Radioactive decay reasoning: a nuclide has a half-life of one month. Using count-rate (e.g., alpha counts per second) as a proxy for activity, which statement best describes the amount remaining after two and four months?

Introduction / Context:
Half-life is the characteristic time over which a radioactive sample’s number of undecayed nuclei falls by half. Activity (count rate in a Geiger–Müller counter) is proportional to the number of undecayed nuclei, so it decays at the same exponential rate. This question tests correct qualitative reasoning about successive half-lives.

Given Data / Assumptions:

• Half-life t1/2 = 1 month.
• Time intervals of interest: 2 months and 4 months.
• We assume a single nuclide with no replenishment and that count rate is proportional to the number of atoms remaining.

Concept / Approach:
After n half-lives, the remaining fraction is (1/2)^n. Disintegrated fraction = 1 − remaining fraction. For two months, n = 2; for four months, n = 4. Complete disintegration never occurs in a finite time for pure exponential decay.

Step-by-Step Solution:
Two months: n = 2 → remaining = (1/2)^2 = 1/4; disintegrated = 1 − 1/4 = 3/4.Four months: n = 4 → remaining = (1/2)^4 = 1/16, not 1/8; still not zero.Thus, the only correct statement in the list is that 3/4 will have disintegrated in two months.

Verification / Alternative check:
Successive-halving check: 1 → 1/2 (1 month) → 1/4 (2 months) → 1/8 (3 months) → 1/16 (4 months). This confirms the fractions.

Why Other Options Are Wrong:
Completely disintegrate in two or four months: exponential decay never reaches zero in finite time.1/8 remaining at four months: incorrect; 1/8 corresponds to three months, not four.

Common Pitfalls:
Mistaking linear for exponential behavior; assuming “complete” decay at a multiple of the half-life; mixing up the numbers after three and four half-lives.

Final Answer:
3/4th of it will disintegrate in two months.