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Chemical Equation Balancer

Chemistry

Balance chemical equations automatically. Enter an unbalanced equation and get stoichiometric coefficients, the balanced equation, and atom-by-atom verification for any reaction.

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Balanced Equation

Enter an equation
Stoichiometric Coefficients
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Atom Balance Verification
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Number of Chemical Species
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This calculator computes your Balanced Equation, Stoichiometric Coefficients, Atom Balance Verification, Number of Chemical Species from the values you enter.

Inputs
Unbalanced Chemical Equation
Outputs
Balanced EquationStoichiometric CoefficientsAtom Balance VerificationNumber of Chemical Species

What is a Equation Balancer?

The Chemical Equation Balancer finds stoichiometric coefficients for an unbalanced chemical equation. Enter an equation like "Fe + O2 -> Fe2O3" and get the balanced form (4Fe + 3O₂ → 2Fe₂O₃), all coefficients, and an atom-by-atom balance check.

Chemical equations are balanced by finding integer coefficients that satisfy conservation of mass for every element. The balancer uses Gaussian elimination over rational numbers on the stoichiometric matrix — a systematic algebraic method that works for any number of species and elements, including complex reactions that cannot be balanced by inspection. The formula parser handles nested parentheses (Ca3(PO4)2, Al2(SO4)3) and multi-letter element symbols.

Balanced coefficients directly give stoichiometric ratios for yield calculations — the Percent Yield Calculator and Theoretical Yield Calculator both depend on the balanced equation's mole ratios. For the reaction outcome in solution, the Net Ionic Equation Calculator writes the ionic form of the same reaction.

How to use this Equation Balancer calculator

  1. Write the unbalanced equation with formulas separated by '+' and sides separated by '->': H2 + O2 -> H2O
  2. Use plain numbers as subscripts: H2O (not H₂O), Fe2O3 (not Fe₂O₃).
  3. Use parentheses for polyatomic groups: Ca3(PO4)2, Al2(SO4)3.
  4. Press Calculate and read the Balanced Equation — copy directly for use.
  5. Check the Atom Balance Verification — all elements should show equal counts on both sides.
  6. Use the Coefficients for stoichiometric calculations with the Theoretical Yield Calculator.

Formula & Methodology

Algebraic balancing via stoichiometric matrix:

1. Parse each formula: Ca3(PO4)2 → {Ca:3, P:2, O:8}  2. Build matrix M (n_elements × n_species):    Reactant atoms: negative sign    Product atoms: positive sign        For: Fe + O2 -> Fe2O3    Elements: Fe, O; Species: Fe, O2, Fe2O3 (reactants negative)        M = [ [-1,  0,  2],   ← Fe row          [  0, -2,  3] ] ← O row  3. Find null space of M (vector x such that Mx = 0):    Row reduce to RREF using rational arithmetic    Free variable = 1 → solve for pivot variables        Solution: Fe=4, O2=3, Fe2O3=2  4. Verify: Fe: 4×1 = 2×2 (4=4 ✓); O: 3×2 = 2×3 (6=6 ✓)

Worked example — thermite reaction (iron oxide reduction):

Fe₂O₃ + Al → Al₂O₃ + Fe (used in thermite welding of railway tracks)

Entered: Fe2O3 + Al -> Al2O3 + Fe  Matrix:      [Fe2O3, Al, Al2O3, Fe] Fe:  [-2,    0,   0,    1] O:   [-3,    0,   3,    0] Al:  [ 0,   -1,   2,    0]  Null space solution: Fe2O3=1, Al=2, Al2O3=1, Fe=2 Balanced: Fe₂O₃ + 2Al → Al₂O₃ + 2Fe Check: Fe 2=2 ✓; O 3=3 ✓; Al 2=2 ✓  Coefficients: Fe2O3: 1, Al: 2, Al2O3: 1, Fe: 2

Thermite welding (aluminothermic welding) is the standard method for joining railway tracks in India — used by Indian Railways (RDSO specification IRS-T-19) for 400+ km of new track welding per year. The highly exothermic reaction (ΔH = −852 kJ/mol) generates liquid iron at ~2500°C that fills the gap between rails. Indian Railways is the world's 4th largest rail network (68,000 km) — thermite welding is performed by IRCON, Kalindee Rail Nirman, and Dedicated Freight Corridor Corporation contractors.

Frequently Asked Questions

Balancing a chemical equation means finding the smallest set of integer coefficients for each species so that the number of atoms of each element is identical on both sides of the arrow — satisfying conservation of mass. For Fe + O₂ → Fe₂O₃: unbalanced, Fe appears once on left and twice on right. Balanced: 4Fe + 3O₂ → 2Fe₂O₃ (4 Fe on each side, 6 O on each side). Balancing does not change the formulas — only the coefficients (the large numbers before each formula). The law of conservation of mass (discovered by Lavoisier in 1789) requires all atoms present in reactants to appear in products — none created or destroyed.
Enter the unbalanced equation in the format: A + B -> C + D. Use element symbols with subscripts as plain numbers (H2O, Fe2O3, Ca3(PO4)2). Separate species with '+'. Use '->' or '→' for the arrow. Examples: 'H2 + O2 -> H2O' → balances to 2H₂ + O₂ → 2H₂O. 'Fe + O2 -> Fe2O3' → balances to 4Fe + 3O₂ → 2Fe₂O₃. The calculator returns the balanced equation, stoichiometric coefficients for each species, atom-by-atom verification, and number of species.
The balancer uses a matrix/linear algebra approach: (1) Parse all species formulas into element-count maps. (2) Build an m×n matrix where m = number of unique elements, n = number of species. Reactant columns are negative (atoms consumed), product columns are positive (atoms produced). (3) Find the null space of this matrix using Gaussian elimination over rationals (fraction arithmetic). (4) Scale the null space vector to the smallest positive integers using LCM of denominators. (5) Verify by checking atom balance. This algebraic method handles any number of species and elements — superior to trial-and-error or inspection for complex equations.
Enter the formula with parentheses exactly as written: Ca3(PO4)2, Al2(SO4)3, Fe(OH)3, Cu(NO3)2. The parser handles nested parentheses with subscripts: Ca3(PO4)2 is parsed as Ca:3, P:2, O:8. Example: Ca3(PO4)2 + H2SO4 -> CaSO4 + H3PO4. Entered as 'Ca3(PO4)2 + H2SO4 -> CaSO4 + H3PO4'. Balanced: Ca₃(PO₄)₂ + 3H₂SO₄ → 3CaSO₄ + 2H₃PO₄ (checks: Ca 3=3, P 2=2, O 8+12=12+8, S 3=3, H 6=6 ✓). This reaction is used in India for superphosphate fertiliser production (PPL Paradeep, IFFCO Kandla).
Simple inspection (trial and error) works for equations with 2–3 species but fails for complex reactions: Multiple transition metal oxidation states (e.g., KMnO₄ + H₂C₂O₄ + H₂SO₄ → MnSO₄ + CO₂ + K₂SO₄ + H₂O — 7 species, requires algebra). Organic combustion with large molecules (C₆H₁₂O₆ + O₂ → CO₂ + H₂O — glucose combustion: balanced 1:6:6:6). Redox reactions in acidic or basic media with OH⁻ or H⁺ adjustment. In these cases, the algebraic (matrix) method is the only systematic approach. Indian NCERT Class 10 and 11 introduce 'hit and trial' (inspection), but JEE Advanced and university-level chemistry require the half-reaction (ion-electron) method for redox equations.
The half-reaction (ion-electron) method separates the overall redox equation into oxidation and reduction half-reactions, balances each separately, then combines by multiplying to equalise electron transfer. Steps: (1) Identify oxidation states and changes. (2) Write oxidation half-reaction and reduction half-reaction. (3) Balance atoms (O with H₂O, H with H⁺). (4) Balance charge with electrons. (5) Multiply to equalise electrons and add. Example: MnO₄⁻ + Fe²⁺ → Mn²⁺ + Fe³⁺ (acidic). Reduction: MnO₄⁻ + 8H⁺ + 5e⁻ → Mn²⁺ + 4H₂O. Oxidation: Fe²⁺ → Fe³⁺ + e⁻ (×5). Combined: MnO₄⁻ + 8H⁺ + 5Fe²⁺ → Mn²⁺ + 5Fe³⁺ + 4H₂O. This method is standard in NCERT Class 11 Chapter 8 (Redox Reactions) and JEE.
Key industrial reactions in India: Haber-Bosch (IFFCO, Rashtriya Chemicals, Chambal Fertilisers): N₂ + 3H₂ → 2NH₃. Contact process for H₂SO₄ (TATA Chemicals, GSFC Vadodara): 2SO₂ + O₂ → 2SO₃; SO₃ + H₂O → H₂SO₄. Chlor-alkali (GACL Gujarat, DCHL): 2NaCl + 2H₂O → 2NaOH + Cl₂ + H₂. Iron smelting (SAIL, Tata Steel, JSW): Fe₂O₃ + 3CO → 2Fe + 3CO₂. Cement production (UltraTech, ACC): CaCO₃ → CaO + CO₂; CaO + SiO₂ → CaSiO₃. Soda ash (GHCL Dhrangadhra): NaCl + NH₃ + CO₂ + H₂O → NaHCO₃ + NH₄Cl; 2NaHCO₃ → Na₂CO₃ + H₂O + CO₂.
Combustion of CₓHᵧ fuel: CₓHᵧ + (x + y/4)O₂ → xCO₂ + (y/2)H₂O. If y/4 or y/2 is not integer, multiply through by 2. Methane (CH₄): CH₄ + 2O₂ → CO₂ + 2H₂O. Ethane (C₂H₆): 2C₂H₆ + 7O₂ → 4CO₂ + 6H₂O. Ethanol (C₂H₅OH): C₂H₅OH + 3O₂ → 2CO₂ + 3H₂O. Octane (C₈H₁₈, petrol): 2C₈H₁₈ + 25O₂ → 16CO₂ + 18H₂O. Glucose (C₆H₁₂O₆): C₆H₁₂O₆ + 6O₂ → 6CO₂ + 6H₂O. The [AFR Calculator](/afr-calculator/) uses these stoichiometric coefficients to calculate air-fuel ratio for engine combustion.
Chemical equation balancing: conserve atoms of each element; coefficients are integers ≥ 1. Nuclear equation balancing: conserve mass number (A, total nucleons) and atomic number (Z, protons) — not atoms of elements. Example: ²³⁸U → ²³⁴Th + ⁴He (α decay). Check: A: 238=234+4 ✓; Z: 92=90+2 ✓. Nuclear equations do NOT conserve atom count — one element transforms into another. Beta decay: ¹⁴C → ¹⁴N + ⁰e (beta-minus). Fission: ²³⁵U + ¹n → ⁹²Kr + ¹⁴¹Ba + 3¹n. BARC (Bhabha Atomic Research Centre, Mumbai) and NPCIL (Nuclear Power Corporation of India) deal with nuclear equations for the Pressurised Heavy Water Reactors (PHWRs) and the proposed Thorium Molten Salt Reactors.
The number of chemical species is the total count of distinct compounds or elements on both sides of the equation (reactants + products, not counting coefficients). H₂ + O₂ → H₂O: 3 species. Fe + O₂ → Fe₂O₃: 3 species. CaCO₃ + HCl → CaCl₂ + H₂O + CO₂: 5 species. N₂ + H₂ → NH₃: 3 species. More species generally means a more complex balancing problem. The matrix rank determines how many free variables exist — for most simple reactions, there is 1 free variable (null space dimension = 1), giving a unique set of smallest integer coefficients.