How to Balance Equations with Smallest Whole-Number Coefficients

A balanced equation conserves every type of atom

A chemical equation is balanced when each element has the same number of atoms on the reactant and product sides. Coefficients multiply entire chemical formulas and express relative amounts of substances.

Subscripts have a different job: they define the composition of a substance. Changing $mathrm{H_2O}$ to $mathrm{H_2O_2}$ would change water into hydrogen peroxide, so subscripts must not be edited to make atom counts match.

Balance systematically

1. Write the correct formulas for every reactant and product.
2. Count each element on both sides.
3. Change one coefficient at a time, beginning with an element that appears in relatively few formulas.
4. Recount all elements after each change.
5. Clear any fractional coefficients and reduce the final set to the smallest whole-number ratio.

Consider the combustion of ethane:

$$\mathrm{C_2H_6+O_2\longrightarrow CO_2+H_2O}.$$

Balance carbon first by placing 2 before carbon dioxide, then hydrogen by placing 3 before water:

$$\mathrm{C_2H_6+O_2\longrightarrow2CO_2+3H_2O}.$$

The products now contain seven oxygen atoms, so a temporary coefficient of $7/2$ balances oxygen:

$$\mathrm{C_2H_6+\frac72O_2\longrightarrow2CO_2+3H_2O}.$$

Multiply every coefficient by 2 to obtain whole numbers:

$$\boxed{2\mathrm{C_2H_6}+7\mathrm{O_2}\longrightarrow4\mathrm{CO_2}+6\mathrm{H_2O}}.$$

Verify instead of trusting inspection

The final equation has four carbon atoms, twelve hydrogen atoms, and fourteen oxygen atoms on each side. Its coefficients $(2,7,4,6)$ share no common factor greater than 1, so the ratio is minimal.

This final atom inventory catches two common errors: balancing one element while unbalancing another, and reporting a balanced but nonminimal multiple such as $(4,14,8,12)$.

Know what balancing does not tell you

A balanced equation enforces conservation of matter and supplies stoichiometric ratios. It does not by itself establish whether a reaction occurs, how fast it proceeds, or what yield an experiment will achieve.