Chemical Equation Balancer

Balancing a chemical equation means finding integer coefficients (the small numbers in front of each species) so that every element has the same total atom count on both sides. This is a direct consequence of the law of conservation of mass — atoms are not created or destroyed during a chemical reaction.

This calculator solves the balancing problem algebraically using matrix null spaces. Each chemical species becomes a column; each element becomes a row; the entry at row i, column j is the count of element i in species j. Reactant entries are positive, product entries negative. A vector x of coefficients balances the equation exactly when M x = 0 — that is, x is in the null space of M.

For most chemical equations the null space is one-dimensional, giving a unique balancing (up to scaling). The calculator finds the basis vector, scales it to smallest positive integers, and reports the coefficients.

Worked example: propane combustion, C₃H₈ + O₂ = CO₂ + H₂O. Elements: C, H, O. Species: C₃H₈, O₂, CO₂, H₂O. Matrix:

• C row: 3, 0, -1, 0
• H row: 8, 0, 0, -2
• O row: 0, 2, -2, -1

Null space gives coefficients (1, 5, 3, 4): 1 C₃H₈ + 5 O₂ = 3 CO₂ + 4 H₂O. Verify: 3 C left, 3 right ✓; 8 H left, 8 right ✓; 10 O left, 10 right ✓.

Input format:

• Use + between species on the same side
• Use = or -> or → between reactant and product sides
• Subscripts go directly after element symbols, no superscripts
• Parentheses with subscripts are supported: Al2(SO4)3
• Capitalization matters — Co (cobalt) is different from CO (carbon monoxide)

Examples to try:

• H2 + O2 = H2O (the classic)
• Fe + O2 = Fe2O3 (rust formation)
• C2H6 + O2 = CO2 + H2O (ethane combustion)
• KMnO4 + HCl = KCl + MnCl2 + H2O + Cl2 (a tougher one with five products)

If the calculator returns an error like “underdetermined” or “no solution,” the equation likely has a typo, a missing species, or contains compounds that genuinely cannot react in the given proportions. The math reflects the chemistry — when you cannot balance, conservation is violated.

For redox reactions in solution, you may need to add H⁺, OH⁻, or H₂O explicitly as separate species. The calculator does not auto-add water or proton balancers.