Nuclear Equations

When an unstable nucleus decays, it transforms into a different nucleus by emitting particles or radiation. Just as chemical equations must balance atoms on both sides, nuclear equations must balance two conserved quantities: the total mass number (A, the count of protons plus neutrons) and the total atomic number (Z, the count of protons). These conservation laws are not arbitrary bookkeeping rules — they reflect fundamental physics: charge cannot appear from nowhere, and the total number of nucleons (protons and neutrons) is conserved in all nuclear reactions at everyday energies. Nuclide notation gives us a compact way to write any nucleus: the chemical symbol carries the mass number as a superscript and the atomic number as a subscript, written as $^A_Z\text{X}$ . For example, uranium-238 is written $^{238}_{92}\text{U}$ , telling us it has 238 nucleons total and 92 protons (hence 146 neutrons). Emitted particles like alpha and beta particles have their own nuclide symbols with fixed A and Z values, so they slot naturally into the equation framework. There are three main types of radioactive decay covered at this level: alpha (α) decay, beta-minus (β⁻) decay, and beta-plus (β⁺) decay. Each produces a characteristic particle, transforms the parent nucleus into a daughter nucleus, and follows the same conservation rules. Mastering nuclear equations means learning to identify the unknown daughter nucleus or emitted particle by applying these two balancing rules — a process very similar to solving a simple algebra problem.

Key takeaways

Write Down What You Know
Balance the Mass Numbers (Top Numbers)
Balance the Atomic Numbers (Bottom Numbers)
Identify the Daughter Element

Write Down What You Know

Start with the parent nuclide on the left side of the arrow and the emitted particle on the right side (or vice versa if you are finding the particle). For alpha decay, the emitted particle is $^4_2\text{He}$ (a helium-4 nucleus: 2 protons, 2 neutrons). For beta-minus decay, the emitted particle is $^0_{-1}\text{e}$ (an electron: mass number 0, charge −1). For beta-plus decay, it is $^0_{+1}\text{e}$ (a positron: mass number 0, charge +1). Write a placeholder $^A_Z\text{X}$ for the unknown daughter nucleus.

Balance the Mass Numbers (Top Numbers)

The sum of all mass numbers on the left must equal the sum of all mass numbers on the right. Mass numbers simply add or subtract like regular integers. For alpha decay: $A_{\text{daughter}} = A_{\text{parent}} - 4$ . For beta decay (either type): $A_{\text{daughter}} = A_{\text{parent}} - 0 = A_{\text{parent}}$ , because the electron or positron carries no nucleons. This step gives you the daughter's mass number.

Balance the Atomic Numbers (Bottom Numbers)

The sum of all atomic numbers (charges) on the left must equal the sum on the right. For alpha decay: $Z_{\text{daughter}} = Z_{\text{parent}} - 2$ , since the alpha particle carries away 2 protons. For beta-minus decay: $Z_{\text{daughter}} = Z_{\text{parent}} + 1$ , because a neutron converts to a proton plus an electron (the electron is emitted, but the proton stays, increasing Z by 1). For beta-plus decay: $Z_{\text{daughter}} = Z_{\text{parent}} - 1$ , because a proton converts to a neutron plus a positron.

Identify the Daughter Element

Once you know Z for the daughter, use the periodic table to find the element symbol. The mass number A you already found in step 2 goes as the superscript. Write the complete daughter nuclide. Double-check both conservation laws by adding all superscripts on each side and all subscripts on each side — they must match exactly. This is your verification step, like checking arithmetic by working backwards.

The Key Equation

^A_Z\text{X} \;\longrightarrow\; ^{A-4}_{Z-2}\text{Y} \;+\; ^4_2\text{He}

^A_Z\text{X} — Parent nuclide (dimensionless): The original unstable nucleus before decay. A is its total nucleon count; Z is its proton count.
^{A-4}_{Z-2}\text{Y} — Daughter nuclide (dimensionless): The new nucleus formed after decay. It has 4 fewer nucleons and 2 fewer protons than the parent.
^4_2\text{He} — Alpha particle (dimensionless): A helium-4 nucleus consisting of 2 protons and 2 neutrons, emitted at high speed during alpha decay.

Worked Example

Problem. Radium-226 ( $^{226}_{88}\text{Ra}$ ) undergoes alpha decay. Write the complete balanced nuclear equation and identify the daughter nucleus.

^{226}_{88}\text{Ra} \;\longrightarrow\; ^{222}_{86}\text{Rn} \;+\; ^4_2\text{He}

Answer. The daughter nucleus is radon-222.

Figure

Diagram

Origins · 1 minMarie Curie and how this was discovered

Marie Curie

1867–1934 · Polish-French

Marie Curie pioneered the scientific study of radioactivity — a term she herself coined. Working with uranium, thorium, and the new elements polonium and radium she discovered, she demonstrated that radioactivity is an atomic property of individual elements rather than a chemical interaction between substances. Her precise measurements of radioactive emissions laid the groundwork for the quantitative nuclear equations we use today. Curie remains the only person to have won Nobel Prizes in two different sciences: Physics (1903, shared with Pierre Curie and Henri Becquerel) and Chemistry (1911). Her original lab notebooks are still so radioactive that researchers must sign a waiver and wear protective gear to view them.

Types of Radioactive Decay — Understanding what alpha, beta, and gamma decay are physically provides essential context for writing and interpreting nuclear equations.
Half-Life and Decay Rate — Half-life describes how quickly a radioactive nucleus undergoes the same decays described by nuclear equations, connecting the equation to real timescales.
Nuclear Stability and Binding Energy — Binding energy explains why certain nuclei are unstable and which type of decay they undergo, giving physical meaning to the equations.
Nuclide Notation and Isotopes — Fluency with the $^A_Z\text{X}$ notation is a prerequisite for reading and writing any nuclear equation correctly.