Half-Life Calculator
Calculate remaining quantity, time elapsed, or half-life for radioactive decay or any exponential decrease. N = N₀ × (1/2)^(t/t½).
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How to use this calculator
N₀ is the initial quantity, t is the time elapsed, t½ is the half-life period, and N is the remaining quantity. Each half-life period reduces the quantity by half.
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Select what you want to calculate: remaining quantity, time elapsed, or the half-life itself.
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Enter the initial quantity N₀ and the other known values.
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The result, percent remaining, half-lives elapsed, and the decay constant λ are shown.
Frequently asked questions
What is a half-life?
A half-life is the time required for exactly half of a quantity to decay, disintegrate, or otherwise decrease. After one half-life, 50% remains; after two half-lives, 25%; after three, 12.5%. The concept applies to radioactive isotopes, drug clearance in the body, capacitor discharge, and any first-order decay process.
How does carbon-14 dating use half-life?
Carbon-14 has a half-life of about 5,730 years. Living organisms continuously absorb C-14 from the atmosphere, maintaining a constant ratio of C-14 to stable C-12. When an organism dies, C-14 decays without replacement. Measuring the remaining C-14 ratio and applying N = N₀ × (1/2)^(t/5730) gives the time since death. This method is accurate for ages up to about 50,000 years.
What is the decay constant λ?
The decay constant λ = ln(2)/t½ ≈ 0.693/t½ relates to the half-life through the exponential decay formula N = N₀ × e^(−λt), which is equivalent to N = N₀ × (1/2)^(t/t½). λ represents the probability per unit time that any given nucleus decays. Higher λ (shorter half-life) means faster decay.
What other processes follow exponential decay?
Many processes follow the same mathematics: drug pharmacokinetics (plasma concentration drops with a drug half-life of hours to days), capacitor discharge in RC circuits (time constant τ = RC, related to half-life by t½ = τ·ln2), caffeine metabolism (half-life ≈ 5 hours), population decline under constant per-capita death rate, and acoustic decay (reverb time in a room).
Half-life calculator — radioactive decay and exponential decrease
The mathematics of exponential decay
Radioactive decay is a random process, but in large populations the statistical behavior is perfectly predictable: the fraction decaying per unit time is constant. This constant fractional rate leads directly to the exponential decay law N(t) = N₀e^(−λt). Rewritten in terms of half-life: N(t) = N₀ × (1/2)^(t/t½), where t½ = ln(2)/λ ≈ 0.693/λ. The half-life is a more intuitive concept than λ because it describes the decay in human-readable time units.
Half-lives across the periodic table
Half-lives span an enormous range: uranium-238 has t½ = 4.5 billion years (making it useful for dating rocks billions of years old), carbon-14 has t½ = 5,730 years (ideal for dating archaeological organic material), iodine-131 has t½ = 8 days (used in thyroid treatment — mostly gone within weeks), and polonium-214 has t½ = 164 microseconds. This range allows different isotopes to be used as geological, archaeological, or medical tools.
Biological half-life vs. physical half-life
In medicine, the "half-life" of a drug refers to how long it takes for the blood plasma concentration to halve. This depends not just on radioactive decay (for radiotracers) but also on metabolism and excretion. The effective half-life combines both: 1/t_eff = 1/t_physical + 1/t_biological. Drug dosing schedules are designed so plasma levels stay within a therapeutic window — understanding half-life determines how often and how much to dose.
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