Determine how much of a radioactive substance remains after a given time period. Enter the initial quantity, the half-life, and the elapsed time to calculate remaining mass and decay metrics.
Starting amount of the substance
Time for substance to decay to 50% (years, hours, seconds, etc.)
Same units as half-life
Remaining Quantity
50.00 units
After 1.00 half-lives (5,730 time units)
Fraction Remaining50.0%
Number of Half-Lives1.000
Decay Constant (λ)0.000121
Mean Lifetime8,266.64
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What is the Half-Life Calculator?
A half-life is the time it takes for a quantity of a radioactive substance to reduce to half its original mass. This calculator uses the exponential decay formula to compute the remaining quantity, decay constant, mean lifetime, and the number of half-lives that have elapsed.
How it works
The calculator applies the radioactive decay formula: Remaining = N₀ × (0.5)^(t/t½). It also derives the decay constant λ = ln(2) / t½ and mean lifetime τ = 1/λ. These values describe how quickly a substance decays at the atomic level.
N(t) is the remaining quantity after time t; N₀ is the initial quantity; t½ is the half-life; λ is the decay constant (rate of exponential decay); τ is the mean lifetime (average time before an atom decays). The formula assumes exponential decay, which governs all radioactive isotopes.
Examples
Input
Result
Notes
N₀ = 100 g, half-life = 5,730 years (Carbon-14), elapsed time = 5,730 years
Remaining = 50 g (50%)
Exactly one half-life has passed, so 50% of the original substance remains. This is why Carbon-14 dating works up to ~60,000 years.
N₀ = 100 g, half-life = 5,730 years, elapsed time = 11,460 years
Remaining = 25 g (25%)
Two half-lives have elapsed. After each half-life, the quantity halves again, so 100 → 50 → 25.
N₀ = 50 mg, half-life = 87.7 days (Strontium-90), elapsed time = 263.1 days
Remaining = 6.25 mg (12.5%)
Three half-lives have passed (263.1 / 87.7 ≈ 3), so the amount decreases to 1/8 of the original.
How to use the Half-Life Calculator
Enter the initial quantity (N₀) of the radioactive substance in any unit (grams, milligrams, counts, etc.).
Enter the half-life of the isotope—the time it takes to decay to 50%. Use consistent time units throughout (years, days, hours, seconds).
Enter the elapsed time since the decay began, using the same time units as the half-life.
The calculator computes the number of half-lives: t / t½.
It applies the decay formula to find remaining quantity: N₀ × (0.5)^(number of half-lives).
Additional metrics (decay constant, mean lifetime, percentage remaining) are calculated for deeper analysis.
Benefits
Understand radioactive decay: See how isotopes degrade over time in a predictable, measurable way.
Carbon dating archaeology: Calculate ages of fossils and artifacts using Carbon-14 half-life (5,730 years).
Medical applications: Determine how long radioactive tracers and treatments remain active in the body.
Nuclear safety: Estimate decay timelines for spent fuel and contamination.
Educational clarity: Learn exponential decay and decay constants with real isotope values.
Tips & common mistakes
Common mistakes
Mixing time units: Always use the same unit for half-life and elapsed time, or convert beforehand.
Assuming linear decay: Radioactive decay is exponential, not linear—quantity doesn't decrease by a fixed amount each period.
Forgetting the decay constant's role: λ describes the instantaneous decay rate; higher λ means faster decay.
Misinterpreting mean lifetime: The mean lifetime τ is not the same as half-life; τ = t½ / ln(2) ≈ 1.44 × t½.
Tips
Use the decay constant (λ) to compare decay rates of different isotopes—higher λ means the isotope is less stable.
Remember that after 10 half-lives, only 0.1% remains; after 20 half-lives, essentially none is left. This is why old samples are hard to date.
For very long half-lives (like Uranium-238 at 4.5 billion years), even over human timescales, the decay is nearly imperceptible.
The mean lifetime τ is useful in nuclear physics calculations; it relates the decay constant to the average survival time of an atom.
Frequently asked questions
What is a half-life?
A half-life is the time required for a radioactive substance to decay to exactly half its original mass or activity. For example, Carbon-14 has a half-life of 5,730 years, meaning after 5,730 years, only 50% of an initial sample remains undecayed.
Why is the decay formula exponential, not linear?
Radioactive decay is a random, probabilistic process at the atomic level. Each atom has the same independent probability of decaying per unit time, leading to exponential decay. The population decreases by a constant fraction per unit time, not a constant amount.
How does this relate to carbon dating?
Carbon-14 decays with a half-life of 5,730 years. By measuring the remaining Carbon-14 in organic samples and comparing it to the expected initial amount, archaeologists can calculate how long ago the organism died. This method works accurately for samples up to ~60,000 years old.
What's the difference between decay constant and half-life?
The decay constant (λ) is the fraction of the substance that decays per unit time; it appears in the formula N(t) = N₀ × e^(−λt). Half-life (t½) is the time for 50% decay. They're related: t½ = ln(2) / λ ≈ 0.693 / λ.
Can I use this for non-radioactive decay?
Yes, the exponential decay formula applies to any exponential process: atmospheric pressure decrease with altitude, cooling of hot objects, bacterial population under antibiotics, or investment depreciation. Just interpret N₀ and t½ accordingly.
Why is mean lifetime larger than half-life?
Mean lifetime (τ = 1/λ) is the average time an individual atom survives before decay. Half-life (t½) is when 50% of a sample has decayed. Since some atoms last much longer than the median, the average (mean lifetime) is higher: τ ≈ 1.44 × t½.