Radioactivity basics: given a radioactive nuclide with half-life T (time for activity to drop to 50%), what is its average life (mean life) expressed in terms of T?

Introduction / Context:
Understanding the relationship between half-life and average life (also called mean life) is fundamental in nuclear engineering, radiological safety, and decay calculations. Half-life T is the time required for the number of undecayed nuclei (or activity) to fall to one-half its initial value, whereas the average life τ represents the statistical average lifetime of radioactive atoms before disintegration. This question checks whether you can convert between these two commonly used time scales in decay processes using the exponential decay model.

Given Data / Assumptions:

• The decay of the nuclide follows first-order kinetics (radioactive decay law).
• Half-life is denoted by T.
• Decay constant is denoted by lambda (λ).
• Average life (mean life) is denoted by tau (τ).

Concept / Approach:
Radioactive decay obeys N(t) = N0 * exp(−λ t). Two standard identities connect λ, T, and τ: T = (ln 2) / λ and τ = 1 / λ. Eliminating λ gives τ = T / ln 2. Since ln 2 ≈ 0.693, τ ≈ 1.4427 T. For practical purposes, this is rounded to 1.44 T in hand calculations and MCQs, which is precise enough for most engineering applications.

Step-by-Step Solution:

Verification / Alternative check:
Consider two half-lives (t = 2T). The fraction remaining is 0.25. The expected average time to decay must exceed T because many atoms survive beyond one half-life; the factor 1.44 aligns with this intuition and with integral definitions of mean life using the exponential distribution.

Why Other Options Are Wrong:

• 0.144 T: off by a factor of 10; contradicts τ = T / ln 2.
• 14.4 T: off by a factor of 10 in the opposite direction.
• 2T: not derived from the decay law; no theoretical basis.
• T / 10: arbitrary and inconsistent with τ = 1 / λ.

Common Pitfalls:
Confusing half-life (median of the distribution) with mean life; using log base 10 instead of natural log; rounding ln 2 poorly. Always remember τ = T / ln 2 ≈ 1.44 T for quick estimates.

Final Answer:
1.44 T