Balance chemical equations using conservation of atoms and coefficients. Interpret balanced equations as particle and mole ratios.
A lesson of Chemistry, in Chemical Reactions and Stoichiometry, about 40 minutes.
By the end of this lesson, you will be able to balance any ordinary equation in a reliable order, know why subscripts may never be touched, handle polyatomic ions as units, and read what the coefficients actually mean.
Count the oxygens: two on the left, one on the right. One oxygen atom has vanished
somewhere between the reactants and the products, which is impossible. A reaction
rearranges atoms; it does not manufacture or delete them.
Four hydrogens and two oxygens on each side. That is conservation of mass, and
balancing is the bookkeeping that enforces it.
Coefficients are the only thing you may change. A coefficient in front
multiplies the whole formula, so $2\mathrm{H_2O}$ means two water molecules,
giving four hydrogens and two oxygens.
Subscripts are NEVER changed, and this is the one absolute rule of the lesson.
Changing $\mathrm{H_2O}$ to $\mathrm{H_2O_2}$ balances the count and changes water
into hydrogen peroxide, a bleach that decomposes violently. You would have
balanced a different reaction, which is worse than not balancing at all, because
the arithmetic looks right.
Sort It: Legal or Not
A Reliable Order
Balancing by staring at it works until it does not. This order works every time.
Count every element on both sides, in a list
Balance the element appearing in the FEWEST formulas first, usually a
metal
Leave hydrogen and oxygen for last, because they turn up everywhere and
adjusting them early undoes your other work
Treat a polyatomic ion that survives intact as ONE unit
Carbon first: one on each side, already done. Hydrogen next: four on the left, two
on the right, so put a $2$ in front of the water. Now oxygen: two on the left,
$2 + 2 = 4$ on the right, so put a $2$ in front of $\mathrm{O_2}$.
Oxygen last is not a stylistic preference. It appears in both products here, so
its total depends on choices you have not made yet. Fixing it first guarantees
having to redo it.
Example 1: Balancing Step by Step
Here is every step, worked all the way through.
$$\mathrm{Al} + \mathrm{Cl_2} \rightarrow \mathrm{AlCl_3}$$
Use the fraction, then clear it. Trying to guess the doubled coefficients
directly is much harder, and this two-stage route is what chemists actually do.
Reduce at the end if everything shares a factor. $4, 2, 4$ should be $2, 1, 2$,
because the convention is the simplest whole-number ratio.
Treat $\mathrm{PO_4}$ as one item. Phosphate: one on the left, two on the right,
so $2\mathrm{H_3PO_4}$. Calcium: one on the left, three on the right, so
$3\mathrm{Ca(OH)_2}$.
Read it as moles, not as molecules. Two moles of hydrogen react with one mole
of oxygen to give two moles of water. The ratio is the same either way, and moles
are what a laboratory can actually measure.
Coefficients are NOT a mass ratio, and this is the trap. Two grams of hydrogen
plus one gram of oxygen gives nothing sensible. Convert to moles first, always.
Mass IS conserved, and the check is that the masses add up: $4.04 + 32.00 =
36.04$. Atom counts balance and gram totals balance; only the coefficients
themselves are not grams.
And volumes of gases follow the coefficients directly, because a mole of any
gas takes the same volume: two liters of hydrogen react with one liter of oxygen.
That balances a different reaction. Coefficients only.
Mistake 2: Balancing oxygen first
Leave hydrogen and oxygen until last.
Mistake 3: Leaving a fraction in the final answer
Multiply through to clear it.
Mistake 4: Not reducing
$4, 2, 4$ should be $2, 1, 2$.
Mistake 5: Treating coefficients as masses
They are mole ratios. Convert first.
Where This Goes Next
Conservation of mass is nearly true rather than exactly true, and the exception is
worth knowing.
$$E = mc^2$$
Every chemical reaction that releases energy loses a little mass, because the
energy came from somewhere. Burning a kilogram of gasoline releases about $46$ MJ,
which corresponds to a mass loss of roughly $5 \times 10^{-10}$ kg. That is half a
nanogram out of a kilogram, far below what any balance could detect, which is why
chemists treat mass as exactly conserved and are right to.
Nuclear reactions are a different matter. The mass defect there is measurable
and large: a uranium fission releases about a million times more energy per atom,
and the missing mass is a real, weighable fraction. That is why nuclear equations
balance protons and neutrons rather than atoms, and why chemical intuition does
not transfer to them.
For balancing, there is also a systematic method when trial and error stalls.
Assign an unknown coefficient to each species, write one linear equation per
element, and solve the system. Balancing is literally solving simultaneous
equations, which is why complicated redox equations can be handed to a computer.
The reason all of this matters starts in the next lesson. Coefficients are the
conversion factors of stoichiometry, and an unbalanced equation gives wrong
answers with no warning at all.
Check Your Understanding
Five of them, no instructions this time.
Summary
Coefficients only, never subscripts. Changing a subscript balances a different
reaction.
Balance in order: the element in fewest formulas first, hydrogen and oxygen
last, polyatomic ions as units.
Fractions are a fine halfway step, then multiply through and reduce.
Coefficients are MOLE ratios, not mass ratios, and they apply directly to gas
volumes.