Counting Atoms

Chapter 2 Atoms in, atoms out

Balancing chemical equations

A chemical equation is an account of atoms. Nothing is created and nothing is lost, so every element must come out exactly as it went in.

9 min read · 5 interactive figures · 1 question

Propane burning in a gas grill, written with one molecule of everything. Each row of the ledger is one element: atoms in on the left, atoms out on the right, the difference in red when they disagree. Use − and + to change how many molecules of each substance take part, and balance every row.

To balance a chemical equation, change the numbers in front of the formulas, the coefficients, until every element has the same number of atoms on both sides. Never change the small numbers inside a formula: those subscripts say which substance it is. Propane burning, C3H8 + O2 → CO2 + H2O, balances as C3H8 + 5O2 → 3CO2 + 4H2O.[1]

The ledger above lets you do it by hand. Each row is an element, with the atoms going in on the left and those coming out on the right. Three carbons go in, so three carbon dioxides must come out. Eight hydrogens go in, so four waters. That makes ten oxygen atoms on the right, so five oxygen molecules on the left. Every row now reads the same on both sides, and the mass row agrees with them: 204.087 g of propane and oxygen become 204.087 g of carbon dioxide and water, for the amounts in moles that the coefficients name.

Why an equation has to balance

A reaction rearranges atoms; it does not make or destroy them. Antoine Lavoisier put this at the foundation of chemistry in his Traité élémentaire de chimie of 1789: “in all the operations of art and nature, nothing is created; an equal quantity of matter exists both before and after the experiment; the quality and quantity of the elements remain precisely the same”.[2] He used it as an accounting rule. Weighing what went into a fermentation and what came out, he wrote that one could treat them “as forming an algebraic equation”, and use each element in turn to check the experiment by calculation.[2] documented

A balanced equation is that rule written down. Each element is a separate account that must close. Because each atom of an element has (on average) the same mass, closing every atom account also closes the mass account: the total mass of the reactants equals the total mass of the products. You can watch the two accounts in a sealed flask as a reaction runs.

Figure 1 A sealed flask holding exactly the amounts in the balanced equation (1 mol of methane and 2 mol of oxygen to start). Slide the reaction from 0 % to 100 %. The mass column changes its make-up, reactants (blue) giving way to products (umber), but its total never moves; the atom account on the right does not change at all.#

Coefficients count molecules; subscripts define them

Question 1 Prediction: fixing the oxygen count#

The temptation to fix the oxygen count by writing H2O2 is natural, and wrong. A subscript is part of a substance's identity: H2O is water, H2O2 is hydrogen peroxide, a bleach. A coefficient is a count of whole molecules (or formula units) of a substance that is already fixed. Balancing is the job of the coefficients alone.

The distinction is easy to recite and hard to hold. William Yarroch interviewed high-school students who balanced equations correctly and found that many could not draw the molecules their balanced equations described, mixing up what the coefficient and the subscript each count.[3] Later studies found the same gap with particle drawings: students who solve the numerical problems often cannot pick the drawing that matches the equation.[4][5] That is why the figures on this site draw molecules, not just numbers.

Figure 2 The simplest case. Hydrogen and oxygen start one of each, so the oxygen row is one short on the products side. Add a water molecule, then a hydrogen molecule, and both rows close: 2H₂ + O₂ → 2H₂O. Notice that you only ever add whole molecules; the drawing never changes what a molecule is.#

A method that works by inspection

For most equations you meet in a course, a few habits make balancing quick:[1]

  1. Count each element on both sides. Write the ledger, even in your head.
  2. Start with an element that appears in only one substance on each side, and leave elements that appear everywhere (often oxygen and hydrogen) for last. In propane's combustion, carbon appears only in C3H8 and CO2, so it fixes the CO2 coefficient at once.
  3. Balance free elements last. O2, H2 or Fe on their own can absorb any count without disturbing another element.
  4. Clear fractions. If you end with 3½ O2, double everything.
  5. Check every row, and check that the coefficients have no common factor.

Ethane shows step 4. Two carbons give 2CO2; six hydrogens give 3H2O; that makes 4 + 3 = 7 oxygen atoms on the right, so 3½ O2. Doubling gives 2C2H6 + 7O2 → 4CO2 + 6H2O. Try it in the ledger at the top of the page by choosing “Ethane burns”.

Balancing as algebra

Inspection is guesswork with good habits. Lavoisier's phrase points to something better: each element gives one equation in the unknown coefficients. For methane burning, call the coefficients a, b, c, d in aCH4 + bO2 → cCO2 + dH2O. Then:

C: a = c H: 4a = 2d O: 2b = 2c + d(1)

Three equations, four unknowns. A balanced equation has no definite size (2CH4 + 4O2 → 2CO2 + 4H2O balances just as well), so one unknown is always free; set d = 2 and the others follow: a = 1, c = 1, b = 2. Written as a table, with one row per element, one column per substance, and products entered with a minus sign, the equations become a single statement: the matrix times the column of coefficients is zero.[6][7] The method that solves it, Gauss–Jordan elimination, is the same row-by-row bookkeeping as the ledger, done systematically.

Figure 3 The element-by-substance table for a reaction, eliminated step by step in exact fractions. Slide through the steps: each one replaces a row with a combination of rows, which keeps the same solutions. At the end, every column but the last has a single 1, and the last column holds the coefficients (with their signs flipped) relative to the last substance. Choose the permanganate reaction for a longer run, or the water-and-peroxide mixture for a table with two free columns.#

Two facts about this method matter for what follows. First, it is exact: the entries stay as fractions, nothing is rounded, and the coefficients come out as the smallest whole numbers in the right ratio. Second, it tells you how many free choices there are. That count is the difference between the number of substances and the number of independent rows (the rank of the table).[6] One free choice means one answer up to scale, which is the ordinary case. The others are where inspection gets lost.

When an equation has no answer, or more than one

No answer. CO2 + H2O → C6H12O6 looks like photosynthesis but cannot be balanced: the only coefficients that close every account are all zero. The missing piece is the oxygen gas that photosynthesis releases; with O2 added on the right, it balances as 6CO2 + 6H2O → C6H12O6 + 6O2. An unbalanceable equation always means a wrong or missing formula.

An unwanted zero, or the wrong side. If a substance can only balance with coefficient zero, it takes no part. If it would need a negative coefficient, it belongs on the other side of the arrow.

More than one answer. H2 + O2 → H2O + H2O2 balances as H2 + O2 → H2O2, as 2H2 + O2 → 2H2O, and as any mix of the two, such as 3H2 + 2O2 → 2H2O + H2O2. The table has two free choices because the “equation” is really two reactions written as one. No amount of atom counting can pick one answer, because each conserves every atom.[7]

The classic example from redox chemistry is permanganate oxidising hydrogen peroxide in acid: KMnO4 + H2O2 + H2SO4 → K2SO4 + MnSO4 + O2 + H2O. The textbook answer, 2, 5, 3 → 1, 2, 5, 8, conserves every atom; so does 2, 1, 3 → 1, 2, 3, 4, and infinitely many others. Counting electrons does not help. For neutral substances, every atom-balanced version is automatically electron-balanced as well: in 2, 1, 3 → 1, 2, 3, 4 the two manganese atoms gain ten electrons, and the oxygen that ends up as O2 (two atoms from the peroxide, four from the permanganate) gives up exactly ten. The alternatives are the textbook reaction mixed with other reactions, such as peroxide decomposing on its own (2H2O2 → 2H2O + O2) or permanganate's own oxygen leaving as O2. The textbook answer is the one in which the peroxide is the only thing oxidised, so that every O2 molecule comes from a peroxide: then each peroxide gives up two electrons and each manganese takes five, two permanganates for every five peroxides. Whether that is what happens is a question about the reaction, answered by chemistry and experiment, not by counting. In the same way, how much water and how much peroxide hydrogen and oxygen make, or how much carbon monoxide and dioxide burning carbon makes, depends on the conditions, and only a measurement of the products can say. A balancer that offers one of these answers without saying so is guessing. The one on this site says that the answer is not unique and lists the independent reactions instead. established

Charge is an account too

When ions appear, charge must balance as well as atoms. Permanganate oxidising iron(II) in acid, MnO4− + Fe2+ + H+ → Mn2+ + Fe3+ + H2O, has four elements and a charge, five accounts in all. Adding a charge row to the table is all it takes: MnO4− + 5Fe2+ + 8H+ → Mn2+ + 5Fe3+ + 4H2O carries a total charge of +17 on each side. Electrons themselves can be entered as a substance, e−, with no atoms and a charge of −1, which is how half-reactions such as Fe3+ + e− → Fe2+ are balanced.

The balancer

The calculator below does all of this: it builds the table, eliminates it in exact fractions, and reports what it finds, including the honest answers “cannot be balanced” and “more than one answer”. It has its own page, the Equation balancer, with the rules for typing formulas and charges.

Figure 4 Type an equation, with species separated by + and the sides by -> (charges with a caret: Fe^3+). The result shows the balanced equation, the atom ledger, and under “How this was calculated”, the table and the size of its null space.#

With a balanced equation in hand, the coefficients become a recipe: one methane needs two oxygens. Stoichiometry and the limiting reactant uses that recipe with real amounts, in grams, and asks which ingredient runs out first.