Chemistry 2e · Nuclear Chemistry

Radioactive Decay

9 min read

Constants: carbon-14 half-life 5730 yr; iodine-131 half-life 8.02 days; technetium-99m half-life 6.0 h; 1 Ci = 3.7 × 10¹⁰ Bq (standard published values).

On this page 9 sections

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 process in which an unstable nucleus emits particles or high-energy photons and transforms into a different nuclear state — often a different element. The trigger is built into the nucleus itself: when the balance between protons and neutrons falls outside the stable range, or when the nucleus carries surplus energy, it reorganizes toward stability and the excess escapes as radiation.

Every radioactive isotope has its own signature: the type of radiation it emits and the rate at which it decays. Some eject helium nuclei (alpha), some emit electrons or positrons (beta), some emit only photons (gamma), and many emit a combination. For heavy elements, decay continues step by step through a long chain until a stable nucleus forms. Because decay follows a predictable statistical clock, the same rules that date ancient artifacts also govern how long a medical tracer stays active in the body.

Why this matters

Decay is the engine behind nearly every application in this chapter. Doctors choose imaging isotopes by and radiation type; archaeologists date artifacts by measuring remaining carbon-14; regulators set safety limits based on how far and how fast radiation travels. On exams, decay problems test two skills repeatedly: writing balanced nuclear equations and applying first-order kinetics. Nuclear equations conserve nucleon number and charge, and the math is the same exponential-decay model used for drug clearance and cooling problems.

The college version

Core Concepts

Why some nuclei decay

Stable nuclei cluster in a region of the neutron-versus-proton chart called the belt of stability. Light stable nuclei hold roughly equal numbers of protons and neutrons; as nuclei grow heavier, extra neutrons are needed to spread out the repulsive electric forces among protons. When a nucleus has too many neutrons, too few neutrons, or too much internal energy, it is radioactive. All nuclei with more than 83 protons (bismuth) are radioactive, and lighter elements can have radioactive isotopes too — carbon-14 is a famous example.

Alpha emission

An is a helium-4 nucleus. Heavy nuclei above the belt of stability often shed alpha particles to reduce mass and charge dramatically. In alpha decay, the mass number drops by 4 and the atomic number by 2:

$23892 U \to 23490 Th + 42 He$

The daughter is a different element, two places left in the periodic table. Alpha particles are heavy and doubly charged, so they collide readily with matter: a sheet of paper or a few centimeters of air stops them. Their short range is offset by intense ionization along the path — the reason alpha emitters are dangerous when inhaled or ingested.

Beta-minus emission

Nuclides with too many neutrons convert a neutron into a proton, releasing an electron from the nucleus — the — plus an antineutrino that carries off part of the energy:

$13153 I \to 13154 Xe + 0 -1 e$

The mass number stays the same, but the atomic number rises by 1, moving the element one place right. The beta particle is created inside the nucleus at the moment of decay, not pulled from the electron cloud. Beta particles penetrate farther than alpha but are stopped by a few millimeters of aluminum.

Positron emission and electron capture

Nuclides with too few protons can raise their neutron-to-proton ratio two ways. In positron emission, a proton converts into a neutron and releases a positively charged electron (a positron) plus a neutrino:

$116 C \to 115 B + 0 +1 e$

In electron capture, the nucleus pulls an inner-shell electron inward, where it combines with a proton to form a neutron. Both processes lower the atomic number by 1 with the mass number unchanged. Positron emitters such as fluorine-18 drive PET imaging: each positron annihilates with an electron, producing two gamma photons that scanners detect.

Gamma emission

A is a high-energy photon emitted when a nucleus drops from an excited state to a lower-energy state. It changes neither the mass number nor the atomic number — only the nucleus's energy. Gamma rays accompany many alpha and beta decays and are emitted alone by metastable nuclides such as technetium-99m (the "m" for metastable), which relaxes to technetium-99 by gamma emission. Because gamma photons carry no charge, they penetrate deeply and need dense shielding such as lead or concrete.

Decay series

When a decay produces a daughter that is itself radioactive, the process continues. Heavy nuclides run through long : uranium-238 passes through 14 steps (including radon-222, a gas that can migrate into buildings) before reaching stable lead-206; uranium-235 ends at lead-207 and thorium-232 at lead-208.

Half-life and first-order kinetics

Radioactive decay is first order: the number of decays per second 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. Equivalently,

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

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

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

The half-life is constant: after one half-life, half of whatever is present remains; after two, one quarter; after three, one eighth. The amount remaining after n half-lives is N₀/2ⁿ, regardless of starting quantity.

Activity

The of a sample is its decay rate, A = kN, and it follows the same exponential decay:

$A t = A 0 e -kt$

The SI unit of activity is the becquerel (Bq), equal to one decay per second. The older curie (Ci) remains common in medicine: 1 Ci = 3.7 × 10¹⁰ Bq. Medical tracer doses are typically stated in millicuries (mCi) or megabecquerels (MBq).

How It Works / Step-by-Step Process

Identify the nuclide's Z and A, 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 decay equation, balancing mass numbers and charges on both sides.
Convert the daughter's Z to an element symbol; if the daughter is still unstable, expect further decay.
For time problems, write the exponential formula, substitute k or t_1/2, and solve.

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 "safe"	After one half-life exactly half remains; safety depends on type and dose, not just time
Positron	Proton or ordinary electron	A positron is an antielectron (charge +1); it annihilates on contact with an electron
Gamma ray	X-ray	Both are photons; gamma rays come from the nucleus, X-rays from electron energy levels
Alpha decay	Chemical change	Alpha decay changes the element itself; chemical reactions only rearrange electrons
Radioactive	Always glowing or hazardous	Hazard depends on type, energy, and dose of radiation

Eli explains

The same idea, in plain words

Explain it like I’m 10

Radioactive decay is like a giant bowl of popcorn popping. Each kernel has its own "pop time," but we can only talk about averages: in one half-life, half the kernels have popped, no matter how many you started with. The popped kernels are new, smaller pieces — new elements! — and the heat, light, and noise they make are like the radiation coming out.

Worked example

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

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

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

Now find the fraction remaining after 24 days:

$ln (N t N 0) = -kt = -(0.0864 days -1)(24 days) = -2.07$

Taking the exponential of both sides:

$N t N 0 = e -2.07 = 0.126$

About 12.6% of the original iodine-131 remains after 24 days. Sanity check: 24 days is almost exactly three half-lives, and (1/2)³ = 0.125, so the answer is consistent.

Example 2: Dating a bone fragment with carbon-14

Living organisms maintain a carbon-14 activity of about 15.3 decays per minute per gram of carbon; a bone fragment shows 1.91. The half-life of carbon-14 is 5730 years. First compute the decay constant:

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

The activity ratio is 1.91/15.3 = 0.125. Substitute into the linear form:

$ln (0.125) = -(1.21 \times 10 -4 yr -1)t$

$-2.08 = -(1.21 \times 10 -4 yr -1)t \Rightarrow t = 2.08 1.21 \times 10 -4 yr -1 = 1.72 \times 104 yr$

The bone is about 17,200 years old. Note 0.125 = (1/2)³, so the answer is three half-lives: 3 × 5730 = 17,190 years — matching the exponential result.

Example 3: Activity decay of technetium-99m

A patient dose of technetium-99m (half-life 6.0 h) is prepared at 40 mCi. How much activity remains after 24 hours? With n = 24/6 = 4 half-lives:

$A t = A 0 (12) n = 40 mCi (12) 4 = 40 mCi \times 0.0625 = 2.5 mCi$

In becquerels, 2.5 mCi = 2.5 × 10^-3 Ci × 3.7 × 10¹⁰ Bq/Ci = 9.3 × 10⁷ Bq. This rapid decay is why 99mTc suits imaging: it delivers its diagnostic signal and largely disappears within a day.

Key takeaways

Alpha decay: A down 4, Z down 2; daughter moves two places left.
Beta-minus decay: Z up 1, A unchanged; a neutron becomes a proton plus an electron.
Positron emission and electron capture both drop Z by 1 with A unchanged.
Gamma emission changes neither A nor Z; it only removes excess nuclear energy.
First-order math: ln(N_t/N₀) = -kt and t_1/2 = 0.693/k.
After n half-lives, the fraction remaining is 1/2ⁿ.
Activity A = kN in Bq (decays/s) or Ci (1 Ci = 3.7 × 10¹⁰ Bq).
Uranium-238 decays through 14 steps to stable lead-206.

Check yourself

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

Write the balanced equation for the alpha decay of polonium-210 (Z = 84, A = 210).
Show answer
²¹⁰₈₄Po → ²⁰⁶₈₂Pb + ⁴₂He. Check: 210 = 206 + 4 and 84 = 82 + 2.
A sample's activity drops from 800 Bq to 100 Bq in 30 days. What is the half-life?
Show answer
100 Bq is 1/8 = (1/2)³ of 800 Bq, so 30 days is three half-lives: t_1/2 = 10 days. Alternatively solve 0.125 = e^-k(30) for k, then t_1/2 = 0.693/k.
Why does beta-minus decay raise the atomic number by 1 while the mass number is 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.
What happens when a positron meets an electron?
Show answer
The positron annihilates with the electron, converting both masses into two gamma photons traveling in opposite directions.
Why is carbon-14 dating unreliable 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, and measurement uncertainty swamps the signal.
Convert an activity of 0.50 mCi into becquerels.
Show answer
0.50 mCi × (10^-3 Ci/mCi) × (3.7 × 10¹⁰ Bq/Ci) = 1.85 × 10⁷ Bq.

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Study tools & related lessonsKey vocabulary · Related

Key vocabulary

Radioactive decay: Spontaneous emission of particles or photons from an unstable nucleus
Alpha particle: A helium-4 nucleus (2 protons + 2 neutrons)
Beta particle: An electron (or positron) created in the nucleus during decay
Gamma ray: High-energy photon from an excited nucleus
Half-life: Time for half a radioactive sample to decay
Decay constant (k): First-order rate constant, k = 0.693/t_1/2
Activity: Decay rate, A = kN
Decay series: Chain of decays linking a heavy nuclide to a stable end product

Sources & references

openstax.org — Chemistry 2e

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

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In 30 seconds

Why this matters

The college version

Core Concepts

Why some nuclei decay

Alpha emission

Beta-minus emission

Positron emission and electron capture

Gamma emission

Decay series

Half-life and first-order kinetics

Activity

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: Dating a bone fragment with carbon-14

Example 3: Activity decay of technetium-99m

Key takeaways

Check yourself

Keep learning

Key vocabulary

Related lessons

Sources & references