Half-life application: a radioactive element has a half-life of 12 years. How long will it take for the sample to reduce to 50% of its original mass?

Introduction / Context:
The half-life is the time required for a radioactive sample to decay to half of its initial quantity. This definition is frequently tested to ensure students can directly connect decay fractions to elapsed time without unnecessary computation.

Given Data / Assumptions:

• Half-life t_half = 12 years.
• Simple first-order decay with no production.
• We seek the time for exactly 50% remaining.

Concept / Approach:

By definition, after one half-life, the remaining quantity is 1/2 of the original. Therefore, if the half-life is 12 years, it takes precisely 12 years to reach 50% of the original mass. No further calculations are needed beyond recalling the definition.

Step-by-Step Solution:

Verification / Alternative check:

Using N = N0*(1/2)^(t/t_half): set N/N0 = 0.5 → (1/2)^(t/12) = 1/2 → t/12 = 1 → t = 12 years.

Why Other Options Are Wrong:

6 years: Would leave about 70.7%, not 50%. 18 years: Between one and two half-lives; not 50%. 24 or 36 years: Two or three half-lives (25% or 12.5% remaining), not 50%.

Common Pitfalls:

Overcomplicating with logarithms; confusing the time to 50% with the time to 25% or other fractions.

Final Answer:

12 years