Half-life And Mean Life

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A particular radioactive isotope is measured to have a half-life of 6.93 hours. Based on the relationship between half-life and average lifetime, what is the mean life of a nucleus of this isotope?

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The mean (average) life $	au$ of a radioactive nucleus and its half-life $T_{1/2}$ are related through the decay constant $\lambda$, since $	au = \dfrac{1}{\lambda}$ and $T_{1/2} = \dfrac{0.693}{\lambda}$. Eliminating $\lambda$ yields $	au = \dfrac{T_{1/2}}{0.693}$, meaning the mean life is always longer than the half-life by the factor $1/0.693 \approx 1.44$. Substituting $T_{1/2} = 6.93$ hours gives $	au = \dfrac{6.93}{0.693} = 10$ hours. The value 4.8 hours is wrong because it multiplies by 0.693 instead of dividing, making the mean life shorter than the half-life. The value 6.93 hours mistakenly equates mean life with half-life. The value 13.86 hours simply doubles the half-life, which has no physical basis. The decay constant $\lambda$ represents the probability per unit time that any single nucleus decays, and its reciprocal naturally gives the average survival time, while the half-life marks when exactly half the population has decayed, which happens earlier than the mean. This reproduces the NCERT relation between half-life and mean life. A plausibility check confirms the mean life of 10 hours exceeds the 6.93-hour half-life, exactly as the factor 1.44 requires, since a few long-lived nuclei pull the average above the median.

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