Nuclear Equation Balancer – Conserve A and Z, Name the Decay
This nuclear equation balancer solves the bookkeeping behind alpha, beta, gamma and electron-capture decay. Type an equation with one unknown, such as U-238 → ? + α, and it finds the missing nuclide or particle, shows the conservation of mass number (A) and atomic number (Z), names the process and, when the masses are in its table, gives the Q-value.
Why nuclear equations balance differently
A chemical equation balances atoms of each element, and the elements never change. A nuclear equation balances nucleons and charge. The element can change, because a nucleus that gains or loses protons becomes a different element. The rule is simple: the sum of A and the sum of Z must be equal on both sides. Electrons and positrons count in Z as −1 and +1, with A = 0.
The decay modes and their A/Z rules
- Alpha decay: A − 4, Z − 2. Example:
Ra-226 → Rn-222 + α. - Beta-minus decay: A unchanged, Z + 1. Example:
C-14 → N-14 + e⁻ + ν̄. - Beta-plus decay: A unchanged, Z − 1. Example:
F-18 → O-18 + e⁺ + ν. - Electron capture: A unchanged, Z − 1. Example:
Be-7 + e⁻ → Li-7 + ν. - Gamma emission: A and Z unchanged. Example:
Tc-99m → Tc-99 + γ.
Solving for the missing particle
Add up A and Z for the side that is complete, then subtract the known species on the other side. What is left is the missing species. For U-238 → ? + α the left side has A = 238 and Z = 92, the alpha particle takes A = 4 and Z = 2, so the unknown has A = 234 and Z = 90: thorium-234. A result of A = 0, Z = 0 could be a gamma photon or a neutrino, and the tool offers both. An unknown multiplier such as ?n is solved the same way, as long as one whole number fits.
Transmutation and shorthand notation
When a light projectile hits a target, physicists write the reaction as target(projectile, ejectile)product. Rutherford's 1919 experiment, N-14 + α → O-17 + p, is written ¹⁴N(α,p)¹⁷O. You can type either form, and the tool shows the other.
Decay chains and the four series
Heavy nuclei rarely reach stability in one step. The decay-chain tab applies one step at a time, or loads a natural series. Because an alpha decay changes A by 4 and a beta decay changes it by 0, A mod 4 never changes along a chain, which gives four classes: the thorium series (4n), the neptunium series (4n + 1), the uranium series (4n + 2) and the actinium series (4n + 3). Three end at lead because the closed proton shell at Z = 82 and the closed neutron shell at N = 126 make lead-206, lead-207 and lead-208 especially stable. The neptunium series is extinct in nature and ends at bismuth-209. The α/β counter answers the classic question of how many alpha and beta decays take U-238 to Pb-206: 8 and 6.
The Q-value in one paragraph
The Q-value is the energy released when the products are lighter than the reactants: Q = (m_reactants − m_products) · c². A positive Q means energy is released, and a spontaneous decay needs Q > 0. For the mass-defect detail and the binding-energy curve, use the Nuclear Binding Energy Calculator. For how much of a sample is left after a given time, use the Half-Life Calculator or the Radioactive Decay Simulator.