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Decay Constant Calculator

Enter the half-life in any consistent time unit.

Result

λ = 0.115525 /unit
8.6562 units
6.0000 units

How it is calculated

The tool applies λ = ln(2) ÷ T½ and τ = 1 ÷ λ = T½ ÷ ln(2), where T½ is the half-life, λ is the decay constant and τ is the mean lifetime. ln(2) is about 0.693147. The tool has no unit selector: whatever time unit you use for the half-life (seconds, hours, years) is the unit of the answer, so λ is per that unit and τ is in that unit. The labels in the output say "unit" for that reason. The half-life must be above zero. Since τ ≈ 1.4427 × T½, the mean lifetime is always about 44% longer than the half-life. The formulas apply to simple first-order exponential decay of a single species.

Worked example

Take a half-life of 6 (hours, say). The decay constant is λ = 0.693147 ÷ 6 = λ = 0.115525 /unit, meaning about 11.55% of the nuclei are lost per hour in the instantaneous-rate sense. The mean lifetime is τ = 1 ÷ 0.115525 = 8.6562 hours. The half-life itself is echoed back as 6.0000, and τ is longer than it because the average includes nuclei that survive well past the half-life.

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Common mistakes

  • Mixing time units, such as a half-life in years and then reading λ as per day. λ and τ come out in whatever unit the half-life was entered in.
  • Confusing mean lifetime with half-life. After one half-life 50% remains, but after one mean lifetime about 36.8% remains, so the two are not interchangeable.
  • Treating λ as a plain percentage lost per unit time. It is an instantaneous rate; over a full unit the fraction lost is 1 − e−λ, slightly less than λ.
  • Entering a decay constant instead of a half-life. The only input here is the half-life, and entering λ would give a meaningless answer.
  • Using the result for a mixture of isotopes or a decay chain, where several rates act together and a single λ does not describe the sample.

Sources

About this calculator

The decay constant λ describes how quickly a radioactive isotope decays: it's the probability per unit time that any given nucleus will decay, and it relates to the more commonly cited half-life through λ = ln(2)/half-life. This calculator also returns the mean lifetime τ = 1/λ, the average time a given nucleus survives before decaying, which is longer than the half-life since it's weighted by the exponential decay curve rather than the 50% mark.

Decay constants are fundamental in nuclear physics, radiometric dating and nuclear medicine. Geologists and archaeologists use known decay constants (for carbon-14, uranium-238 and other isotopes) to date rock and organic samples from measured remaining activity. In medicine, the decay constant of a radiopharmaceutical isotope like technetium-99m determines both how quickly a diagnostic tracer's signal fades and how soon a patient's radiation exposure drops to a safe level, directly shaping dosing and imaging-window decisions.

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Comments (1)

  • Editorial team

    The answer is in whatever time unit you typed, so keep that same unit when you use lambda later, for instance in a remaining-fraction formula e to the power of minus lambda times t. A useful check: after one mean lifetime about 36.8% of the nuclei remain, not 50%, which is the half-life mark. Which isotope are you working with, and would a remaining-fraction option help?

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