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.

Worked Examples

Particle Physics

What decay constant corresponds to the muon's 2.197 μs mean lifetime?

Muons created by cosmic-ray showers in the upper atmosphere have a measured mean lifetime of τ ≈ 2.197 μs in their rest frame — a textbook test of special relativity, since their measured ground-level flux is only consistent with relativistic time dilation. Invert τ = 1/λ to express the same physics as a decay constant.

  • Knowns: τ = 2.197 μs = 2.197 × 10⁻⁶ s
  • Formula: λ = 1 / τ
  • λ = 1 / (2.197 × 10⁻⁶)

λ ≈ 4.55 × 10⁵ s⁻¹

This is the rest-frame decay constant; the lab-frame value is suppressed by the Lorentz factor γ, which is why fast-moving muons reach sea level instead of decaying high in the atmosphere.

Nuclear Medicine

What is the mean lifetime of a Tc-99m radiotracer with λ = 3.21 × 10⁻⁵ s⁻¹?

Technetium-99m is dominant in SPECT imaging because its 6-hour half-life is short enough to limit patient dose and long enough to image a delivered tracer. From the decay constant, find τ — the average time an individual Tc-99m nucleus survives.

  • Knowns: λ = 3.21 × 10⁻⁵ s⁻¹
  • Formula: τ = 1 / λ
  • τ = 1 / (3.21 × 10⁻⁵)

τ ≈ 3.12 × 10⁴ s ≈ 8.66 hours

As expected, the mean lifetime is τ / t½ = 8.66 / 6.0 ≈ 1.44 times longer than the half-life — the constant 1/ln(2) ≈ 1.443 that relates τ to t½ for any exponential decay.

Radiocarbon Dating

How long is the mean lifetime of carbon-14 given its decay constant?

Carbon-14 underpins radiocarbon dating of organic material younger than about 50,000 years. Using λ ≈ 3.84 × 10⁻¹² s⁻¹ for C-14, find the mean lifetime — the average time an individual ¹⁴C nucleus persists.

  • Knowns: λ = 3.84 × 10⁻¹² s⁻¹
  • Formula: τ = 1 / λ
  • τ = 1 / (3.84 × 10⁻¹²)

τ ≈ 2.60 × 10¹¹ s ≈ 8,260 years

C-14's mean lifetime of ~8,260 years is again 1.443 × its half-life of ~5,730 years. The mean lifetime is a more natural decay scale for first-order kinetics, but radiocarbon dating papers traditionally report the half-life.

Mean Lifetime Formula

Mean lifetime is the average time an unstable atom (or particle) survives before decaying. It's the reciprocal of the decay constant:

τ = 1 / λ
τ = t½ / ln(2) ≈ t½ × 1.4427Relationship to half-life

Where:

  • τ (tau) — mean lifetime, in seconds (or any consistent time unit)
  • λ (lambda) — decay constant, units of inverse time (s⁻¹). Equal to ln(2) / t½
  • t½ — half-life, the time at which 50% of the original atoms remain

Mean lifetime is always longer than half-life by a factor of 1/ln(2) ≈ 1.4427 — because the exponential decay curve has a long tail. Particle physicists usually quote mean lifetime (τ for the muon is 2.197 µs), while nuclear medicine and radiochemistry usually quote half-life (t½ for the muon is 1.523 µs). They describe the same decay; they're just different summary statistics of the underlying exponential distribution.

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