Moles, Stoichiometry and Concentration

Long before anyone could see an atom, five carefully measured experimental laws established that matter combines in fixed, predictable, whole-number patterns, and every calculation in this chapter, moles, formulas, stoichiometry, ultimately rests on these same five laws. They were discovered independently, by different chemists working decades apart, yet they all point toward the same underlying truth: matter is not infinitely divisible or arbitrarily combinable, it is built from discrete units that combine in fixed ratios, exactly the picture Dalton's atomic theory would soon make explicit.

Lavoisier's Law of Conservation of Mass, established in 1789 from careful combustion experiments, states that matter is neither created nor destroyed in any physical or chemical change, the total mass before and after always equal. Proust's Law of Definite Proportions found that a given compound always contains exactly the same elements in exactly the same proportion by mass regardless of its source, natural or synthetic cupric carbonate both giving 51.35 percent copper, 9.74 percent carbon and 38.91 percent oxygen, without exception. Dalton's own Law of Multiple Proportions, from 1803, covers what happens when two elements form more than one compound between them: the masses of one element that combine with a fixed mass of the other always stand in a simple whole-number ratio, hydrogen combining with 16 g of oxygen to form water but with 32 g to form hydrogen peroxide, a clean 1:2 ratio. Gay-Lussac's Law of Gaseous Volumes, from 1808, found that gases combining or forming in a reaction do so in simple whole-number volume ratios at the same temperature and pressure, 100 mL hydrogen combining with exactly 50 mL oxygen to give 100 mL water vapour, a 2:1 ratio. And Avogadro's Law, in 1811, explained why: equal volumes of any gas, at the same temperature and pressure, contain equal numbers of molecules, a proposal correct but genuinely ignored for nearly fifty years until Cannizzaro championed it at the first international chemistry conference in 1860.

Building on these laws, Dalton's 1808 atomic theory proposed that matter consists of indivisible atoms, that all atoms of a given element are identical, including in mass, while atoms of different elements differ, that compounds form when atoms of different elements combine in a fixed ratio, and that chemical reactions simply reorganise atoms, never creating or destroying them. This picture explained the laws of chemical combination cleanly, conservation, definite and multiple proportions all follow directly from atoms combining in fixed whole-number ratios, but it genuinely could not explain Gay-Lussac's own gaseous-volume law, since Dalton assumed atoms of the same element could never combine with each other, ruling out diatomic molecules like H2 or O2 entirely; only Avogadro's later distinction between atoms and molecules resolved the gap, matching the observed gas volumes exactly once hydrogen and oxygen were correctly understood as diatomic.

Since 1961, atomic mass has been measured relative to carbon-12 specifically, one atomic mass unit, u, defined as exactly one-twelfth the mass of a single carbon-12 atom, and every other atom's mass reported relative to that fixed standard. Because most elements occur naturally as a mix of isotopes, the atomic mass actually used in calculations is an average atomic mass, weighted by each isotope's own natural abundance: carbon's own three isotopes, 12C at 98.892 percent abundance, 13C at 1.108 percent, and a vanishingly rare 14C, combine to give carbon's familiar average atomic mass of 12.011 u. Molecular mass simply sums the atomic masses of every atom in a molecule, methane's CH4 giving 12.011 plus four times 1.008, or 16.043 u, water's H2O giving two times 1.008 plus 16.00, or 18.02 u; but some substances, sodium chloride among them, exist as an extended ionic lattice rather than discrete molecules, so chemists calculate a formula mass instead, simply summing the atomic masses in the compound's own formula unit, sodium's 23.0 u plus chlorine's 35.5 u giving NaCl a formula mass of 58.5 u.

Just as a dozen always means 12 items regardless of what is being counted, a mole always means exactly 6.02214076 times ten to the 23rd elementary entities, atoms, molecules, ions, or any other specified particle, this fixed number, the Avogadro constant, chosen specifically because it connects a countable number of particles to a mass you can actually weigh on a balance. Written out in full, without any power of ten, it reads 602,213,670,000,000,000,000,000, a number so large it was historically determined by dividing the known mass of one mole of carbon-12, exactly 12 g, by the independently measured mass of a single carbon-12 atom, 1.992648 times ten to the minus 23 g, giving back that same 6.022 times ten to the 23rd. Once a mole is defined, molar mass follows immediately: the mass of one mole of any substance, in grams, is numerically identical to its own atomic, molecular or formula mass in u, water's molar mass 18.02 g per mole, sodium chloride's 58.5 g per mole, letting a chemist move freely between counting particles and weighing them on a scale.

Knowing a compound's mass percentage of each element lets a chemist check purity or confirm identity directly: water's own 18.02 g molar mass splits into 11.18 percent hydrogen and 88.79 percent oxygen by mass, calculated simply as an element's own mass in the compound divided by the compound's molar mass, times 100. Running the same logic in reverse recovers the compound's own formula: given that an unknown compound is 4.07 percent hydrogen, 24.27 percent carbon and 71.65 percent chlorine by mass, with a molar mass of 98.96 g, treating a 100 g sample gives 4.07 g hydrogen, 24.27 g carbon and 71.65 g chlorine directly; dividing each mass by its own atomic mass gives 4.04 mol hydrogen, 2.021 mol carbon and 2.021 mol chlorine; dividing every value by the smallest, 2.021, gives a clean 2:1:1 ratio, the empirical formula CH2Cl. The empirical formula alone only gives the simplest ratio, though, not necessarily the true molecular formula, so a final check against the known molar mass is essential: CH2Cl's own formula mass, 49.48 g, divides into the actual 98.96 g molar mass to give exactly 2, meaning the real molecular formula doubles every subscript, C2H4Cl2.

A balanced chemical equation carries the same number of atoms of every element on both sides, exactly what conservation of mass demands, and balancing generally proceeds one element at a time: for propane's combustion, C3H8 plus O2 giving CO2 plus H2O, matching three carbons first forces three CO2 molecules, matching propane's eight hydrogens next forces four water molecules, and only then does oxygen get balanced last, ten oxygen atoms needed on the right, five O2 molecules supplying them on the left, C3H8 plus 5O2 giving 3CO2 plus 4H2O, with subscripts inside a formula never touched, only the coefficients in front of each formula. Once balanced, those coefficients, called stoichiometric coefficients, unlock four equivalent ways of reading the same equation at once: one mole of methane reacts with two moles of oxygen to give one mole of carbon dioxide and two moles of water; equally, one molecule reacts with two molecules to give one plus two molecules; equally, at the same temperature and pressure, 22.7 L of methane reacts with 45.4 L of oxygen to give 22.7 L of carbon dioxide and 45.4 L of water vapour; and equally, 16 g of methane reacts with 64 g of oxygen to give 44 g of carbon dioxide and 36 g of water, mole ratios, molecule ratios, volume ratios and mass ratios all locked together by the very same coefficients.

Real reactions rarely start with exactly the stoichiometric ratio of reactants a balanced equation calls for, and whichever reactant is present in less than its own required amount runs out first, halting the reaction regardless of how much of the other reactant remains, the limiting reagent. Mixing 50.0 kg N2 with 10.0 kg H2 to make ammonia, N2 plus 3H2 giving 2NH3, illustrates the calculation directly: converting to moles gives 1786 mol N2 and 4960 mol H2; since the equation needs 3 mol H2 for every 1 mol N2, the full 1786 mol N2 would require 5360 mol H2, more than the 4960 mol actually available, so hydrogen, not nitrogen, is the limiting reagent, and every subsequent calculation must be based on the 4960 mol H2 actually present, not the larger nitrogen supply; using the equation's own 3:2 ratio of H2 to NH3 gives 3300 mol NH3, converting to 56.1 kg, the genuine, achievable product yield, with nitrogen left over unreacted.

A majority of real chemistry happens in solution, so knowing exactly how much solute a given volume or mass of solution contains matters constantly, and four common ways express that concentration. Mass percent, solute mass divided by total solution mass times 100, is the simplest; mole fraction, moles of one component divided by total moles of every component present, works well for mixtures without singling out a solvent. Molarity, moles of solute per litre of solution, is the most widely used unit of all, and dilution problems use the compact relationship M1V1 equals M2V2, since the total moles of solute stay fixed while only the solvent volume changes, preparing 200 mL of 1 M sodium hydroxide from stock and diluting it to exactly 1 litre giving precisely the 0.2 M solution wanted; molarity does shift slightly with temperature, though, since a solution's own volume itself changes as temperature does. Molality, moles of solute per kilogram of solvent, avoids that temperature dependence entirely, since mass, unlike volume, does not change with temperature, and converting between molarity and molality needs a solution's own density: a 3 M sodium chloride solution with density 1.25 g per mL contains 175.5 g NaCl and 1250 g of total solution per litre, leaving 1074.5 g of water once the salt's own mass is subtracted, giving a molality of 3 mol divided by 1.0745 kg, or 2.79 m.