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In 1789 Antoine Lavoisier stated, after years of careful weighing, that matter is neither created nor destroyed in a chemical reaction. Atoms rearrange; they don’t appear or vanish. A balanced chemical equation is simply that law written down: every atom on the left must show up on the right.
The one rule you can’t break
To balance an equation you change the coefficients — the big numbers in front of each formula. You never change the subscripts — the small numbers inside a formula.
Changing H₂O to H₂O₂ to “fix” an oxygen count would turn water into hydrogen peroxide, a different substance. Changing H₂O to 2H₂O just means “two molecules of water”, which is perfectly fine.
The inspection method
- Write the unbalanced equation with correct formulas.
- Count atoms of each element on both sides.
- Start with the element that appears in the fewest formulas, usually leaving hydrogen and oxygen until later. Save any element that appears on its own (like O₂ or Fe) for last, because you can adjust it without disturbing anything else.
- Adjust coefficients one element at a time, recounting after each change.
- Reduce to the smallest whole numbers and do a final count.
Example 1: making water
H₂ + O₂ → H₂O
- H: 2 left, 2 right. O: 2 left, 1 right.
- Fix oxygen: put 2 in front of water. H₂ + O₂ → 2H₂O
- Now H: 2 left, 4 right. Fix hydrogen: 2H₂ + O₂ → 2H₂O
- Check: H 4 = 4, O 2 = 2. ✓
Example 2: rust
Fe + O₂ → Fe₂O₃
- Oxygen: 2 on the left, 3 on the right. The lowest common multiple is 6, so 3O₂ and 2Fe₂O₃. Fe + 3O₂ → 2Fe₂O₃
- Iron: now 4 on the right, so 4Fe. 4Fe + 3O₂ → 2Fe₂O₃
- Check: Fe 4 = 4, O 6 = 6. ✓
The lowest-common-multiple trick is useful whenever an element has an odd number on one side and an even number on the other.
Example 3: combustion of a hydrocarbon
Burning hydrocarbons always follows the same pattern, and a fixed order makes it easy: carbon, then hydrogen, then oxygen last.
C₃H₈ + O₂ → CO₂ + H₂O
- Carbon: 3 on the left → 3CO₂.
- Hydrogen: 8 on the left → 4H₂O.
- Oxygen on the right is now 3 × 2 + 4 × 1 = 10 → 5O₂.
C₃H₈ + 5O₂ → 3CO₂ + 4H₂O ✓
When the oxygen comes out odd — for example ethane, C₂H₆, gives 2CO₂ + 3H₂O = 7 oxygen atoms — use 7/2 O₂ temporarily, then double everything:
2C₂H₆ + 7O₂ → 4CO₂ + 6H₂O
Example 4: keep polyatomic ions together
If a polyatomic ion like sulfate (SO₄²⁻) or nitrate (NO₃⁻) appears unchanged on both sides, balance it as a single unit instead of counting its atoms separately.
Al + CuSO₄ → Al₂(SO₄)₃ + Cu
- Treat SO₄ as one unit: 1 on the left, 3 on the right → 3CuSO₄.
- Copper: now 3 on the left → 3Cu.
- Aluminium: 2 on the right → 2Al.
2Al + 3CuSO₄ → Al₂(SO₄)₃ + 3Cu ✓
When inspection gets hard: the algebraic method
Some equations defeat trial and error. For those, assign an unknown to each coefficient and write one equation per element.
a KMnO₄ + b HCl → c KCl + d MnCl₂ + e H₂O + f Cl₂
- K: a = c
- Mn: a = d
- O: 4a = e
- H: b = 2e
- Cl: b = c + 2d + 2f
Set a = 2 to avoid fractions: then c = 2, d = 2, e = 8, b = 16, and 16 = 2 + 4 + 2f gives f = 5.
2KMnO₄ + 16HCl → 2KCl + 2MnCl₂ + 8H₂O + 5Cl₂
This is essentially what computer equation balancers do, using matrix algebra to solve the whole system at once.
Final checks
- Count every element on both sides one last time.
- Make sure the coefficients have no common factor (4, 2, 4 should become 2, 1, 2).
- For ionic equations, check that the total charge also balances.
Quick answers
Can a coefficient be a fraction? Temporarily, as a working step. In a final answer, convert to whole numbers by multiplying through.
Why is it called “balancing”? Because, as Lavoisier showed with a balance, the total mass of the reactants equals the total mass of the products.
What does a coefficient of 1 look like? It isn’t written. O₂ means 1O₂.
Let a tool check your work
The chemical equation balancer uses exact fraction arithmetic to balance any equation and re-verifies every answer. Once balanced, the coefficients feed straight into stoichiometry and limiting reagent calculations.
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