Radioactive decay and half-life: A certain radioisotope has a half-life of 100 years. If a sample is stored for 400 years (i.e., four half-life periods), what fraction of the original mass will remain at the end of 400 years?

Introduction / Context:
Radioactive decay follows first-order kinetics. The concept of half-life tells us how long it takes for half of the nuclei in a sample to undergo decay. This question checks whether you can translate a given half-life into the remaining fraction after multiple half-life intervals, a core skill in nuclear chemistry, radiological safety, and environmental dose assessment.

Given Data / Assumptions:

• Half-life (t1/2) = 100 years.
• Total elapsed time = 400 years.
• Decay is purely radioactive (no chemical loss or addition) and follows first-order decay law.
• We report the remaining mass as a fraction of the initial mass.

Concept / Approach:
The fraction of material remaining after n half-lives is (1/2)^n. Alternatively, the decay law is N(t) = N0 * 2^(−t / t1/2). Because only fractions are asked, absolute amounts cancel out. This avoids the need for decay constants in this simple case.

Step-by-Step Solution:
Compute number of half-lives: n = 400 / 100 = 4.Remaining fraction after n half-lives: f = (1/2)^n = (1/2)^4.Evaluate: (1/2)^4 = 1/16.Thus, after 400 years, one-sixteenth of the original mass remains.

Verification / Alternative check:
Use the exponential form: N/N0 = 2^(−t/t1/2) = 2^(−4) = 1/16, which confirms the result.

Why Other Options Are Wrong:
1/2 and 1/4 correspond to 1 and 2 half-lives, not 4. 1/8 corresponds to 3 half-lives (300 years), not 400 years.

Common Pitfalls:

• Dividing the mass by 4 instead of halving it four times (linear versus exponential thinking).
• Confusing the decay constant with half-life; here the half-life method is faster.
• Using percent instead of fraction; the problem explicitly asks for a fractional answer.

Final Answer:
1/16