8,266.642584
time units
0.00012097
per time unit
367,879.4412
atoms
0.367879
1.442695
8,266.642584
time units
0.00012097
per time unit
367,879.4412
atoms
0.367879
1.442695
The Mean Life Calculator computes the mean lifetime (τ) of a radioactive isotope—the average time an individual atom survives before decaying. While the half-life tells you when 50% of the sample has decayed, the mean lifetime gives the statistical expectation value for the survival time of a single atom, making it a more natural parameter in many theoretical contexts.
The mean lifetime is related to the half-life by a simple factor: \(\tau = t_{1/2} / \ln 2 \approx 1.4427 \times t_{1/2}\). This means the mean lifetime is always about 44.3% longer than the half-life. The reason is that exponential decay has a long tail—while most atoms decay within a few half-lives, some atoms survive much longer than average, pulling the mean above the median.
At time \(t = \tau\), the fraction of atoms remaining is \(e^{-1} \approx 0.3679\), or about 36.8%. This is a universal constant, independent of the isotope. In contrast, at \(t = t_{1/2}\), exactly 50% remains. The mean lifetime also equals the reciprocal of the decay constant: \(\tau = 1/\lambda\).
Mean lifetime is the preferred parameter in particle physics, where unstable particles like muons (\(\tau = 2.2\) μs), pions, kaons, and B mesons are characterized by their mean lifetimes rather than half-lives. It also appears naturally in the Breit-Wigner resonance formula, the energy-time uncertainty relation (\(\Delta E \cdot \tau \geq \hbar/2\)), and in the exponential form of the decay law \(N = N_0 e^{-t/\tau}\).
In quantum optics, the mean lifetime of an excited atomic state determines the natural linewidth of spectral lines through the uncertainty principle. In nuclear physics, mean lifetimes help characterize nuclear excited states and their electromagnetic transition rates. Understanding both the mean lifetime and half-life provides a complete picture of the decay timescale.
This calculator takes the half-life as input and provides the mean lifetime, decay constant, and the number of atoms remaining at \(t = \tau\), offering a comprehensive view of the relationship between these fundamental decay parameters.
The mean lifetime is computed from the half-life:
$$\tau = \frac{t_{1/2}}{\ln 2} = \frac{1}{\lambda}$$
This can be derived by computing the expectation value of the survival time:
$$\tau = \int_0^{\infty} t \cdot \lambda e^{-\lambda t} \, dt = \frac{1}{\lambda}$$
Decay constant:
$$\lambda = \frac{\ln 2}{t_{1/2}}$$
Fraction remaining at mean lifetime:
$$\frac{N(\tau)}{N_0} = e^{-\lambda \tau} = e^{-1} \approx 0.36788$$
Ratio of mean lifetime to half-life:
$$\frac{\tau}{t_{1/2}} = \frac{1}{\ln 2} \approx 1.44269$$
The mean lifetime is always 44.3% longer than the half-life. At t = τ, about 36.8% of the original atoms remain—not 50% as at the half-life. This difference is important when estimating average decay times or when working with the natural exponential form of the decay law. In particle physics, mean lifetime determines the average distance a particle travels before decaying: d = vτ (or γvτ with relativistic time dilation).
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Carbon-14's mean lifetime is 8,267 years (compared to its half-life of 5,730 years). After one mean lifetime, about 36.8% of the original C-14 remains.
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The muon's half-life of 1.523 μs gives a mean lifetime of 2.197 μs—the well-known muon lifetime used throughout particle physics.
Half-life (t½) is the time for half the atoms to decay (the median survival time). Mean lifetime (τ) is the average survival time. Because the exponential distribution has a long right tail, τ > t½ by a factor of 1/ln(2) ≈ 1.443. Both describe the same decay process but from different statistical perspectives.
Mean lifetime appears naturally in the decay law N = N₀e^(-t/τ), in the Breit-Wigner resonance formula (Γ = ħ/τ), and in the energy-time uncertainty relation. It simplifies many theoretical expressions. Additionally, particle physicists often measure the average flight distance of particles, which directly gives the mean lifetime through d = βγcτ.
Exactly e⁻¹ ≈ 36.788% of the original atoms remain at t = τ. This is a universal constant for all exponential decay processes, regardless of the specific isotope or half-life. It follows directly from N(τ)/N₀ = e^(-λτ) = e^(-λ/λ) = e⁻¹.
The energy width (Γ) of a nuclear or particle state is related to its mean lifetime by the uncertainty principle: Γ = ħ/τ, where ħ is the reduced Planck constant. Short-lived states have broad energy widths (resonances), while long-lived states have narrow widths. This relationship is fundamental to resonance scattering theory.
Yes, mean lifetime and average lifetime are the same thing. It is the first moment (expectation value) of the decay time probability distribution p(t) = λe^(-λt). The term 'mean life' is also commonly used, particularly in nuclear physics literature.
Yes. In particle physics, the mean lifetime is measured by recording the decay times of many individual particles and computing the average (or fitting an exponential to the distribution). For nuclear isotopes, it can also be obtained from the measured half-life using τ = t½/ln(2), since half-life is often easier to measure directly.
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