Chemical Equation Balancer

Coefficients change; subscripts never do

A balanced equation asserts that no atoms were created or destroyed, and — for an ionic equation — that no charge was either. The only thing you are allowed to adjust in service of that is the number in front of each species.

This is worth stating first because the commonest error in the whole topic is fixing an imbalance by editing a subscript. Turning H₂O into H₂O₂ to find another oxygen produces a balanced-looking line describing a reaction that makes hydrogen peroxide instead of water. The subscripts are part of the substance's identity; the coefficients are the recipe quantities.

Underneath, balancing is linear algebra rather than puzzle-solving. Each element gives one equation, each species gives one unknown, and the balanced coefficients are the smallest whole-number solution of that homogeneous system. Solving it exactly with integer arithmetic is why a machine never produces the almost-balanced answers that trial and error tends to.

Ethanol, balanced in the order that works

Order matters to a person and not at all to the algebra. Solving the system treats every element's constraint simultaneously, so there is no first element and no last one. Working on paper you have to choose a sequence, and the choice is the difference between two minutes and ten; the reasoning behind the sequence used below is set out in the guide to balancing equations.

Take the combustion of ethanol: C₂H₅OH + O₂ → CO₂ + H₂O.

Carbon. Two carbons on the left, so two carbon dioxides on the right.

C₂H₅OH + O₂ → 2 CO₂ + H₂O

Hydrogen. Six hydrogens in ethanol — five in the chain and one in the hydroxyl — so three waters.

C₂H₅OH + O₂ → 2 CO₂ + 3 H₂O

Oxygen, last. The right-hand side now has 2 × 2 + 3 = 7 oxygen atoms. The left has one oxygen already, sitting inside the ethanol, so the oxygen gas must supply six: three O₂.

C₂H₅OH + 3 O₂ → 2 CO₂ + 3 H₂O

Check across: carbon 2 and 2, hydrogen 6 and 6, oxygen 7 and 7.

That embedded oxygen atom is the trap in this example. The habit of counting only the O₂ on the left survives every hydrocarbon question and fails the first time an alcohol, an ether or a sugar appears.

Odd hydrogen counts and the fraction you clear at the end

Butane, C₄H₁₀, will not balance in whole numbers on the first pass. Four carbons give four carbon dioxides; ten hydrogens give five waters; the right-hand side then needs 8 + 5 = 13 oxygen atoms, and 13 is odd while O₂ supplies them in pairs.

Do not fight it. Write the fraction, then clear it:

C₄H₁₀ + 13/2 O₂ → 4 CO₂ + 5 H₂O
2 C₄H₁₀ + 13 O₂ → 8 CO₂ + 10 H₂O

Doubling every coefficient is always legal, because an equation is a ratio. Any alkane with an even number of hydrogens per molecule and an odd oxygen requirement behaves this way, and the fraction is a step in the method rather than a sign of a mistake.

Why inspection collapses on a redox equation

Now try the reaction between permanganate and iron(II) in acid, written as the skeleton a question would give you:

MnO₄⁻ + Fe²⁺ + H⁺ → Mn²⁺ + Fe³⁺ + H₂O

Inspection has nothing to grip. Hydrogen and oxygen appear in species — H⁺ and H₂O — whose coefficients you can raise together without breaking anything, so the atom counts alone do not pin the answer down. And nothing in an atom count notices that the left side carries a different total charge from the right.

The missing constraint is the electrons. Manganese goes from an oxidation state of +7 to +2, a gain of five electrons; iron goes from +2 to +3, a loss of one. Every electron released has to be collected, and that ratio — five to one — is the number no amount of staring at atoms will produce.

Half-equations, and the column inspection ignores

Split the reaction into the two things actually happening, balance each for atoms, then for oxygen with water, then for hydrogen with H⁺, and finally for charge with electrons.

Reduction:

MnO₄⁻ + 8 H⁺ + 5 e⁻ → Mn²⁺ + 4 H₂O

Oxidation:

Fe²⁺ → Fe³⁺ + e⁻

Multiply the second by five so the electrons cancel, and add:

MnO₄⁻ + 5 Fe²⁺ + 8 H⁺ → Mn²⁺ + 5 Fe³⁺ + 4 H₂O

Then check both columns, which is the habit that separates a reliable answer from a plausible one. Atoms: Mn 1 and 1, Fe 5 and 5, O 4 and 4, H 8 and 8. Charge: the left side is −1 + 10 + 8 = +17, the right is +2 + 15 = +17. A redox equation that balances for atoms and fails for charge is wrong, and the charge check catches errors nothing else will.

The same procedure handles dichromate with iron(II) — Cr₂O₇²⁻ + 6 Fe²⁺ + 14 H⁺ → 2 Cr³⁺ + 6 Fe³⁺ + 7 H₂O, with 24 units of charge on each side — and it handles disproportionation, where one element is oxidised and reduced at once. Chlorine in alkali gives Cl₂ + 2 OH⁻ → Cl⁻ + ClO⁻ + H₂O, which inspection finds baffling and half-equations resolve immediately, because the two halves simply happen to involve the same element.

For a reaction in alkaline conditions, balance it in acid first and then add hydroxide ions to both sides to convert the H⁺ into water. Trying to balance directly in base is possible and consistently more error-prone.

Equations with more than one answer, and equations with none

Two outcomes surprise people, and both are properties of the underlying system rather than failures of arithmetic.

Some skeleton equations have no solution at all. If an element appears on only one side, or the species given cannot conserve every element simultaneously, no set of coefficients exists. The correct response is to say so rather than to force a near-miss, because the fault is in the chemistry that was written down.

Others have infinitely many independent solutions. This happens when one species can be expressed as a combination of the others — a reaction skeleton that is really two reactions superimposed. The system's solution space then has more than one dimension, and there is no single smallest-integer answer, only a family of them. A balancer that silently returns one member of that family is hiding something the chemist needs to know: the equation as written is ambiguous and should be split.

A balanced equation is a ratio, not a prediction

The last point is the one most worth carrying away. Balancing establishes the stoichiometry of a reaction you have asserted. It says nothing about whether that reaction occurs, how fast, at what temperature, or by what mechanism.

Any conservation-respecting rearrangement can be balanced, including plenty that no laboratory could ever run. The equation is also silent about mechanism: a balanced line showing five iron ions reacting with one permanganate does not mean six particles collide at once, and they certainly do not. What the coefficients give you is the ratio to use for a limiting-reagent calculation, a yield, or a titration — and for that, they need to be exactly right.