Stoichiometry and the mole

Every chemical process an engineer designs, from a natural gas furnace to a fertilizer plant to a water treatment system, starts with the same question: how much? How much fuel and air does the burner need? How much ammonia will a tonne of hydrogen make? How much lime must be added to soften a day's water supply? Stoichiometry is the arithmetic that answers these questions, and the mole is its unit.

This chapter defines the mole and molar mass, shows how to balance chemical equations, and builds the step-by-step method for converting between masses, moles, gas volumes and solution concentrations. It then covers limiting reactants, percent yield, empirical formulas and titration, all with industrial examples.

Atoms, molecules and the mole

Atoms are far too small to count individually. A single drop of water contains about \(10^{21}\) molecules. Chemists therefore count in bundles called moles. One mole is exactly \(6.022\,140\,76 \times 10^{23}\) particles, a number called Avogadro's number, \(N_A\). The particles can be atoms, molecules, ions or electrons, whichever the problem is about.

The mole is useful because of how it was chosen: one mole of a substance has a mass in grams equal to its formula mass in atomic mass units. Carbon-12 atoms have a mass of 12 u, so a mole of carbon-12 weighs 12 g. The mass of one mole is the molar mass \(M\), in g/mol, found by adding the atomic masses from the periodic table.

Some molar masses you will use often: methane CH₄ 16.04, propane C₃H₈ 44.10, carbon dioxide CO₂ 44.01, oxygen O₂ 32.00, nitrogen N₂ 28.01, ammonia NH₃ 17.03, calcium carbonate CaCO₃ 100.09 and iron(III) oxide Fe₂O₃ 159.69 g/mol.

Percent composition

The mass fraction of an element in a compound comes straight from the molar mass. This is how metallurgists decide how much metal an ore can yield.

Balancing chemical equations

A chemical equation must obey conservation of mass: every atom on the left must appear on the right. Balancing means choosing whole-number coefficients so each element balances. The coefficients count moles (or molecules), never grams.

A practical order of work: balance the element that appears in the fewest compounds first, leave free elements (such as O₂) until last, and adjust for any fractions at the end by multiplying through.

The stoichiometry method

Every stoichiometry problem follows the same road: convert what you know to moles, use the mole ratio from the balanced equation, then convert moles to what you want.

Notice that burning 1 kg of fuel produces 3 kg of CO₂. The extra mass is oxygen taken from the air. This is why emission factors for fuels look surprisingly large.

Limiting reactants and yield

Reactants are rarely supplied in exactly the ratio the equation needs. The one that runs out first is the limiting reactant, and it sets how much product can form. The others are in excess.

To find the limiting reactant, convert each reactant to moles and compare with the mole ratio. Whichever would make less product limits the reaction.

Percent yield

The amount calculated from the limiting reactant is the theoretical yield. Real processes give less, because of side reactions, incomplete conversion and losses in separation. The percent yield is

If the reactor above produced 45.0 kg of ammonia, the yield would be \(45.0/56.3 = 79.9\%\). Industrial ammonia converters convert only about 15% to 25% of the feed per pass, so unreacted gas is separated and recycled, and the overall yield of the plant ends up well above 95%.

Empirical and molecular formulas

Laboratory analysis gives the mass percentage of each element in a compound. Converting those percentages to a mole ratio gives the empirical formula, the simplest whole-number ratio of atoms.

Solutions and concentration

Most industrial reactions happen in solution. The usual concentration unit is molarity, moles of solute per litre of solution:

The dilution equation holds because adding water does not change the number of moles of solute. To make 2.0 L of 0.50 M sulfuric acid from 12 M stock, you need \(V_1 = (0.50)(2.0)/12 = 0.083\) L, or 83 mL of stock. Tip: always add acid to water, never water to acid, because the heat released can make concentrated acid boil and spatter.

Environmental and water engineers often use mass concentrations instead: milligrams per litre (mg/L), which for dilute water solutions is the same as parts per million (ppm).

Titration

A titration measures an unknown concentration by adding a solution of known concentration until the reaction is just complete, shown by an indicator colour change or a pH meter. At that equivalence point the moles of the two reactants match the equation's ratio.

Plant laboratories run titrations daily to check boiler water alkalinity, the acidity of process streams and the strength of cleaning solutions.

Mass balances

Stoichiometry is one half of the most important tool in chemical and environmental engineering: the mass balance. For any process unit, a tank, a reactor or a whole plant, mass is conserved:

For a process running steadily, accumulation is zero. Applied to total mass, nothing is generated or consumed. Applied to individual chemical species, the generation and consumption terms come from the reaction stoichiometry. Engineers draw a box around the unit, label every stream entering and leaving, and write one balance per unknown.