Applications — radiocarbon dating, medical (Tc-99m, I-131), RTG (Pu-238 in spacecraft): Question bank

Prerequisites worth re-reading if a line stumps you: Radioactive Decay Law, Half-life and Decay Constant, Types of Radioactive Decay (alpha, beta, gamma), Nuclear Reactions and Transmutation, Activity and Units (Becquerel, Curie), Seebeck Effect / Thermocouples.

True or false — justify

A radioactive sample's decay constant

λ

changes if you heat it or apply pressure.

False —

λ

is set by nuclear forces inside the nucleus, untouched by temperature, pressure, or chemistry; that invariance is exactly why decay makes a reliable clock.

Doubling the amount of

^{14} C

in a sample doubles its half-life.

False — half-life

t_{1/2} = ln 2/ λ

depends only on

λ

, not on how many nuclei you have; more nuclei just means more decays per second, not a slower clock.

A living tree and a freshly cut plank of the same species have the same

^{14} C /^{12} C

ratio.

True — both were alive recently enough that intake only just stopped; the ratio still matches the atmosphere, so a fresh plank is a valid

A_{0}

reference.

Because Tc-99m emits only

γ

, it deposits no dose in the patient at all.

False —

γ

photons deposit little dose because most pass through, but "little" is not "none"; some scatter and are absorbed, which is why the short 6 h half-life still matters for limiting exposure.

An RTG's electrical output stays perfectly constant until the Pu-238 is gone, then drops to zero.

False — heat output tracks

N \propto e^{- λ t}

, so the power fades smoothly (about 0.8%/yr); RTGs slowly dim, they never suddenly die.

I-131 works as a thyroid therapy because doctors surgically inject it into the gland.

False — the thyroid naturally concentrates iodine to build hormones, so swallowed I-131 homes there on its own; biology does the targeting for free.

A shorter half-life always means a more dangerous isotope.

False — a short half-life means the activity clears fast, which for medical isotopes is usually safer; danger depends on radiation type, energy, and how the body handles it, not half-life alone.

Pu-238 and Pu-239 are interchangeable in a spacecraft power source.

False — Pu-238 is a strong

α

/heat source with an 87.7 yr life for power; Pu-239 is the fissile bomb/reactor isotope, a completely different job.

Spot the error

"We can radiocarbon-date a granite rock to find when it formed."

Granite is not organic and never took in atmospheric carbon, so it has no starting

^{14} C

clock to read; carbon dating only works on once-living material.

"Tc-99m is great for therapy because its gamma rays destroy tumour cells."

γ

passes through tissue depositing little energy, so it is a poor killer; therapy needs short-range

β

α

that dump all their energy locally —

γ

is for seeing, not slaying.

"We date a 60,000-year-old bone easily with C-14 by just counting more carefully."

After ~10 half-lives (~57,000 yr) so little

^{14} C

is left that it is swamped by background — better counting cannot recover a signal that has essentially vanished.

"The 'm' in Tc-99m stands for 'medical'."

The "m" means metastable — a long-lived excited nuclear state that relaxes by emitting a 140 keV

γ

; it is a physics label, not a usage label.

"Because I-131 has an 8-day half-life, a treated patient is radioactive for exactly 8 days then perfectly clean."

After 8 days half the I-131 remains, not zero; activity keeps falling exponentially for weeks, and biological excretion removes some even faster — there is no clean cut-off.

"An RTG converts alpha particles directly into electricity."

The

α

particles are stopped and their kinetic energy becomes heat first; the Seebeck effect in thermocouples then turns the hot–cold temperature difference into electricity.

"Radiocarbon dating measures the amount of

^{12} C

left in the sample."

^{12} C

is stable and does not decay; dating tracks the depletion of $^{14} C$ (via its activity) relative to a living reference.

Why questions

Why is the decay rate proportional to

N

in the first place?

Each nucleus decays independently with the same fixed chance per second, so the total number of decays per second scales with how many nuclei are present — that is the meaning of

d N / d t = - λ N

in the Radioactive Decay Law.

Why do we need a modern "fresh sample" reference

A_{0}

to date something?

We measure only the present activity

A

; to solve

A = A_{0} e^{- λ t}

for time we must know the starting activity, which a living sample of the same size supplies.

Why is a

γ

emitter ideal for imaging but useless for therapy?

γ

escapes the body to reach the camera (good for seeing) precisely because it deposits little energy along the way (bad for killing); imaging wants penetration, therapy wants local energy dump.

Why is Mo-99, not Tc-99m, shipped to hospitals?

Tc-99m's 6 h half-life is too short to survive transport; the longer-lived Mo-99 (

t_{1/2} = 66

h) is shipped and Tc-99m is "milked" from it on-site as needed.

Why choose an

α

emitter (Pu-238) for a power source rather than a

β

γ

emitter?

α

decays release large energy per event yet the particle is stopped by paper, so almost all energy becomes usable heat with minimal shielding — dense, safe-ish power.

Why does the RTG on Voyager still produce power decades after 1977?

Pu-238's 87.7 yr half-life means most nuclei have not yet decayed after ~45 yr, so heat output has only fallen by roughly a third, not vanished.

Why can't solar panels replace an RTG on outer-planet probes?

Far from the Sun sunlight is far too weak for panels, and RTGs need no sunlight and have no moving parts, giving steady power for decades.

Edge cases

If a sample shows higher

^{14} C

activity than the modern reference, what does that imply?

The naive age formula gives a negative time — it means the "reference" assumption is wrong (e.g. contamination, or nuclear-era carbon), not that the object predicts the future.

At exactly

t = 0

(moment of death), what is the sample's activity in the dating formula?

It equals

A_{0}

, the living-organism activity, since

e^{- λ \cdot 0} = 1

; this is the anchor point from which all elapsed time is measured.

After 4 half-lives of Tc-99m (~one day), roughly what fraction of activity remains, and why does that matter?

About

(1/2)^{4} = 1/16 \approx 6%

; the dose self-limits quickly, so a full diagnostic scan does not leave the patient irradiated for days.

What happens to the "clock" if a fossil is contaminated with modern carbon?

Added modern

^{14} C

raises the measured activity, making the sample look younger than it is — contamination biases dates toward the present.

t \to \infty

for any decaying isotope, does

N

ever truly reach zero?

Mathematically

N = N_{0} e^{- λ t}

only approaches zero, but physically once counts drop below background the signal is unmeasurable — the practical limit (~10 half-lives) is why old fossils need long-lived clocks.

For an isotope with an extremely long half-life like U-238 (4.5 Gyr), why is it useless for medical imaging?

Its activity per gram is tiny (few decays per second) and it lingers in the body essentially forever, so it gives a weak signal and unacceptable long-term dose — the opposite of what a 6 h imaging agent needs.

Recall One-line self-test before you close

Cover this and say it out loud: dating counts what has decayed, imaging wants penetrating $γ$ , therapy wants short-range $β / α$ , power wants dense $α$ -heat over decades. If you can justify each in a sentence, you have the traps beaten.