Half-life and sample size: a 4 g sample of a radioisotope has a half-life of 10 days. What is the half-life if we instead take a 2 g sample of the same isotope?

Introduction / Context:
This question tests the key principle that half-life is an intrinsic nuclear property of a nuclide and does not depend on how much of the substance you start with. While the activity (decays per second) scales with the number of atoms, the half-life remains constant for a given isotope under ordinary conditions.

Given Data / Assumptions:

• Isotope half-life T_1/2 = 10 days.
• Two different initial masses: 4 g and 2 g.
• Same physical and chemical environment (no exotic conditions).

Concept / Approach:
The decay law N(t) = N0 * (1/2)^(t/T_1/2) shows that the characteristic time T_1/2 depends only on the decay constant lambda via T_1/2 = ln 2 / lambda. Changing N0 (initial amount) alters only the starting activity A0 = lambda * N0, but it does not alter lambda or T_1/2. Therefore, halving the sample mass halves the initial activity but leaves the half-life unchanged.

Step-by-Step Solution:

Verification / Alternative check:
Compute activity ratio: A0(2 g) = (2/4) * A0(4 g) = 0.5 * A0(4 g). Despite different A0 values, the time to halve each sample is the same (10 days), confirming independence from the amount.

Why Other Options Are Wrong:

• 5, 20, 30 days: suggest mass-dependent half-lives, which is incorrect.
• “Depends on geometry”: ordinary geometry has no effect on nuclear decay constants.

Common Pitfalls:
Confusing activity (amount-dependent) with half-life (amount-independent); assuming dilution or concentration changes the nuclear decay rate.

Final Answer:
10 days