Equation balancer

Every element and the charge give one conservation equation, with one unknown coefficient per species. The balancer solves this system exactly, with whole-number fractions of unlimited size rather than floating point, and takes the null space of the matrix: one independent solution means one balanced equation, scaled to the smallest whole numbers. No solution means some element or the charge cannot be conserved, and the tool says which. More than one independent solution means the equation hides several reactions, which it lists.

In acidic or basic solution, H₂O and H⁺ or OH⁻ are added as free species that may land on either side. If the charge still cannot balance, electrons are added and the result is a half-reaction.

A reaction is treated as a redox reaction when an element's oxidation state, assigned with the usual rules (F −1, O −2, H +1 or −1 in metal hydrides, alkali and alkaline earth metals +1 and +2, halides −1, and the common oxoanions such as sulfate, nitrate and phosphate in metal salts), differs between the two sides. The average oxidation state of carbon in organic compounds is not used as evidence, because it also changes in reactions such as esterifications that transfer no electrons. When a formula alone does not fix the oxidation states, as in K₄[Fe(CN)₆], the tool makes no claim either way.

For a redox reaction the species are split into the oxidation and reduction couples. Each couple is balanced on its own, first in the elements other than oxygen and hydrogen, then oxygen with H₂O, hydrogen with H⁺ or OH⁻, and the charge with electrons; the two halves are scaled until the electrons cancel. This is the half-reaction method from general chemistry, and it also decides equations such as permanganate with hydrogen peroxide, where atoms and charge alone allow infinitely many balances.