IMAT Chemistry · Atomic Structure

Radioactive Decay, Nuclear Equations and Half-Life

Unstable nuclei decay by emitting alpha, beta or gamma radiation; nuclear equations balance mass and atomic numbers, and the half-life is the time for half the nuclei to decay.

All Atomic Structure notes

Why this matters

No past paper in this chapter has asked about radioactivity yet, but it is in the official syllabus. Expect a short calculation (a half-life, or the nucleus left after a decay) rather than a long one.

Concept 1 of 3: Alpha, beta and gamma radiation and what each does to the nucleus

A nucleus with the wrong balance of protons and neutrons is unstable. A heavy nucleus can shed a compact package of two protons and two neutrons (alpha). A nucleus with too many neutrons turns a neutron into a proton and throws out an electron (beta-minus). Gamma rays carry away leftover energy without changing which nucleus it is.

Definition

Radioactive decay is random and spontaneous: it cannot be sped up by heat, pressure or chemical reaction.
Ionising power (damage done along the path) is greatest for alpha and least for gamma; penetrating power is the reverse.
A beta-minus electron is made in the nucleus ( $\mathrm{n} \rightarrow \mathrm{p} + e^-$ ); it is not one of the atom's shell electrons.
Alpha and beta particles are deflected by electric and magnetic fields in opposite directions; gamma rays are not deflected.

Radiation	What it is	Change to the nucleus	Charge	Stopped by
Alpha (α)	Helium nucleus ${}^{4}_{2}\mathrm{He}$ : 2 protons and 2 neutrons	A falls by 4, Z falls by 2	+2	A sheet of paper or a few cm of air
Beta-minus (β⁻)	Fast electron ${}^{0}_{-1}e$ , made when a neutron becomes a proton	A unchanged, Z rises by 1	−1	A few mm of aluminium
Beta-plus (β⁺)	Positron ${}^{0}_{+1}e$ , made when a proton becomes a neutron	A unchanged, Z falls by 1	+1	A few mm of aluminium
Gamma (γ)	High-energy electromagnetic radiation	No change to A or Z	0	Only reduced, by thick lead or concrete

Practice this conceptself-check · 3 quick reps

The beta electron comes from the nucleus

In beta-minus decay a neutron in the nucleus changes into a proton and an electron, and the electron is emitted. The atom's shell electrons are not involved. That is why the atomic number rises by one.

Concept 2 of 3: Balancing nuclear equations

Nucleons and charge are both conserved in a decay. So the mass numbers on the top line must add up to the same total on both sides, and so must the atomic numbers on the bottom line. Find the unknown by subtraction, then name it from its atomic number.

Definition

In a nuclear equation:

The mass numbers (top) add up to the same total on each side.
The atomic numbers (bottom) add up to the same total on each side.
Write particles in the same style: alpha ${}^{4}_{2}\mathrm{He}$ , beta-minus ${}^{0}_{-1}e$ , positron ${}^{0}_{+1}e$ , neutron ${}^{1}_{0}\mathrm{n}$ .
The new atomic number tells you the new element, so a decay changes one element into another (except gamma).

Conservation in a nuclear equation

\sum A_{\text{before}} = \sum A_{\text{after}} \qquad \sum Z_{\text{before}} = \sum Z_{\text{after}}

$A$ mass number of each particle
$Z$ atomic number (charge number) of each particle

Worked example

Uranium-238 (

Z = 92

) emits an alpha particle, and the product then emits a beta-minus particle. Write both equations. (Th has

Z = 90

; Pa has

Z = 91

Practice this conceptself-check · 3 quick reps

Beta decay does not change the mass number

An electron has mass number 0, so beta emission leaves A unchanged and changes Z by one. Only alpha decay lowers the mass number (by 4). Options that lower A after a beta decay are wrong.

Concept 3 of 3: Half-life calculations

Each unstable nucleus has the same chance of decaying in a given time, whatever has happened before. So in one half-life, half of whatever is left decays: a sample never runs out in a fixed time, it keeps halving. Count the half-lives, then halve that many times.

Definition

The half-life $T_{1/2}$ is the time taken for half the radioactive nuclei in a sample to decay. The activity (decays per second, in becquerels, Bq) and the mass of the radioactive isotope halve in the same time.

After 1, 2, 3, 4 half-lives the fraction left is $\tfrac{1}{2}, \tfrac{1}{4}, \tfrac{1}{8}, \tfrac{1}{16}$ .
The half-life is fixed for each isotope (carbon-14: about 5730 years; iodine-131: about 8 days). It does not depend on the amount, temperature or chemical form.
The fraction decayed is 1 minus the fraction left.

Amount left after time t

N = N_0 \left(\tfrac{1}{2}\right)^{t/T_{1/2}}

$N_0$ starting amount (nuclei, mass or activity)
$N$ amount left after time $t$
$T_{1/2}$ half-life, in the same unit as $t$

Worked example

A sample contains 80 g of an isotope with a half-life of 5 days. How much of the isotope is left after 20 days?

Practice this conceptself-check · 4 quick reps

Two half-lives leave a quarter, not nothing

Each half-life halves what is LEFT, not the original amount. After two half-lives a quarter remains, after three an eighth. An option saying the sample is gone after two half-lives is wrong.

Heating or reacting a sample does not change its half-life

Radioactive decay happens in the nucleus, which chemical reactions and temperature do not reach. The half-life of an isotope is the same in any compound, at any temperature and for any sample size.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Balancing nuclear equations
Conservation in a nuclear equation
$\sum A_{\text{before}} = \sum A_{\text{after}} \qquad \sum Z_{\text{before}} = \sum Z_{\text{after}}$
Half-life calculations
Amount left after time t
$N = N_0 \left(\tfrac{1}{2}\right)^{t/T_{1/2}}$

Reference tables (1)

Alpha, beta and gamma radiation and what each does to the nucleus4 rows

Radiation	What it is	Change to the nucleus	Charge	Stopped by
Alpha (α)	Helium nucleus ${}^{4}_{2}\mathrm{He}$ : 2 protons and 2 neutrons	A falls by 4, Z falls by 2	+2	A sheet of paper or a few cm of air
Beta-minus (β⁻)	Fast electron ${}^{0}_{-1}e$ , made when a neutron becomes a proton	A unchanged, Z rises by 1	−1	A few mm of aluminium
Beta-plus (β⁺)	Positron ${}^{0}_{+1}e$ , made when a proton becomes a neutron	A unchanged, Z falls by 1	+1	A few mm of aluminium
Gamma (γ)	High-energy electromagnetic radiation	No change to A or Z	0	Only reduced, by thick lead or concrete

Watch out for (4)

The beta electron comes from the nucleus
→ Alpha, beta and gamma radiation and what each does to the nucleus
Beta decay does not change the mass number
→ Balancing nuclear equations
Two half-lives leave a quarter, not nothing
→ Half-life calculations
Heating or reacting a sample does not change its half-life
→ Half-life calculations

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