Half-Life Calculator
Half-life is the time required for exactly half of a substance to decay or transform, and this exponential relationship governs radioactive decay, first-order chemical reactions, and pharmacokinetics. By entering the initial amount, elapsed time, and half-life, this calculator shows both the remaining quantity and the fraction remaining.
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Formula
N(t) = N₀ × (0.5)^(t / t₁/₂)
N(t) is the amount remaining at time t. N₀ is the initial amount. t is the elapsed time. t₁/₂ is the half-life. The exponent t/t₁/₂ counts the number of half-lives that have passed. Each complete half-life reduces the remaining amount by 50%. After two half-lives, 25% remains; after three, 12.5%, and so on. Time and half-life must be in the same units for the formula to work correctly.
How to use the Half-Life Calculator
- 1
Enter your initial amount
Initial quantity in any consistent unit (g, mol, atoms, etc.)
- 2
Enter your time elapsed
Time elapsed in the same unit as half-life
- 3
Enter your half-life
Time for half the substance to decay
- 4
Read your results instantly
Results update in real time as you type.
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Half-life and exponential decay
Radioactive decay and first-order chemical reactions both obey exponential decay kinetics, which means the rate of decay at any moment is proportional to how much substance is still present. This self-similar property is what makes half-life a useful constant: no matter when you start your clock, the time for the amount to halve is always the same. Carbon-14 has a half-life of about 5,730 years — the basis of radiocarbon dating. Technetium-99m, used in medical imaging, has a half-life of just 6 hours, which limits radiation exposure in patients. Uranium-238 has a half-life of 4.47 billion years, making it useful for dating very ancient geological formations.
Applications beyond radioactive decay
The half-life concept applies broadly to any process that follows first-order kinetics. In pharmacokinetics, a drug's half-life determines dosing intervals — after approximately five half-lives, a drug is considered fully eliminated from the body (less than 3% remains). In environmental chemistry, the half-life of pollutants in soil or water determines how long remediation efforts must continue. In electronics, the RC time constant of a capacitor-resistor circuit behaves analogously. In each case, the mathematics is identical: the fraction remaining at any time equals 0.5 raised to the power of time divided by half-life, or equivalently, e raised to the power of negative lambda times time, where lambda is the decay constant ln(2)/t₁/₂.
Tips & Insights
Use consistent time units
If your half-life is in days, your elapsed time must also be in days. Mixing hours and days is a common error that changes the result by orders of magnitude.
Five half-lives means ~97% gone
After five half-lives, only (0.5)⁵ = 3.125% of the original amount remains. This is a useful rule of thumb in pharmacology and nuclear medicine for determining when a substance is effectively gone.
Convert decay constant to half-life
If you know the first-order rate constant k, the half-life is t₁/₂ = ln(2)/k ≈ 0.693/k. This lets you use published rate constants directly in this calculator.
Worked Examples
Radiocarbon dating sample
Remaining = 50.00 — after exactly one half-life of carbon-14, 50% of the original C-14 remains.
Drug elimination after 24 hours
Remaining = 12.50 — a drug with a 6-hour half-life reduces to 6.25% of the initial dose after 24 hours (4 half-lives).
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Frequently Asked Questions
What is a half-life?
A half-life is the time required for exactly half of a given quantity to decay or transform. It is a constant for any given isotope or first-order reaction — independent of how much material is present.
Does the amount remaining ever reach zero?
Mathematically, exponential decay never reaches absolute zero — the function approaches zero asymptotically. In practice, when fewer than one atom remains, the substance is considered completely decayed.
How is half-life related to the decay constant?
The decay constant λ and half-life t₁/₂ are related by t₁/₂ = ln(2)/λ ≈ 0.693/λ. A larger decay constant means a shorter half-life and faster decay.
Can I use this calculator for drug half-lives?
Yes. Enter the initial dose as the initial amount, the time since dosing as elapsed time, and the drug's half-life (available in the prescribing information). The result shows how much drug remains in the body.
What is the difference between half-life and mean lifetime?
The mean lifetime (τ) is the average time a single nucleus survives before decaying. It equals t₁/₂/ln(2) ≈ 1.443 × t₁/₂. Half-life is more commonly used in chemistry and medicine.
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