Chemical Equation Balancer

Reviewed by Hamza Gujjar, Algorithm Architect & Software Developer Last updated: August 20, 2026

📚 Confused about how this is calculated? Read the full Conservation of Mass, and Why You Balance Coefficients (Never Subscripts) →

Matter Doesn't Appear or Vanish Mid-Reaction

Every atom present in the reactants of a chemical equation has to show up somewhere in the products — that's the law of conservation of mass, and it's why chemical equations need coefficients in front of each formula to balance out. Getting those coefficients right by trial and error works for simple reactions, but larger ones benefit from solving the atom-conservation system directly.

The Method

For each element: sum of atoms on the reactant side = sum of atoms on the product side

This calculator sets up one conservation equation per distinct element across the reaction and solves the resulting linear system with exact fractions, then scales the result to the smallest set of whole-number coefficients — avoiding the floating-point rounding errors a decimal-based solver could introduce.

Where This Calculation Matters

Stoichiometry calculations — every mole-ratio calculation depends on starting from a correctly balanced equation.
Verifying a proposed reaction — if an equation can't be balanced with sensible coefficients, that's a signal the reaction as written is incomplete or wrong.
Combustion and energy calculations — balanced combustion equations are the starting point for calculating fuel efficiency and emissions.
Teaching reaction stoichiometry — balancing equations is a foundational skill before any quantitative chemistry can be attempted.

A Worked Example

The unbalanced combustion of methane, CH4 + O2 -> CO2 + H2O, balances to:

CH4 + 2 O2 → CO2 + 2 H2O

Checking each element confirms it: 1 carbon and 4 hydrogens on each side, and 4 oxygens on each side (2 from O2 × 2, matching 2 from CO2 and 2 from the two H2O molecules).

How to Use This Calculator

Enter the Unbalanced Equation, separating reactants and products with an arrow (e.g. CH4 + O2 -> CO2 + H2O).
Separate multiple species on each side with a plus sign.
Select Calculate to get the balanced equation with whole-number coefficients.

Note: This balancer supports up to three reactants, three products, and five distinct elements, and it solves for a single unique set of whole-number coefficients — equations with multiple independent solutions or more elements than that are outside its scope.

Related Calculations

Once an equation is balanced, use the Stoichiometry Calculator to find the mass of product formed from a given amount of reactant. If more than one reactant is limited, check the Limiting Reactant Calculator next.

Principles of Chemical Stoichiometry and Mass Balance Conservation

A chemical equation balancer balances chemical reaction equations by determining lowest integer stoichiometric coefficients that satisfy the Law of Conservation of Mass (Antoine Lavoisier, 1789). In chemical engineering, stoichiometry, and industrial synthesis, balanced reaction equations dictate reactant feed ratios, limiting reagent calculations, and thermodynamic heat of reaction enthalpy modeling.

The Mathematical Linear Algebra Balancing Method

Balancing any chemical reaction is mathematically equivalent to solving a Homogeneous System of Linear Diophantine Equations:

General Reaction: c₁ · Reactant₁ + c₂ · Reactant₂ → c₃ · Product₁ + c₄ · Product₂
Atomic Conservation Constraint: For every chemical element j: ∑ c_i · a_i,j = 0
Where a_i,j is the atom count of element j in molecule i (positive for reactants, negative for products).

Balancing Redox Reactions via the Half-Reaction Method

For reduction-oxidation (redox) reactions in aqueous solutions:

Step 1: Split the unbalanced skeletal equation into separate Oxidation and Reduction Half-Reactions.
Step 2: Balance all elements except Hydrogen and Oxygen.
Step 3: Balance Oxygen atoms by adding H₂O molecules.
Step 4: Balance Hydrogen atoms by adding H⁺ ions (in acidic solution).
Step 5: Balance electrostatic charges by adding electrons (e^-).
Step 6: Multiply half-reactions by integers to equalize electron transfer, then add half-reactions together.

Step-by-Step Worked Calculation Example

Example: Balancing the Combustion of Propane Gas (C₃H₈)

Problem: Balance the complete combustion reaction: _ C₃H₈ + _ O₂ → _ CO₂ + _ H₂O.

Step 1: Set up stoichiometric variables: a C₃H₈ + b O₂ → c CO₂ + d H₂O.

Step 2: Write element conservation balance equations:

Carbon (C): 3a = c &implies; c = 3a

Hydrogen (H): 8a = 2d &implies; d = 4a

Oxygen (O): 2b = 2c + d

Step 3: Solve for lowest whole integers by setting a = 1:

If a = 1 &implies; c = 3 | d = 4

Substitute c and d into Oxygen equation: 2b = 2(3) + 4 = 6 + 4 = 10 &implies; b = 5

Step 4: Write finalized balanced equation:

1 C₃H₈ + 5 O₂ → 3 CO₂ + 4 H₂O

Verification: Reactants (3 C, 8 H, 10 O) = Products (3 C, 8 H, 10 O) &implies; Perfectly Balanced!

Matrix Nullspace Algorithm for Multi-Reagent Complex Reactions

In modern computational chemistry software, chemical equation balancing is performed by constructing an m×n Chemical Composition Matrix (A):

A · x = 0
Where rows correspond to chemical elements, columns represent reactant and product chemical formulas, and vector x contains stoichiometric coefficients.

Finding the Nullspace (Kernel) Basis of Matrix A via Gaussian Elimination yields the unique smallest integer stoichiometric solution vector, effortlessly balancing 10-reagent organic synthesis and complex metallurgical extraction equations.

Disproportionation and Auto-Redox Reaction Balancing

In advanced inorganic chemistry, Disproportionation Reactions occur when a single chemical species is simultaneously oxidized and reduced:

For example, in the decomposition of hydrogen peroxide (2 H₂O₂ → 2 H₂O + O₂), oxygen atoms in peroxide (oxidation state -1) simultaneously reduce to water (oxidation state -2) and oxidize to diatomic oxygen gas (oxidation state 0). Tracking split oxidation states ensures precise stoichiometric mass and charge balancing.

Balancing Nuclear Transmutation Reactions

In nuclear physics and radioisotope medicine (radiopharmacy):

Nuclear equations must satisfy two fundamental conservation laws:

Conservation of Atomic Number (Total Nuclear Charge Z): Sum of proton atomic numbers on reactant side equals product side.
Conservation of Mass Number (Nucleon Count A): Total neutrons plus protons is conserved across alpha decay, beta emission, and neutron capture processes.

Spectator Ions in Net Ionic Equations

In aqueous precipitation reactions, dissolved ionic compounds dissociate completely into hydrated aqueous ions.

Chemical equations are simplified by canceling identical Spectator Ions on both reactant and product sides, leaving the fundamental Net Ionic Equation (e.g., Ag⁺_(aq) + Cl^-_(aq) → AgCl_(s)).

Balancing Complex Coordination Complexes

In coordination chemistry (such as cisplatin synthesis or hemoglobin binding), balancing equations requires tracking polydentate chelating ligands (e.g., EDTA^4-) and coordination sphere complex counter-ions.

Fractional Stoichiometric Coefficients in Thermochemistry

In thermochemical equations defining standard standard enthalpy of formation (ΔH°_f per 1 mole of product), fractional coefficients (e.g., H₂ + ½ O₂ → H₂O) are officially accepted conventions.

Physical State Descriptors in Chemical Equations

Balanced chemical equations explicitly declare physical state annotations: solid (s), liquid (l), gas (g), and aqueous solution (aq).