator determines how much of a radioactive or decaying substance remains after a given time period, using the exponential decay formula N = N₀ × (1/2)^(t/t½), where a substance’s half-life is fixed regardless of the starting quantity. This guide breaks down the formula, worked examples across multiple half-life periods, and how to rearrange the equation to solve for elapsed time or half-life itself.
What Is a Half-Life Calculator?
A half-life calculator estimates how much of a radioactive isotope, drug compound, or other exponentially decaying substance remains after a specific amount of time, based on that substance’s known half-life — the fixed time it takes for exactly half of any given quantity to decay.
The Half-Life Formula
N = N₀ × (1/2)^(t / t½)
N = remaining quantity
N₀ = initial quantity
t = elapsed time
t½ = half-life (time for half the substance to decay)
Rearranging to solve for elapsed time
t = t½ × log₂(N₀ ÷ N)
Rearranging to solve for half-life
t½ = t ÷ log₂(N₀ ÷ N)
Worked Examples
Example 1: Remaining quantity after exactly 3 half-lives
A radioactive sample starts at 800 grams, with a half-life of 10 years. How much remains after 30 years?
Number of half-lives elapsed = 30 ÷ 10 = 3
N = 800 × (1/2)³ = 800 × 0.125 = 100 grams
Example 2: Remaining quantity at a non-whole-number half-life count
The same 800-gram sample, but checking the remaining amount after 25 years instead.
Number of half-lives elapsed = 25 ÷ 10 = 2.5
N = 800 × (1/2)^2.5 = 800 × 0.1768 ≈ 141.4 grams
Notice this doesn’t fall neatly between the 3-half-life result (100g) and the 2-half-life result (200g) by simple linear interpolation — decay is exponential, not linear, so the remaining amount at the halfway point between two half-life intervals isn’t the arithmetic average of the two endpoints.
Example 3: Solving for elapsed time
A substance with a half-life of 5,730 years (the well-known half-life of Carbon-14, used in radiocarbon dating) has decayed from an original quantity down to 25% remaining. How much time has elapsed?
t = 5,730 × log₂(100 ÷ 25)
= 5,730 × log₂(4)
= 5,730 × 2
= 11,460 years
This confirms what should be intuitive: since 25% remaining means exactly two half-lives have passed (100% → 50% → 25%), the elapsed time equals exactly 2 × 5,730 years.
Example 4: Solving for the half-life itself
A substance starts at 500 mg and after 8 hours has decayed to 125 mg. Find its half-life.
t½ = 8 ÷ log₂(500 ÷ 125)
= 8 ÷ log₂(4)
= 8 ÷ 2
= 4 hours
Step-by-Step: How to Use a Half-Life Calculator
- Determine which value you’re solving for — remaining quantity, elapsed time, or the half-life itself.
- Gather the known values: initial quantity, half-life (if known), and elapsed time (if known).
- For remaining quantity: apply N = N₀ × (1/2)^(t/t½) directly.
- For elapsed time: rearrange to t = t½ × log₂(N₀/N).
- For half-life: rearrange to t½ = t ÷ log₂(N₀/N).
Why Half-Life Stays Constant Regardless of Starting Amount
A defining feature of exponential decay is that half-life doesn’t depend on how much substance you start with — whether you begin with 1 gram or 1,000 kilograms of the same isotope, exactly half of it will always decay within one half-life period. This is fundamentally different from linear processes, where the rate of change might stay constant in absolute terms; radioactive decay instead has a constant proportional rate, meaning the actual amount decaying per unit time shrinks continuously as the remaining quantity shrinks, while the percentage decaying per half-life period always stays at exactly 50%.
Real-World Applications of Half-Life Calculations
- Radiocarbon dating — archaeologists and geologists use Carbon-14’s 5,730-year half-life to estimate the age of organic materials by measuring how much C-14 remains relative to the stable Carbon-12 isotope.
- Nuclear medicine — medical imaging and treatment isotopes are selected partly based on half-life, balancing a short enough half-life to limit patient radiation exposure against a long enough one to complete the necessary procedure.
- Nuclear waste management — understanding decay timelines for different radioactive isotopes is essential for safely storing nuclear waste until radioactivity drops to safe levels.
- Pharmacology — many drugs follow similar exponential decay kinetics in the body, and understanding a drug’s biological half-life helps determine appropriate dosing intervals.
Applying a Half-Life Calculator to Real Scientific Problems
A half-life calculator becomes especially useful once you move beyond textbook exercises into problems where you’re solving for an unknown variable rather than simply plugging in all three known values.
Practical ways to use a half-life calculator
- Identify which variable is actually unknown before choosing a formula version. A half-life calculator needs the remaining-quantity version, the elapsed-time version, or the half-life version depending on which single value is missing — misidentifying this is a common source of setup errors.
- Double-check units are consistent throughout. Whether working in years, hours, or seconds, a half-life calculator’s time inputs (elapsed time and half-life itself) must use the same unit, or the ratio inside the formula becomes meaningless.
- Use radiocarbon dating problems to build intuition for the log-based formulas. Since Carbon-14 problems often involve clean fractions like 25% or 12.5% remaining, they’re a useful way to verify a half-life calculator is correctly applying the elapsed-time formula before tackling messier, non-round-number scenarios.
- Sanity-check results against the “10 half-lives ≈ negligible” rule of thumb. If a half-life calculator returns a remaining quantity that seems too large after many elapsed half-lives, or too small after very few, that’s a signal to recheck the formula setup rather than trusting the output blindly.
Frequently Asked Questions
Mathematically, no — the exponential decay formula approaches zero but never technically reaches it, since you’re always left with half of whatever remained. In practice, after roughly 10 half-lives, the remaining quantity becomes negligible (less than 0.1% of the original), which is generally considered effectively decayed for most practical purposes.
No — different isotopes of the same element (elements with the same number of protons but different numbers of neutrons) can have vastly different half-lives, ranging from fractions of a second to billions of years, since half-life depends on the specific nuclear stability of each isotope individually.
Yes — any process following the same exponential decay pattern (a constant proportional rate of decrease) can be described using half-life math, including certain drug elimination processes in pharmacology and some chemical reaction kinetics.
Half-life is the time for half a quantity to decay, while mean lifetime (sometimes called the decay constant’s reciprocal) represents the average time a single atom or particle exists before decaying — mean lifetime is mathematically related to half-life but is a slightly larger number (mean lifetime = half-life ÷ ln(2) ≈ half-life × 1.443).
Related Calculators
Solar Saving Calculator By State
Scientific Notation Calculator
Greatest Common Factor Calculator
In summary, a half-life calculator applies the exponential decay formula to determine remaining quantity, elapsed time, or half-life itself, and understanding why half-life stays constant regardless of starting amount — a defining feature of proportional decay processes — explains why this single formula works identically whether you’re dating an ancient artifact or timing a medical isotope’s decay.
About the author: Mathew is a Financial Tools & Calculation Specialist focused on building and fact-checking online calculators across science, mathematics, and technical education topics.
Note: This calculator and article are provided for general educational and informational purposes only. Always double-check results for critical academic, scientific, medical, or safety-related applications.