Subjects and lessons to explore.

Law

Chemistry: Atoms First 2e · Nuclear Chemistry

Nuclear Equations

8 min read
11 key termsSourcesReport an error
Science note: Nuclide identities, decay modes, and the uranium-238 decay chain are standard published facts; all worked equations balance from the stated inputs. Shielding statements are general principles for educational context, not operational safety instructions.
Want it in plain words first? Jump to Eli explains — the same idea, no jargon.
On this page 9 sections
  1. In 30 seconds
  2. Why this matters
  3. The college version
  4. Eli explains
  5. Worked example
  6. Key takeaway
  7. Check yourself
  8. Study tools
  9. Sources & references

In 30 seconds

A nuclear equation is a before-and-after record of a change inside an atomic nucleus. The starting nucleus is the , the nucleus left behind is the , and the difference shows up as an emitted particle or photon. Every nuclide is written with its chemical symbol, the mass number A (protons + neutrons) as a superscript, and the atomic number Z (protons) as a subscript, as in 23892U.

Nuclear equations are balanced by two bookkeeping rules only: the total mass number on each side must match, and the total atomic number on each side must match. That is far simpler than balancing a chemical equation — no coefficients — but it demands precision. An 42He, a 0-1e, a 0+1e, and a 00γ each change the balance differently. Once you can write and complete these equations, you can predict what element a sample turns into, what it emits, and how fast — the foundation for the rest of this chapter.

Why this matters

Nuclear equations are the working language of nuclear medicine, radiometric dating, and nuclear power. A technologist preparing a technetium-99m scan, a geologist measuring a rock's uranium-to-lead ratio, and an engineer tracking fission products all rely on the same balancing rules. For everyday decisions, the equation answers practical questions: does this isotope emit penetrating gamma rays (needing shielding) or short-range alpha particles (dangerous only if inhaled or swallowed)? Balancing skills are also the prerequisite for every calculation that follows, since half-lives, activities, and ages all start from a correctly written decay equation.

The college version

Core Concepts

The two bookkeeping rules

Every nuclear equation must conserve the total mass number A and the total atomic number Z. The left side (parent, plus any captured particle) and the right side (daughter, plus any emitted particles) must show equal sums for both quantities:

∑Aleft = ∑Aright   and   ∑Zleft = ∑Zright

Charge is conserved automatically, because the bottom numbers track protons: a particle with Z = -1 (the beta particle) carries one unit of negative charge. The nucleon count (protons + neutrons) never changes in alpha and beta processes; the nucleus merely rearranges which particles are protons.

The particles that appear in decay equations

  • Alpha particle, 42He — a helium-4 nucleus (2 protons, 2 neutrons) ejected by heavy nuclei; it lowers A by 4 and Z by 2.
  • Beta particle, 0-1e — a high-speed electron created inside the nucleus when a neutron converts to a proton; it raises Z by 1 and leaves A unchanged.
  • Positron, 0+1e — an antimatter electron produced when a proton converts to a neutron; it lowers Z by 1 and leaves A unchanged.
  • Gamma ray, 00γ — a high-energy photon carrying away excess nuclear energy; it changes neither A nor Z.
  • — the nucleus pulls in an inner-shell electron, 0-1e, which combines with a proton to form a neutron; like positron emission, Z falls by 1 with A unchanged.

Writing and completing a decay equation

Write the parent nuclide on the left and the known emitted particle on the right, leave the daughter as the unknown, and solve the two balance equations for its A and Z. Then look up the element with that Z on the periodic table. Finally, verify both sums — remembering that a beta particle contributes -1 to the charge sum. If the daughter is itself radioactive, the process continues into a .

Decay series

One decay step often leaves a still-unstable daughter, so heavy nuclides travel through decay series — chains of alternating alpha and beta steps — until a stable nucleus forms. Uranium-238 passes through 14 steps (including the gas radon-222) before reaching stable lead-206. Every step obeys the same balance rules: alpha drops A by 4, beta keeps A the same.

How It Works / Step-by-Step Process

  1. Read the parent's symbol to identify its A and Z.
  2. Identify the emitted particle from the decay type (alpha, beta, positron, gamma, or electron capture).
  3. Write the skeleton equation with the daughter as an unknown.
  4. Balance mass numbers: Adaughter = Aparent - Aparticle.
  5. Balance atomic numbers: Zdaughter = Zparent - Zparticle (watch the sign: beta contributes -1).
  6. Identify the element from Z, write the balanced equation, and check both sums.

Common Confusions

Do Not ConfuseWithDifference
Beta particleAn electron orbiting the nucleusA beta particle forms inside the nucleus at decay; it is not an orbital electron
Positron emissionBeta emissionPositron emission lowers Z by 1; beta emission raises Z by 1
Nuclear equation balancingChemical equation balancingNuclear equations conserve A and Z only, not atoms or coefficients
Mass number AAtomic mass in gramsA counts nucleons (unitless); atomic mass is a mass in amu or grams
Gamma emissionParticle emissionGamma rays carry no mass or charge, so the nuclide keeps its identity
Electron captureBeta decayBoth lower Z by 1; electron capture consumes an inner electron instead of emitting a positron
Eli, the EliExplains learning guide

Eli explains

The same idea, in plain words

Explain it like I’m 10

A nuclear equation is like a seesaw with two scoreboards. The top scoreboard counts the total pieces in the nucleus, and the bottom one counts the protons — and both must show the same total on both sides. When a big nucleus spits out an alpha particle, it loses 4 pieces and 2 protons, so a new element is born. When a neutron turns into a proton, a tiny electron called a beta particle flies out and the element moves one step right on the periodic table.

Worked example

Example 1: Alpha decay of radium-226

Radium-226, 22688Ra, is an alpha emitter found in the uranium decay series. Use the balance formulas before substituting:

Adaughter = Aparent - Aalpha = 226 - 4 = 222

Zdaughter = Zparent - Zalpha = 88 - 2 = 86

The element with Z = 86 is radon, so:

22688Ra → 22286Rn + 42He

Check: 226 = 222 + 4 and 88 = 86 + 2 — both sides balance. The daughter, radon-222, is a radioactive gas whose decay products are the main source of residential radon exposure.

Example 2: Beta decay of carbon-14

Carbon-14, 146C, the isotope used to date once-living material, decays by beta emission. A beta particle is 0-1e, so the mass number is unchanged:

Adaughter = 14 - 0 = 14

Balance the atomic numbers, remembering the beta particle contributes -1:

6 = Zdaughter + (-1)   ⇒  Zdaughter = 7

The element with Z = 7 is nitrogen:

146C → 147N + 0-1e

A neutron became a proton, so the daughter is the element one place to the right with the same mass number. This equation is the "clock" behind radiocarbon dating.

Example 3: Positron emission of fluorine-18

Fluorine-18, 189F, is a positron emitter used in PET imaging. A positron is 0+1e. The mass number is unchanged:

Adaughter = 18 - 0 = 18

Balance the atomic numbers:

9 = Zdaughter + 1   ⇒  Zdaughter = 8

The element with Z = 8 is oxygen:

189F → 188O + 0+1e

Each emitted positron soon meets an ordinary electron; the pair annihilates into two gamma photons that PET scanners detect. Positron emission and beta emission move Z in opposite directions while both leave A untouched.

Key takeaways

  • Nuclear equations conserve mass number A and atomic number Z; nothing else needs balancing.
  • Alpha decay: AZX → A−4Z-2Y + 42He.
  • Beta decay: AZX → AZ+1Y + 0-1e — a neutron becomes a proton.
  • Positron emission and electron capture: AZX → AZ-1Y + 0+1e (or captured electron) — a proton becomes a neutron.
  • Gamma emission changes neither A nor Z; it only removes energy.
  • To find an unknown daughter, subtract the emitted particle's A and Z from the parent's.
  • Decay series continue until a stable nuclide forms; each step must balance.

Check yourself

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

  1. What two quantities must balance in every nuclear equation?

    Show answer

    The total mass number A and the total atomic number Z on each side of the equation.

  2. Write the alpha decay of polonium-210, 21084Po, and identify the daughter.

    Show answer

    21084Po → 20682Pb + 42He; the daughter is lead-206. Check: 210 = 206 + 4 and 84 = 82 + 2.

  3. In beta decay, what happens inside the nucleus, and how does Z change?

    Show answer

    A neutron becomes a proton and an electron (the beta particle) is emitted; Z increases by 1 while A is unchanged.

  4. Why does gamma emission not change the identity of a nuclide?

    Show answer

    Gamma rays carry energy but no mass or charge, so both A and Z stay the same and the nuclide keeps its identity.

  5. What do the superscript and subscript in 13153I represent?

    Show answer

    The superscript 131 is the mass number (protons + neutrons); the subscript 53 is the atomic number (protons), identifying iodine.

  6. Write the positron emission of carbon-11, 116C, and identify the daughter.

    Show answer

    116C → 115B + 0+1e; the daughter is boron-11.

Keep learning

Ready to build on this? Continue to the next lesson.

Study tools & related lessonsKey vocabulary · Related

Key vocabulary

parent nuclide
The unstable nucleus that undergoes decay
daughter nuclide
The nucleus produced by the decay
mass number A
Total protons + neutrons in a nucleus
atomic number Z
Number of protons; defines the element
alpha particle
A helium-4 nucleus, 42He
beta particle
An electron created in the nucleus, 0-1e
positron
An antimatter electron, 0+1e
gamma ray
High-energy photon, 00γ
electron capture
Nucleus absorbs an inner electron to make a neutron
decay series
A chain of decays from a heavy nuclide to a stable end
Mass number (A)
The total number of protons + neutrons

Sources & references

  1. 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.