8,266.642584
5,730
0.00012097
per unit time
1.442695
8,266.642584
5,730
0.00012097
per unit time
1.442695
The Mean Life Calculator computes the average lifetime of a radioactive atom before it undergoes decay. While the half-life tells us when half a population has decayed, the mean life $$\tau$$ represents the statistical average time any single atom survives. Related by the factor $$\tau = t_{1/2} / \ln(2)$$, the mean life is always approximately 44.3% longer than the half-life due to the asymmetric nature of exponential decay distributions.
Mean life is particularly important in particle physics, where it characterizes the lifetime of subatomic particles, and in nuclear engineering, where it helps calculate the total number of decays over time. This calculator converts between mean life, half-life, and decay constant instantly.
The mean life is the expected value of the decay time distribution. For exponential decay with rate $$\lambda$$, the probability density function for decay at time $$t$$ is:
$$f(t) = \lambda \cdot e^{-\lambda t}$$
The mean (expected) life is the integral:
$$\tau = \int_0^{\infty} t \cdot \lambda e^{-\lambda t} \, dt = \frac{1}{\lambda}$$
Since $$\lambda = \ln(2)/t_{1/2}$$, we obtain:
$$\tau = \frac{1}{\lambda} = \frac{t_{1/2}}{\ln(2)} \approx 1.4427 \times t_{1/2}$$
An important physical interpretation: the mean life is the time at which the number of remaining atoms has decreased to $$1/e \approx 36.8\%$$ of the original value, compared to 50% at the half-life.
The Mean Life (τ) is the average survival time of an atom in the sample. If you could track each atom individually, the average of all their lifetimes would converge to this value. The Ratio τ/t½ should always equal $$1/\ln(2) \approx 1.4427$$, serving as a consistency check. In particle physics, mean life is the standard reported quantity because it directly relates to the decay width $$\Gamma = \hbar/\tau$$.
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Carbon-14 with a half-life of 5,730 years has a mean life of about 8,267 years. The average C-14 atom survives 44.3% longer than the median atom.
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Radon-222 with λ = 0.1814 day⁻¹ has a mean life of about 5.51 days and a half-life of 3.82 days.
Mean life ($$\tau$$) is the average time a radioactive atom exists before decaying. It equals $$1/\lambda$$ where $$\lambda$$ is the decay constant. Unlike half-life which is the median of the distribution, mean life is the arithmetic mean of all individual atom lifetimes in a large sample.
The exponential decay distribution is right-skewed: while 50% of atoms decay before the half-life, the remaining atoms can survive much longer. These long-surviving atoms pull the average (mean life) above the median (half-life). The ratio is always $$\tau/t_{1/2} = 1/\ln(2) \approx 1.4427$$.
At time $$t = \tau$$, the fraction remaining is $$e^{-1} \approx 0.3679$$ or about 36.8%. This compares to 50% at the half-life. The mean life marks the $$1/e$$ time of the exponential decay curve.
In particle physics, mean life is the standard measure of particle stability. It relates to the decay width (energy uncertainty) through $$\Gamma = \hbar/\tau$$. Shorter-lived particles have larger decay widths, appearing as broader resonance peaks in collision experiments.
For particles with very short lifetimes, mean life can be inferred from the distance traveled before decay: $$\tau = d/(\gamma v)$$ where $$\gamma$$ is the Lorentz factor. For longer-lived isotopes, it is calculated from the measured half-life or decay constant rather than directly averaged.
A free neutron has a mean life of approximately 879.4 seconds (about 14.7 minutes), corresponding to a half-life of about 609.8 seconds (10.2 minutes). Free neutrons undergo beta decay into a proton, electron, and antineutrino.
The total number of decays over all time equals the initial number of atoms $$N_0$$. The mean life helps calculate intermediate totals: the number of decays in one mean life is $$N_0(1 - 1/e) \approx 0.632 N_0$$, or about 63.2% of all atoms that will eventually decay.
Yes, in physics "lifetime" and "mean life" are synonymous. Both refer to $$\tau = 1/\lambda$$. The Particle Data Group reports particle "lifetimes" which are mean lives. Be careful not to confuse with half-life, which is a different quantity.
The decay width $$\Gamma$$ is an energy quantity related to mean life by the uncertainty principle: $$\Gamma = \hbar/\tau$$. A short mean life implies a large energy width (uncertainty), which manifests as a broad resonance peak. For example, the Z boson has $$\tau \approx 3 \times 10^{-25}$$ s and $$\Gamma \approx 2.5$$ GeV.
Yes, mean life applies to any first-order exponential process: fluorescence lifetimes, RC circuit discharge time constants, pharmacological drug clearance, and more. In all cases, $$\tau = 1/k$$ where $$k$$ is the first-order rate constant. The mathematical framework is identical.
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