Chemical Equation Balancer

About the Chemical Equation Balancer

Type a reaction such as Fe + O2 = Fe2O3 or C3H8 + O2 -> CO2 + H2O and get it balanced with the smallest whole-number coefficients: 4Fe + 3O₂ → 2Fe₂O₃. Ionic equations and redox half-reactions work too — charges (MnO4-, Fe3+, Cr2O7^2-) and electrons (e-) are balanced along with the atoms — and so do states such as (s), (aq) and (g), hydrates, isotopes and common names.

Instead of trial and error, the balancer solves the equations exactly: one equation per element and one for charge, solved by rational Gaussian elimination for the null space of the composition matrix. It shows every step, checks each element and the charge on both sides, confirms that mass is conserved, and explains clearly when an equation cannot be balanced or can be balanced in more than one way.

Common uses

• Chemistry homework: balancing molecular, ionic and redox equations, with the method shown.
• Checking coefficients you balanced by hand — the page tells you whether yours already balance.
• Preparing stoichiometry: the balanced coefficients are the mole ratios — Calculate amounts opens the equation in the stoichiometry calculator.
• Writing up reactions in LaTeX with the mhchem package.

How the balancing works

Give each substance an unknown coefficient — for x₁ Fe + x₂ O₂ → x₃ Fe₂O₃ — and write one equation per element saying it is conserved: iron x₁ − 2x₃ = 0, oxygen 2x₂ − 3x₃ = 0. Ions add one more equation for charge. These are a homogeneous linear system A·x = 0, where A is the composition matrix (reactants counted positive, products negative) — the formula-matrix method of Smith and Missen.

The balancer reduces A to reduced row echelon form with exact fractions and reads off its null space, as L. R. Thorne describes (Chem. Educator 15, 304, 2010): here x₃ is free, and x₃ = 1 gives x = (2, 3/2, 1). Multiplying by the lowest common denominator, 2, gives the smallest whole numbers (4, 3, 2). Because no decimals are ever rounded, even large equations come out exact.

When an equation cannot be balanced, or balances many ways

• No solution: if the null space is empty, only zero coefficients satisfy every element — usually a reactant or product is missing (often H₂O, O₂, H⁺ or OH⁻) or a formula is wrong. The page names elements that appear on one side only, and says when the atoms could balance but the charge cannot (add electrons for a half-reaction).
• One solution: the usual case; if a substance can only balance on the other side of the arrow it is moved there, and one that cannot take part (coefficient 0) is left out — both with a warning.
• Several solutions: an equation such as H₂ + O₂ → H₂O + H₂O₂ mixes independent reactions (2H₂ + O₂ → 2H₂O and H₂ + O₂ → H₂O₂). Any positive mix balances; the page shows the one with the smallest whole numbers that uses every substance, lists the independent reactions, and confirms your own coefficients if they balance.

Redox reactions and half-reactions

The balancer balances the substances you list, so include what the reaction medium supplies: in acidic solution add H⁺ and H₂O, in basic solution OH⁻ and H₂O. For a half-reaction, add e- on the side that gains electrons; the charge row then fixes their number, as in Cr₂O₇²⁻ + 14H⁺ + 6e⁻ → 2Cr³⁺ + 7H₂O.

How to type equations

• Arrows: =, ->, →, ⟶ (also –> or —> as word processors write -->), and <=> or ⇌ for equilibria (shown as ⇌).
• Separate substances with +; spaces around it are best: Fe3+ + e- = Fe2+.
• Charges at the end of a formula: SO4^2-, SO4{2-}, SO4²⁻, NH4+, Fe3+, Fe+++.
• Electrons: e-, e^- or e⁻.
• States are kept: H2O(l), NaCl(aq), CO2(g), AgCl(s).
• Conditions written as terms, such as + heat, + light or + energy, are left out of the balancing (with a note).
• Oxidation states in Roman numerals are read and left out: Fe(III)Cl3 is FeCl₃.
• Brackets, hydrates (CuSO4·5H2O), isotopes (D2O, ^13C) and common names (water, carbon dioxide) work as in the molar mass calculator. A leading number is a coefficient: 13CO2 means 13 CO₂, so write carbon-13 as ^13CO2.

Sources

• L. R. Thorne, “An innovative approach to balancing chemical-reaction equations: a simplified matrix-inversion technique for determining the matrix null space”, Chem. Educator 15, 304–308 (2010), arXiv:1110.4321.
• W. R. Smith and R. W. Missen, Chemical Reaction Equilibrium Analysis: Theory and Algorithms (Wiley, 1982) — the formula-matrix description of reactions.
• OpenStax, Chemistry 2e (CC BY 4.0), §4.1 Writing and Balancing Chemical Equations and §4.3 Reaction Stoichiometry.
• IUPAC CIAAW, Standard Atomic Weights 2024 (for the mass check).

Limitations

• Balancing is purely algebraic: it does not know whether a reaction really happens, which products form or what the reaction conditions are.
• Organic abbreviations (Me, Et, Ph) are not expanded — write the full formulas.
• For equations that combine several independent reactions, the coefficients shown are one valid choice among many; the real proportions depend on the chemistry.
• Up to 600 characters per equation.

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