Balancing Nuclear Equations Calculator
A nuclear equation balances on two counts at once: the mass numbers (the top numbers, A) must add up the same on both sides, and the atomic numbers (the bottom numbers, Z) must add up the same on both sides. Get both sums right and any single missing particle in the equation is forced: it is whatever makes up the difference. The Balancing Nuclear Equations Calculator on this page does exactly that subtraction for you. Enter the mass number and atomic number of the starting nuclide and of the one known product, and it returns the missing particle’s A and Z, identifies the particle, and writes out the full balanced equation.
The tool is built for the most common textbook shape of the problem, one nuclide breaking into one known product plus one unknown particle, which covers alpha decay, beta decay, positron emission, gamma emission, and single neutron or proton emission. Every worked example in this article balances on both counts, and you can check each one by hand in seconds: the two sums either match or the equation is wrong.
Starting nuclide (reactant)
Known product nuclide
The tool solves for one missing particle in a single-step equation with one known product. Equations with several products, such as most fission and fusion reactions, need the sums of all known products subtracted instead; the method section below works one of those by hand.
Quick Answer
Key takeaways
- Two sums must match: total A on the left equals total A on the right, and total Z on the left equals total Z on the right. Nothing else about the equation can rescue a failed sum.
- The missing particle is found by plain subtraction, reactant minus known products, once for A and once for Z.
- Six signatures cover most textbook problems: alpha (4, 2), beta-minus (0, -1), positron (0, 1), gamma (0, 0), neutron (1, 0), proton (1, 1).
- This tool handles one missing particle. Multi-product fission and fusion equations use the same rule with the sums of all known products.
- Quick Answer
- The two conservation rules that do all the work
- The six particle signatures worth memorizing
- Worked example: uranium-235 alpha decay, checked twice
- When the equation has more than one product: fission worked by hand
- What competing resources miss
- Frequently asked questions
- Add This Calculator to Your Website
- References
The two conservation rules that do all the work
Mass number and atomic number are both conserved in every nuclear reaction. The mass number counts nucleons, protons plus neutrons, and nucleons are neither created nor destroyed in a decay or a bombardment reaction; they are only rearranged between nuclei and free particles. The atomic number counts protons, and its conservation is really charge conservation: total charge before equals total charge after. Chemistry LibreTexts states the pair of rules in exactly this form, and every balancing method, from this calculator to a graduate problem set, is those two rules applied patiently.
This is the sharp difference from ordinary chemical equations. In chemistry, atoms keep their identity and you balance element by element. In nuclear reactions the elements themselves change: a neutron inside the nucleus converts to a proton during beta-minus decay, the atomic number climbs by one, and a new element exists where the old one stood. The bookkeeping still works because A and Z, not element symbols, are the conserved quantities.
The six particle signatures worth memorizing
Most missing-particle problems resolve to one of six particles. Learn their (A, Z) signatures and you can read a decay type straight off the change in the parent nuclide, before any calculator comes out.
| Particle | A | Z | What happens to the parent |
|---|---|---|---|
| Alpha particle (helium-4 nucleus) | 4 | 2 | A drops by 4, Z drops by 2, new element two places lower |
| Beta-minus (electron) | 0 | -1 | A unchanged, Z rises by 1 in the product: a neutron became a proton |
| Positron (beta-plus) | 0 | 1 | A unchanged, Z drops by 1 in the product: a proton became a neutron |
| Gamma ray | 0 | 0 | Nothing changes but energy: an excited nucleus settling down |
| Neutron | 1 | 0 | A drops by 1, Z unchanged, same element, lighter isotope |
| Proton | 1 | 1 | A drops by 1, Z drops by 1, new element one place lower |
Any other (A, Z) result means the missing piece is a whole nuclide rather than a fundamental particle, and the calculator reports it as a custom particle with its A and Z, from which you name the element by Z. That case is common in bombardment reactions, for instance when an alpha particle strikes a nucleus and a new nuclide plus a neutron come out. Working out which nuclide a reaction produces starts from its Z, exactly the habit the average atomic mass calculator builds from the isotope side.
Worked example: uranium-235 alpha decay, checked twice
The calculator is pre-filled with this case: reactant A 235, Z 92 (uranium-235); known product A 231, Z 90 (thorium-231).
Subtract for A. Missing A = 235 minus 231 = 4.
Subtract for Z. Missing Z = 92 minus 90 = 2.
Match the signature. (4, 2) is the alpha particle, a helium-4 nucleus. The equation reads: uranium-235 goes to thorium-231 plus an alpha particle.
Check both sums. Mass numbers: 231 + 4 = 235, matching the reactant. Atomic numbers: 90 + 2 = 92, matching the reactant. Balanced on both counts.
A second check with a different decay type: carbon-14 (A 14, Z 6) decaying to nitrogen-14 (A 14, Z 7). Enter reactant 14 and 6, product 14 and 7, and the missing pair is A = 0, Z = -1, a beta-minus particle. Check: 14 + 0 = 14 and 7 + (-1) = 6. Balanced. Students often stumble on that negative Z; it is correct, because the emitted electron carries charge -1, and charge conservation is what Z bookkeeping tracks. Averaging exercises like these pair naturally with the average mass calculator when a course moves from single decays to isotope mixtures.
When the equation has more than one product: fission worked by hand
Real fission and fusion equations carry several products, and the subtraction must use the sums of everything known. Take a classic: uranium-235 absorbs a neutron and splits into barium-141, krypton-92, and some neutrons. Reactant side totals: A = 235 + 1 = 236, Z = 92 + 0 = 92. Known products: A = 141 + 92 = 233, Z = 56 + 36 = 92. Missing A = 236 minus 233 = 3, missing Z = 92 minus 92 = 0, so the missing piece is 3 neutrons, each (1, 0). Final check: A, 141 + 92 + 3 = 236; Z, 56 + 36 + 0 = 92. Balanced.
One more honesty point: the tool’s conservation check can never fail, because the missing values are defined as the difference. The check exists to display the two sums so you can verify them yourself, which is the actual skill exams test. If your two subtractions give a particle that makes no physical sense for the reaction in front of you (a positron from a neutron-rich fission fragment, say), the arithmetic is still balanced, but the reaction as written probably is not the one your problem intended.
What competing resources miss
The pages ranking for this topic are mostly homework-answer sites. Pearson and Vaia work textbook problems step by step, including multi-particle cases, but they answer their problems, not yours: there is no tool to run your numbers through. LibreTexts teaches the conservation rules cleanly and is the source this article follows, but it offers no calculator either. AI answer engines return solutions that are sometimes garbled mid-derivation, with symbols dropped exactly where a student needs them.
This page pairs the instant subtraction with the full signature table and a hand-worked fission case, so a student can move from running the tool to doing the method unaided. That progression, tool first and method second, is what homework help usually skips.
Frequently asked questions
Add up the mass numbers (A) and the atomic numbers (Z) on each side separately. A balanced equation has equal A totals and equal Z totals on both sides. To find a single missing particle, subtract the known products’ totals from the reactants’ totals, once for A and once for Z, then identify the particle from its (A, Z) pair.
An alpha particle, which is a helium-4 nucleus with A = 4 and Z = 2. Example: uranium-235 (A 235, Z 92) decays to thorium-231 (A 231, Z 90) plus an alpha particle. Check the sums: 231 + 4 = 235 and 90 + 2 = 92.
Because Z bookkeeping tracks charge. The emitted electron carries charge -1, so it is written with A = 0 and Z = -1. When carbon-14 (Z 6) emits a beta-minus particle, the product’s Z is 7, and 7 + (-1) = 6 conserves charge. Inside the nucleus, a neutron has converted into a proton.
It solves one missing particle when there is one known product, which covers most decay problems. Fission and fusion usually have several products; use the same rule with the sums of all known products. Example: if the known products total A 233 and Z 92 against reactants totalling A 236 and Z 92, the missing piece is 3 neutrons, as the fission example in this article shows.
Then the missing piece is a whole nuclide, and the calculator reports it as a custom particle with its A and Z. Read the element off Z (for example Z = 8 is oxygen) and write it as that element’s isotope with mass number A. This happens often in bombardment reactions.
Chemical equations conserve each element’s atom count, and elements never change identity. Nuclear equations conserve total A and total Z, while elements routinely change: alpha emission drops Z by 2, beta-minus emission raises the product’s Z by 1. You balance nucleon and charge totals, not element counts.
No. A gamma ray carries energy away with A = 0 and Z = 0, so the nuclide keeps the same mass number and atomic number and stays the same isotope, just in a lower energy state. Gamma emission often follows alpha or beta decay, which is why it appears alongside them in decay chains.
Because the missing A and Z are computed as the exact differences, the two sums match by construction. The check is shown so you can see both sums and verify the method yourself, which is the skill worth learning. A physically sensible particle for the reaction type is a separate judgement the arithmetic cannot make.
Add This Calculator to Your Website
References
- Chemistry LibreTexts, “Balancing Nuclear Equations” – states the two conservation rules and works alpha and beta decay examples with both sums checked. LibreTexts: Balancing Nuclear Equations. Accessed October 2026.
- Chemistry LibreTexts, “Nuclear Equations” – covers particle symbols for protons, neutrons, alpha, beta, positron, and gamma, with the magnesium-25 bombardment example. LibreTexts: Nuclear Equations. Accessed October 2026.
- OpenStax (Rice University), “Chemistry 2e: Nuclear Equations” – textbook treatment of balancing nuclear reactions by mass number and charge conservation. OpenStax Chemistry 2e: Nuclear Equations. Accessed October 2026.
- Chemistry LibreTexts, “Nuclear Equations (General Chemistry II)” – worked identification of unknown nuclides from A and Z balance, including the magnesium-25 plus alpha example. LibreTexts: Nuclear Equations and Unknown Nuclides. Accessed October 2026.
- International Atomic Energy Agency, “Nuclear chemistry teaching resources” – agency background material on radioactivity, decay types, and nuclear reactions for students. IAEA: Nuclear Science. Accessed October 2026.
- US Nuclear Regulatory Commission, “Science 101: what is a chain reaction” – explains neutron multiplication in fission in plain language, the context for the multi-neutron fission example. NRC: What Is a Chain Reaction. Accessed October 2026.