Questions: Half-Life and the Radioactive Decay Law

Question 1

A radioactive sample has undergone exactly 3 half-lives since it was created. What fraction of the original nuclei remains undecayed?

• A. None — after 3 half-lives the material has fully decayed
• B. 1/3 — one-third remains after dividing by the number of half-lives elapsed
• C. 1/6 — each half-life removes another sixth of the original amount
• D. 1/8 — each half-life halves the remaining amount, so (1/2)³ = 1/8

Answer: (D)

After each half-life, exactly half of *what remains* decays — not half of the original. After 1 half-life: 1/2 remains. After 2: 1/4. After 3: 1/8. Option A is the most common misconception: radioactive material never fully disappears in finite time. Options B and C misapply arithmetic division rather than repeated halving. The key formula is (1/2)^n after n half-lives, which follows directly from N(t) = N₀ e^(−λt) evaluated at t = nT½.

Question 2

What does the decay constant λ physically represent for a radioactive nucleus?

• A. The time required for exactly one nucleus in the sample to decay
• B. The probability per unit time that any given nucleus will decay, independent of how long it has already survived
• C. The total number of decays that will occur before the sample is exhausted
• D. The average time between successive decays in a large sample

Answer: (B)

λ is a probability rate — units of inverse time (e.g., per second). Each nucleus has a fixed probability λ·dt of decaying in any small time interval dt, regardless of its age. This memoryless property is what makes the decay law exponential: the fraction decaying per unit time is always the same constant λ, so the *number* decaying per unit time is proportional to N, giving dN/dt = −λN. Option A confuses λ with the mean lifetime τ = 1/λ. Options C and D describe the activity, not the decay constant itself.

Question 3

After two half-lives have elapsed, very few of the original radioactive material remains.

• T. True
• F. False

Answer: False

After two half-lives, one-quarter (1/4) of the original nuclei remains undecayed. The material never fully disappears in finite time — exponential decay is asymptotic. Each half-life halves *whatever is currently present*, so you always have something left, even if the amount becomes negligibly small. After 10 half-lives, about 0.1% remains; after 100 half-lives, an extraordinarily tiny fraction persists. This is the most common misconception about half-lives.

Question 4

Because radioactive decay is a quantum process with a fixed probability per unit time, a nucleus that has survived undecayed for a million years is no more likely to decay in the next second than a freshly created nucleus of the same species.

• T. True
• F. False

Answer: True

This is the memoryless (Markov) property of radioactive decay. λ is constant in time — the nucleus does not 'age,' accumulate internal stress, or become more likely to decay the longer it survives. This is what makes the decay law exponential: the future decay probability depends only on the current state (undecayed), not on history. It is also why the half-life is constant regardless of how many nuclei remain — each surviving nucleus has the same λ as it always did.

Question 5

Why is the half-life constant regardless of how many nuclei remain, and what does this reveal about radioactive decay at the level of individual nuclei?

Model answer: The half-life is constant because each nucleus decays independently with a fixed probability per unit time (λ), regardless of how many other nuclei surround it or how long it has already survived. Whether 10²³ nuclei remain or just 10, each one has the same probability of decaying in the next second. This reveals that radioactive decay is a memoryless quantum process: a nucleus carries no internal clock and does not become 'due' to decay. The exponential law N(t) = N₀e^(−λt) describes the average behavior of large ensembles — it cannot predict when any individual nucleus will decay, which is fundamentally random.

The constant half-life is a direct signature of the memoryless property. If nuclei 'aged,' the decay rate would change over time and the law would not be exponential. Applications depend on this constancy: carbon-14 dating works because the ratio of decayed to undecayed carbon follows a predictable exponential curve that does not shift based on the size of the sample or its history, only on the elapsed time.