In $2\,\text{H}_2\text{O}$ the big number $2$ in front is the coefficient: it says how many molecules (or moles) there are. The small number $_2$ after H is a subscript: it is part of the formula and says each molecule has two hydrogen atoms. So $2\,\text{H}_2\text{O}$ contains $2 \times 2 = 4$ hydrogen atoms and $2 \times 1 = 2$ oxygen atoms.
To balance an equation you may change only the coefficients. Changing a subscript would change the substance itself: $\text{H}_2\text{O}_2$ is hydrogen peroxide, not water. For example, $\text{H}_2 + \text{O}_2 \rightarrow \text{H}_2\text{O}$ has 2 oxygen atoms on the left but only 1 on the right. Put a $2$ in front of water to fix oxygen, which gives 4 hydrogen atoms on the right, then put a $2$ in front of $\text{H}_2$: $2\,\text{H}_2 + \text{O}_2 \rightarrow 2\,\text{H}_2\text{O}$.
Balance metals first, then other non-metals, then hydrogen, and oxygen last, because O and H usually appear in several formulas. Treat a polyatomic ion such as $\text{SO}_4$ that appears unchanged on both sides as one unit. Finish by checking every element and by making sure the coefficients are the smallest whole numbers (divide them all if they share a factor).
Start with Level 1, $\text{H}_2 + \text{O}_2 \rightarrow \text{H}_2\text{O}$: raise the water coefficient and watch the oxygen row of the atom counter turn green while hydrogen turns red, then fix hydrogen. Look at the molecule picture: the number of red O atoms is the same on both sides. Then switch off the atom counter and try the Level 2 game, or type an equation from your homework into Type any equation.
| Step | What to do | Propane example |
|---|---|---|
| 1 | Write the correct formulas and count the atoms of each element on both sides | Left: C 3, H 8, O 2. Right: C 1, H 2, O 3 |
| 2 | Balance an element that appears in only one formula on each side (carbon here) | $3\,\text{CO}_2$ → C 3 = 3 |
| 3 | Balance hydrogen | $4\,\text{H}_2\text{O}$ → H 8 = 8 |
| 4 | Balance oxygen last | Right now has $3\times2 + 4\times1 = 10$ O, so $5\,\text{O}_2$ |
| 5 | Check every element and use the smallest whole numbers | $\text{C}_3\text{H}_8 + 5\,\text{O}_2 \rightarrow 3\,\text{CO}_2 + 4\,\text{H}_2\text{O}$ |
A number after a bracket multiplies everything inside it. $\text{Al}_2(\text{SO}_4)_3$ contains 2 aluminium atoms, $1\times3 = 3$ sulfur atoms and $4\times3 = 12$ oxygen atoms. With a coefficient in front, multiply again: $3\,\text{Ca(OH)}_2$ contains 3 Ca, 6 O and 6 H.
| Type | Pattern | Example |
|---|---|---|
| Synthesis | A + B → AB | $2\,\text{Mg} + \text{O}_2 \rightarrow 2\,\text{MgO}$ |
| Decomposition | AB → A + B | $2\,\text{KClO}_3 \rightarrow 2\,\text{KCl} + 3\,\text{O}_2$ |
| Combustion | fuel + O2 → CO2 + H2O | $\text{CH}_4 + 2\,\text{O}_2 \rightarrow \text{CO}_2 + 2\,\text{H}_2\text{O}$ |
| Single replacement | A + BC → AC + B | $\text{Zn} + 2\,\text{HCl} \rightarrow \text{ZnCl}_2 + \text{H}_2$ |
| Double replacement | AB + CD → AD + CB | $\text{Pb(NO}_3)_2 + 2\,\text{KI} \rightarrow \text{PbI}_2 + 2\,\text{KNO}_3$ |
| Neutralisation | acid + base → salt + water | $2\,\text{NaOH} + \text{H}_2\text{SO}_4 \rightarrow \text{Na}_2\text{SO}_4 + 2\,\text{H}_2\text{O}$ |
Each element gives one equation: the atoms of that element on the left must equal those on the right. For $a\,\text{Fe} + b\,\text{O}_2 \rightarrow c\,\text{Fe}_2\text{O}_3$: iron gives $a = 2c$ and oxygen gives $2b = 3c$. The calculator solves these equations exactly (with fractions, using linear algebra) and scales the answer to the smallest whole numbers: $c = 2$, $a = 4$, $b = 3$. It also tells you when an equation cannot be balanced, for example when an element appears on only one side.
Count the atoms of each element on both sides, then change the coefficients (the numbers in front of the formulas) until every element has the same number of atoms on the left and the right. A good order is metals first, then other non-metals, then hydrogen, and oxygen last. Finally check every element and make sure the coefficients are the smallest whole numbers.
Key takeaway: change only the coefficients until every element matches on both sides.Because of the law of conservation of mass: in a chemical reaction atoms are only rearranged, never created or destroyed, so the reactants and products contain exactly the same atoms. A balanced equation shows this and gives the correct ratio of substances for calculations.
Key takeaway: atoms are conserved, so both sides must contain the same atoms.A coefficient is the large number in front of a formula and tells you how many molecules or formula units there are. A subscript is the small number inside a formula and tells you how many atoms of that element are in one molecule. In 3 H2O the coefficient is 3 and the subscript 2 means each water molecule has two hydrogen atoms, so there are 6 hydrogen atoms in total.
Key takeaway: coefficients count molecules; subscripts are part of the formula.Changing a subscript changes the substance. H2O is water, but H2O2 is hydrogen peroxide, a different chemical with different properties. The formulas are fixed by the substances that actually react, so only the coefficients may change.
Key takeaway: subscripts define the substance; never change them.Balance carbon first by putting the number of carbon atoms in front of CO2, then hydrogen by putting half the number of hydrogen atoms in front of H2O, and finally count the oxygen atoms on the right and put half that number in front of O2. If you get a fraction such as 13/2, multiply every coefficient by 2: 2 C4H10 + 13 O2 → 8 CO2 + 10 H2O.
Key takeaway: C, then H, then O; double everything if O2 needs a half.It tells you the ratio in which substances react and form, in molecules or in moles. For 2 H2 + O2 → 2 H2O, two moles of hydrogen react with one mole of oxygen to give two moles of water. These mole ratios are the starting point for stoichiometry calculations of masses and volumes.
Key takeaway: the coefficients give the mole ratio for calculations.Only if the formulas are correct and the same elements appear on both sides. If an element appears on only one side, or the products are wrong, no set of coefficients works. Some combinations of substances can also be balanced in more than one way because they describe two reactions at once; then you need to know the actual reaction.
Key takeaway: wrong formulas or missing products make balancing impossible.It writes one equation per element (atoms on the left = atoms on the right), solves the set exactly with fractions using linear algebra, and scales the result to the smallest whole numbers. It also reports when an equation cannot be balanced or can be balanced in more than one way.
Key takeaway: one equation per element, solved exactly, then scaled to whole numbers.