Chemistry: Atoms First 2e · Nuclear Chemistry

Radioactive Decay

8 min read

Constants: carbon-14 half-life 5730 yr; iodine-131 half-life 8.02 days; modern carbon activity ≈15.3 decays/min/g; 1 Ci = 3.7 × 10¹⁰ Bq (standard published values; activity figure is an approximate textbook value used for ratio calculations).

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In 30 seconds
Why this matters
The college version
Eli explains
Worked example
Key takeaway
Check yourself
Study tools
Sources & references

In 30 seconds

is the spontaneous transformation of an unstable nucleus. When a nucleus has too many neutrons relative to protons, too few, or simply too much internal energy, it reorganizes toward a more stable arrangement and the excess escapes as radiation — an alpha particle, a beta particle, a positron, or a gamma ray. No chemical trigger is needed; each unstable nuclide decays on its own schedule.

Although we cannot predict which individual atom decays next, a large collection decays with clockwork statistical regularity. The fraction remaining after a given time depends only on the isotope's — the time for half of any sample to decay — and the mathematics is identical to first-order chemical kinetics. The same exponential equations that describe drug clearance from the body also describe carbon-14 leaving a dead organism.

Why this matters

Decay is the engine behind nearly every application in this chapter. A physician choosing a tracer isotope weighs half-life against imaging time; an archaeologist dates a bone by its remaining carbon-14; a regulator sets exposure limits based on how much and how fast radiation is emitted. On exams, decay problems test two connected skills: writing balanced nuclear equations and applying the exponential decay equations. Half-life arithmetic and unit conversions (becquerels and curies) pay off immediately in the topics that follow.

The college version

Core Concepts

Why some nuclei decay and others do not

Stable nuclei occupy a region of the neutron-versus-proton chart called the . Light stable nuclei hold roughly equal numbers of protons and neutrons; heavier stable nuclei need extra neutrons to spread out proton repulsion, so their neutron-to-proton ratio rises above 1. A nucleus above the belt (too many neutrons) tends to undergo beta decay; one below it (too few neutrons) tends to undergo positron emission or electron capture; a very heavy nucleus may shed an alpha particle. Every element with more than 83 protons has only radioactive isotopes, and many lighter elements do too — carbon-14 is the classic example.

The decay modes and what they change

Alpha decay: a helium-4 nucleus, ⁴₂He, leaves the parent. The mass number drops by 4 and the atomic number by 2, moving the element two places left on the periodic table. Alpha particles are short-range — a sheet of paper or a few centimeters of air stops them — but they deposit intense energy along their path, which is why alpha emitters are hazardous when inhaled or ingested.
Beta-minus decay: a neutron becomes a proton and emits an electron (the beta particle) plus an antineutrino. The atomic number rises by 1 and the mass number is unchanged. Beta particles penetrate farther than alpha but are stopped by a few millimeters of aluminum.
Positron emission and electron capture: a proton becomes a neutron by emitting a positron, ⁰₊₁e, or by capturing an inner-shell electron. Both lower Z by 1 with A unchanged.
Gamma emission: an excited nucleus releases a high-energy photon, ⁰₀γ, changing neither A nor Z. Gamma rays penetrate deeply and require dense shielding such as lead or concrete.

Decay series

A decay often produces a daughter that is itself radioactive, so heavy nuclides travel stepwise down a until a stable nucleus forms: uranium-238 decays through 14 steps (including the gas radon-222) to stable lead-206; uranium-235 ends at lead-207; thorium-232 at lead-208.

Half-life and first-order kinetics

Radioactive decay is first order: the decay rate at any instant is proportional to the number of radioactive nuclei present. If N₀ nuclei are present at time zero, the number remaining after time t is

$N t = N 0 e -kt$

where k is the decay constant (units of inverse time). Taking the natural logarithm gives the linear form used in calculations:

$ln (N t N 0) = -kt$

The half-life, t_1/2, is the time for half the nuclei to decay, and it relates to the decay constant by

$t 1/2 = ln 2 k = 0.693 k$

Because half-life is constant, the fraction remaining after n half-lives is simply

$N t N 0 = (12) n$

After one half-life, half remains; after two, one quarter; after three, one eighth.

Activity and its units

The of a sample is its decay rate, A = kN, which decays exponentially just like N:

$A t = A 0 e -kt$

The SI unit of activity is the (Bq), equal to one decay per second. The older (Ci) unit is still common in medicine, with the exact definition

$1 Ci = 3.7 \times 1010 Bq$

Medical doses are usually stated in millicuries (mCi) or megabecquerels (MBq), with 1 mCi = 37 MBq.

How It Works / Step-by-Step Process

Identify the nuclide's A and Z, and locate it relative to the belt of stability.
Choose the decay mode: too heavy → alpha; too many neutrons → beta-minus; too few neutrons → positron emission or electron capture.
Write the balanced decay equation.
For time problems, write the exponential formula, substitute k (from t_1/2 if needed), and solve for the unknown.
Sanity-check with the half-life shortcut: compare n = t/t_1/2 with the result.

Common Confusions

Do Not Confuse	With	Difference
Beta particle	Electron from the electron cloud	Beta particles are created inside the nucleus when a neutron converts to a proton
Half-life	Time until the sample is harmless	After one half-life exactly half remains; safety depends on type, energy, and dose, not just time
Positron	Proton or ordinary electron	A positron is an antielectron (charge +1); it annihilates on contact with an electron
Activity	Radioactive mass	Activity is a decay rate (Bq or Ci); the number of nuclei falls as activity falls, but the element's identity changes too
First-order decay	Zero-order or second-order decay	In first-order decay the rate is proportional to the number of nuclei present; half-life is constant, not dose-dependent
Alpha particle range	Alpha particle hazard	Alpha stops in paper or skin but is intensely damaging if an emitter is inhaled or ingested

Eli explains

The same idea, in plain words

Explain it like I’m 10

Radioactive atoms are like a jar full of bouncy balls, each with a hidden timer. We never know which ball pops next, but after one "half-life" about half have popped, after two half-lives only a quarter are left, and so on — the pattern is the same no matter how many balls you start with. Each popped ball is a new, smaller thing (a new element!), and the pop itself is the radiation.

Worked example

Example 1: Decay constant and remaining fraction for iodine-131

Iodine-131 has a half-life of 8.02 days. Find its decay constant using the formula before substituting:

$k = 0.693 t 1/2 = 0.693 8.02 days = 0.0864 days -1$

Find the fraction remaining after 24.06 days (three half-lives) using the linear form:

$ln (N t N 0) = -kt = -(0.0864 days -1)(24.06 days) = -2.08$

Exponentiate both sides:

$N t N 0 = e -2.08 = 0.125$

Exactly 12.5% remains. Sanity check: 24.06/8.02 = 3 half-lives, and (1/2)³ = 0.125 — the exponential result matches the half-life shortcut.

Example 2: Activity decay with unit conversion

A hospital prepares a 10 mCi dose of iodine-131. What activity remains after 32.08 days, in megabecquerels? First, 32.08/8.02 = 4 half-lives, so apply the shortcut:

$A t = A 0 (12) n = 10 mCi (12) 4 = 10 mCi \times 0.0625 = 0.625 mCi$

Convert to megabecquerels by dimensional analysis, using 1 mCi = 37 MBq:

$0.625 mCi \times 37 MBq 1 mCi = 23 MBq$

After four half-lives the dose has fallen to about 23 MBq — a reminder that activity, not mass, changes with decay.

Example 3: Carbon-14 dating a bone fragment

Living organisms maintain a roughly constant carbon-14 activity of about 15.3 decays per minute per gram of carbon; a bone fragment shows 3.82. The half-life of carbon-14 is 5730 years. Find the decay constant first:

$k = 0.693 t 1/2 = 0.693 5730 yr = 1.21 \times 10 -4 yr -1$

The activity ratio is 3.82/15.3 = 0.250 = (1/2)², so exactly two half-lives have elapsed:

$t = 2 \times 5730 yr = 11,460 yr$

The bone is roughly 11,500 years old. As a check, ln(0.250) = -(1.21 × 10^-4 yr^-1)t gives t = 11,460 yr — the same answer within rounding.

Key takeaways

Decay modes: alpha (A down 4, Z down 2); beta-minus (Z up 1, A unchanged); positron/electron capture (Z down 1, A unchanged); gamma (no change).
First-order math: N_t = N₀ e^-kt and ln(N_t/N₀) = -kt.
Half-life: t_1/2 = 0.693/k; after n half-lives the fraction remaining is 1/2ⁿ.
Activity A = kN is measured in Bq (decays/s) or Ci, with 1 Ci = 3.7 × 10¹⁰ Bq.
Heavy nuclides decay in series (U-238 → ... → Pb-206, 14 steps).
Alpha particles are short-range but intensely ionizing; gamma rays penetrate deeply.

Check yourself

6 review questions from the chapter. Try each one, then open the answer.

Write the balanced equation for the alpha decay of thorium-230, ²³⁰₉₀Th.
Show answer
²³⁰₉₀Th → ²²⁶₈₈Ra + ⁴₂He. Check: 230 = 226 + 4 and 90 = 88 + 2.
A sample's activity falls from 1200 Bq to 150 Bq in 18 days. What is the half-life?
Show answer
150/1200 = 0.125 = (1/2)³, so 18 days is three half-lives: t_1/2 = 6 days.
Why does beta-minus decay raise the atomic number by 1 while leaving the mass number unchanged?
Show answer
A neutron (charge 0) becomes a proton (charge +1) plus an electron (charge −1); the nucleon count A is unchanged while Z rises by 1.
Convert 0.25 mCi to becquerels.
Show answer
0.25 mCi × (10^-3 Ci/mCi) × (3.7 × 10¹⁰ Bq/Ci) = 9.25 × 10⁶ Bq (about 9.3 MBq).
Why is carbon-14 dating impractical for samples older than about 50,000 years?
Show answer
After about 50,000 years (roughly 8–9 half-lives), less than about 0.4% of the original carbon-14 remains, so measurement uncertainty swamps the signal.
What happens when a positron meets an electron, and why does it matter for PET imaging?
Show answer
The positron annihilates with an electron, converting both masses into two gamma photons traveling in opposite directions; PET scanners detect the paired photons to locate the tracer.

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Study tools & related lessonsKey vocabulary · Related

Key vocabulary

radioactive decay: Spontaneous emission of particles or photons from an unstable nucleus
half-life: Time for half of any sample of an isotope to decay
decay constant k: First-order rate constant; k = 0.693/t_1/2
belt of stability: Region of the N-vs-Z chart where nuclei are stable
activity: Decay rate, A = kN
becquerel: One decay per second
curie: 3.7 × 10¹⁰ decays per second
decay series: Chain of decays from a heavy nuclide to a stable end product

Sources & references

openstax.org — Chemistry Atoms First 2e

This lesson was adapted from the open educational references above; their licenses and attributions are preserved. See Copyright & Licensing.

Educational content only. It is not medical, legal or professional advice. Found an error? Tell us.

In 30 seconds

Why this matters

The college version

Core Concepts

Why some nuclei decay and others do not

The decay modes and what they change

Decay series

Half-life and first-order kinetics

Activity and its units

How It Works / Step-by-Step Process

Common Confusions

The same idea, in plain words

Worked example

Example 1: Decay constant and remaining fraction for iodine-131

Example 2: Activity decay with unit conversion

Example 3: Carbon-14 dating a bone fragment

Key takeaways

Check yourself

Keep learning

Key vocabulary

Related lessons

Sources & references