Chemical equation balancer
each element → one conservation equation
Enter the equation without coefficients (H2 + O2 -> H2O, parentheses accepted): the tool computes the smallest integer coefficients through exact linear algebra, the null space of the composition matrix, solved in big-integer fractions, never by trial and error nor floats. The element-by-element check is displayed as proof, and impossibilities are named: an element on one side only, a reaction that does not conserve atoms, or an equation superposing several independent reactions.
4 Fe + 3 O2 → 2 Fe2O3
- Method
- Null space of the composition matrix, solved in exact fractions (Gaussian elimination over rationals), reduced to the smallest integers.
- Element-by-element check
- Fe: 4 on the left = 4 on the right-O: 6 on the left = 6 on the right
Balanced: each of the 2 elements counts exactly as many atoms on both sides, the check table is the proof. Recounted independently of the solver.
Scientific dossier
What the tool computes, what it assumes, where it stops being valid, and where its data comes from.
Method & formulaseach element → one conservation equation
each element → one conservation equation
each species → one unknown coefficient
coefficients = null space of the composition matrix
solved in exact fractions → smallest integers (LCM, gcd)
Balancing an equation is solving a linear system: for each element, the number of atoms on the left must equal the right. The tool builds that system and solves it by Gaussian elimination over exact fractions (big integers) (no rounding possible) then reduces the solution to the smallest integer coefficients. It is the general algebraic method: it balances at once what trial and error takes minutes to find, like KMnO4 + HCl -> KCl + MnCl2 + H2O + Cl2.
- Stoichiometric coefficients
- · the numbers placed before each species that make the equation obey atom conservation. They are given as smallest integers, multiplying everything by 2 stays correct, but is no longer canonical.
- Conservation of atoms
- · a chemical reaction rearranges bonds; it neither creates nor destroys atoms (Lavoisier): each element counts exactly as many atoms on both sides.
- Symbol case
- · CO is one carbon and one oxygen; Co is cobalt. An uppercase letter opens a symbol, a lowercase letter completes it, case mistakes are the first cause of “impossible” equations.
Validity domainThe tool balances neutral species by atom conservation: ionic equations and redox half-equations (which also require charge and electron conservation) are not covered.
The tool balances neutral species by atom conservation: ionic equations and redox half-equations (which also require charge and electron conservation) are not covered. Hydrates (CuSO4·5H2O) can be written expanded: CuSO4(H2O)5 works. Twelve species at most, far beyond coursework equations. Balancing says atoms are conserved, not that the reaction exists: thermodynamics and kinetics are different questions.
Common pitfall: coefficients that do not always existNot every equation balances, and the reasons differ.
Not every equation balances, and the reasons differ. An element present on one side only can never be conserved, a species is missing. An equation like C + O2 -> CO + CO2 admits infinitely many solutions: it superposes two independent reactions (C + O2 -> CO2 and 2C + O2 -> 2CO), and no unique coefficients exist. The real proportions depend on experimental conditions, not algebra. The tool distinguishes these cases instead of returning a silent failure or an arbitrary answer.