A nuclear equation shows how an unstable nucleus changes during radioactive decay. Unlike chemical equations, nuclear equations track changes in mass number (the total number of nucleons) and atomic number (the number of protons), not atoms. Every nuclear equation must balance on both sides: the total mass numbers must match, and the total atomic numbers must match.
What notation is used in nuclear equations?
Every nucleus in a nuclear equation is written using nuclide notation:
$$\large{^A_Z X}$$
Where:
- X is the chemical symbol of the element
- A is the mass number (total number of protons + neutrons; top left)
- Z is the atomic number (number of protons; bottom left)
Examples:
- Uranium-238: ²³⁸₉₂U (92 protons, 238 nucleons total, so 146 neutrons)
- Carbon-14: ¹⁴₆C (6 protons, 8 neutrons)
- Helium-4: ⁴₂He (2 protons, 2 neutrons — this is also an alpha particle)
The rule for balancing is: the sum of all mass numbers on the left = the sum of all mass numbers on the right, AND the sum of all atomic numbers on the left = the sum of all atomic numbers on the right.
What is alpha decay?
Alpha decay occurs when an unstable nucleus ejects an alpha particle (α). An alpha particle consists of 2 protons and 2 neutrons — identical to a helium-4 nucleus.
Alpha particle notation: ⁴₂He (or written as ⁴₂α)
Effect on the nucleus:
- Mass number decreases by 4
- Atomic number decreases by 2
- The element changes (fewer protons = different element)
Worked example — alpha decay of uranium-238:
²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He
Check — mass numbers: 238 = 234 + 4 ✓ Check — atomic numbers: 92 = 90 + 2 ✓
Uranium-238 decays to thorium-234 by emitting an alpha particle.
Worked example — alpha decay of radium-226:
²²⁶₈₈Ra → ²²²₈₆Rn + ⁴₂He
Check: mass 226 = 222 + 4 ✓; atomic 88 = 86 + 2 ✓
What is beta-minus decay?
Beta-minus decay (β⁻) occurs when a neutron in the nucleus changes into a proton, ejecting a fast-moving electron (the beta-minus particle) and an antineutrino.
Beta-minus particle notation: ⁰₋₁e (mass number 0, atomic number −1)
Effect on the nucleus:
- Mass number unchanged (a neutron becomes a proton — same nucleon count)
- Atomic number increases by 1
Worked example — beta-minus decay of carbon-14:
¹⁴₆C → ¹⁴₇N + ⁰₋₁e
Check — mass numbers: 14 = 14 + 0 ✓ Check — atomic numbers: 6 = 7 + (−1) ✓
Carbon-14 decays to nitrogen-14 by emitting a beta-minus particle. (This reaction is the basis of radiocarbon dating — carbon-14 has a half-life of 5,730 years.)
Worked example — beta-minus decay of thorium-234:
²³⁴₉₀Th → ²³⁴₉₁Pa + ⁰₋₁e
Check: mass 234 = 234 + 0 ✓; atomic 90 = 91 + (−1) ✓
What is gamma emission?
Gamma radiation (γ) is high-energy electromagnetic radiation emitted when a nucleus releases excess energy after alpha or beta decay. It accompanies alpha or beta decay rather than occurring independently.
Gamma notation: ⁰₀γ (mass number 0, atomic number 0)
Effect on the nucleus:
- No change in mass number
- No change in atomic number
- The element is unchanged; only energy is released
Because gamma emission does not change any nucleon numbers, it is often shown alongside an alpha or beta decay:
²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He + ⁰₀γ
Gamma radiation is the most penetrating type of ionising radiation and is stopped only by thick lead or concrete.
Summary of the three types of decay
| Type | Symbol | Particle emitted | Change in A | Change in Z | Penetration |
|---|---|---|---|---|---|
| Alpha (α) | ⁴₂He | Helium nucleus (2p + 2n) | −4 | −2 | Stopped by paper or a few cm of air |
| Beta-minus (β⁻) | ⁰₋₁e | Fast electron | 0 | +1 | Stopped by a few mm of aluminium |
| Gamma (γ) | ⁰₀γ | Electromagnetic radiation | 0 | 0 | Reduced (not stopped) by thick lead |
How to balance a nuclear equation when one product is unknown
Worked problem: an isotope of polonium (Po, Z = 84) undergoes alpha decay. The daughter nucleus has mass number 206. Identify the daughter element.
Step 1 — set up the equation with the unknown daughter X:
²¹⁰₈₄Po → ᴬ_ZX + ⁴₂He
Step 2 — balance mass numbers:
210 = A + 4 → A = 206
Step 3 — balance atomic numbers:
84 = Z + 2 → Z = 82
Step 4 — identify the element with atomic number 82:
Z = 82 is lead (Pb).
Answer: ²¹⁰₈₄Po → ²⁰⁶₈₂Pb + ⁴₂He
Frequently asked questions
How do you balance a nuclear equation?
Write the nuclear equation with the known nuclei and the unknown product. Then use two conservation rules: (1) the sum of all mass numbers (top numbers) must be the same on both sides; (2) the sum of all atomic numbers (bottom numbers) must be the same on both sides. Work out the mass number and atomic number of the unknown product by subtraction, then look up the element with that atomic number in the periodic table.
Why does beta-minus decay increase the atomic number?
In beta-minus decay, a neutron inside the nucleus converts into a proton and releases a fast-moving electron (the beta particle). The proton stays in the nucleus, so the number of protons (atomic number) increases by one. The total number of nucleons (neutrons + protons) stays the same because one neutron becomes one proton — so the mass number is unchanged. The daughter nucleus has the same mass number but belongs to the next element in the periodic table.
What is the difference between alpha, beta, and gamma in terms of charge?
An alpha particle carries a charge of +2 (two protons). A beta-minus particle carries a charge of −1 (one electron). Gamma radiation is electromagnetic and carries no charge. The different charges affect how each type of radiation is deflected by electric and magnetic fields: alpha deflects one way, beta deflects the other way (opposite charge), and gamma is undeflected. The degree of ionisation they cause also differs: alpha is the most strongly ionising, gamma the least, because charge and mass determine how readily each type interacts with atoms in matter.
Can a nucleus emit both alpha and beta particles?
Not in a single decay event — a given decay event produces one type: alpha, beta-minus, or (rarely) beta-plus. However, a nucleus may undergo a decay series — a chain of successive decays producing multiple daughter products. For example, uranium-238 undergoes a long series of alpha and beta-minus decays before eventually reaching stable lead-206. Each step in the chain is a separate decay event, and the chain can include different types of decay at different steps.
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