Chemical Equation Balancer

What this balancer does

Type a chemical equation — reactants on the left, products on the right, separated by an arrow (→ or ->) — and the balancer returns the smallest set of positive integer coefficients that make every element's atom count equal on both sides. It also shows the per-element check so you can verify the balance by eye.

Unlike trial-and-error, the algorithm is deterministic: for any balanceable equation it returns the unique smallest-integer answer, and for non-balanceable equations it tells you so instead of looping forever.

How it works — the algebraic method

Every compound in the equation gets a coefficient (call them a, b, c, …). The constraint that each element must balance on both sides gives one linear equation per element. So if your equation has n compounds and k elements, you have k equations in n unknowns. Writing this as a matrix:

1. Each row of the matrix represents one element.
2. Each column represents one compound.
3. The entry is the number of atoms of that element in that compound — positive for reactants, negative for products.
4. The balanced equation is a nontrivial solution to Ax = 0 — i.e. a vector in the matrix's null space.

The balancer solves this via Gaussian elimination with partial pivoting, treats the last unknown as a free variable (= 1), and back-substitutes. The resulting rational coefficients are converted to integers by taking the LCM of fractional denominators, then reduced by the GCD to ensure the answer is in lowest terms.

Worked example: KMnO₄ + HCl → KCl + MnCl₂ + H₂O + Cl₂

Six compounds, four elements (K, Mn, O, H, Cl — five elements, actually). The balancer constructs a 5×6 matrix:

• K:   +1 (KMnO₄) − 1 (KCl)  →  coefficients must satisfy 1a − 1c = 0
• Mn:   +1 (KMnO₄) − 1 (MnCl₂)  →  1a − 1d = 0
• O:   +4 (KMnO₄) − 1 (H₂O)  →  4a − 1e = 0
• H:   +1 (HCl) − 2 (H₂O)  →  1b − 2e = 0
• Cl:   +1 (HCl) − 1 (KCl) − 2 (MnCl₂) − 2 (Cl₂)  →  1b − 1c − 2d − 2f = 0

Solving this system gives 2 KMnO₄ + 16 HCl → 2 KCl + 2 MnCl₂ + 8 H₂O + 5 Cl₂, which the atom check then verifies: 2 K on each side, 2 Mn on each side, 8 O on each side, 16 H on each side, 16 Cl on each side. ✓

What this balancer can and can't do

Can:

• Any equation balanceable from atom counts alone — combustion, double-displacement, precipitation, acid-base neutralization, most synthesis and decomposition reactions.
• Polyatomic ions in parentheses: Ca₃(PO₄)₂, Al₂(SO₄)₃, Cu(NO₃)₂.
• Hydrate notation: CuSO₄·5H₂O.
• Equations with many compounds: the algorithm scales — KMnO₄ + HCl with 6 compounds and 5 elements solves as quickly as H₂ + O₂.

Can't:

• Half-reactions (e.g. MnO₄⁻ + e⁻ → Mn²⁺) — electron transfer isn't expressed as an atom count. Use the half-reaction method by hand for those.
• Incomplete redox equations where the algebraic method admits multiple solutions. The balancer reports "could not balance" in that case.
• Charge balancing for ionic equations (no +/− charges in formulas).
• Net ionic equations as written (spectator ions removed) — balance the molecular form first.

Common errors

Case matters. Co is cobalt, CO is carbon monoxide. NACL won't parse — it has to be NaCl.

Use → or -> between sides. Equal signs work too (A + B = C), but use exactly one separator. Multi-step reactions need to be balanced one step at a time.

If the balancer can't solve it, check the chemistry, not the math. The most common cause is a missing product or reactant. The balancer is honest: if the atom counts can't balance with positive coefficients, the equation as written is wrong.