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Mean Lifetime Calculator

Mean lifetime equals 1 divided by decay constant

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How It Works

The mean lifetime τ is the average time an individual atom in a radioactive sample survives before decaying. It is the reciprocal of the decay constant: τ = 1/λ. Mean lifetime is always longer than the half-life by a factor of 1/ln(2) ≈ 1.443, because the long tail of late decays pulls the average up beyond the population median.

Example Problem

A radioactive isotope has a decay constant λ = 0.05 s⁻¹. Calculate its mean lifetime.

  1. Start with the mean-lifetime formula: τ = 1 / λ.
  2. Substitute the given value: τ = 1 / 0.05 s⁻¹.
  3. Compute: τ = 20 seconds.
  4. Cross-check via half-life: t½ = ln(2)/λ ≈ 0.6931 / 0.05 ≈ 13.86 s.
  5. Verify the ratio: τ / t½ = 20 / 13.86 ≈ 1.443 ≈ 1/ln(2). Consistent.

Key Concepts

Mean lifetime answers a different question than half-life: instead of 'how long before half the sample is gone?' it asks 'on average, how long does one atom last?' For an exponential decay process, the integral 〈t〉 = ∫₀^∞ t × λe^(-λt) dt evaluates to 1/λ. The mean lifetime appears naturally in expressions for total energy released over a sample's lifetime, in particle-physics detector live-time corrections, and in any rate equation where you need the time constant of an exponential.

Applications

  • Particle physics detector design — predicting how many particles will decay inside the fiducial volume given their mean lifetime.
  • Reactor decay-heat modeling — the integrated heat release scales with τ for each fission product.
  • Pharmacokinetics analog — elimination half-life and mean residence time use the same τ = 1/k relationship.
  • Capacitor RC time-constant calculations — the same exponential math applies; τ = RC is the mean lifetime of stored charge.
  • Astrophysics — mean lifetimes of unstable nuclei govern stellar nucleosynthesis branching ratios.

Common Mistakes

  • Substituting mean lifetime for half-life in dating equations — the numerical answer will be off by a factor of ln(2) ≈ 0.693.
  • Forgetting that τ has the same time unit as the reciprocal of λ — if λ is in s⁻¹, τ is in seconds; convert before reporting.
  • Confusing 'mean' with 'median' — for exponential decay the median lifetime is the half-life, not the mean.
  • Quoting one decimal of τ when the underlying λ measurement only justifies two significant figures — propagate uncertainty.

Frequently Asked Questions

How do you calculate mean lifetime?

Take the reciprocal of the decay constant: τ = 1 / λ. If you know the half-life instead, multiply by 1/ln(2): τ = t½ / ln(2) ≈ 1.443 × t½.

What is the formula for mean lifetime?

τ = 1 / λ, where λ is the decay constant in inverse time units. Equivalently, τ = t½ / ln(2).

What is the difference between mean lifetime and half-life?

Half-life is the time for half of a sample to decay; mean lifetime is the average time one atom survives. Because exponential decay has a long tail, the mean is always larger: τ = t½ / ln(2) ≈ 1.443 × t½.

Why is mean lifetime longer than half-life?

Some atoms decay quickly, but a small fraction survive much longer than the half-life — the long tail of the exponential distribution drags the average above the median. Mathematically, integrating t × λe^(-λt) from 0 to ∞ gives 1/λ, which exceeds ln(2)/λ.

What units are mean lifetime in?

Mean lifetime carries the same time unit as the inverse of the decay constant. λ in s⁻¹ → τ in seconds; λ in yr⁻¹ → τ in years; etc. The calculator handles the conversion between time and timeInverse units automatically.

Is mean lifetime used in particle physics?

Yes — particle-physics literature usually quotes mean lifetime τ (e.g., the muon's τ ≈ 2.197 μs) rather than half-life because rate equations and decay-volume calculations use the reciprocal form 1/τ = λ directly.

Reference: Lindeburg, Michael R. 1992. Engineer In Training Reference Manual. Professional Publication, Inc. 8th Edition.

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