Chemical equation balancer

Enter the reactants and products; the coefficients appear as you type. Redox and ionic equations work too: charge is balanced along with the atoms, and electrons can be written as e-.

Method: the null space of the composition matrix

Each species becomes a column and each element a row (plus a row for charge when ions are present). Entries are atom counts, positive for reactants and negative for products. Balancing means finding whole numbers x with A·x = 0. Gaussian elimination in exact fractions (no rounding) finds every solution:

• One independent solution with all coefficients positive: that is the balanced equation, scaled to the smallest whole numbers.
• No solution: some element (or the charge) cannot be conserved, usually because of a typo or a missing species. If an element appears on one side only, the tool names it.
• More than one independent solution: the equation combines separate reactions (for example H2 + O2 = H2O2 + H2O), so any mixture of them balances and there is no single answer.
• A solution that needs a zero or negative coefficient: a species cannot take part, or belongs on the other side.

A·x = 0, x = smallest positive integer vector of the one-dimensional null space

Limits

Balancing checks conservation only. It cannot say whether a reaction happens, what its products are, or whether the species exist. Write charges with ^ for anything other than ±1: Fe2+ is read as Fe₂ with a charge of one, so write Fe^2+.

Worked example

KMnO4 + HCl = KCl + MnCl2 + H2O + Cl2 balances as 2 KMnO₄ + 16 HCl → 2 KCl + 2 MnCl₂ + 8 H₂O + 5 Cl₂. The ionic form MnO4^- + Fe^2+ + H^+ = Mn^2+ + Fe^3+ + H2O gives MnO₄⁻ + 5 Fe²⁺ + 8 H⁺ → Mn²⁺ + 5 Fe³⁺ + 4 H₂O, with a charge of +17 on each side.

References

• Blakley, Chemical equation balancing: a general method which is quick, simple, and has unexpected applications, J. Chem. Educ. 59, 728 (1982)
• IUPAC, Quantities, Units and Symbols in Physical Chemistry (Green Book), 3rd edition (2007)