Questions: Radioactive Decay Constant and Half-Life

Question 1

A hospital has 10¹² atoms of technetium-99m (half-life 6 hours); a research lab has only 10⁶ atoms of the same isotope. After 6 hours, what fraction of each sample remains?

• A. Half of both samples — the half-life is a property of the isotope, independent of sample size
• B. The hospital sample retains more than half because large samples have lower activity per atom
• C. The research lab sample retains more because fewer atoms means less inter-nuclear interaction
• D. The half-life predicts nothing without knowing the initial activity of each sample

Answer: (A)

The half-life is a property of the isotope, not the sample. Each nucleus independently has the same fixed probability per unit time (λ) of decaying, regardless of how many other nuclei are present. After one half-life, exactly half the nuclei in any sample — large or small — will have decayed, on average. The hospital's 10¹² atoms reduce to ~5×10¹¹; the lab's 10⁶ reduce to ~5×10⁵. The fraction remaining is (1/2) for both. This is a direct consequence of the memoryless, probabilistic nature of radioactive decay.

Question 2

Carbon-14 has a half-life of 5,730 years. An ancient organic artifact contains 1/8 of the C-14 found in living organisms. Approximately how old is the artifact?

• A. About 5,730 years — one half-life accounts for the reduction
• B. About 11,460 years — 1/8 requires two halvings
• C. About 17,190 years — three half-lives, since (1/2)³ = 1/8
• D. Cannot be determined without knowing the original C-14 concentration

Answer: (C)

After n half-lives, the fraction remaining is (1/2)^n. If 1/8 remains, then (1/2)^n = 1/8 = (1/2)³, so n = 3 half-lives. The artifact is 3 × 5,730 = 17,190 years old. The original C-14 concentration does not need to be measured independently — it is assumed to equal that of living organisms (set by atmospheric equilibrium), and the ratio of C-14 to stable C-12 in the artifact provides the fraction remaining.

Question 3

A radioactive nucleus that has existed for 100 years without decaying is more likely to decay in the next second than a freshly created nucleus of the same isotope.

• T. True
• F. False

Answer: False

Radioactive decay is a memoryless quantum process: the probability per unit time of decaying (the decay constant λ) is fixed by nuclear structure and does not change with the age of the nucleus. A nucleus that has 'survived' for 100 years has the same probability per second of decaying as a nucleus created one second ago. This memorylessness is a fundamental property of quantum tunneling and exponential distributions — there is no accumulation of 'pressure to decay' over time. This is why the half-life is constant and independent of the sample's history.

Question 4

An isotope with a shorter half-life has higher activity (more decays per second) per atom than an isotope with a longer half-life, given the same number of atoms.

• T. True
• F. False

Answer: True

Activity A = λN = (ln 2 / t₁/₂) × N. For a fixed number of atoms N, shorter half-life means larger λ, hence higher activity per atom. This is why polonium-213 (half-life ~4 microseconds) is extraordinarily radioactive — each atom decays almost immediately — while uranium-238 (half-life 4.5 billion years) has nearly negligible activity per atom but persists indefinitely. This trade-off governs medical isotope selection: technetium-99m (6-hour half-life) is intense enough to detect but short-lived enough to be safe.

Question 5

Why does the half-life of a radioactive isotope remain constant regardless of sample size, temperature, or how old the nuclei are?

Model answer: Because radioactive decay is a probabilistic, memoryless quantum process. Each nucleus has an intrinsic, fixed probability per unit time (λ) of decaying, determined solely by nuclear structure. This probability does not change with time (no memory), is unaffected by neighboring nuclei (no collective effect), and is independent of temperature or pressure (no classical trigger needed). The half-life t₁/₂ = ln2/λ depends only on λ, which is a property of the isotope. Since all nuclei of a given isotope share the same λ, the half-life is the same regardless of how many there are, how old they are, or what conditions they are in.

This contrasts with macroscopic processes like chemical reactions, where temperature and concentration dramatically affect rates. Radioactive decay is governed by quantum mechanics (tunneling through the nuclear potential barrier), not classical thermodynamics. The only way to change a nucleus's decay rate is to change its nuclear structure — which is why chemical or thermal manipulation cannot speed or slow nuclear decay, a fact that was once considered deeply surprising.